Method and system for ultrasonically characterizing a medium

Abstract

The invention relates to a method for ultrasonically characterizing a medium, comprising a step of generating a series of incident ultrasonic waves, a step of generating an experimental reflection matrix Rui(t) defined between an emission basis (i) as input and a reception basis (u) as output, a step of determining a de-scanned reflection matrix R' of the medium between an input virtual transducer calculated on the basis of an input focus of the experimental reflection matrix and an output virtual transducer calculated on the basis of an output focus of the experimental reflection matrix, the virtual transducers being at the same depth. The method comprises a step of extracting an optimal position deviation or an optimal speed-of-sound model from the focal-spot image.

Description

[0001] METHOD AND SYSTEM FOR ULTRASONIC CHARACTERIZATION OF A MEDIUM

[0002] TECHNICAL FIELD

[0003] This description relates to methods and systems for ultrasonic characterization of a medium, and applies in particular to medical imaging as well as to non-destructive testing of a medium and more generally to all fields which can use a multi-element network of transducers to measure the reflection of a diffusing medium.

[0004] STATE OF THE ART

[0005] Limitations of ultrasound imaging

[0006] In the field of ultrasound imaging, we seek to construct an image of the reflectivity of a medium from echoes backscattered by heterogeneities in the medium. This is the principle of the ultrasound used in medical imaging, which allows in particular to visualize the internal anatomy of an individual. Most of the time, to construct an ultrasound image, the medium is considered homogeneous, with a constant speed of sound. However, the hypothesis of a homogeneous medium is often not verified in the majority of concrete cases, in particular when it comes to imaging parts of human or animal bodies. For example, in the case of liver ultrasound, acoustic waves pass through a succession of layers of fat and muscles before reaching the targeted organ. The speed of sound can in this case vary for example between 1450 m / s for adipose tissue and 1600 m / s for the liver.The disagreement between the actual distribution of the speed of sound in the medium and the speed model used to construct the ultrasound image can induce fairly strong aberrations in the image thus obtained, compromising for example the practitioner's diagnosis during a medical examination.

[0007] As illustrated in Figure 1, conventional ultrasound methods use an array of piezoelectric transducers that can emit and / or receive ultrasound signals independently, each transducer being at a position u in the array supporting said array. The array of transducers, placed opposite a medium, makes it possible to insonify and image the medium in different ways. The conventional method consists of insonifying the medium using focused emissions by a technique called beamforming. As shown in Figure 1(a), this method consists of applying to the signals emitted by each transducer a set of appropriate delays T(u in , x in , z, co) based on a homogeneous velocity model co, in order to interfere the wavelets produced by each transducer at the targeted focal point of spatial position (x in, z). Due to the physical limitations of diffraction, ultrasound is emitted through the aperture of the ultrasound probe, concentrated in an area often called a "focal spot", with a lateral width of <5x.

[0008] In order to then possibly construct an ultrasound image, a digital focusing step is also carried out at reception. The echoes captured by the network transducers, as shown in Figure 1(b), are put back into phase by shifting them in time. The delays T(uout, x mit , z, co) are identical to those applied to the emission, with the variable Uout designating the position of each transducer. In the emission phase, all signals interfere at the position point (x in , z) at ballistic time t = z / co if the velocity model co corresponds to reality. In reception, the signals coming from this same point (x ou t = x in) interfere by summing the signals at the echo time t = 2z / co. This summation gives the final result of the reception focusing. This confocal method with double focusing at emission and reception, illustrated in Figure 1, makes it possible to directly image the reflectivity of the medium with a lateral resolution ôx and good contrast. However, this method is time-consuming because it requires physically focusing at emission at each point of the medium or at least at a given depth, on each of the lines of the constructed image representative of the medium. Furthermore, this image is often of a low qualitative level when the medium is heterogeneous, in particular with a speed of sound that is not constant between the various zones constituting the observed medium. Figure 1(c) and Figure 1(d) show a medium that includes an aberrator layer with a speed of sound different from the speed of sound observed in the rest of the medium.This results in an aberration of the acoustic wavefront which leads to a distortion of the resulting ultrasound image and therefore to a degradation of its resolution and contrast, which is particularly annoying during a medical examination.

[0009] Matrix approach to ultrasound imaging

[0010] To compensate for these aberration problems, a matrix approach to ultrasound imaging has been developed in recent years. This approach is based on the measurement by the ultrasound probe of an experimental reflection matrix of the medium studied.

[0011] A first proposal to measure this experimental reflection matrix is ​​to successively emit an ultrasonic pulse from each transducer in the array whose position is identified by the coordinate Um, as shown schematically in Figure 2(a). This gives rise to a diverging cylindrical (or spherical) incident wave. This wave is reflected by the diffusers in the medium and the backscattered field is recorded by each of the transducers as a function of time as shown in Figure 2(b). By repeating this operation with each transducer used successively as a source, the experimental reflection matrix R is determined. uu (t) expressed in the base of the transducers, composed of the set of impulse responses R(M OH Z, u in, t) between each transducer. This matrix is ​​then rich in quantity of information on the medium studied. However, the method assumes that the medium remains fixed throughout the duration of the measurements. In addition, the recorded signals have a poor signal-to-noise ratio because the medium is insonified by a single transducer.

[0012] A second way to construct this experimental reflection matrix consists of insonifying the medium with a series of plane waves. This method overcomes the previous problems. Figure 2(c) illustrates the principle of this plane wave illumination. A delay law T' is applied to each signal at transmission to form a wavefront inclined at an angle 6L relative to the transducer array. At reception, illustrated in Figure 2(d), the field backscattered by the medium, R(u ou t, 0m, t), is measured by all position sensors u out for each incident plane wave. All of these responses form an experimental reflection matrix Rue(t). The double focusing method described above can be implemented numerically by temporally shifting the measured signals before summing them coherently. This method gave rise to ultrafast imaging and elastography, and is described, for example, in the document:

[0013] “Coherent plane-wave compounding for very high frame rate ultrasound and transient elastography”, G. Montaldo et al. (IEEE Trans. Ultrason., Ferroelect. Freq. Control 56 489-506, 2009).

[0014] A third way to create this experimental reflection matrix is ​​to insonify the medium with a basis of diverging waves, as shown in Figure 2(e) and Figure 2(f), which allows illuminating the acoustic field more broadly than by using plane waves. This technique, used in particular in superresolution imaging, is explained in the document:

[0015] “Ultrafast imaging of the heart using Circular Wave Synthetic Imaging with Phased Arrays”, Couade et Al., IEEE International Ultrasonics Symposium (2009).

[0016] Conventional imaging therefore consists of double focusing on transmission and reception at the same focal point for each pixel of the image (x« = x out), as shown in Figure 3(a). Matrix imaging, on the contrary, consists of dissociating the focal points at emission and reception, for the same echo time / , as shown in Figure 3(b). Thus, a focused reflection matrix Rir( / , co) is determined, composed of the responses between virtual input transducers and virtual output transducers of respective spatial positions n n = (x in , z in ) and r O ut = (x ou t, z ou t) corresponding to focal spots synthesized at the input and output at different spatial positions, and whose depth is dictated by the echo time t. This matrix is ​​obtained by channel formations from the experimental reflection matrix R ui(t). This matrix provides access to much more information than the confocal image of conventional imaging. This focused reflection matrix can be synthesized in the time domain by delay laws or in the frequency domain by applying appropriate phase shifts.

[0017] This matrix imaging was described in the document:

[0018] “Reflection Matrix Approach for Quantitative Imaging of Scattering Media”, William Lambert, et al., Phys. Rev. X 10, 021048, (2020), then in the document

[0019] “Distortion matrix approach for ultrasound imaging of random scattering media”, W Lambert, et Al. - Proceedings of the National Academy of Sciences, (2020).

[0020] It should be noted that in these publications the focused reflection matrix Rn Ç, co) was considered between virtual transducers at the expected ballistic depth: Zin = Zout = z t = cot / 2, z tbeing the depth of the isochronous volume expected for a sound speed co, the isochronous volume being the set of points in the middle contributing to the signals received at echo time t. This sub-part of the matrix Rrr( / , co) is noted Rxx(z / , t co). While the diagonal coefficients (x oll t = x in ) of this matrix Rxx(z?, t, co) give the synthetic confocal image at depth z t , its off-diagonal coefficients inform us about potential aberration and multiple diffusion problems likely to alter the quality of this same image.

[0021] The reflection matrix focused at ballistic depth, Rxx(z?, t, co), plays a pivotal role for ultrasound matrix imaging. It was initially studied to probe the transverse evolution of the input and output focal spots, then to compensate for aberrations by determining optimal delay laws (or phase shifts) to be applied at the input and output of the reflection matrix. However, this approach is limited because it only allows aberrations to be probed transversely and does not allow for "repositioning the scatterers at their true depth". Thus, any discrepancy between the sound speed model co and its true distribution c(r) implies a poor positioning of the imaged scatterers at depth as shown in Figure 1.

[0022] In particular, as mentioned in patent applications US 2022 / 082693, US 2022 / 084496 and US 2022 / 082527, the focused reflection matrix was studied by the applicant beyond the ballistic depth by scanning in depth the relative position of the virtual transducers. By placing oneself at a constant echo time t (fixed isochronous volume), the focal spot at the input (or output) of the device can be probed locally by scanning the spatial position r O ut (or n n ) of the virtual transducer at the output (or input). A focus defect Az Zout Z in can thus be measured at each point of spatial position n n of the image and an integrated sound speed map between the probe and each point nn can be deduced from this measurement.

[0023] However, this approach is sometimes not optimal because the virtual spatial position transducer r out (or r™) is also sensitive to aberrations, which biases the estimator of the input (or output) focal spot.

[0024] SUMMARY

[0025] The present description relates, according to a first aspect, to a method for characterizing a medium by ultrasound, the method comprising:

[0026] - a step of generating a series of incident ultrasonic waves USm in an area of ​​said medium, by means of a network of transducers, said series of incident ultrasonic waves being an emission base i; and

[0027] - a step of generating an experimental reflection matrix Rui(t) defined between the emission base i at the input and a reception base u at the output, the coefficients of this matrix corresponding to the signals received by the transducers induced by the reflected ultrasonic waves.

[0028] The method further comprises:

[0029] - a step of determining a de-scanned reflection matrix R', which comprises, for a set of points in the middle of abscissa x and echo time / , responses of the middle calculated by channel formations from the experimental reflection matrix Rui(t), between a virtual input transducer of spatial position n n = (x, z t +Az) and a virtual spatial position output transducer r ou t = (x+Ax, z t +Az), the two virtual transducers being at the same depth z t +Az, z tbeing the depth of the isochronous volume expected for a sound speed co, the isochronous volume being the set of points in the medium contributing to the signals received at echo time t, the responses of the unscanned reflection matrix R' being determined for a set of lateral deviation values ​​Ax comprising at least Ax = 0 and for a set of depth deviation values ​​Az, said unscanned reflection matrix R' being formed with each row corresponding to an element of a pair Ar = {Ax, Az} which corresponds to a focal spot image and each column corresponding to an element of a pair {x, t}, which is noted:

[0030] R' = [R'({Ax, AZ}5{®3})]

[0031] - an extraction step in which an optimal position deviation dr is determined from each focal spot image O pt, said optimal position deviation zlr op t = {Ax opt , Az op t} close to the maximum in said focal spot image.

[0032] Thanks to these provisions, the method advantageously allows the medium to be probed locally at any point and in any direction relative to a ballistic time of propagation of the ultrasound background in the medium.

[0033] This method provides access to a lot of local information about the medium, which makes it possible to characterize the process of focusing the ultrasonic wave locally, and thus to estimate the focusing defect in depth. p! .

[0034] This estimation is then very useful for improving the focus and quality of the ultrasound image. This optimization of ultrasound images is performed by calculation without requiring iteration of new emissions and / or acquisitions, which is a significant advantage especially during in vivo measurements. Optimizations can indeed be performed in real time or later, possibly on a different and optionally remote system.

[0035] In various embodiments of the method, one may optionally further use fune and / or the other of the following arrangements.

[0036] According to one variant, the method further comprises:

[0037] - a step of forming an ultrasound image by calculating the reflectivity at a plurality of points in the medium, from the unscanned reflection matrix R', the reflectivity of each of these points in the medium corresponding to the response of the medium between the virtual input transducer of spatial position n n = (x, z t +Az opt ) and the virtual spatial position output transducer r O ut= r + Ar op t = (x+Ax op t, z t +Az op t), said reflectivity being defined by:

[0038] Alternatively, the ultrasound image is transformed into the depth dimension, using the sound speed model co by: in which z t is the depth of a pixel of the ultrasound image or depth of the isochornic volume.

[0039] In one embodiment, the ultrasound image is displayed on a display device.

[0040] According to one variant, the method further comprises before the extraction step:

[0041] - a spatio-temporal averaging step of the unscanned reflection matrix R' in which the values ​​of the unscanned reflection matrix R' are locally averaged, along the lateral dimension x and along the echo time dimension / , for each element of the pair Ar = {Ax, Az}, the focal spot images then being averaged.

