Method, device and computer program for testing a workpiece by means of X-ray radiography

Description

[0001] The invention relates to a method, a device and a computer program for the non-destructive testing of a part by X-ray radiography.

[0002] The field of the invention relates to aeronautical parts, in particular turbomachine blades, in particular their turbine blades.

[0003] Non-Destructive Testing (NDT) of aeronautical parts is an essential element of aircraft operational safety, the objective of which is to avoid any defects that could cause failure in flight. Among the NDT methods, X-ray radiography stands out for its ability to visualize the interior of the part in a minimally / non-intrusive manner and to resolve details down to a micrometric size. These performances have made it popular in the aeronautical sector. X-ray radiography makes it possible to image the internal structure of the part in transmission. Tomography consists of acquiring a large number of X-rays during a rotation, most of the time complete, of a part in order to calculate a complete three-dimensional image of a part.The significant acquisition time of these tomographic images leads manufacturers to consider only a limited number of radiographic images (called radiographs or projections thereafter) to carry out material and dimensional health NDT (three-dimensional geometric indications). However, image artifacts due to beam hardening and Compton scattering affect the radiographs and make the metrological analysis carried out during NDT delicate or even uncertain. The document . " Beam hardening correction with an iterative scheme using an exact backward projector and a polychromatic forward projector" by R. Bock et al, XP055915705 discloses a treatment of these artifacts.

[0004] When a large number of projections are considered, for example in tomography, these artifacts can be neglected. However, they must be carefully considered when NDT is performed from few views.

[0005] Digital radiography NDT booths (registered trademarks such as Yxlon, GE, Nikon) or operating software (Vg Studio, Aviso, RX solution) and image processing software obtained by X-ray radiography all have filters to assist the inspector in his sanction task. A (normalized) radiograph is interpreted as an image of the attenuation of X-rays during passage through the part, attenuation itself linked to the thickness by a law that is often approximated by an exponential function (Beer-Lambert law). The quantities of interest in the characterization of geometric indications are three-dimensional in nature, and their estimation from radiographs (two-dimensional images) is partial and unreliable. This can induce significant errors in the inspection, and makes the sanction procedure sensitive to the variation of the viewing angle during the acquisition of each image.The analysis therefore requires precise knowledge of the geometry of the cabin-room system, and the faithful correction of image artifacts. The use of X-ray tomography involves the acquisition of one or several thousand projections, which is time-consuming and also involves processing steps to consider. Thus, it is preferable to acquire only a small number (of the order of ten) of projections, typically a hundred times fewer.

[0006] X-ray part sanctioning from a small number of views is usually performed manually: inspectors, specialist technicians trained for this task, analyze the images produced by the system in search of any anomalies. Anomalies are represented by an abnormal variation in the gray levels of the images. However, image artifacts alter these gray levels, making the sanction uncertain and difficult: when a small number of images is considered, image artifacts have a high weight and cannot be neglected. Thus, the reduction of imaging artifacts for the sanctioning of aeronautical parts from a limited number of projections is a crucial element for this application. For accelerating voltages around 350 keV (typically used to acquire turbine blade images), the artifacts to be treated are essentially beam hardening and Compton scattering.

[0007] There is variability between and within inspectors, reducing the reliability of the sanction. Furthermore, the careful analysis of images by inspectors is laborious and tiring work, both physically and psychologically.

[0008] An objective of the invention is to obtain a method, a device and a computer program for the non-destructive testing of a part by X-ray radiography, which overcome the drawbacks mentioned above, by minimizing one or more types of image artifacts, thus allowing the use of a small number of acquired projections to carry out the NDT.

[0009] To this end, a first object of the invention is a method for non-destructive testing of a part by transmission radiography, comprising the following steps, executed by a computer: acquisition of N projections of the part from a transmission radiography device according to N different and predetermined viewing angles of the part, where N is a prescribed natural number, generation of N calculated images of the part from a reference model of the part corresponding to the N viewing angles and from a vector p of parameters characterizing the projection geometry of the acquisition for the N viewing angles at each of several successive iterations, estimation, by the successive iterations, of the vector p from an initial vector p = pin and at least one of a vector c of parameters from an initial vector c = c ini and a vector α of parameters from an initial vector α = α This , where the vector c parameters accounts for the beam hardening of the radiation in the room and the vector α parameters characterize the Compton scattering of the radiation in the room, by minimizing the sum of the norms of the squared differences between the N acquired projections and the N calculated images, processing of the N projections and / or the N calculated images comprising a first processing and / or a second processing, the first processing comprising a correction of the beam hardening on the N projections from the vector c having been estimated or a generation of the beam hardening on the N images calculated from the vector c having been estimated, the second treatment comprising a correction of the Compton diffusion on the N projections from the vector α having been estimated or a generation of Compton diffusion on the N images calculated from the vector α having been estimated, identification of defects in the part by comparing the N processed projections to the N processed calculated images.

[0010] The invention thus allows the non-destructive testing of aeronautical parts by X-ray radiography, from a limited number of acquired projections. The invention allows the sanctioning of aeronautical parts using a limited number of acquired projections (which may be for example less than or equal to 1000) of the inspected part. Indeed, after application of the algorithmic processing described in the invention, the images acquired by the system and those reproduced by simulation can be quantitatively compared, and a sanction based on their differences is carried out. The method, the device and the control program according to the invention help in the sanctioning of the inspected part. The procedure being systematically parameterized, this sanction is then made reliable.

[0011] The invention presents a NDT method from a small number of projections. The invention makes it possible to increase the reliability of the control method and device, but also its performance by allowing the sanctioning of the part with a limited number of projections. For this, one or more types of artifact are estimated from their modeling. This estimation is used either to correct the artifacts in the acquired images, or to reproduce them in simulated images from the model of the ideal part (for example its computer-aided design model or CAD model). This is made possible by the reproduction of changes in pixel intensity, called artifacts, due to physical phenomena which are responsible for deviation from the simple Beer-Lambert law, such as beam hardening or Compton scattering.This consideration allows a more relevant comparison of gray levels between acquired and simulated, and therefore a more precise sanction. Once these artifacts are considered, the proposed NDT method makes it possible to highlight all the indications regardless of their nature (material health, geometric), or their position in the part, independently of the acquisition conditions used (by following the NDT sanction process defined in an industrial context), or even the geometry of the part. The gain in reliability is not accompanied by any other cost, except the modest one of modeling the projection.