[0042] Alternatively, at the spatio-temporal averaging stage, an incoherent point spread function PSF is calculated. inc by the following formula: for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of depth deviation values ​​Az, denoted in the form of the pair Ar = {Ax, Az}, to obtain the focal spot image for the lateral position x and the echo time t, in which:

[0043] < > is an average operator according to the parameters of the pair {xt '}

[0044] R' is the unscanned reflection matrix,

[0045] W(xt ') is a spatio-temporal weighting window according to the same parameters, and at the extraction stage, we search for a maximum point of the focal spot image corresponding to a maximum of the modulus of the points of the focal spot image, and the optimal position deviation ZJr<, pl is the position of said maximum point in said focal spot image.

[0046] According to a variant: at the spatio-temporal averaging step, a singular value decomposition SVD of a local focal spot matrix R is calculated (,) , said local focal spot matrix being expressed from the unscanned reflection matrix Æ' by: in which: R' is the unscanned reflection matrix,

[0047] W(xt ') is a spatio-temporal weighting window according to the same parameters. The singular value decomposition SVD of said local matrix is ​​noted: in which

[0048] Âp are the singular values ​​of said local matrix,

[0049] U P are the output eigenvectors,

[0050] V P are the input eigenvectors, we calculate a coherent point spread function PSF co h by the following formula: for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of depth deviation values ​​Az, denoted in the form of the pair zlr = {Ax, Az}, to obtain the focal spot image for the lateral position x and the echo time / , and in the extraction step, a maximum point of the focal spot image is sought corresponding to: a maximum of the modulus of the points of the focal spot image, or to a maximum of the modulus of the derivative according to the depth direction of the points of the focal spot image, or to a maximum of the modulus of the derivative according to the depth direction of the imaginary part of the points of the focal spot image, and the optimal position deviation zlr op t is the position of said maximum point in said focal spot image.

[0051] According to a variant, at the extraction stage, it is considered that the lateral deviation Ax opt is zero, Axopt 0.

[0052] According to a variant, in the step of determining a de-scanned reflection matrix R', the de-scanned reflection matrix R' is formed only for lateral deviation values ​​Ax of zero value, Ax = 0.

[0053] According to a variant, at the spatio-temporal averaging step, the values ​​of the unscanned reflection matrix R' are averaged according to the lateral dimension x and the echo time t, only for focal spot images corresponding to a zero lateral deviation Ax.

[0054] Alternatively, at the extraction stage, the optimal depth difference Az opt is determined from the points of the focal spot image having a lateral deviation Ax of zero value

[0055] According to a variant, an integrated speed of sound c is determined opt from the optimal depth deviation Az op t, from the optimal position this integrated sound speed corresponding to an average sound speed between the transducer network and the depth of the isochronous volume.

[0056] According to a variant: we calculate local sound speeds c(r), for a set of pairs {x, / }, from integrated sound speed values ​​c opt obtained for the focal spot images of said pairs {x, / },

[0057] According to a variant, the method further comprises: a step of forming a local sound speed image of the medium from said set of local sound speeds.

[0058] Alternatively, a de-scanned reflection matrix R' is recalculated using the local sound speed values.

[0059] According to a variant, the calculations of path formations between the virtual input transducer and the virtual output transducer are carried out from the experimental reflection matrix Rui(t), using a forward delay time of the ultrasonic waves between the transmission base (i) and the virtual input transducer, and using a return delay time of the ultrasonic waves between the virtual output transducer and the transducers of the reception base (u).

[0060] Alternatively, the unscanned reflection matrix R' is calculated from the experimental reflection matrix Rui(t) by the following formula: in which:

[0061] Nm is the number of elements in the emission base (i)

[0062] Nom is the number of elements of the reception base (u), Rui(t) is the experimental reflection matrix, of which Ruifaout, lin, t) is the element of the experimental reflection matrix Rui / 7; recorded by the spatial position transducer Uout following the emission of index iin in the emission base (i), at a time t , co being the expected speed of sound.

[0063] Alternatively, the unscanned reflection matrix R' is calculated from a focused reflection matrix R by the following formula:

[0064] R'({^x, AzS}, {æ, t}) =R(x + Ax, x, t, Zt + Az) in which: the unscanned reflection matrix R' is formed with each row formed by elements of a pair {Ax, Az} and each column formed by elements of a pair {x, t}, and the focused reflection matrix R is calculated from the experimental reflection matrix Rui(t) by the following formula: in which:

[0065] Nin is the number of elements in the emission base (i)

[0066] N O ut is the number of elements in the receiving base (u),

[0067] Rui(t) is the experimental reflection matrix, whose

[0068] Rui(Uout, lin, t) is the element of the experimental reflection matrix Rui(t) recorded by the spatial position transducer Uout following the emission of index iin in the emission base (i) and at time / , co being the expected sound speed.

[0069] According to a variant, a film is formed from a succession of ultrasound images by a first number N of iterations of the steps of the method.

[0070] According to a variant: among the N iterations, a second number P, less than N of iterations, does not include the steps of determining the unscanned reflection matrix R', the spatio-temporal averaging steps, and the extraction steps, and for these P iterations, at the step of forming the ultrasound image, the ultrasound image is obtained from an optimal position deviation extracted during a previous iteration. The present description relates, according to a second aspect, to a system for ultrasonic characterization of a medium, and configured for the implementation of methods as described previously. The system according to the second aspect comprises:

[0071] - an array of transducers adapted to generate a series of incident ultrasonic waves in an area of ​​the medium, and to record as a function of time the ultrasonic waves backscattered by said area; and

[0072] - a computing unit associated with the transducer network and adapted to implement the method according to the first aspect.

[0073] The present description relates, according to a third aspect, to a method for characterizing a medium by ultrasound, the method comprising:

[0074] - a step of generating a series of incident ultrasonic waves USm in an area of ​​said medium, by means of a network of transducers, said series of incident ultrasonic waves being an emission base i; and

[0075] - a step of generating an experimental reflection matrix Ruifiy defined between the emission base i at the input and a reception base u at the output, the coefficients of this matrix corresponding to the signals received by the transducers induced by the reflected ultrasonic waves.

[0076] The method being characterized in that it further comprises

[0077] - a step of determining a de-scanned reflection matrix R', which comprises, for a set of points in the middle of abscissa x and echo time / , responses of the middle calculated by channel formations following a sound speed model co from the experimental reflection matrix Rui, between a virtual input transducer of spatial position n n = (x, z t ) and a virtual spatial position output transducer r O ut = (x+Ax, z t ), the two virtual transducers being at the same depth z t , z tbeing the depth of the isochronous volume expected for the sound speed co, the isochronous volume being the set of points in the medium contributing to the signals received at time / , the responses of the unscanned reflection matrix R' being determined for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed models co, said unscanned reflection matrix being formed with each row corresponding to an element of a pair {Ax, co} which corresponds to a focal spot image and each column corresponding to an element of a pair {x, t}, which is noted:

[0078] - an extraction step in which an optimal lateral deviation Ax is determined from each focal spot opt and an optimal sound speed model c opt , close to the maximum in said focal spot image, .

[0079] Thanks to these provisions, the method advantageously allows the medium to be probed locally at any point and in any direction relative to a ballistic time of propagation of the ultrasound background in the medium.

[0080] This method provides access to local information about the medium, which makes it possible to characterize the process of focusing the ultrasound wave locally, and thus to estimate the distribution of sound speed in the medium. This physical parameter is a particularly relevant biomarker in ultrasound imaging because it allows the diagnosis of certain diseases, such as, for example, liver steatosis.

[0081] This estimation of the speed of sound is also very useful for improving the focusing and quality of the ultrasound image. This optimization of ultrasound images is performed computationally without requiring iteration of new emissions and / or acquisitions, which is an important advantage especially during in vivo measurements.

[0082] In various embodiments of the method, one and / or the other of the following provisions may optionally be used in addition.

[0083] According to one variant, the method further comprises:

[0084] - a step of forming the ultrasound image by calculating the reflectivity at a plurality of points in the medium, from the unscanned reflection matrix R', the reflectivity of each of these points in the medium corresponding to the response of the medium between the virtual input transducer of spatial position n n= (x, zi) and the virtual spatial position output transducer r O ut= (x+Ax op t, z t ), z t being the depth of the isochronous volume expected for the optimal sound speed model c opt and said reflectivity being defined by:

[0085] Alternatively, the ultrasound image is transformed into the depth dimension, using the optimal sound speed model c opt by :

[0086] Z2(æ, 5) = I2(x, t = 25 / c O pt (æ, É)) in which z is the estimate of the depth z of a pixel of the ultrasound image.

[0087] According to one variant, the ultrasound image is displayed on a display device. According to one variant:

[0088] - we calculate local sound speeds c(r), for a set of pairs {x, t}, from the optimal sound speed values ​​c optobtained for the focal spot images of said pairs {x, t}.

[0089] According to a variant, the method further comprises a step of forming a local sound speed image of the medium from said set of local sound speeds, and displaying said local sound speed image on a display device.

[0090] Alternatively, a de-scanned reflection matrix R' is recalculated using local sound speed values.

[0091] According to one variant, the method further comprises, before the extraction step:

[0092] - a spatio-temporal averaging step of the unscanned reflection matrix R' in which the values ​​of the unscanned reflection matrix are locally averaged along the lateral dimension x and the echo time / , for each element of a pair {Ax, co\, the focal spot images then being averaged.

[0093] Alternatively, at the spatio-temporal averaging stage, an incoherent point spread function PSF is calculated. inc by the following formula:

[0094] > for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed model values ​​co, to obtain the focal spot image for lateral position x and echo time t, in which:

[0095] <.> is an average operator according to the parameters of the pair {xt '}

[0096] R' is the unscanned reflection matrix,

[0097] W(xt ') is a spatio-temporal weighting window according to the same parameters, and at the extraction stage, we search for a maximum point of the focal spot image corresponding to a maximum of the modulus of the points of the focal spot image, and the optimal sound speed model c opt is the ordinate of said maximum point in said focal spot image.

[0098] According to a variant: at the spatio-temporal averaging step, a singular value decomposition SVD of a local focal spot matrix R is calculated (1) , said local focal spot matrix being expressed from the unscanned reflection matrix R' by: in which:

[0099] R' is the unscanned reflection matrix,

[0100] W(xt ') is a spatio-temporal weighting window according to the same parameters. The singular value decomposition SVD of said local matrix is ​​noted: in which

[0101] Âp are the singular values ​​of said local matrix,

[0102] U P are the output eigenvectors,

[0103] V P are the input eigenvectors, we calculate a coherent point spread function PSF co h by the following formula:

[0104] PSFcohCfAx, co}, {x, t}) = Pi({Az, co}, {x, t}) for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed model values ​​co, to obtain the focal spot image for the lateral position x and the echo time / , and in the extraction step, a maximum point of the focal spot image is sought corresponding to: a maximum of the modulus of the points of the focal spot image, or to a maximum of the modulus of the derivative according to the sound speed co of the points of the focal spot image, or to a maximum of the modulus of the derivative according to the sound speed co of the imaginary part of the points of the focal spot image, and the optimal sound speed model c opt is the ordinate of said maximum point in said focal spot image.

[0105] According to a variant, at the extraction stage, it is considered that the lateral deviation Ax opt is zero, Axopt 0.

[0106] According to a variant, in the step of determining a de-scanned reflection matrix R', the de-scanned reflection matrix R' is formed only for lateral deviation values ​​Ax of zero value, Ax = 0.

[0107] According to a variant, at the spatio-temporal averaging step, the values ​​of the unscanned reflection matrix R' are averaged to determine a focal spot image, for each element of a pair {Ax = 0, Az}, according to the lateral dimension x and the echo time t.

[0108] Alternatively, at the extraction stage, the optimal sound speed model c opt is determined from the points of the focal spot image having a lateral deviation Ax of zero value, Ax = 0.

[0109] According to a variant, the calculations of path formations between the virtual input transducer and the virtual output transducer are carried out from the experimental reflection matrix Rui(t), using a forward delay time of the ultrasonic waves between the transmission base (i) and the virtual input transducer, and using a return delay time of the ultrasonic waves between the virtual output transducer and the transducers of the reception base (u).

[0110] Alternatively, the unscanned reflection matrix R' is calculated from the experimental reflection matrix Rui by the following formula:

[0111] R'({Ax, co}, {x, i}) = in which:

[0112] Nm is the number of elements in the emission base (i)

[0113] Name is the number of elements in the receiving base (u),

[0114] Rui(t) is the experimental reflection matrix, whose

[0115] Rui(Uout, lin, t) is the element of the experimental reflection matrix Rui / 7> recorded by the spatial position transducer Uout following the emission of index iin in the emission base (i), at a time t , co being the expected speed of sound.

[0116] Alternatively, the unscanned reflection matrix R' is calculated from a focused reflection matrix R by the following formula: in which: the unscanned reflection matrix R' is formed with each row formed by elements of a pair {Ax, Az} and each column formed by elements of a pair {x, t}, and the focused reflection matrix R is calculated from the experimental reflection matrix Rui(t) by the following formula: i'(*Eout » "-l'in, Q) in which:

[0117] Nm is the number of elements in the emission base (i)

[0118] Nout is the number of elements in the receiving base (u),

[0119] Rui(t) is the experimental reflection matrix, whose

[0120] Rui(Uout, lin, t) is the element of the experimental reflection matrix Rui(t) recorded by the spatial position transducer Uout following the emission of index iin in the emission base (i) and at time / , co being the expected sound speed.

[0121] According to a variant, a film is formed from a succession of ultrasound images by a first number N of iterations of the steps of the method.

[0122] According to a variant, among the N iterations, a second number P strictly less than N of iterations does not include the steps of determining the de-scanned reflection matrix R', the steps of spatio-temporal averaging, and the steps of extracting the optimal sound speed model, and for these P iterations, at the step of forming the ultrasound image, the ultrasound image is obtained from an optimal sound speed extracted during a previous iteration.

[0123] The present description relates, according to a fourth aspect, to a system for ultrasonic characterization of a medium, and configured for the implementation of methods as described previously. The system according to the fourth aspect comprises:

[0124] - an array of transducers adapted to generate a series of incident ultrasonic waves in an area of ​​the medium, and to record as a function of time the ultrasonic waves backscattered by said area; and

[0125] - a computing unit associated with the transducer network and adapted to implement the method according to the first aspect. BRIEF DESCRIPTION OF THE FIGURES

[0126] Other advantages and characteristics of the technique presented above will appear on reading the detailed description below, presented in a non-limiting manner for the purposes of illustration, made with reference to the figures in which:

[0127] Figures 1(a) to 1(d) (already described) illustrate the impact of aberrations in ultrasound imaging, according to the prior art;

[0128] Figures 2(a) to 2(f) (already described) illustrate known transmit / receive sequences for ultrasound imaging and quantification;

[0129] Figure 3 illustrates usual confocal imaging and matrix imaging;

[0130] Figure 4 illustrates an example of an ultrasonic characterization system for implementing the methods according to the present description;

[0131] Figure 5 illustrates the definitions used in the method of forming an ultrasound image according to the present disclosure;

[0132] Figures 6(A) to 6(D) illustrate a focusing defect due to an error in the sound speed value;

[0133] Figures 7(a) to 7(d) illustrate the impact of a sound velocity value error on an ultrasound image from simulated ultrasound data;

[0134] Figures 8(al) to 8(a3) show three experimental configurations with different media, a first without an aberrator layer, a second with an aberrator water layer, and a third with an aberrator plexiglass layer;

[0135] Figures 8(bl) to 8(b3) show the ultrasound images obtained without the method of the present disclosure, and corresponding respectively to the middles of Figures 8(al) to 8(a3);

[0136] Figure 8(c) shows the histogram of the estimated focus defects for the different experimental configurations of Figures 8(al) to 8(a3);

[0137] Figures 8(dl) to 8(d3) show the ultrasound images obtained with a first embodiment of the method which corrects a focus defect, and corresponding respectively to the middles of Figures 8(al) to 8(a3);

[0138] Figures 9(a) to 9(d) illustrate the method of correcting the defocusing error;

[0139] Figures 10(a) to 10(c) show focal spot images (modules) obtained without spatio-temporal averaging in Figure 10(a), then with various spatio-temporal averaging variants for the first embodiment of the focus correction method; Figure 10(d) shows the imaginary part of the spatio-temporal averaging variant of Figure 10(c);

[0140] Figure 10(e) shows an axial phase evolution curve of the focal spot image of Figure 10(c);

[0141] Figure 10(f) shows an axial evolution curve of the imaginary part of the focal spot image of Figure 10(c);

[0142] Figure 10(g) shows axial defocus search curves on the focal spot images of Figures 10(b) to 10(d);

[0143] Figure 11(a) shows an ultrasound image of a liver without implementing the method of the present disclosure;

[0144] Figure 11(b) shows a focus defect map (depth / optimal axial deviation) determined by the method of the first embodiment corresponding to the image of Figure 11(a);

[0145] Figure 11(c) shows an optimal integrated sound speed map determined by the method of the second embodiment corresponding to the image of Figure 11(a);

[0146] Figure 11(d) shows a local sound speed map obtained from the map in Figure 11(c);

[0147] Figure 12(a) shows an ultrasound image of a liver without implementing the method of the present disclosure;

[0148] Figure 12(b) shows an ultrasound image corrected from the image of Figure 12(a) and obtained with the method of the first embodiment;

[0149] Figures 12(c) and 12(d) show enlarged areas of the image of Figure 12(a);

[0150] Figures 12(e) and 12(f) show the same enlarged areas in the corrected image of Figure 12(b);

[0151] Figure 13(a) shows an ultrasound image of a liver without implementing the method of the present disclosure;

[0152] Figure 13(b) shows the focused reflection matrix associated with the isochronous volume represented by the gray arc in the ultrasound image of Figure 13(a);

[0153] Figure 13(c) shows a corrected ultrasound image of the liver of the image of Figure 13(a) and obtained by the method of the first focus correction embodiment;

[0154] Figure 13(d) shows the focused reflection matrix associated with the isochronous volume represented by the gray arc on the corrected ultrasound image of Figure 13(c);

[0155] Figure 13(e) shows the transverse evolution of the focal spot deduced from the focused reflection matrices shown in Figures 13(b) and 13(d);

[0156] Figures 14(a) to 14(b) show focal spot images (modules) obtained with various spatio-temporal averaging variants for the second embodiment of the sound speed correction method;

[0157] Figure 14(c) shows the imaginary part of the focal spot image shown in Figure 14(b);

[0158] Figure 14(d) shows axial maximum search curves on the focal spot images shown in Figures 14(a) to 14(c);

[0159] Figure 15(a) shows an ultrasound image with the focus defects superimposed (axial / depth deviations obtained);

[0160] Figure 15(b) shows a corrected ultrasound image of the liver established from the method of the second embodiment of sound velocity correction;

[0161] Figure 15(c) shows a corrected and repositioned ultrasound image of the image of Figure 15(a) from the method of the second embodiment of sound velocity correction;

[0162] Figure 16(a) shows an ultrasound image of a liver without implementing the method of the present disclosure;

[0163] Figure 16(b) shows the focused reflection matrix associated with the isochronous volume represented by the gray arc in the ultrasound image of Figure 16(a);

[0164] Figure 16(c) shows a corrected ultrasound image of the liver obtained by the method of the second embodiment of sound velocity correction;

[0165] Figure 16(d) shows the focused reflection matrix associated with the isochronous volume represented by the gray arc on the corrected ultrasound image of Figure 16(c);

[0166] In the various embodiments described with reference to the figures, similar or identical elements bear the same references unless otherwise stipulated.

[0167] DETAILED DESCRIPTION

[0168] In the following detailed description, only certain embodiments are described in detail to ensure clarity of the disclosure, but these examples are not intended to limit the general scope of the principles emerging from this description.

[0169] The various embodiments and aspects described in this description may be combined or simplified in multiple ways. In particular, the steps of the various methods may be repeated, interchanged, and / or executed in parallel, unless otherwise specified.

[0170] The present description relates to methods and systems for ultrasonic characterization of an environment, and applies in particular to medical imaging of living or non-living tissues. The environment is for example a heterogeneous environment that one seeks to characterize in order, for example, to identify and / or characterize heterogeneities. Optionally, these methods and systems can be applied to the non-destructive testing of products, such as mineral, metallic or other parts. These characterization techniques are notoriously non-invasive for the environment, which is then preserved.

[0171] Figure 4 illustrates an example of an ultrasound imaging system 40 for implementing ultrasound imaging methods of a medium such as a heterogeneous medium 20, according to the present description. This system and method is useful for enabling the formation of an ultrasound image of at least a portion (area of ​​interest or field of vision) of the medium. The system 40 comprises at least one network 10 of transducers 11, for example a linear or two-dimensional or matrix network; the transducers are for example ultrasonic piezoelectric transducers which may conventionally be in the form of a rigid bar placed in direct or indirect contact with the medium 20.The transducer array is for example part of a probing device 41 (usually called a probe); the transducer array is connected to a computing unit 42, which can itself be connected or associated with a display device 43; the computing unit emits and records electrical signals to and / or from each of the transducers 11. The ultrasonic transducers then transform these electrical signals into ultrasonic waves and vice versa. By "connection" or "link" between the probing device 41, the computing unit 42 and the display device 43, we mean any type of wired connection of electrical or optical type, or any type of wireless connection using any protocol such as WiFi™, Bluetooth™ or others. These connections or links are one-way or two-way. The associated display device 43 can be of any type, such as a touchscreen or non-touchscreen, connected or not.

[0172] The calculation unit 42 is configured for the implementation of calculation or processing steps, in particular for the implementation of method steps according to the present description. By convention, a spatial reference frame of the medium 20 is defined, by taking a first axis X and a second axis Z perpendicular thereto. For simplification, the first axis X corresponds to the transverse direction in which the transducers 11 are aligned for a linear array, and the second axis Z corresponds to the depth of the medium 20 relative to this array 10 of transducers 11. This definition can be adapted to the context and thus for example extended to a three-axis spatial reference frame in the case of a two-dimensional array 10, or to a polar reference frame in the case of a curved array 10, or to any other reference frame adapted and / or dependent on the shape of the array 10 of ultrasonic transducers.Thus, in the remainder of this description, we will use a Cartesian XZ reference frame, corresponding to a linear probe, for simplicity in the description, but a specialist in the field would generalize or apply the results to any type of reference frame.

[0173] In Figure 4 as in the rest of the description, reference is made to a network of transducers for transmission and reception, it being understood that, in a more general case, several networks of transducers may be used simultaneously. Similarly, a network may consist of one (1) to N translators, of identical type or of different natures. The transducers may be both transmitters and receivers, or only transmitters for some and only receivers for others.

[0174] The transducer array serves, for example, as both a transmitter and a receiver, or is made up of several sub-arrays of transducers, some dedicated to the transmission, others to the reception of ultrasonic waves. A transducer array means at least one transducer, an aligned or non-aligned sequence of transducers, or a matrix of transducers.

[0175] When in this description, reference is made to calculation or processing steps for the implementation in particular of method steps, it is understood that each calculation or processing step can be implemented by software, hardware, firmware, microcode or any appropriate combination of these technologies or related technologies. When software is used, each calculation or processing step can be implemented by computer program instructions or code which can be, for example, interpreted or executed. These instructions can be stored or transmitted to a storage medium readable by a computer (or computing unit) and / or be executed by a computer (or computing unit) in order to implement these calculation or processing steps.

[0176] Analysis of a midpoint by focused reflection matrix

[0177] This description describes methods and systems for ultrasonic characterization of a medium. In practical cases, the medium is assumed to be heterogeneous. These methods and systems are based on definitions shown in Figure 5:

[0178] We define in the environment:

[0179] - a first point PI of expected spatial position n n in the spatial reference of the middle, and

[0180] - a second point P2 of expected spatial position r ou t in the spatial reference of the middle.

[0181] These spatial positions n n and r O ut are noted in bold, to signify that these elements are position vectors, vectors taken in the spatial reference of the medium (X, Z). Other representations and definitions of the positions of the points are possible and accessible to any ultrasound technician.

[0182] In the present disclosure, the first point PI has a lateral position denoted, x in . The second point P2 has a lateral position noted x ollt . Both points Pl, P2 have the same expected depth z. n = z out , also noted z which is controlled by the echo time t considered. For a medium of homogeneous sound speed, this depth z would be equal to the expected depth zt = cot / 2. ​​Thus, the spatial positions of the points PI and P2 are respectively n n = (x zn , z) and r ou t = (x ou t, z).

[0183] These two points PI and P2 are chosen at a relatively short distance from each other, that is to say a few millimeters from each other, and for example twenty (20) millimeters or less.

[0184] As shown in Figure 5, the ultrasonic characterization method implemented by the computing unit 42 of the system 40 comprises:

[0185] - a step of generating a series of incident ultrasonic waves USm in an area of ​​said medium, by means of an array 10 of transducers 11, said series of incident ultrasonic waves being an emission base i; and

[0186] - a step of generation (construction / measurement) of an experimental reflection matrix Rui(t) defined between the emission base i at the input and a reception base u at the output; - a step of determination of a focused reflection matrix R xx (t, z, co) which includes responses of the medium between a virtual input transducer TVm of spatial position n n and a virtual TV output transducer ou t of spatial position r O ut.

[0187] This focused reflection matrix R xx can be expressed in different ways. In the expressions of the present disclosure, the first point PI of spatial position (x in, z) is taken as a reference. But, we can establish these expressions with respect to the second point P2 of spatial position (x out , z) or relative to a midpoint between PI and P2, of spatial position ((x in +x out ) / 2, z) or relative to any reference point. The domain technician will be able to make the necessary variable changes in the expressions presented here.