[0012] The invention relates in particular to the implementation of a control system consisting of a device for acquiring X-ray radiographs of a part composed of a single known material, coupled with a sanction and validation algorithm. The algorithmic processing makes it possible to minimize one or more types of image artifacts expected for the material and the power of the X-ray beam used, or to reproduce them, so as to make the acquired images and the simulated images quantitatively comparable.

[0013] Beam hardening can be corrected by the computer by performing an attenuation calibration and then a correction of the image intensities.

[0014] According to one embodiment of the invention, the method comprises at each iteration the estimation of the vector p pi parameters of the projection geometry of the acquisition for the N viewing angles by the calculator, this estimation including: the calculation of projection residuals ρ p n = P n − P ^ n from the vector of initial values p = p ini field calculation s p i n of sensitivity according to s p i n = ∂ P ^ n ∂ p i p from the vector of initial values p = pini, Or P̂ (n)< are the N images calculated for the N viewing angles, P̂ (n)< are the N acquired projections of the part, p is the column vector of the parameters pi of projection geometry, pin is the column vector of the prescribed initial values of the parameters pi of projection geometry, n is a natural integer designating the number of the viewing angle and ranging from 1 to N, p is updated according to p* = p + δp*, Or p * is the column vector of parameters pi updated projection geometry, δp* is the column vector of variation of the parameters piof the projection geometry and is calculated as the δp minimizing the sum of the norms of the squared differences between the projection residuals ρ p n and the product of δp by s p n For n ranging from 1 to N, δ p * = arg min δ p ∑ n ρ p n − s p n δ p 2 Or δp is a column vector, s p n is the matrix of fields s p i n of sensitivity.

[0015] According to one embodiment of the invention, the method comprises at each iteration the estimation of the vector c parameters ck beam hardening calibration of the radiation in the room by the calculator, this estimate including: the calculation of projection residuals ρ c n = P n − P ⌣ n from the vector of initial values c = c ini , field calculation s c k n of sensitivity according to s c k n = ∂ P ⌣ n ∂ c k c from the vector of initial values c = c ini , Or P ⌣ n x = u P ^ n x is the image obtained by applying the function u(y) to the intensity y of each pixel x of P̂ (n)< , P̂ (n)< are the N images calculated for the N viewing angles, P(n)< are the N acquired projections of the part, c is the column vector of the parameters ck beam hardening calibration, c ini is a column vector of the prescribed initial values of the parameters ck radiation beam hardening calibration in the room, φ k ( y ) is a basis of prescribed form functions, u y = ∑ k = 1 K 3 c k φ k y K 3 is a prescribed natural number greater than or equal to 1, k is a natural integer ranging from 1 to K 3 , c is updated according to c * = c+ δc*, Or c * is the column vector of parameters ck radiation beam hardening calibration in the room, having been updated, δ c* is the column vector of variation of the parameters ck beam hardening calibration of radiation in the room and is calculated as the δc minimizing the sum of the norms of the squared differences between the projection residuals ρ c n and the product of δc by s c n for n ranging from 1 to N, δ c * = arg min δ c ∑ n ρ c n − s c n δ c 2 where δc is a column vector, s c n is the matrix of fields s c k n of sensitivity.

[0016] According to one embodiment of the invention, the method comprises at each iteration the estimation of the vector α parameters α j Compton scattering of the radiation in the room by the calculator, this estimate including: the calculation of projection residuals ρ α n = P n − P ˜ n from a vector of initial values α = a ini , the calculation of fields s α j n of sensitivity according to s α j n = ∂ P ˜ n ∂ α j α from the vector of initial values α = a ini , Or P(n)< are the N acquired projections of the part, P̂ (n)< are the N images calculated for the N viewing angles, P ˜ n = P ⌣ n + P ⌣ n ∗ K is the image obtained by convolving images simulated, having been obtained from at least the N images P̂ (n)< calculated, with the kernel d + K , α is the column vector of the parameters α j Compton scattering of radiation in the room, a ini is a column vector of the prescribed initial values of the parameters α j Compton scattering of radiation in the room, K is the convolution kernel defined by K x = ∑ j = 1 K 2 α j g σ j x − δ x K 2 is a prescribed natural number greater than or equal to 1, j is a natural number ranging from 1 to K 2 , g σj , a two-dimensional Gaussian kernel of standard deviation s j prescribed, d ( x ) is the Dirac function at the pixel x, α is updated according to α* = α + da* , Or α * is the column vector of the parameters α j Compton scattering of the radiation in the room, having been updated, yes * is the column vector of variation of the parameters α j Compton scattering of radiation in the room and is calculated as the yes minimizing the sum of the norms of the squared differences between the projection residuals ρ α n and the product of yes by s α n For n ranging from 1 to N, δ α * = arg min δ α ∑ n ρ α n − s α n δ α 2 Or yes is a column vector, s α n is the matrix fields s α j n of sensitivity.

[0017] According to one embodiment of the invention, P ⌣ n x = u P ^ n x is the simulated image, obtained by applying the function u ( y ) to the intensity of each pixel x of the calculated image P̂ (n)< .

[0018] According to one embodiment of the invention, N is less than or equal to 1000.

[0019] A second subject of the invention is a computer program, comprising code instructions for implementing the following steps of a method of non-destructive testing of a part by transmission radiography, when executed by a computer: receiving N projections of the part from a transmission radiography device according to N different and predetermined viewing angles of the part, where N is a prescribed natural number, generating N calculated images of the part from a reference model of the part corresponding to the N viewing angles and from a vector p of parameters characterizing the projection geometry of the acquisition for the N viewing angles at each of several successive iterations, estimation, by the successive iterations, of the vector p from an initial vector p = pin and at least one of a vector c of parameters from an initial vector c = ci,, i and a vector α of parameters from an initial vector α = a ini , where the vector c parameters accounts for the beam hardening of the radiation in the room and the vector α parameters characterize the Compton scattering of the radiation in the room, by minimizing the sum of the norms of the squared differences between the N acquired projections and the N calculated images, processing of the N projections and / or the N calculated images comprising a first processing and / or a second processing, the first processing comprising a correction of the beam hardening on the N projections from the vector c having been estimated or a generation of the beam hardening on the N images calculated from the vector chaving been estimated, the second treatment comprising a correction of the Compton diffusion on the N projections from the vector α having been estimated or a generation of Compton diffusion on the N images calculated from the vector α having been estimated, identification of defects in the part by comparing the N processed projections to the N processed calculated images.

[0020] A third object of the invention is a device for non-destructive testing of a part by transmission radiography, comprising: a transmission radiography device, for acquiring N projections of the part according to N different and predetermined viewing angles of the part, where N is a prescribed natural number, a computer configured to carry out the following steps: generation of N calculated images of the part from a reference model of the part corresponding to the N viewing angles and from a vector p of parameters characterizing the projection geometry of the acquisition for the N viewing angles at each of several successive iterations, estimation, by the successive iterations, of the vector p from an initial vector p = pin and at least one of a vector c of parameters from an initial vector c = c ini and a vector α of parameters from an initial vector α = a ini , where the vector c parameters accounts for the beam hardening of the radiation in the room and the vector α parameters characterize the Compton scattering of the radiation in the room, by minimizing the sum of the norms of the squared differences between the N acquired projections and the N calculated images, processing of the N projections and / or the N calculated images comprising a first processing and / or a second processing, the first processing comprising a correction of the beam hardening on the N projections from the vector c having been estimated or a generation of the beam hardening on the N images calculated from the vector c having been estimated, the second treatment comprising a correction of the Compton diffusion on the N projections from the vector α having been estimated or a generation of Compton diffusion on the N images calculated from the vector α having been estimated, identification of defects in the part by comparing the N processed projections to the N processed calculated images.

[0021] The invention will be better understood upon reading the description which follows, given solely as a non-limiting example with reference to the figures below in the attached drawings. [ Fig. 1 ] schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to an embodiment of the invention. Fig. 2 ] schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to one embodiment of the invention. Fig. 3 ] schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to one embodiment of the invention. Fig. 4] schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to one embodiment of the invention. Fig. 5 ] schematically represents a flowchart of a method for non-destructive testing of a part by X-ray radiography according to an embodiment of the invention. Fig. 6 ] schematically represents a curve of the intensity of the pixels of a projection P ( x ) in the ideal case and in a practical case altered by beam hardening artifacts of a projection obtained by a state-of-the-art device. Fig. 7 ] represents an explanatory diagram of Compton scattering during the acquisition of an X-ray projection. [ Fig. 8] represents the difference between, on the one hand, the inverse of the exponential of the intensity of an acquired projection, and on the other hand, the inverse of the exponential of the intensity of a simulated projection, before correction on a part of a turbomachine turbine blade, obtained by a state-of-the-art device. Fig. 9 ] represents the difference between, on the one hand, the inverse of the exponential of the intensity of an acquired projection, and on the other hand, the inverse of the exponential of the intensity of a simulated projection, before correction on a part of a turbine blade root of a turbomachine, obtained by a device of the state of the art. Fig. 10 ] represents the difference between, on the one hand, the inverse of the exponential of the intensity of an acquired projection, and on the other hand, the inverse of the exponential of the intensity of a simulated projection, after correction on a part of the same turbomachine turbine blade as at figure 8, obtained by the method, a device and a computer program for the non-destructive testing of a part by X-ray radiography according to an embodiment of the invention. Fig. 11 ] represents the difference between, on the one hand, the inverse of the exponential of the intensity of an acquired projection, and on the other hand, the inverse of the exponential of the intensity of a simulated projection, after correction on a part of the same turbine blade root as at the figure 9 , obtained by the method, a device and a computer program for the non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0022] As illustrated in figures 1 has 5 , the device 1 for non-destructive testing of a real part 200 comprises a transmission radiography device 100 making it possible to acquire, during step E1 of the method, projections P ( n)< , also called raw acquired projections, of the part 200. In the following, the radiation is X-rays. The radiography device 100 makes it possible to inspect the part 200 under controlled acquisition conditions with a known acquisition geometry and makes it possible to image in transmission the internal structure of the part 200. In the following, the projections P ( n )< and the images mentioned below are defined by their gray levels (or intensity) for their different pixels x.

[0023] In the embodiments described below, the projections P ( n )< are images P ( n)< radiographic in X-ray transmission, in the method of non-destructive testing of a part by X-ray radiography, the device 1 for non-destructive testing of a part by X-ray radiography and the computer program for non-destructive testing of a part by X-ray radiography.

[0024] The part 200 is a mechanical part, and may be, in a non-limiting manner, an aeronautical part, in particular a turbomachine blade (for example a turbojet), for example one of their turbine blades (turbine blade), or perhaps other. The part 200 is made of a material that is semi-transparent to radiation, here to X-rays, and may be made of a single material (mono-material). The part 200 may be a metal part or a composite part with a ceramic matrix, or other. The method, the device 1 and the computer program for the non-destructive testing of the part 200 by X-ray radiography are used to check the part 200 during its manufacture or during maintenance operations, in order to detect defects in this part, which may for example cause a failure in flight in the case of an aeronautical part.

[0025] The device 1 for non-destructive testing of the part 200 comprises and the method of the device for non-destructive testing of the part 200 uses one (or more) CAL calculator. The CAL calculator may be or comprise one or more computers, one or more servers, one or more machines, one or more processors, one or more microprocessors, one or more permanent memories MEM, one or more random access memories MEM. The CAL calculator may comprise one or more physical data input interfaces INT1, one or more physical data output interfaces INT2. This or these physical data input interfaces INT1 may be or comprise one or more computer keyboards, one or more physical data communication ports, one or more touch screens, or others.This or these physical data output interfaces INT2 may be or comprise one or more physical data communication ports, one or more screens, or others. A computer program may be recorded and executed on the CAL computer of the device 1 for non-destructive testing of the part 200 and comprise code instructions, which when executed thereon, implement all or part of the method for non-destructive testing of the part 200 according to the invention (including the reception of the N projections during step E1).

[0026] The X-ray radiography device 100 comprises an X-ray source 101, a support 102 on which the part 200 is located, a control mechanism 104 for rotating the support 102 and the source 101 relative to each other 101 around an axis 103 of rotation, which may be for example vertical (for example the source 101 is fixed and the support 102 is rotated around the axis 103), a detector 105 of the X-rays passing through the part 200, the part 200 therefore being in the path of the X-rays between the source 101 and the detector 105. The source 101, the support 102 and the detector 105 are arranged in a high-power X-ray cabin. The detector 105 provides the projections P ( n )< of the part 200 during the first step E1. The control mechanism 104 is controlled for the acquisition at the N viewing angles ANG (n)<, different from each other, of the part 200 with respect to the X-rays, by the detector 105, of N projectionsP ( n )< . N is a prescribed natural number, greater than or equal to 1. The natural number n ranges from 1 to N and denotes the number of the respective viewing angle ANG (n)< and therefore the number of the projection P ( n )< acquired. The radiography device 100 thus makes it possible to acquire N projections (i.e. N projections P ( n )< ) of the volume of the part 200 under respectively the N angles ANG (n)< of view. The N angles ANG (n)< of view have been predetermined to be used for the subsequent step E5 of analysis of the part. The significant acquisition time of the projections acquired by X-rays leads to considering only a limited number N of projections P ( n )< acquired, less than 1000 or even 100, or other. The X-ray radiography device 100 thus provides all the projections P ( n )< acquired during stage E1.