[0188] The responses of the medium are calculated by forming channels from the experimental reflection matrix Rui(t).

[0189] The answers of the focused reflection matrix R xx ( / , z, co) correspond to an acoustic pressure field calculated between all midpoints of lateral positions x in , and x ou t, located at the expected depth z and at an echo time / , and for an assumed sound speed co. In other words, this focused reflection matrix R xx ( / , z, co) is defined by: RXX ( / , Z, co) = [RfXout, Xin, t, Z, co)}

[0190] The parameters of depth z in the medium and sound speed co control the delay laws used in the path formation process (i.e. focusing process).

[0191] The input emission base i is for example a base of waves each generated by a single one of the transducers 11 of the network 10 or a base of plane waves of angular inclination 0 relative to the X axis or a base of virtual sources, as described previously in the description of figures 2(a) to 2(f).

[0192] The reception base u is for example the base of the transducers 11. Optionally, another reception base can be used for reception, for example a frequency or spectral base.

[0193] Thus, the ultrasonic wave generation step is understood between the emission base i and the reception base u. This ultrasonic generation step is therefore defined for any type of focused or unfocused ultrasonic waves, such as plane waves.

[0194] In the matrix generation step, the experimental reflection matrix Rui(t) is defined between the emission base i as input and a reception base u as output. This matrix contains all the temporal responses of the medium, measured at time t by each transducer 11 of spatial coordinate Uout and for each emission iin. It is understood that the elements denoted with the index "in" refer to the emission (i.e. the input) and the elements denoted with the index "out" refer to the reception (i.e. the output). This experimental matrix can also be recorded and / or stored, for example in the memory of the computing unit, or on any other medium, removable or not, allowing permanent or temporary storage.

[0195] More precisely, in the step of determining the focused reflection matrix R xx (t, z, c0), we apply:

[0196] - an input focusing process from the experimental reflection matrix Rui(t) which uses a time of flight on the outward journey of the waves between the emission base i and the virtual input transducer TVm and which creates a so-called input focal spot around the first point PI of spatial position n n , said input focal spot corresponding to the virtual input transducer TVin,

[0197] - an output focusing process from the experimental reflection matrix Rui(t) which uses a return flight time of the waves between the virtual output transducer TV ou t) and the transducers of the reception base u and which creates a focal spot called output around the second point P2 of spatial position r ou t, said output focal spot corresponding to the virtual output transducer TVout.

[0198] These input and output focusing processes actually form an input-output focusing process, referred to in the remainder of this description as the focusing process.

[0199] In other words, in this ultrasonic characterization process, the virtual input transducer TVm corresponds to an ultrasonic "virtual source" located at the spatial position n n in the medium and the virtual output transducer TVout corresponds to an ultrasonic "virtual sensor" located at the spatial position r O ut. This virtual source and sensor are spatially separated by the difference of their spatial positions Ar = r Out - n». In the present case, they are simply separated laterally by Ax = Xout X / n- Their expected depth is the parameter z used in the focusing law for a sound speed model co. Their actual depth is dictated by the axial (depth) position of the isochronous volume, i.e. by the echo time t and by the sound speed distribution c(r) in the medium. The lateral dimension of the virtual transducers is dictated by the focal spot produced by focusing at this actual depth. For example, a calculation of the focused reflection matrix R xx (t, z, co) of the medium between the virtual input transducer TVm and the virtual output transducer TVout by said input and output focusing processes, is an improved path forming method, which can be expressed by the following simplified formula: in which

[0200] Nm is the number of elements of the emission base i,

[0201] Nout is the number of elements of the receiving base u at output, is the element of the experimental reflection matrix Rui(t) recorded by the transducer Uout following the emission iin to which the delay times r and r' have been applied, and z, CQ) is the time delay or time shift that must be applied to the emission to make each incident wave iin constructively interfere at the first focusing point of expected spatial position (x in , z) for a sound speed model co. This time shift is deduced from the outward flight time of the ultrasonic wave between the transducers of the emission base i and the virtual input transducer TVm of expected spatial position n n (first point PI). z. CQ) is the delay time or time shift that must be applied to the signals measured by each output transducer u out to constructively interfere with echoes from a scatterer located at the expected spatial position (x ou t, z) for the same velocity model co. This time shift is deduced from the return flight time of the ultrasonic wave between the virtual output transducer TV ou t of spatial position r O ut (second point P2) and the transducers of the receiving base u.

[0202] These delay times r and r' are calculated from a sound speed model. The simplest assumption is to assume a homogeneous medium with a constant sound speed co. In this case, the flight times are directly obtained from the distances between the probe transducers and the virtual transducers.

[0203] For example, in the special case of a plane wave with an emission angle 6L - the emission delay time r' can be obtained by:

[0204] Delay Flight time in which z in / co is the time to reach the virtual input transducer.

[0205] - the reception delay time r can be obtained by:

[0206] Delay Flight time in which z out / co is the time to reach the virtual output transducer.

[0207] Thus, these examples of delay time calculations clearly show that they are a function of the type of wave and the speed of sound, assumed here to be constant in the medium.

[0208] The number of elements of the transmission base Nm is for example greater than or equal to one (1), and advantageously greater than or equal to two (2). The number of elements of the reception base N ou t is for example greater than or equal to two (2).

[0209] This improved path-forming formula is therefore a double sum of the temporal responses recorded in the experimental reflection matrix Rui, a first sum according to the emission basis i translating a focusing at emission and a second sum according to the reception basis u linked to a focusing at reception, this calculation being carried out for the spatial coordinates of the two points PI and P2, that is to say the respective points of expected spatial positions n n = (x !n , z) and r ou t = (x out , z). The result of this improved channel formation formula is therefore a time signal for these two spatial coordinates (n n , r ou t) or for these two lateral positions •V / w, Xout-

[0210] Such a channel-forming formulation can also be supplemented by input and output weighting terms, often called transmit and / or receive apodization.

[0211] The recorded experimental reflection matrix Rui(t) may be a "real" matrix, i.e. composed of real coefficients in the time domain, the electrical signals recorded by each of the transducers being real numbers. Alternatively, this matrix may be a "complex" matrix, i.e. composed of complex values, for example in the case of demodulation for in-phase and quadrature channel formation (known in English as "IQ beamforming").

[0212] Thus, it is possible to deduce from this temporal response after channel formation, a line of the ultrasound image of the medium by considering the absolute value of the confocal signals characterized by identical lateral positions at the input and output (x in = x out ) and at the ballistic depth z = z t= cot / 2. ​​Thus, to construct an ultrasound image, one can scan or choose a set of lateral positions X Xin Xout that correspond to a set of lines in the ultrasound image. The ultrasound image / can then be constructed from the diagonal coefficients of the focused reflection matrix, by:

[0213] The axial dimension of the ultrasound image is thus dictated by the time of flight t of the echoes. If the velocity model co coincides with the speed of sound in the medium, the ultrasound image illustrates a reliable estimate of the reflectivity of the medium allowing the depth of the scatterers to be assessed.

[0214] However, in real conditions (in-vivo in medical imaging), the sound speed model co often does not coincide with the sound speed distribution c(r) in the medium, which can also be often heterogeneous or even very heterogeneous. To understand the impact that the mismatch between co and c can have, the analogy between the formation of channels and a digital time reversal experiment can be enlightening. The latter is described in Figure 6 where we seek to re-propagate by time reversal the wave backscattered by a diffuser s in a medium of sound speed c, as illustrated in Figure 6(a) and Figure 6(b), in a new medium of sound speed co.

[0215] If c = co, the refocusing plane, z = Zf, of the temporally returned wave coincides with the depth of the diffuser z s and with the position of the isochronous volume, z t = cot / 2, as shown in Figure 6(c).

[0216] For c < co (or conversely c > co), the focusing of the wave takes place upstream (or conversely downstream) at a depth zy, such that: under the paraxial approximation while the isochronous volume associated with the target's time of flight appears downstream (or reciprocally upstream) of the focal plane, at a depth z t , such as :

[0217] Thus the ultrasound image shows the diffuser at the wrong depth (z t z$) and its image is degraded because the wave is totally defocused at the level of the isochronous volume (zf^ z t ) as seen in Figure 6(d). In other words, the depth of the isochronous volume z twhich depends on the flight time t and the sound speed cO is different from the focusing depth zy of the ultrasonic wave in this specific example. Figure 7 illustrates the impact of a bad sound speed model on the ultrasound image by means of a k-wave simulation of an acoustic phantom (c = 1700 m / s) mimicking the properties of soft tissues (ultrasonic speckle) and presenting echogenic diffusers allowing to appreciate the contrast and the resolution of the formed image.

[0218] If the sound speed model co is incorrect, as shown in Figure 7(b) where a sound speed co of 1850 m / s is used, the scatterers appear at the wrong depth compared to Figure 7(a) which represents the simulated medium model. Furthermore, the dissociation between the focusing depth zf and the isochronous volume depth z tgenerates an enlarged and distorted focal spot, hence the poor transverse / lateral resolution of each bright point in this image 7(b).

[0219] Thus, there is an uncertainty for each ultrasound image on the axial dimension of the image which is governed by the echo time t and which makes any distance measurement hazardous. Furthermore, any disagreement between the sound speed model co and the actual sound speed distribution c(r) generates significant aberrations on the ultrasound image. The resolution and contrast of the latter are then unsatisfactory for use of the image, particularly in medical imaging.

[0220] To solve these problems, we will consider two methods:

[0221] 1) According to a first embodiment of the method for forming an ultrasound image, a first approach is used which consists of dissociating the depth z of the virtual transducers from the echo time t in order to make the focal plane and the isochronous volume coincide. The coincidence is obtained for each point of the image by optimizing the confocal spot with respect to the parameter z which controls the curvature of the applied delay laws.

[0222] 2) According to a second embodiment of the method for forming an ultrasound image, a second approach is used which consists of varying (scanning) the sound speed model co. The coincidence between the focusing plane and the isochronous volume is obtained by optimizing the confocal spot with respect to the sound speed model co. 1) Compensation for a focus defect / focus position defect

[0223] The first approach of the first embodiment of the method for characterizing the medium by ultrasound is for example adapted to the following situation: the imaging of a medium whose tissues have a homogeneous sound speed c and equal to the speed model (c = co) but whose ultrasound image is degraded by the presence of several layers of tissues of sound speed c{ l> different upstream of these tissues; (i) is an index in a plurality of layers, and (1) means layer or “layer” in English.

[0224] Figure 8 illustrates the problem we are facing. The results of three experiments are reported. The first experiment in Figure 8(ai) consists of a reference experiment on an ultrasonic phantom mimicking the acoustic properties of a medium whose effective sound speed is substantially homogeneous (c = 1540 m / s). This phantom contains a random distribution of under-resolved scatterers generating an ultrasonic speckle, such as in a soft tissue (agar-agar in our example), as well as several echogenic scatterers (nylon threads) and a cylinder with acoustic characteristics different from the surrounding speckle (symbolized by the 5 points and the black disk in Figures 8(ai) to 8(as)).

[0225] In the other two experiments (Figures 8(az) and Figure 8(as)), a first aberrator layer is introduced between the transducer array (the probe) and the phantom (the medium).

[0226] In figure 8(az) there is a single layer of water with a uniform sound speed a = 1480 m / s and in figure 8(a3) there is a single layer formed from a plexiglass plate with a uniform sound speed ci = 2690 m / s.

[0227] Although the sound velocity pattern co coincides with the sound velocity in the phantom in the present case, the first layer (water) degrades the resolution of the ultrasound image compared to the reference image, as evidenced by the impaired resolution of the bright spots and the degraded speckle contrast of the corresponding ultrasound image in Figure 8(bz) compared to the reference image in Figure 8(bi). Due to a significantly different sound velocity than the phantom, the Plexiglas plate causes a stronger aberration on the ultrasound image and spurious echoes related to multiple reverberations as represented by the corresponding ultrasound image in Figure 8(bs).

[0228] To understand the nature of the aberration induced by a layer with a different sound speed, we take the analogy in Figure 6 between channel formation and digital time reversal, and we represent this illustration of the associated reasoning in Figure 9 in the case of a multi-layer medium (two layers).

[0229] A diffuser s at depth z s is associated with the echo time t s = 2. [fz s - z / ) / co + zi / a]

[0230] It therefore appears at depth z t in the image of figure 9(c):

[0231] The diffuser image is highly aberrated due to the mismatch between the sound speed model used and the sound speed distribution upstream of the diffuser due to the first aberrator layer &. Under a paraxial approximation, it is known that the position of the focusing plane is indeed given by:

[0232] Zf = zt + Azf (6) with with zi the thickness of the first aberrator layer.

[0233] The parameter Az / can be considered as a defocusing defect represented in Figure 9(c). Remarkably, the latter does not depend on the position z s of the diffuser in the second middle layer if the sound speed model is exact and equal to co: c =co.