[0027] During a second step E2, after the first step E1, the CAL calculator generates N images P̂ ( n )< calculated from part 200 from a reference digital MODP model of part 200, as illustrated in figures 1 has 5 . These N images P̂ ( n )< calculated correspond to the N angles ANG (n)< of view, that is to say adopt the same angle ANG (n)< of view as respectively the N projections P ( n )< .This MODP model is a geometric reference of the part 200, is recorded in advance in a memory of the CAL calculator and can be for example a computer-aided design model (or CAD for short) of the part 200, reproducing an ideal part 200 without defects, having an external shape of defined three-dimensional coordinates. This MODP model can take into account the composition of the material of the part 200. At each step E2, the CAL calculator generates N images P̂ ( n )< calculated from part 200 from the reference MODP model of part 200 corresponding to the N angles ANG (n)< of view and from a vector p of parameters pi characterizing the projection geometry of the acquisition for the N angles ANG (n)< of view at each of several successive iterations.

[0028] The CAL calculator performs one or more or all of the steps E3a, E3b, E3c of estimating artifacts of multi-view acquired projections, which will be described below. According to embodiments, the determination of the parameters described below is done by an optimization procedure which exploits sensitivity fields to minimize projection residuals.

[0029] During the third step E3a, after the first step E1, the CAL calculator estimates the vector p parameters pi of the projection geometry of the acquisition of step E1 for the N angles ANG (n)< of view, by minimizing the sum of the quadratic norms of the squared differences, calculated by the CAL calculator, between the N projections P ( n )< acquired and the N images P̂ ( n )< calculated. During the third step E3a the CAL calculator estimates the vector p describing the geometry of the X-ray radiography device 100 with respect to the MODP model of the ideal part 200 (for example its CAD model). The third step E3a allows the CAL calculator to precisely determine the vector p of the projection geometry used when acquiring the projections P(n)< by the X-ray radiography device 100. This ensures a more reliable simulation of the images. The third step E3a makes it possible to carry out, during the following iteration of step E2, a geometric registration of the N images P̂ ( n )< calculated for the N angles ANG (n)< of view by calculating the N images P̂ ( n )< calculated by projection of the MODP model onto the plane of the detector 105 according to the geometry determined by the vector p N projections P ( n)< acquired. Step E3a can be implemented by a SIMX simulator, as shown in Figure 4. During the third step E3a, the CAL calculator estimates the vector p parameters pi of the projection geometry of the acquisition by successive iterations from an initial vector p = p ini .

[0030] The vector p projection geometry parameters are used to simulate the calculated images P̂ (n)< . So, in step E2, we already use the vector pin parameters of the initial values prescribed for the parameters of the projection geometry during the first iteration; then the estimated values of the vector are used during the following iterations of step E2 p. The calculated images P̂ ( n )< are subsequently used to estimate the vector c and / or α in step E3a and step E3b and / or E3c. The calculated images P̂ ( n )< are not corrected only in step E4, but also in step E2. Thus, step E2 includes the regeneration of the calculated image P̂ ( n )< at each new iteration with the new vector p*, which was estimated in step E3a. The estimation of the vector p during step E3a is an iterative procedure, and therefore at each iteration we use the best possible estimate (until convergence) to generate the calculated images P̂ ( n )< . The vector p is used in step E2 and in each of the iterations of step E3a.

[0031] The reproduction or correction of artifacts is based on the digital simulation of projections P ( n)< acquired by an X-ray system. The artifacts are then estimated based on parametric modeling (steps E3b, E3c) before being either reproduced (step E4) or corrected (step E4).

[0032] The parameters pi projection angles (such as projection angles ANG (n)<, distance from source 101 to detector 105) make it possible to link the ideal MODP model of the part 200 to the acquired projections P (n)< of the part 200 and are thus registration parameters of the ideal MODP model of the part 200 with the acquired projections P (n)< .

[0033] According to an embodiment of the invention, during step E3a, the calculator CAL calculates (for example by the calculator CRES) the residues at the figure 4 ) at each iteration the residues ρ p n of projection, these residuals being the differences between the observed projections P ( n)< and simulated projections P̂ ( n )< according to the equation ρ p n = P n − P ^ n . From the first prescribed approximation, noted pini, of these parameters pi , the CAL calculator estimates (for example by the ESTV estimator at the figure 4 ) the vector p * best describing the observed system, i.e. minimizing the residues ρ p n of projection. The fields of sensitivity s p i n are calculated (for example by finite differences, or other) by the CAL calculator according to the following equation: s p i n = ∂ P ^ n ∂ p i p and translate the effect of the variation of each of the pi parameters on the calculated image. The parameters pi of the projection geometry are calculated by the CAL calculator by optimization from the projection residuals ρ p n , of value pini, and fields of sensitivity s p i n . The CAL calculator calculates the variation δp* vector optimal p of parameters and updates the vector p according to the following equation: p * = p + δ p * Or p * is the column vector of the updated projection geometry pi parameters, δp* is the column vector of variation of the pi parameters of the projection geometry and is calculated by the CAL calculator as the δp minimizing the sum of the norms of the squared differences between the projection residuals ρ p n and the product of δp by s p n For n ranging from 1 to N, δ p * = arg min δ p ∑ n ρ p n − s p n δ p 2 Or δp is a column vector, s p n is the matrix of fields s p i n of sensitivity. The number of rows in the matrix s p n is the number of pixels used in the projections P (n)< , in the calculated images P̂ (n)< and in the residue ρ p n . The number of columns in the matrix s p n is the number of parameters pi. The CAL calculator calculates all the sensitivity fields during step E3a. s p i n For n ranging from 1 to N and i ranging from 1 to K 1 . The estimation of the vector p parameters pi of the projection geometry for the N angles ANG (n)< of view by the CAL calculator is carried out by successive iterations on p which becomes p ini , for: calculating at each current iteration, from the vector p of the previous iteration taking the place of the vector pin mentioned above fields s p i n of sensitivity according to the equation mentioned above, for each current iteration: calculate a new δp* and update p according to p* = p + δp ', with δp* calculated as the δp minimizing the sum of the norms of the squared differences between the projection residuals ρ p n and the product of δp by s p n For nranging from 1 to N, depending on δ p * = arg min δ p ∑ n ρ p n − s p n δ p 2 until p converges to a best estimate (determined for example by the fact that the norm of the difference calculated between the p of the current iteration and the p of the previous iteration becomes lower than a prescribed threshold).

[0034] During the fourth step E3b, after the first step E1, the CAL calculator estimates the vector c parameters ck X-ray beam hardening calibration in room 200 by minimizing the sum of the squared difference norms between the N projections P ( n )< acquired and the N images P̂ ( n)< calculated, calculated by the CAL calculator. Beam hardening refers to the fact that the absorption of a photon by the material constituting the analyzed part depends on its energy. Low energy photons are preferentially absorbed compared to high energy ones. From the deviation between the actual CRI curve given by the gray level P (x) pixels x of a projection P ( n )< measured and the theoretical CTI curve resulting from the modeling used in the projection analyses (Beer-Lambert law), illustrated in figure 6 , emerges from so-called beam hardening artifacts. Beam hardening induces a non-linear relationship between the measured intensity P ( x ) of a projection and the thickness ξ(x) of material crossed (deviation from the Beer-Lambert law) in the case of a single-material part. The thickness ξ(x) is the thickness crossed by the ray before reaching the pixelx . In the case of single-material parts 200, beam hardening artifacts can be exactly corrected by calibration of the beam hardening attenuation. The CAL calculator performs an identification of the calibration function relating the effective attenuation of the X-rays passing through the part 200 and the length of material passed through in the part 200. Step E3b can be implemented by an X-ray beam hardening simulator SMBH, as shown in figure 4 . During step E3b, the CAL calculator estimates, by successive iterations, the vector c from an initial vector c = c ini .

[0035] Embodiments of the fourth step E3b are described below. What is described below is performed at each of the iterations of step E3b.

[0036] The CAL calculator calibrates the beam hardening by identifying the u function changing from P̂( n )< ( x ) has P ( n )< ( x ) , namely p ( n )< ( x ) = u ( P̂ ( n )< ( x )) .

[0037] According to one embodiment of the invention, the calibration of the beam hardening is carried out by applying a function u piecewise linear connecting P ( n )< ( x ) And P̂ ( n )< ( x ).

[0038] According to one embodiment of the invention, the function u is discretized on a basis of shape functions φ k ( y ) prescribed (for example by one-dimensional finite elements, or spline functions, or polynomial functions, or others): u y = ∑ k = 1 K 3 c k φ k y where K 3 is a prescribed natural number greater than or equal to 1, kis a natural integer ranging from 1 to K 3 , y is a dummy variable designating the gray level (intensity) of a pixel in an image P̂ (n)< of reference. The column vector c containing the parameters ck quantifies the X-ray beam hardening calibration function for k ranging from 1 to K 3 . From a prescribed initial estimate c ini of these parameters, the CAL calculator calculates the vector c * best describing the observed effect.

[0039] According to one embodiment of the invention, the CAL calculator calculates (for example by finite differences) the fields s c k n of sensitivity, associated with each degree of freedom ck: s c k n = ∂ P ⌣ n ∂ c k c from the vector c = c ini prescribed initial values of the X-ray beam hardening calibration parameters ck in room 200, with the simulated image, obtained by applying the function u(with the parameters contained in the vector c ini ) to each of the pixels of P̂ ( n )< . The vector c ini of the initial values is such that u(y) = y. We thus have: P ⌣ n x = ∑ k = 1 K 3 c k φ k P ^ n x The fields s c k n sensitivity translates the effect of the variation of each of the parameters ck on the calculated image.

[0040] According to an embodiment of the invention, during step E3b, the CAL calculator calculates (for example by the CRES calculator of residues at the figure 4 ) projection residues ρ c n = P n − P ⌣ n , these residuals being the differences between the observed projections P ( n )< and simulated projections . From a first prescribed approximation c ini of the column vector c of parameters ck , the CAL calculator estimates (for example by the ESTV estimator at the figure 4 ) the vector c *best describing the observed system, i.e. minimizing the residuals ρ c n of projection. The vector c is calculated by the CAL calculator by optimization from the residues ρ c n of projection, of value c ini , and fields s c k n sensitivity. The CAL calculator calculates the variation δc * optimal vector c of parameters and updates the vector c according to the following equation: c * = c + δ c * Or c * is the column vector of parameters ck X-ray beam hardening calibration in room 200 updated, δc * is the column vector of variation of the parameters ck X-ray beam hardening calibration in room 200 and is calculated as the δc minimizing the sum of the norms of the squared differences between the projection residuals ρ c n and the product of δc by s c n For nranging from 1 to N, δ c * = arg min δ c ∑ n ρ c n − s c n δ c 2 Or δc is a column vector, s c n is the matrix of fields s c k n of sensitivity.

[0041] The number of rows in the matrix s c n is the number of pixels used in the projections P ( n )< , in the calculated images P̂ ( n )< and in the residue ρ c n . The number of columns in the matrix s c n is the number of parameters ck .

[0042] The CAL calculator calculates all the fields during step E3b s c k n of sensitivity for n ranging from 1 to Net k ranging from 1 to K 3 .

[0043] Vector estimation c parameters ck beam hardening calibration by the CAL calculator is performed by successive iterations on c who becomes c ini , For : calculate at each current iteration, from the vector c of the previous iteration taking the place of the vector c ini mentioned above fields s c k n of sensitivity according to the equation mentioned above, for each current iteration: calculate a new δc * and update c according to c * = c + δc *, with δc * calculated as the δc minimizing the sum of the norms of the squared differences between the projection residuals ρ c n and the product of δc by s c n For n ranging from 1 to N, depending on δc * = arg min δc ∑ n ρ c n − s c n δc 2 until c converges to a best estimate (determined for example by the fact that the norm of the difference calculated between the c of the current iteration and the c of the previous iteration becomes lower than a prescribed threshold).