[0234] To correct this “focusing defect” or in other words focusing position, the method of the present disclosure proposes to vary the focusing depth z independently of the time of flight t while these two parameters are generally intrinsically linked in a conventional beamforming process.

[0235] We can then construct a de-scanned reflection matrix R', from the focused reflection matrix R by rearranging the rows and columns to obtain a two-dimensional matrix, for the purpose of easier and less expensive matrix calculations, particularly in terms of their duration, which is an important advantage in the context of real-time or quasi-real-time use of the system. Each column of the de-scanned reflection matrix corresponds to an element of a pair {x, t} and each row corresponds to an element of a pair {Ax, Az}. This new de-scanned reflection matrix is ​​therefore defined by: R' = [^({A^ A^jM})]

[0236] In other words, the unscanned reflection matrix R' is expressed in a reference base of the first point PI of spatial position (x in , z in ) with x in = x, z in = z t + dz and with the second point P2 of spatial position (x ou tz out) referenced relative to the first point PI, i.e. Xout = x + Zlx, z O ut = z t + Az. This expression can also be written in another base, such as that of the second point P2 or the point located in the middle of PI and P2, or any other reference point, by a simple change of variable accessible to a technician in the field.

[0237] This de-scanned reflection matrix R' makes it possible to easily extract for each point {x, 1} in the middle, a local image of the focal spot in the dimensions {Ax, Az}, with:

[0238] Ax = x oll t - xtn, the lateral gap between the input and output transducers; and

[0239] Az the out-of-focus value between the depth of focus z and the depth z t expected isochronous volume for a medium of sound speed co.

[0240] Thus, the unscanned reflection matrix R' is constructed from the focused reflection matrix R by:

[0241] R\{Ax, Az}, {x, É}) =R(x + Ax, x, t, z t + Az) (8)

[0242] To simplify the expressions, we can introduce a position deviation as the pair comprising the lateral deviation and the depth deviation: Ar = {Ax, Az}.

[0243] We can also construct the unscanned reflection matrix R', directly from the experimental reflection matrix Rui. Thus, the unscanned reflection matrix includes the path-forming calculation of the focused reflection matrix R, and we can note:

[0244] J? z ({Ax, Az}, {x, t}, CQ) =

[0245] 1

[0246] Ar Kr oiit

[0247] In other words, in all cases, the method according to the first embodiment comprises a step of determining a de-scanned reflection matrix R', which comprises, for a set of points in the middle of abscissa x and echo time t, responses of the middle calculated by channel formations from the experimental reflection matrix Rui(t), between a virtual input transducer of expected spatial position n n = (x, zx Az) and a virtual output transducer of expected spatial position r O ut = (x+Ax, z t +Az), the two virtual transducers being located at the same expected depth z, i Az, z tbeing the depth of the isochronous volume expected for a sound speed co, the isochronous volume being the set of points in the medium contributing to the signals received at echo time / , the responses of the unscanned reflection matrix R' being determined for a set of lateral deviation values ​​Ax comprising at least Ax = 0 and for a set of depth deviation values ​​Az, said unscanned reflection matrix R' being formed in such a way that each row corresponds to an element of a pair Ar = {Ax, Az} which corresponds to a focal spot image and each column corresponding to an element of a pair {x, t}, which is noted:

[0248] Figure 10(a) shows an example image representing the apparent focal spot for a point {x, t} in the field of view that can be visualized from the previous unscanned reflection matrix. This image corresponds to a column of the unscanned reflection matrix R', shown as a two-dimensional image with the lateral deviation Ax on the abscissa and the depth deviation Az on the ordinate.

[0249] The method further comprises:

[0250] An extraction step in which an optimal position deviation zlropt close to the maximum in said focal spot image is determined from each focal spot image.

[0251] This optimal position deviation zlropt = {Ax op t, Az opt} corresponds substantially to the defect of focus or a defect of focus position of the processes of formation of tracks.

[0252] The maximum in the focal spot image is the point in this focal spot image having a maximum modulus value compared to the entire focal spot. This means that this maximum corresponds to the point in this focal spot image with the largest modulus value. By "close to the maximum", we mean that we extract a position close to the position of this maximum point. In fact, we can take a point as close as possible, or for example another point according to different criteria such as a point having a zero lateral deviation Ax (Ax = 0), or another point between these two, or by another criterion of maximum point neighborhood. The neighborhood can be defined for example from a desired spatial resolution. This apparent focal spot for example as illustrated in figure 10(a) is modulated both by the random reflectivity of the medium and noisy by the multiple scattering echoes occurring upstream of the isochronous volume.

[0253] To overcome these speckle problems and to have a more precise estimator of the focal spot and therefore of the focusing quality, an average can be performed on the adjacent speckle grains. For this, we consider a spatio-temporal window of dimension (L x , L t ) on which we will incoherently average the focal spot around each virtual transducer of spatial position n n .

[0254] Thus, the method according to the first embodiment may optionally comprise, before the extraction step: a step of spatio-temporal averaging of the de-scanned reflection matrix R' in which the values ​​of the de-scanned reflection matrix R' are locally averaged, along the lateral dimension x and along the echo time dimension / , for each element of a pair Ar = {Ax, Az}, the focal spot images then being averaged.

[0255] Thus, the spatio-temporal averaging step makes it possible to average or smooth the focal spot image from the unscanned reflection matrix, in order to more correctly extract the position of the maximum in this image, a position which corresponds to a deviation to be used to correct the calculation of the ultrasound image, as will be explained later.

[0256] This spatio-temporal averaging step can be implemented in various ways, based on various weighted or unweighted averaging formulas, with or without normalization.

[0257] According to a first variant of spatio-temporal averaging, represented in figure 10(b), we use an incoherent point spread function PSFmc (or “incoherent point spread function” in English) which can be calculated by: in which

[0258] < > is an averaging operator, which calculates an average based on the parameters {xt '}

[0259] W(xt ') is a spatio-temporal weighting window.

[0260] For example, the spatio-temporal weighting window is rectangular, Hanning, or Gaussian. The rectangular spatio-temporal weighting window can be defined by:

[0261] W(x', t') = 1 for |xj < L x / 2 and \tj < L t / 2, and (10) W(x t') = 0 otherwise

[0262] The previous incoherent point spread function PSF inc provides for a set of lateral deviation values ​​zlx including at least Ax = 0 and for a set of depth deviation values ​​dz, denoted in the form of the pair Zlr = {Ax, Az}, values ​​corresponding to a focal spot image for the lateral position x and the echo time t. This focal spot image is smoothed of speckle and multiple scattering effects, as shown in figure 10(b).

[0263] The depth position of the maximum of this focal spot image, Az opZ , constitutes a first relevant estimator of the defocusing defect Az / sought in equation 7.

[0264] In this case, the position of the maximum in the depth direction, Az op r, can be estimated by the position of the maximum of the modulus of the point spread function PSF inc .

[0265] More generally, the position of the maximum Ar opt (mc) can be estimated by the position of the maximum of the modulus of the point spread function PSF inc .

[0266] Thus, we can extract the position deviation from the maximum AY opt (mc)by this same formula 11, that is to say the position of the maximum in the focal spot image (focal spot image by spatio-temporal averaging). This position of the maximum corresponds to the defect of focusing or defect of focusing position by the processes of formation of paths, because of variation of the speed of the sound in the medium, that is to say medium presenting inhomogeneities in the speed of the sound of the different tissues composing it.

[0267] In this way, during the extraction step, a maximum point of the focal spot image is sought corresponding to or close to a maximum of the modulus of the points of the focal spot image calculated by the incoherent point spread function, and the optimal position deviation Zlr ( , pl is the position of said maximum point in said focal spot image, or a position close to this maximum point, as explained previously.

[0268] Note that in practice it is not always necessary to calculate or prepare the complete image of the focal spot, and in particular it is often not necessary to scan the points along the transverse direction Ax, since the maximum is a priori located on the axis Ax= 0 or very close to this axis Ax= 0. Indeed, to estimate the defocusing error in the depth direction, Az, we can simply study the axial evolution of the confocal spot, that is to say for a set of points with Ax = 0, i.e. for only spatial positions x in = x ou t as illustrated by the first curve G1 of figure 10(g) which represents the evolution of the values ​​of the focal spot image of figure 10(b) on the axis Ax= 0.

[0269] This estimator is however biased by an incoherent background highlighted by a relatively low contrast as seen in figure 10(b) and generates a spread of the peak of the first curve G1 of figure 10(g).

[0270] According to a second variant of spatio-temporal averaging, the confocal spot can be extracted consistently by intelligently recombining the apparent focal spots obtained on each window W. This allows to obtain a more precise measurement of the depth deviation Az, and more generally of the position deviation Ar.

[0271] This recombination is carried out in practice using a singular value decomposition SVD of a local matrix of focal spots, considered for each space-time window W(x, t), and which is expressed by: in which the singular values ​​ / . P are arranged for example in descending order.

[0272] Each of the singular values ​​X p is associated:

[0273] - to an output eigenvector U p ({x, t}) {x, É})] defined in a de-scanned focused basis (Ar), and

[0274] - to an input eigenvector V p ({x, t}') =[V p ( {x / , t'}, {x, t})] defined in the conventional focused basis, i.e. the basis of the pixels of the ultrasound image.

[0275] Among these eigenvectors, the first eigenspace is of particular interest since:

[0276] (i) the coefficients of Vi({x, z'}) directly give the phase shift to be applied to each speckle grain in {xt '} to put them back in phase;

[0277] (ii) the vector Ui({x, / }) gives the coherent confocal spot resulting from the phase recombination of each apparent focal spot measured around each point {x t'}: Then, a coherent point spread function PSFco h (or “coherent point spread function” in English) which can be calculated using:

[0278] PSF„,„(Ar. {. c , Z}) = [ / , (Ar. { :t , t})

[0279] Each coherent confocal spot PSF co h(&r, {x, / }) precisely quantifies the focusing process at each point {x, t} of the acoustic field. Due to its coherent nature, this coherent point spread function PSF co h allows to obtain an image with a much better contrast as illustrated by the image in figure 10(c) (corresponding to the amplitude of the coherent point spread function PSF coh) than an incoherent average of the confocal spots as illustrated by the image in Figure 10(b). The singular value decomposition SVD defined by equation 12 allows to recombine each focal spot by compensating the phase of each speckle grain at the point {x', t '}. In doing so, we drastically reduce the echoes linked to scattering events outside the focal point {x, t}.

[0280] As previously realized from the incoherent point spread function PSF inc (see equation 12), a new estimator of the defocusing error can be constructed by considering the value of the depth deviation Az maximizing the amplitude of the coherent point spread function PSF co h and illustrated on curve G2 in figure 10(g).

[0281] More generally, we consider the position of the maximum Ar op t (coh) of the coherent point spread function by:

[0282] Aiÿl) (x, t) = argmax (| PSF coh (Ar, {rc, t}) |) Ar

[0283] The uncertainty of the Az^pf^ estimator is for example given (in the depth direction) by the ratio between the depth of field of the imaging system and the square root of the number of speckle grains N s covered by the space-time window W, i.e.:

[0284] More generally, we can extract the position deviation from the maximum A r o / , / = {Ax opt , Az op t} by this same formula 14, that is to say the position of the maximum in the focal spot image (focal spot image by spatio-temporal averaging). This position of the maximum corresponds to the defect of focusing or defect of focusing position by the processes of formation of channels, induced by the non-homogeneity of the speed of sound in the medium.

[0285] In this way, during the extraction step, a maximum point of the focal spot image is sought corresponding to or close to a maximum of the modulus of the points of the focal spot image calculated by the coherent point spread function, and the optimal position deviation zlr O pt is the position of said maximum point in said focal spot image, or a position close to this maximum point, as explained previously.

[0286] According to a third variant of spatio-temporal averaging, one can also use the information contained in the phase of the coherent confocal spot as illustrated in Figure 10(d) for example by examining the imaginary part of the coherent point spread function PSF co h. This allows us to obtain a more precise estimator of the depth deviation Az, and more generally of the position deviation Ar.

[0287] The phase of the confocal spot indeed presents a phase jump of n at the level of the focusing plane as visualized in Figure 10(e) which represents the phase of the coherent point spread function PSF co h on the axis Ax= 0. The coherent confocal spot indeed accumulates the Gouy phase jumps (~ n / 2 in a 2D configuration) induced by the input and output focusing processes. These Gouy phase jumps result at the depth z + Az / where the transverse confinement of the focused waves is minimal. It therefore constitutes a more reliable estimator of the defocusing error than the modulus of the coherent point spread functions PSF co h or the incoherent point spread function PSF incas can be seen from the G3 curve in Figure 10(g) which are also sensitive to the geometric decay of the focused wave. To increase the sensitivity of the method, one can also combine the two previous observables, the modulus and the phase of the coherent point spread function PSFcoh, by examining the imaginary part of this function (see curve in Figure 10(f) which represents the imaginary part of the coherent point spread function PSF co h on the axis Ax 0). Its inflection point, i.e. the maximum of its axial derivative or derivative along the depth direction Az, provides a new estimator of the optimal depth deviation kz op t (see curve G4 in figure 10(g)).

[0288] This estimator can be generalized by finding the maximum of the modulus of the complex derivative of the point spread function PSF co h:

[0289] Arfe uy ^(ay t) = argmax (|â^ zPSF co h(Ar, { / :, t}) |)

[0290] Ar

[0291] (16) According to an uncertainty calculation, this estimator significantly improves the determination of Az by a factor y / Q, i.e. approximately ~ 2.5 compared to ^£(coh) of equation 15.

[0292] Figure 8(c) shows the histogram of the estimated defocusing errors Az opt for the three experiments described in Figure 8(a). While the reference experiment results in an estimate of the defocus close to zero (0), Az opt is close to its expected value, whether for the water layer (Az o / ) / = -1.4 mm) or for the plexiglass plate (Az o / ) / = +8.5 mm).

[0293] The scatter in the estimated defocus values ​​highlighted by Figure 8(c) is related to the paraxial approximation used to establish Equation 7 and which is not strictly verified in the experiments of Figure 8(a).

[0294] More generally, we can extract the position deviation from the maximum A r^, / = {Ax opt , Az opt} by this same equation 16, that is to say the position of the maximum in the focal spot image (focal spot image by spatio-temporal averaging). This position of the maximum corresponds to the defect of focusing or defect of focusing position by the processes of formation of channels, because of variation of the speed of sound in the medium.

[0295] In this way, during the extraction step, we search for a maximum point of the focal spot image corresponding to:

[0296] - a maximum of the modulus of the points (values) of the focal spot image, or at

[0297] - a maximum of the modulus of the derivative according to the depth direction (axial direction) of the points (values) of the focal spot image, or at

[0298] - a maximum of the modulus of the derivative along the depth direction (axial direction) of the imaginary part of the points (values) of the focal spot image.

[0299] The optimal position deviation zlropt is the position of the maximum point in said focal spot image, or a position close to this maximum point, as explained previously.

[0300] The optimal position deviation Ar op t or the optimum depth difference \z„ pl can be used to obtain a new correctly focused ultrasound image, i.e. with correction of the focusing position error:

[0301] - either from the focused reflection matrix R:

[0302] - either directly from the unscanned reflection matrix R': Or if we consider that Ax opt = 0 or has a negligible value, by:

[0303] The optimal position deviation A can be advantageously exploited with its two components, as expressed in equation 17 above.

[0304] Thus, the method according to the first embodiment of the method may further comprise: a step of forming an ultrasound image by calculating the reflectivity at a plurality of points in the medium, from the unscanned reflection matrix R', the reflectivity of each of these points in the medium corresponding to the response of the medium between the virtual input transducer of spatial position n n = (x, z t +Az opt ) and the virtual spatial position output transducer r O ut= r + Ar opt = (x+Ax opt , z t +Az opt ), and said reflectivity being defined by:

[0305] The ultrasound image shown above is parameterized as a function of the lateral position x and the echo time t.

[0306] This ultrasound image can be transformed to be parameterized as a function of the lateral position x and depth z using the sound speed model co to replace the echo time t. This transformation consists of a change of variables or parameters. The resulting image is identical, but with dimensions in distances as usually used, and on which the practitioner can make measurements of the sizes of structures. Then, the ultrasound image is of the type: in which z t is the depth of a pixel of the ultrasound image or depth of the isochornic volume.

[0307] Either of the preceding ultrasound images may be displayed on the display device 43.

[0308] According to a variant, a film is formed from a succession of ultrasound images by a first number N of iterations of the steps of the method. Thus, the method is iterated to form ultrasound images at a predetermined rate, and to visualize a substantially real-time evolution of the ultrasound image of the medium.

[0309] According to a variant:

[0310] - among the N iterations, a second number P less than N of iterations does not include the steps of determining the unscanned reflection matrix R', the spatio-temporal averaging steps, and the steps of extracting an optimal position deviation, and for these P (particular) iterations, at the step of forming the ultrasound image, the ultrasound image is obtained from an optimal position deviation extracted during a previous iteration.

[0311] Thus, for a number P of iterations, the calculations of the steps of determining the unscanned reflection matrix R', the spatio-temporal averaging steps, and the steps of extracting an optimal position deviation are advantageously avoided. The method of this variant requires fewer computing resources and / or time while providing a film of a succession of high-quality ultrasound images. Thanks to this arrangement, the rate of determining the ultrasound image can be decoupled from the rate of calculating the optimal position deviation. Thus, a high ultrasound image rate can be maintained despite a reduced number of calculations.

[0312] Furthermore, the previous calculations of the method according to the present description can be carried out only in the axial direction (depth direction z). Thus, in the extraction step, an optimal position deviation ztr is determined op t= {Ax opt , Az opt} . In the present case, we can limit ourselves to determining only the second component of the optimal position deviation, in depth, i.e. the depth deviation Az op t. Thus, according to this variant, we consider that the first component of the optimal position deviation, i.e. the lateral deviation Ax opt is zero, Axopt = 0. This simplifies the calculation of the ultrasound image and reduces the overall cost in time and resources.

[0313] Similarly, in the step of determining a de-scanned reflection matrix R', the de-scanned reflection matrix R' can be formed only for lateral deviation values ​​Ax of zero value, Ax = 0. This avoids calculating a complete de-scanned reflection matrix, which greatly reduces the overall cost in time and resources of calculating this de-scanned reflection matrix, but it also reduces the cost of calculating the steps of the method after this step of determining the de-scanned reflection matrix.

[0314] Optionally, at the spatio-temporal averaging step, one can limit oneself to averaging the values ​​of the unscanned reflection matrix R' along the lateral dimension x and along the echo time dimension / , to the elements of the pairs {Ax = 0, Az} of the focal spot images whose lateral deviation Ax is zero (Ax = 0). Thus, the number of averaged focal spot images to be calculated is reduced, and the computational cost is reduced. In addition, at the extraction step, the optimal depth deviation Az op t can be determined from the points or values ​​of the focal spot image having a lateral deviation Ax of zero value, Ax = 0. Thus, the optimal lateral deviation Ax op t is zero, and the optimal depth gap Az opt has a value determined only with the points of a curve of the focal spot image corresponding to a zero lateral deviation. Thus, the determined optimal spatial position zlr oplis slightly different from that of the maximum of the focal image, but remains close to it because by essence the lateral deviation must be small. The computational cost of this variant is thus advantageously very reduced.

[0315] The ultrasound images corresponding to the results of the method of the present description, correcting the defect in focus or defect in the position of the focusing plane, are illustrated in Figure 8(d). In the case of the first layer of water, the corrected ultrasound image is shown in Figure 8(dz). This ultrasound image regains the resolution of the reference image of Figure 8(di), that is to say the ultrasound image obtained without this first layer of aberration. The application of a single correction on the curvature of the focusing laws (which the present correction achieves) makes it possible to correct the entire ultrasound image. The same effect is observed in the case of the first layer with a plexiglass plate in Figure 8(d3). This result is spectacular when comparing this corrected ultrasound image to the initial image of Figure 8(b3).Note, however, that this correction of the focus defect does not, however, allow for the complete compensation of the reverberations induced by the first aberrator layer.

[0316] The results of corrections of the ultrasound images in Figure 8 are interesting from an academic point of view but they remain limited because these cases only consider a translation-invariant aberrator layer. Figure 11 presents an in-vivo ultrasound imaging experiment on a liver in which a natural irregular arrangement of adipose and muscular tissues, and other vessels, alters the quality of the ultrasound image (see Figure 11(a)). The method described previously is extended to the case of a curved probe, which amounts to replacing the Cartesian coordinates (x, z) by polar coordinates (0, r). Figure 11(b) presents a mapping of the estimated defocus at each point of the image, i.e. a mapping of an axial position deviation Ar calculated with the method. Unlike the previous experiments in Figure 8, this defocus is not homogeneous over the field of view.This is particularly related to the lateral variations (polar Aff) of the sound speed in the aberrator layers which result in a fairly large lateral variation of the defocusing error. The axial variation of the defocusing error in the liver is related to the disagreement between the sound speed model co and the effective sound speed c in the liver.

[0317] Finally, the above method can be extended to further determine an integrated speed of sound c opt from the optimal depth deviation Az opt , from the optimal position zJr opl = {Ax opt , Az opt}, by :

[0318] This integrated sound speed corresponds to an average sound speed between the transducer array and the depth of the isochronous volume. This integrated sound speed can be determined for any pair {x, t}, i.e. for any focal spot image, i.e. for any point in the middle or any point of an ultrasound image.

[0319] The local sound speeds c(r) can then be calculated for a set of pairs {x, t}, from integrated sound speed values ​​c optobtained for the focal spot images of the pairs {x, t} . This calculation allowing the deduction of local sound speeds from an integrated sound speed field is generally carried out by an inversion method. An example of an inversion method of this type will be described in more detail later, at the end of the description concerning the second embodiment of the method, more specifically dedicated to the more precise determination of a sound speed model (integrated sound speed model).

[0320] The method may then further comprise a step of forming a local sound speed image of the medium from the set of previously determined integrated sound speeds, and a step of displaying the local sound speed image on the display device 43. This local sound speed image is important for the practitioner, as it is useful for, combined with other information, identifying certain pathological states or diseases, such as fatty liver.

[0321] Having obtained a local sound speed field, it is then possible to iterate the method described above in order to obtain a more precise ultrasonic characterization of the medium, by recalculating a de-scanned reflection matrix R' using the local sound speed values ​​determined in a previous step. As described above for this first embodiment of the method, each pixel of the ultrasound image can be corrected from the locally estimated defocus (see equation 17). The result is shown in Figure 12. The corrected image of Figure 12(b) advantageously shows a gain in terms of contrast and resolution compared to the initial image of Figure 12(a). Veins that are difficult to discern in the initial image are clearly revealed after a local correction of the defocus on all or part of the pixels of the image.The correction of transverse aberrations is also quantified in Figure 13 which shows an example of focused reflection matrices, for illustrative purposes, before and after correction. While initially the backscattered energy extends well beyond the diagonal of the focused reflection matrix R. xx shown in Figure 13(b), the out-of-focus compensation "brings" most of the backscattered energy back to the diagonal of the focused reflection matrix R xx as illustrated in its representation in Figure 13(d). This is confirmed by the study of the transverse spreading of the incoherent point spread function PSF inc before and after compensation for the defocus as illustrated by the curves in Figure 13(e).

[0322] The presented method is particularly effective in compensating for axial aberrations (along the depth direction) of an ultrasound image. However, it does not allow for the scatterers to be perfectly relocated to their actual depth; the axial direction of the image remains dictated by the time of flight t of the echoes, such that z t = cot / 2. ​​To this end, the following section proposes to no longer scan the curvature of the focusing laws to determine a focusing defect, but to directly scan the sound speed model in order to map the latter and thus reposition each diffuser at its real depth on the ultrasound image.

[0323] 2) Sound speed tomography

[0324] The second approach to the second embodiment of the method for characterizing a medium by ultrasound consists of varying the sound speed co of the sound speed model. A focused reflection matrix R is then constructed xx ,t(co) at ballistic depth z t including the different models of sound speed co. The approach is therefore similar to that of the first approach, with the variation of the parameter of the sound speed co model, and a technician in the field will be able to take up and adapt the calculation methods.

[0325] Similar to equation 1, this focused reflection matrix can be written by removing the depth parameter z, since now this depth is adjusted to the expected depth of the isochronous volume z t :

[0326] An unscanned reflection matrix R' can thus be constructed from the focused reflection matrix R, by rearranging the rows and columns of this matrix to obtain a two-dimensional matrix for easier matrix calculations. Each column of the unscanned reflection matrix corresponds to an element of a pair {x, t} and each row corresponds to an element of a pair {Ax, co}. Each row of the unscanned reflection matrix thus corresponds to a focal spot image. This new unscanned reflection matrix R' is therefore defined by:

[0327] In other words, this unscanned reflection matrix R' is expressed in / according to a reference base of the first point PI of spatial position n n = (x in , z iri ) where x in = x and and with the second point P2 of spatial position r O ut = (x ollt , z otJ t), with x out = x + Ax. We also have Zin = zO ut = z t . This expression can also be written in another base, such as that of the second point P2 or the midpoint or any other reference point, by a simple change of variable.

[0328] This de-scanned reflection matrix R' makes it possible to simply extract for each point {x, t} of the medium, a local image of the focal spot along the dimensions {Ax, co}, with: the lateral distance between the input and output transducers; and co the value of the model of the speed of sound.

[0329] Thus, the unscanned reflection matrix R' is constructed from the focused reflection matrix R (using equation 18) by:

[0330] This unscanned reflection matrix R' contains a set of apparent focal spots for each speckle grain {x, t} .