[0044] During the fifth step E3c, after the first step E1, the CAL calculator estimates the vector α parameters α j of Compton scattering of X-rays in room 200 by minimizing the sum of the norms of the squared differences, calculated by the CAL calculator, between the N projections P ( n )< acquired and the N images simulated, having been calculated from the N images P̂ ( n )< calculated. The parameters α j Compton scattering represents the effect of Compton scattering on projections P ( n )< acquired. As illustrated in the figure 7, Compton scattering consists of the interaction of an X-ray photon with a free electron (or one weakly attached to its atom) of the part 200 being inspected, causing the photon to deviate from its initial trajectory. Upon reaching the detector 105, the photon thus scattered and deviated PDEV induces an increase in the measured intensity (called secondary signal P sec n in the following) in the DEV zone located around the direct and central point of impact CENT of the primary ray PCENT, and a decrease in this intensity (called primary signal P prim n in the following) at this CENT impact point, creating a blur in the projections that degrades the metrological analysis. Unlike known methods using a direct correction of the Compton effect by a deconvolution operation that is numerically unstable, making these reprocessed radiographs unreliable, step E3c plans to reproduce this Compton scattering effect on simulated images, to allow their comparison with the acquired projections. Step E3c can be implemented by an ESDC Compton scattering estimator, as shown in figure 4 . During step E3c, the CAL calculator estimates, by successive iterations, the vector α from an initial vector α = a ini . What is described below is done for each iteration of step E3c.

[0045] According to one embodiment of the invention, each projection P(n)<acquired is modeled by the CAL calculator as the sum of two components: the primary signal P prim n from photons passing through the object without Compton scattering but undergoing absorption by the atoms constituting the material (photoelectric effect); and the secondary signal P sec n which corresponds to the contribution of photons scattered by Compton scattering from room 200, according to the following equation: P n = P prim n + P sec n The secondary, or broadcast, signal P sec n is calculated by the CAL calculator as the result of the convolution of the primary signal P prim n with a parametric kernel K : P n = P prim n + P prim n ∗ K The CAL calculator calculates the parametric kernel K as a weighted sum of two-dimensional Gaussians g σ of widths different from each other (standard deviations σ different from each other): K x = ∑ j = 1 K 2 α j g σ j x − δ x where K 2 is a prescribed natural number greater than or equal to 1, jis a natural number ranging from 1 to K 2 , g s a two-dimensional Gaussian kernel of standard deviation s prescribed (following a sequence sj ), d ( x ) is the Dirac function at the pixel x . Alternatively, the parameters α j of the Gaussian kernel g sj , can be identified experimentally or via Monte Carlo simulations.

[0046] According to one embodiment of the invention, by identifying the primary signal P prim n to digitally simulated images , the CAL calculator calculates the vector α * which best approximates the observed Compton scattering. α * is the column vector of the parameters α j Compton scattering of X-rays in room 200, for j ranging from 1 to K 2 .

[0047] From a vector a ini prescribed initial values of the parameters α j Compton scattering of X-rays in room 200, the CAL calculator calculates images P ˜ n = P ⌣ n + P ⌣ n ∗ K from the images simulated, obtained from the N calculated images P̂ (n)< , according to the equation: P ˜ n = P ⌣ n + P ⌣ n ∗ K where the broadcast signal P ⌣ n ∗ K is simulated and added to the images so as to generate the images P̃ (n)< . The CAL calculator thus calculates all N images during step E3c.

[0048] According to one embodiment of the invention, P ⌣ n x = u P ^ n x is the image obtained by applying the function u to the intensity (gray level) of each pixel x of the calculated image P̂ (n)< . Of course, the simulated images can be obtained in another way from the N calculated images, in the case where step E3c is implemented without step E3b. The vector α in i prescribed initial values of the parameters α j Compton scattering initial values α = α ini is such that P ˜ n = P ⌣ n for α ini .

[0049] According to an embodiment of the invention, during step E3c, the calculator CAL calculates (for example by the calculator CRES) the residues at the figure 4 ) the residues ρ α n of projection, these residues ρ α n projection being the differences between the acquired projections P (n)< and the simulated projections P̃ (n)< according to the equation: ρ α n = P n − P ˜ n

[0050] From the first prescribed approximation of the vector α of parameters, noted a ini , the CAL calculator estimates (for example by the ESTV estimator at the figure 4 ) the vector α * best describing the observed system, i.e. minimizing the residuals ρ α n of projection.

[0051] According to one embodiment of the invention, the CAL calculator calculates (for example by finite differences) the fields s α j n of sensitivity, associated with each degree of freedom α j : s α j n = ∂ P ˜ n ∂ α j α from the column vector a ini prescribed initial values of the parameters α j Compton scattering of X-rays in room 200. The parameters α j Compton scattering of X-rays in room 200 are calculated by the CAL calculator by optimization from the residuals ρ α ini n of projection, of value a ini , and fields s α j n sensitivity. The CAL calculator calculates the variation yes * optimal of the vector α of parameters and updates the vector α according to the following equation: α * = α + δα * Or yes * is the column vector of variation of at least the parameters α j of Compton scattering of X-rays in room 200 and is calculated as the yes minimizing the sum of the norms of the squared differences between the projection residuals ρ α n and the product of yes by s α n For n ranging from 1 to N, δ α * = arg min δ α ∑ n ρ α n − s α n δ α 2 Or yes is a column vector, s α n is the matrix at less than fields s α j n of sensitivity. The number of rows in the matrix s α n is the number of pixels used in the projections P (n)< , in the calculated images P̂ (n)< and in the residue ρ α n . The number of columns in the matrix s α n is the number of parameters α j . The CAL calculator calculates during step E3c all the fields s α j n of sensitivity for n ranging from 1 to N and j ranging from 1 to K 1 . The estimation of the vector α parameters α j Compton scattering of X-rays by the CAL calculator is carried out by successive iterations on α who becomes a ini , for: calculate at each current iteration, from the vector α of the previous iteration taking the place of the vector a ini mentioned above fields s α j n of sensitivity according to the equation mentioned above, for each current iteration: calculate a new yes * and update α according to α * = α + yes *, with yes * calculated as the yes minimizing the sum of the norms of the squared differences between the projection residuals ρ α n and the product of yes by s α n For n ranging from 1 to N, depending on δ α * = arg min δ α ∑ n ρ α n − s α n δ α 2 until α converges to a best estimate (determined for example by the fact that the norm of the difference calculated between the α of the current iteration and the α of the previous iteration becomes lower than a prescribed threshold).

[0052] According to one embodiment of the invention, N is less than or equal to 1000 or 100 or 50 or 10. This reflects the low number of projections acquired. P(n)< despite which the invention is capable of operating. This number is lower than the number of acquired projections required in the state of the art for the tomographic reconstruction of the part 200.