[0331] The unscanned reflection matrix R' can also be constructed directly from the experimental reflection matrix Rui. Thus, the unscanned reflection matrix includes the path-forming calculation of the focused reflection matrix R, and can be written as:

[0332] In other words, the method according to this second embodiment comprises a step of determining a de-scanned reflection matrix R' which comprises, for a set of points in the middle of abscissa x and echo time / , responses of the middle calculated by channel formations following a sound speed model co_from the experimental reflection matrix Rui(t), between a virtual input transducer of spatial position n n = (x, z t ) and a virtual spatial position output transducer r ou t = (x+Ax, z t ), the two virtual transducers being at the same depth z t , z tbeing the depth of the isochronous volume expected for the sound speed co, the isochronous volume being the set of points in the medium contributing to the signals received at time t, the responses of the unscanned reflection matrix R' being determined for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed models co, said unscanned reflection matrix being formed with each row corresponding to an element of a pair {Ax, co} which corresponds to a focal spot image and each column corresponding to an element of a pair {x, / }, which is noted: This unscanned reflection matrix R' makes it possible to obtain an image that represents the apparent focal spot for a point {x, t} in the field of view. This focal spot image corresponds to a column of the unscanned reflection matrix R', shown as a two-dimensional image with the lateral deviation Ax on the abscissa and the sound speed model c0 on the ordinate.

[0333] The method further comprises:

[0334] - an extraction step, in which an optimal lateral deviation Ax is determined from each focal spot op t and an optimal sound speed model c opt close to the maximum in said focal spot image.

[0335] The optimal sound speed model c opt then corresponds to a model of sound speed close to the distribution of sound speed in the medium studied.

[0336] This optimal sound speed model is advantageously much closer to the (true) sound speed of the medium than the commonly used assumption of a constant sound speed in the medium.

[0337] The maximum in the focal spot image is the point in this focal spot image having a maximum modulus value compared to the entire focal spot. This means that this maximum corresponds to the point in this focal spot image with the largest modulus value. By "close to the maximum", we mean that we extract a point close to this maximum point. In fact, we can take a point as close as possible, or for example another point according to different criteria such as a point having a zero lateral deviation Ax (Ax = 0), or another point between these two, or by another criterion of maximum point neighborhood. The neighborhood can be defined for example from a desired spatial resolution and / or a desired sound speed resolution.

[0338] As in the first embodiment of the method, this apparent focal spot is modulated due to the ultrasonic speckle problem and multiple scattering. A statistical average of these focal spots is performed in a similar manner over a spatio-temporal window of dimension (L x , Lî).

[0339] Thus, the method according to the second embodiment also comprises:

[0340] - a spatio-temporal averaging step of the unscanned reflection matrix R ' in which the values ​​of the unscanned reflection matrix are locally averaged along the lateral dimension x and the echo time dimension / , for each element of a pair {Ax, co}, the resulting focal spot images then being averaged.

[0341] This spatio-temporal averaging step allows to correctly average or smooth the focal spot image from the unscanned reflection matrix, in order to more correctly extract the position of the maximum in this image, a position which corresponds to the lateral deviation and the sound speed model to be used to correct the calculation of the ultrasound image.

[0342] This spatio-temporal averaging step can be implemented in various ways, based on various weighted or unweighted averaging formulas, with or without normalization.

[0343] According to a first variant of spatio-temporal averaging, the intensity of the unscanned reflection matrix is ​​locally averaged by calculating the incoherent point spread function PSF. inc on virtual input transducers {x', t'} around point of interest {x, t}, for example by the following expression: in which:

[0344] <.> is an average operator according to the parameters of the pair {x', t'}

[0345] R' is the unscanned reflection matrix,

[0346] W(x', t') is a spatio-temporal weighting window according to the same parameters.

[0347] The spatio-temporal weighting window may be of any type as defined in the presentation of the first embodiment.

[0348] This incoherent point spread function provides for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed model values ​​co, values ​​corresponding to a focal spot image for the lateral position x and the echo time t.

[0349] The incoherent point spread function PSF incobtained from the k-wave simulation data shown in Figure 7 of the first embodiment of the method is now illustrated for the second embodiment of the method in Figure 14(a).

[0350] The position of the maximum of the incoherent point spread function PSF inc , defined by PSF 11 ".({Bow u}, {.r. i}) , gives at its ordinate an estimator of the integrated speed of sound c , ie each point {x, t} of the focal spot image. The abscissa position of this maximum gives an optimum lateral shift. In Figure 14(d) is represented a DI curve showing the evolution of the incoherent point spread function for a zero lateral shift, i.e. Ax = 0; that is to say the evolution of the point spread function on the axis Ax = 0.

[0351] Note that in practice, as for the first embodiment, it is not always necessary to calculate or prepare the complete image of the focal spot image, and it is often not necessary to scan the points along the lateral / transverse direction Ax, since the maximum is a priori located on the axis Ax = 0. We can simply study the axial evolution of the confocal spot, i.e. for Ax 0, which is equivalent to x in = x ollt to estimate the integrated sound speed c as represented by the DI curve of figure 14(d) which represents the evolution of the focal spot values ​​on the axis Ax = 0.

[0352] In this first variant, with extraction step, we search for a maximum point of the focal spot image corresponding to a maximum of the modulus of the points of the focal spot image, and the optimal sound speed model c optis the ordinate of said maximum point in said focal spot image. The abscissa of the maximum point in the focal spot image is then the optimal lateral deviation Ax op r, in a manner equivalent to the first embodiment.

[0353] Then, we obtain the coordinates of the maximum point in the focal spot image by:

[0354] We can extract the optimal sound speed model c opt , of this formula 21, which corresponds to the value of the ordinate of the maximum point in the image of the focal spot produced (focal spot image by spatio-temporal averaging).

[0355] In this way during the extraction step, we search for a maximum point of the focal spot image corresponding to a maximum of the modulus of the points of the focal spot image, and the optimal sound speed model c opt is the ordinate of said maximum point in said focal spot image.

[0356] Optionally, we extract from the maximum point in the image of the focal spot, its abscissa and its ordinate, that is to say respectively the optimal lateral deviation Ax opt and the optimal sound speed model c opt .

[0357] According to a second variant of spatio-temporal averaging, the intensity of the unscanned reflection matrix R' is locally averaged by a calculation of a coherent point spread function PSF co h. This provides a better estimate of the sound speed model, which is an integrated sound speed model c .

[0358] To do this, as in the first embodiment, a singular value decomposition SVD of a local matrix of focal spots is carried out, by the following expression: we obtain an expression similar to the previous equation 13.

[0359] The output eigenvectors are then:

[0360] The first output eigenvector, Ui({x, / }) = [Ui({Ax, co}, {x, / })] directly gives the evolution of the coherent focal spot with respect to the model sound speed co around each point {x, / }, that is:

[0361] A new estimator of the integrated velocity ë can be deduced from the maximum of the coherent spreading function, i.e. by: argmax (|PSF co h({Aæ, co}, {æ, 4)|)

[0362] {co.Aœ} (23)

[0363] More generally, the optimal sound speed model can be extracted by formula 23, using the position of the maximum in the focal spot image (focal spot image by spatio-temporal averaging).

[0364] In this way, during the extraction step, we search for a maximum point of the focal spot image corresponding to a maximum of the modulus of the points of the focal spot image, and the optimal sound speed model c optis the ordinate of said maximum point in said focal spot image. The abscissa of the maximum point in the focal spot image is then the optimal lateral deviation kx opt .

[0365] The focal spot image obtained by the coherent point spread function PSF co h is shown in Figure 14(b). This image advantageously has better sensitivity than that obtained by the incoherent point spread function of Figure 14(a).

[0366] Indeed, the precision of this estimator of the integrated speed of sound c is inversely proportional to the second derivative of the point spread function PSF around its following maximum co, that is to say:

[0367] Figure 14(d) shows the values ​​of the focal spot images for zero lateral deviation, i.e. Ax = 0, from Figures 14(a) to 14(c). In particular, the DI curve in Figure 14(d) corresponds to the evolution of the values ​​of the point spread function PSF inc at zero lateral deviation of Figure 14(a). Curve D2 corresponds to the evolution of the values ​​of the coherent point spread function PSF co h at zero lateral deviation of Figure 14(b). We note the sharper or tighter character of this last curve of the coherent confocal spot around its maximum. In other words, curve D2 has a greater curvature at its maximum point and with a greater signal-to-noise ratio, which is representative of a better precision of estimation of the integrated speed of sound with this estimation variant:

[0368] According to a third variant of spatio-temporal averaging, we consider the axial derivative of the coherent point spread function PSF co h as shown in Figure 14(c), which allows us to obtain an even more reliable estimator of the integrated sound speed c . We exploit the Gouy double phase jump at the focal plane. For this, a new estimator of the integrated sound speed ë can be constructed from the maximum of the axial derivative of the coherent point spread function PSFcoh, i.e. by:

[0369] This estimator is an excellent estimator of the position of the focal plane, as illustrated by curve D3 in Figure 14(d). The second derivative of this observable around its maximum is indeed much larger than that of the modulus of the coherent point spread function PSF co h and the modulus of the incoherent point spread function PSF inc .

[0370] More generally, the optimal sound speed model can be advantageously extracted by formula 25, thanks to the position of the maximum in the image of the focal spot (image of the focal spot by spatio-temporal averaging).

[0371] In this way, during the step of extracting an optimal sound speed model, we search for a maximum point of the focal spot image corresponding to: - a maximum of the modulus of the points (of the values) of the focal spot image, or to

[0372] - a maximum of the modulus of the derivative according to the speed of sound co of the points (of the values) of the focal spot image, or at

[0373] - a maximum of the modulus of the derivative according to the speed of sound co of the imaginary part of the points (values) of the focal spot image.

[0374] The optimal sound speed model c opt is then the ordinate of said maximum point in said focal spot image.

[0375] The value of the optimal (integrated) sound speed model can be used to obtain a new corrected ultrasound image, i.e. with a correction of the sound speed model:

[0376] - either from the focused reflection matrix R:

[0377] - either from the unscanned reflection matrix R':

[0378] Optionally, it is possible to use both coordinates (abscissa and ordinate), i.e. the optimal lateral deviation Ax opt and the optimal sound speed model c opt The ultrasound image is then obtained:

[0379] - either from the focused reflection matrix R:

[0380] - either from the unscanned reflection matrix R':

[0381] Thus, the method of the present disclosure also comprises: a step of forming the ultrasound image by calculating the reflectivity at a plurality of points in the medium, from the unscanned reflection matrix R', the reflectivity of each of these points in the medium corresponding to the response of the medium between the virtual input transducer of spatial position n n = (x, zî) and the virtual spatial position output transducer r O ut= (x+Ax opt , z t ), z t being the depth of the isochronous volume expected for the optimal sound speed model c opt and said reflectivity being defined by: The ultrasound image above is parameterized according to the lateral position x and the echo time t.

[0382] This ultrasound image can be transformed to be parameterized as a function of lateral position x and depth z, using the sound speed model copt to replace the echo time t. This transformation is a change of variables or parameters. The resulting image is identical, but with dimensions in distances as usually used, and on which the practitioner can make measurements of the sizes of structures. Then, the ultrasound image is of the type: where z is the depth of a pixel of the ultrasound image and the estimated depth of the isochornic volume.

[0383] Either of the preceding ultrasound images may be displayed on the display device 43.

[0384] According to a variant, a film is formed from a succession of ultrasound images by a first number N of iterations of the steps of the method. Thus, the method is iterated to form ultrasound images at a predetermined rate, and to visualize a substantially real-time evolution of the ultrasound image of the medium.

[0385] According to a variant:

[0386] - among the N iterations, a second number P less than N of iterations does not include the steps of determining the unscanned reflection matrix R', the spatio-temporal averaging steps, and the steps of extracting an optimal sound speed model, and for these P (particular) iterations, at the step of forming the ultrasound image, the ultrasound image is obtained from an optimal sound speed model extracted during a previous iteration.

[0387] Thus, for a number P of iterations, the calculations of the steps of determining the unscanned reflection matrix R', the spatio-temporal averaging steps, and the steps of extracting an optimal sound speed model are avoided. The method of this variant requires less computing resources, while providing a film of a succession of high-quality ultrasound images. Thanks to this arrangement, the rate of determining the ultrasound image can be decoupled from the rate of calculating the optimal position deviation. Thus, a high ultrasound image rate can be maintained with a drastically reduced number of calculations.

[0388] The various variants for reducing the computational cost of the first embodiment are also usable and / or adaptable to the second embodiment, concerning the search for an optimal sound speed model.

[0389] Notably :

[0390] - at the extraction stage we can consider that the optimal lateral deviation Ax opt is zero, and / or

[0391] - at the stage of determining the unscanned reflection matrix R', we can calculate only the values ​​for a zero lateral deviation Ax; and / or

[0392] - in the spatio-temporal averaging step, we average the focal spot images corresponding to the pairs {Ax = 0, Az}, and / or

[0393] - at the extraction stage, the optimal velocity model will be determined from the points of the focal spot image having a lateral deviation Ax of zero value.