[0053] During the seventh step E4, after the iterations of step E3a and step E3b and / or E3c, the CAL calculator processes the N projections P ( n )< obtained in step E1 and / or the N images P̂ ( n )< calculated, having been obtained after the iterations of step E2. The CAL calculator performs a first processing and / or a second processing.

[0054] The first treatment involves a beam hardening correction on the N projections P(n)< from the vector chaving been estimated or a generation of beam hardening on the N images P̂ ( n )< calculated from the vector c having been estimated.

[0055] The second treatment involves a correction of the Compton scattering on the N projections P ( n )< from the vector α having been estimated or a generation of Compton scattering on the N images P̂ ( n )< calculated from the vector α having been estimated.

[0056] The calculator thus obtains N acquired projections P a n , which are equal to or corrected from the N acquired projections P ( n )< of part 200, and N simulated images P s n , which are equal to or generated from the N images P̂ ( n )< calculated from part 200, these acquired projections P a n being directly comparable to the N simulated images Ps ( n )<, to then be able to carry out the analysis of part 200 during step E5.

[0057] For example, the CAL calculator first generates the N images during step E2 P̂ ( n )< calculated by applying the geometric recalibration for the N angles ANG (n)< of view, and this by using the parameters pi projection to simulate the N images P̂ ( n )< calculated. These are the images P̂ ( n )< calculated thus modified (recalibrated) which are used as images P̂ ( n )< calculated in the treatments below. This modification by recalibration using the parameters pi projection is carried out beforehand on the N images P̂ ( n )< calculated in the first and second embodiments of step E4, described below.

[0058] According to a first embodiment of step E4, the CAL calculator corrects the acquired projections P (n) < by applying to them the inverse of the function u depending on the X-ray beam hardening calibration parameters ck at each of the pixels of P ( n )< (to thus perform a beam hardening correction), to obtain the N projections P a n acquired corrected according to the following equation: P a n = u − 1 P n The CAL calculator calculates the function u depending on the parameters ck X-ray beam hardening calibration according to the following equation: u y = ∑ k = 1 K 3 c k ∗ φ k y Or c k ∗ designates the parameters ck X-ray beam hardening calibration values, contained in the vector c* and having been calculated. The CAL calculator processes the N images P̂ ( n )< calculated by convolving them with the kernel δ + K having been calculated based on the parametersα j Compton scattering (reproduction of Compton scattering on the N images) P̂ ( n )< calculated), to obtain the N simulated images P s n according to the following equations: P s n = P ^ n ∗ δ + K K = ∑ j = 1 K 2 α j ∗ g σ j − δ Or α j ∗ designates the parameters α j Compton scattering, contained in the vector α * and having been calculated.

[0059] According to a second embodiment of step E4, the CAL calculator calculates the projections P a n acquired according to the following equation: P a n = P n The calculator processes the N images P̂ (n)< calculated by applying the function to them u (calculated according to the calculation described above) depending on the parameters ck X-ray beam hardening calibration (to thus reproduce beam hardening), then convolving them with the kernel d+ K having been calculated (calculated according to the calculation described above) based on the parametersα j Compton scattering (reproduction of Compton scattering on the N images) P̂ ( n )< calculated), to obtain the N simulated images P s n according to the following equations: P s n = u P ^ n ∗ δ + K

[0060] During the eighth step E5 of analysis of the actual part 200, after step E4, the CAL calculator identifies defects in the part 200 by comparing the N projections P ( n )< processed to N images P̂ ( n )< calculated processed. The CAL calculator identifies defects in part 200 by comparing the N acquired projections P a n to the N simulated images P s n according to one of the embodiments described above. The CAL calculator carries out an identification of defects in the part by comparing the acquired images P ( n )< and simulated P̂ ( n)< having taken into account the correction or the treatment either on one or on the other. The CAL calculator calculates N residues ρ ( n )< of projection, equal to the difference between the acquired projections P a n and simulated images P s n following the equation: ρ n = P a n − P s n The CAL calculator records the N acquired projections in the MEM memory P a n and / or the N simulated images P s n and / or the N projection residues ρ (n)<. The CAL computer analyzes these N projection residues ρ (n)< in order to identify defects inherent in the inspected part 200. The CAL computer can provide on the output interface INT2 these N projection residues ρ (n)< and / or the defects identified from these N projection residues ρ (n)<. The CAL computer can provide on the output interface INT2 a certificate of validity or invalidity of the part, determined by the CAL computer from the N projection residues ρ (n)<. The CAL computer or the user can, on the basis of the N acquired projections P a n and / or N simulated images P s n and / or N projection residues ρ (n)<, sanction part 200 as being invalid because it has too many defects, or validate part 200 as being valid because it does not have too many defects.

[0061] According to one embodiment of the invention, the vector p groups together one after the other all the K 1 parameters pi of projection geometry, K 3 parameters ck beam hardening of radiation in room 200 and K 2 parameters α j of Compton scattering of the radiation and therefore has a dimension of K 1 + K 2 + K 3 . Correspondingly, the vector s p n groups together one after the other all the K 1 sensitivity fields s p i n , K 3 fields s c k n sensitivity and K 2 fields s α j n of sensitivity and therefore has a dimension of K 1 + K 2 + K 3 . Correspondingly, the vector δp groups together one after the other all the variations of the K 1 parameters pi of the projection geometry, of the K 3 parameters ck beam hardening of radiation in room 200 and K 2 parameters α j of Compton scattering of the radiation and therefore has a dimension of K 1 + K 2 + K 3 .

[0062] There figure 10 shows an example of the difference between, on the one hand, the corrected acquired intensity image I a n = I 0 exp − P a n having been obtained according to the invention from the acquired projection P ( n )< , where I 0 is a prescribed setting from the projections P ( n )< acquired, and on the other hand the corrected simulated intensity image I s n = exp P s n having been obtained from the simulated projection P s n for a turbomachine turbine blade. The figure 11 shows an example of the difference between, on the one hand, the corrected acquired intensity image I a n = I 0 exp − P a n having been obtained according to the invention from the acquired projection P (n)< , where I 0 is a prescribed setting from the projections P ( n )< acquired, and on the other hand the simulated intensity image I s n = exp P s n having been obtained from the simulated projection P s n for a turbomachine turbine blade root. The figures 10 and 11 show that the analysis of the part 200 is facilitated by the invention taking into account the artifacts. The figures 8 and 9 show I s - I a , without correction on I a , and where I am is generated with pin , c ini And a ini . THE figures 10 and 11 show I s - I a , without correction on I a , and where I am is generated with p*, c * And α *.