[0394] These variants have advantages similar or even identical to those stated in the first embodiment, and make it possible to reduce the calculations at different stages of the process.

[0395] The set of optimal sound speed model values ​​c optcan be assembled to form an integrated sound speed image. This integrated sound speed image can be displayed on the display device 43. However, it is preferable to calculate the local sound speed values ​​c(r) as explained later.

[0396] Figure 11(c) shows a map of the integrated velocity c opt (x1t') obtained in a liver imaging experiment. This integrated sound speed map can be used to obtain a new ultrasound image whose axial dimension is no longer given by the echo time t but by the actual depth of the scatterers.

[0397] The result of the method according to this second embodiment of the method is illustrated in Figure 15. Figure 15(a) shows the initial ultrasound image in which the axial dimension is an effective depth dictated by the echo time t and the sound velocity model co.

[0398] The first step of the correction process consists of considering for each pixel (x, t) of the image the value of the confocal signal obtained for the integrated sound speed model corresponding to c opt t) . The image obtained in Figure 15(b) demonstrates a much better contrast than the initial image.

[0399] The second step of the process consists of repositioning each pixel in depth by shifting it axially by a depth Az = (c opt (.E, t) — co)t / 2 - The corresponding displacement map is given in Figure 15(a). This image obtained by equation 26 is here a function of the depth and no longer of the echo time t. The interfaces between tissues show better lateral coherence, particularly at shallow depths where transverse variations in sound speed are strongest and their impact on the quality of the ultrasound image is greatest.

[0400] The correction of axial aberrations of the image is also accompanied by a drastic reduction of transverse aberrations and therefore a significant improvement in the transverse resolution and contrast of the image presented in Figure 16(c).

[0401] The resulting integrated sound speed map or image can also be used to estimate a local sound speed map as shown for example in Figure 11(d).

[0402] Thus, the method may comprise a step in which local sound speeds c(r) are calculated for a set of pairs {x, t} (i.e. positions in the medium) from the values ​​of the optimal sound speed models c opt obtained for the focal spot images of the pairs {x, t}.

[0403] A local sound speed image of the medium can then be formed, and it is possible to display it on the display device 43. This local sound speed image is important to the practitioner because it is useful, in combination with other information, for identifying certain pathological conditions or diseases, such as fatty liver disease.

[0404] Finally, having a local sound speed field, it is then possible to iterate the process described above in order to obtain a more precise ultrasonic characterization of the medium, by recalculating a de-scanned reflection matrix R' using the local sound speed values ​​determined in a previous step.

[0405] We will now describe in more detail an example of a method for finding a map or image of the local sound speed c(r) at each point in the image from an integrated sound speed map or image. To do this, we must invert the following equation: for example, by the method developed in the document

[0406] “Local speed of sound estimation in tissue using pulse-echo ultrasound: Model-based approach. ”, Jakovljevic M, et Al., Journal of the Acoustical Society of America. 2018 Jul;144(l):254.

[0407] Under a paraxial approximation, the previous equation can be rewritten in the following matrix form:

[0408] A x S = S (27)

[0409] Which can be explained in terms of matrix coefficients as follows:

[0410] With z) and CT — c 1(x, z) showing the local and integrated slowness maps. Inversion of matrix A leads to the following relation:

[0411] S = A - 1 x S

[0412] (28)

[0413] Which is rewritten in terms of matrix coefficients in the following form:

[0414] The reversal of the integrated care speed map c O pt (®; 7) shown in Figure 11(c) gives an estimator t) of the local sound speed map c(x, t) shown in Figure 11(d). This local sound speed map reveals the different tissue layers present in the liver imaging experiment, including the fat layer and muscle tissues which exhibit particularly contrasting sound speeds. Accurate and reliable measurement of sound speed in the liver is crucial for the detection of diseases such as steatosis. The sound speed measured here in the patient's liver is particularly low: <c>= 1450 m / s while the speed of sound in a healthy liver is more like 1600 m / s. This patient is in fact suffering from steatosis, which has been confirmed by other, more expensive, long and / or invasive examination methods, a disease which could ultimately be revealed by a simple ultrasound measurement of the speed of sound in the liver according to the disclosure.

[0415] Note that the inversion method mentioned above is based on a very strong approximation according to which the speed of sound integrated at a point c(x, z) depends only on the value of the local speed in plumb c(æ, z' < z).

[0416] Other more sophisticated methods can take into account oblique trajectories and refraction phenomena experienced by incident and reflected waves. They can be used to invert the integrated sound speed map obtained with the present method.

[0417] The local sound speed map can be exploited to construct a more sophisticated propagation model than the initial speed model in order to obtain a finer estimate of the reflectivity of the medium via an ultrasound image obtained using delay laws based on the sound speed map.

[0418] Also, the whole method described previously for a uniform velocity model co can also be used to optimize a more complex velocity model co(r) from more sophisticated optimization algorithms seeking to maximize the coherent confocal signal and / or the gradient of the Gouy phase.< / c>

Claims

CLAIMS 1. Method for ultrasonic characterization of a medium comprising: - a step of generating a series of incident ultrasonic waves (USm) in an area of ​​said medium, by means of an array (10) of transducers (11), said series of incident ultrasonic waves being an emission base (i); and - a step of generating an experimental reflection matrix Rui(t) defined between the emission base (i) at the input and a reception base (u) at the output, the coefficients of this matrix corresponding to the signals received by the transducers induced by the reflected ultrasonic waves; said method being characterized in that it further comprises - a step of determining a de-scanned reflection matrix R', which comprises, for a set of points in the middle of abscissa x and echo time / , responses of the middle calculated by channel formations following a sound speed model cp from the experimental reflection matrix Rui(t), between a virtual input transducer of spatial position n n = (x, z t ) and a virtual spatial position output transducer r O ut = (x+Ax, z t ), the two virtual transducers being at the same depth z / , z tbeing the depth of the isochronous volume expected for the sound speed co, the isochronous volume being the set of points in the medium contributing to the signals received at time / , the responses of the unscanned reflection matrix R' being determined for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed models co, said unscanned reflection matrix being formed with each row corresponding to an element of a pair {Ax, co} which corresponds to a focal spot image and each column corresponding to an element of a pair {x, / }, which is noted: R' =[B'({A;c, c0}, {a:, 4)] et - an extraction step in which an optimal lateral deviation Ax is determined from each focal spot O pz and an optimal sound speed model c opt close to the maximum in said focal spot image.

2. The method of claim 1, further comprising: - a step of forming the ultrasound image by calculating the reflectivity at a plurality of points in the medium, from the unscanned reflection matrix R', the reflectivity of each of these midpoints corresponding to the response of the midpoint between the virtual spatial position input transducer n n = (x, z,) and the virtual spatial position output transducer r ou t= (x+Ax opt , z t ), z t being the depth of the isochronous volume expected for the optimal sound speed model c opt and said reflectivity being defined by:

3. The method of claim 2, wherein the ultrasound image is transformed into the depth dimension, using the optimal sound speed model c opt by : where z is the estimate of the depth z of a pixel in the ultrasound image.

4. A method according to claim 2 or claim 3, wherein the ultrasound image is displayed on a display device.

5. Method according to claim 1, in which: - we calculate local sound speeds c(r), for a set of pairs {x, t}, from the optimal sound speed values ​​c opt obtained for the focal spot images of said pairs {x, t}.

6. The method of claim 5, further comprising a step of forming a local sound speed image of the medium from said local sound speed set, and displaying said local sound speed image on a display device.

7. Method according to claim 5, in which a de-scanned reflection matrix R' is recalculated using the local sound speed values.

8. The method of claim 1, further comprising, before the extraction step: - a spatio-temporal averaging step of the unscanned reflection matrix R' in which the values ​​of the unscanned reflection matrix are locally averaged along the lateral dimension x and the echo time / , for each element of a pair {Ax, CO}, the focal spot images then being averaged.

9. Method according to claim 8, in which in the spatio-temporal averaging step, an incoherent point spread function PSF is calculated. inc by the following formula: > for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed model values ​​co, to obtain the focal spot image for lateral position x and echo time t, in which: < > is an average operator according to the parameters of the pair {x', t'} R' is the unscanned reflection matrix, W(xt ') is a spatio-temporal weighting window according to the same parameters, and at the extraction stage, we search for a maximum point of the focal spot image corresponding to a maximum of the modulus of the points of the focal spot image, and the optimal sound speed model c opt is the ordinate of said maximum point in said focal spot image.

10. Method according to claim 8, wherein: in the spatio-temporal averaging step, a singular value decomposition SVD of a local focal spot matrix R is calculated (1) , said local focal spot matrix being expressed from the unscanned reflection matrix R' by: in which: R' is the unscanned reflection matrix, W(xt ') is a spatio-temporal weighting window according to the same parameters. The singular value decomposition SVD of said local matrix is ​​noted: in which are the singular values ​​of said local matrix, U P are the output eigenvectors, V P are the input eigenvectors, we calculate a coherent point spread function PSF co h by the formula next: PSF coh({Ax, co}, {x, t}) = Ui({^x, c0}, {x, i}) for a set of lateral deviation values ​​Ax including at least Ax = 0 and for a set of sound speed model values ​​co, to obtain the focal spot image for the lateral position x and the echo time t, and in the extraction step, a maximum point of the focal spot image is sought corresponding to: a maximum of the modulus of the points of the focal spot image, or to a maximum of the modulus of the derivative according to the sound speed co of the points of the focal spot image, or to a maximum of the modulus of the derivative according to the sound speed co of the imaginary part of the points of the focal spot image, and the optimal sound speed model c opt is the ordinate of said maximum point in said focal spot image.

11. Method according to one of claims 1 to 10, in which in the extraction step, it is considered that the lateral deviation Ax opt is zero, Ax opt = 0.

12. Method according to one of claims 1 to 11, wherein in the step of determining a de-scanned reflection matrix R', the de-scanned reflection matrix R' is formed only for lateral deviation values ​​Ax of zero value, Ax = 0.

13. Method according to claim 8, in which in the spatio-temporal averaging step, the values ​​of the unscanned reflection matrix R' are averaged to determine a focal spot image, for each element of a pair {Ax = 0, Az}, according to the lateral dimension x and the echo time t.

14. Method according to one of claims 1 to 13, wherein in the extraction step, the optimal sound speed model c opt is determined from the points of the focal spot image having a lateral deviation Ax of zero value, Ax = 0.

15. Method according to one of claims 1 to 14, in which the calculations of path formations between the virtual input transducer and the virtual output transducer are carried out from the experimental reflection matrix Rui(t), using a forward delay time ultrasonic waves between the transmitting base (i) and the virtual input transducer, and using a return delay time of the ultrasonic waves between the virtual output transducer and the transducers of the receiving base (u).

16. A method according to claim 1 or claim 15, wherein the unscanned reflection matrix R' is calculated from the experimental reflection matrix Rui by the following formula: R' (JJ\x, CQ}, {te, t}) in which: Nin is the number of elements in the emission base (i) N mi t is the number of elements in the receiving base (u), Rui(t) is the experimental reflection matrix, whose Rui(Uout, lin, t) is the element of the experimental reflection matrix Ruif / > recorded by the spatial position transducer Uout following the emission of index iin in the emission base (i), at a time t , co being the expected speed of sound.

17. A method according to claim 1 or claim 15, wherein the unscanned reflection matrix R' is calculated from a focused reflection matrix R by the following formula: in which: the unscanned reflection matrix R' is formed with each row formed by elements of a pair {Ax, Az} and each column formed by elements of a pair {x, t}, and the focused reflection matrix R is calculated from the experimental reflection matrix Rui(t) by the following formula: in which: Nin is the number of elements in the emission base (i) Name is the number of elements in the receiving base (u), Rui(t) is the experimental reflection matrix, whose Rui(ltout, lin, t) is the element of the experimental reflection matrix Rui(t) recorded by the spatial position transducer Uout following the emission of index iin in the emission base (i) and at time / , co being the expected speed of sound.

18. Method according to one of claims 1 to 17, in which a film is formed from a succession of ultrasound images by a first number N of iterations of the steps of the method.

19. The method of claim 18, wherein among the N iterations, a second number P strictly less than N of iterations does not include the steps of determining the unscanned reflection matrix R', the spatio-temporal averaging steps, and the steps of extracting the optimal sound speed model, and for these P iterations, in the step of forming the ultrasound image, the ultrasound image is obtained from an optimal sound speed extracted during a previous iteration.

20. System for ultrasonic characterization of a medium (20), the system comprising: - an array (10) of transducers adapted to generate a series of incident ultrasonic waves in a zone of the medium, and to record as a function of time the ultrasonic waves backscattered by said zone; and - a calculation unit (42) connected to the transducer network and adapted to implement the method according to one of claims 1 to 19.

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