[0063] In the case of the combination of steps E3a, E3b and E3c, taking into account the knowledge on the acquisition system (cabin and part, step E3a) and coupling it to an adapted algorithmic method E3a, E3b, E3c of image processing allows a significant gain on the quality of the information contained in the images produced by the system and in those reproduced by simulation, and therefore of their difference. This allows the estimation of the artifacts (1) of beam hardening and (2) of Compton scattering, in the context of the analysis of aeronautical parts from a limited number of multi-view projections. Step E3a allows the estimation of the geometric parameters making it possible to link the MODP model of the ideal part to the acquired projections of the part to be controlled. This step E3a makes it possible to find the parameters in order, in particular, to use them to numerically simulate the projections and reproduce artifacts in the simulated images.The invention also makes it possible, in step E4, either to correct the artifacts due to beam hardening (E3b) in the acquired projections, or to reproduce them in the simulated images; and to reproduce the artifacts due to Compton scattering (E3c) in the simulated images or to correct them in the acquired projections. The image acquisition system and the algorithmic suite make the validation and sanction of the material health indications and the indications of the three-dimensional geometry of the part 200 reliable from a limited number N of X-ray radiographic views.

[0064] According to one embodiment of the invention, the method, the device and the program according to the invention can be implemented on one or more regions of interest, selected by a selection module of the CAL calculator, in the projection P ( n )< .

[0065] Of course, the embodiments, features, possibilities and examples described above can be combined with each other or selected independently of each other.

Claims

1. A method of non-destructive testing of a part (200) by transmission radiography, comprising the following steps, executed by a calculator (CAL): acquiring (E1) of N projections (P(n)) of the part (200) using a transmission radiography device (100) from N different and predetermined angles (ANG(n)) of view of the part (200), where N is a given natural integer, generating (E2) of N computed images (P̂(n)) of the part (200) from a reference model (MODP) of the part (200) corresponding to the N angles (ANG(n)) of view and from a vector p of parameters (pi) characterizing a projection geometry of acquisition for the N angles (ANG(n)) of view at each of several successive iterations, estimating (E3a, E3b, E3c), by the successive iterations, of the vector p from an initial vector p = pini and of at least one of a vector c of parameters from an initial vector c = cini and of a vector α of parameters from an initial vector α = αini, where the vector c of the parameters (ck) accounts for beam hardening of radiation in the part (200) and the vector α of the parameters (αj) characterizes Compton scattering of the radiation in the part (200), by minimizing the sum of the squared norms of the differences between the N projections (P(n)) having been acquired and the N computed images (P̂(n)), processing (E4) of the N projections (P(n)) and / or of the N computed images (P̂(n)) comprising a first processing and / or a second processing, the first processing comprising a correction of the beam hardening over the N projections (P(n)) from the vector c having been estimated or a generation of the beam hardening over the N computed images (P̂(n)) from the vector c having been estimated, the second processing comprising a correction of the Compton scattering over the N projections (P(n)) from the vector α having been estimated or a generation of the Compton scattering over the N computed images (P̂(n)) from the vector α having been estimated, identifying (E5) of defects of the part (200) by comparison of the N projections (P(n)) having been processed with the N computed images (P̂(n)) having been processed.

2. The method as claimed in claim 1, characterized in that it comprises at each iteration estimating (E3a) of the vector p of the parameters pi of the projection geometry of the acquisition for the N angles (ANG(n)) of view by the calculator (CAL), the estimating comprising: computing of projection residuals ρ p n = P n − P ^ n from the initial vector p = pini of initial values, computing of sensitivity fields s p i n according to s p i n = ∂ P ^ n ∂ p i p from the initial vector p = pini of the initial values, where P̂(n) are the N computed images for the N angles (ANG(n)) of view, P(n) are the N projections having been acquired of the part (200), p is a column vector of the parameters pi of the projection geometry, pini is a column vector of the initial values of the parameters pi of the projection geometry, n is a natural integer denoting the number of the angle (ANG(n)) of view and ranging from 1 to N, p is updated according to p* = p + δp*, where p* is a column vector of the parameters pi of the projection geometry having been updated, δp* is a column vector of variation of the parameters pi of the projection geometry and is calculated as a δp minimizing the sum of the squared norms of the differences between the projection residuals ρ p n and the product of δp by s p n for n ranging from 1 to N, δ p * = arg min δ p ∑ n ρ p n − s p n δ p 2 where δp is a column vector, s p n is a matrix of the sensitivity fields s p i n .

3. The method as claimed in any one of the preceding claims, characterized in that it comprises at each iteration estimating (E3b) of the vector c of the parameters ck of calibration of the beam hardening of the radiation in the part (200) by the calculator (CAL), the estimating comprising: computing of projection residuals ρ c n = P n − P ⌣ n from the initial vector c = cini of initial values, computing of sensitivity fields s c k n according to s c k n = ∂ P ⌣ n ∂ c k c from the initial vector c = cini of the initial values, where P ⌣ n x = u P ^ n x is the image obtained by applying a function u(y) to an intensity y of each of pixels x of P̂(n), P̂(n) are the N computed images for the N angles (ANG(n)) of view, P(n) are the N projections having been acquired of the part (200), c is a column vector of the parameters ck of calibration of the beam hardening, cini is a column vector of the initial prescribed values of the parameters ck of calibration of the beam hardening of the radiation in the part (200), φk(y) is a base of given form functions, u y = ∑ k = 1 K 3 c k φ k y K3 is a given natural integer greater than or equal to 1, k is a natural integer ranging from 1 to K3, c is updated according to c* = c + δc*, where c* is a column vector of the parameters ck of calibration of the beam hardening of the radiation in the part (200), having been updated, δc* is a column vector of variation of the parameters ck of calibration of the beam hardening of the radiation in the part (200) and is computed as a δc minimizing the sum of the squared norms of the differences between the projection residuals ρ c n and the product of δc by s c n for n ranging from 1 to N, δ c * = arg min δ c ∑ n ρ c n − s c n δ c 2 where δc is a column vector, s c n is a matrix of the sensitivity fields s c k n .

4. The method as claimed in any of the preceding claims, characterized in that it comprises at each iteration estimating (E3c) of the vector α of the parameters αj of Compton scattering of the radiation in the part (200) by the calculator (CAL), the estimating comprising: computing of projection residuals ρ α n = P n − P ˜ n from the initial vector α = αini of the initial values, computing of sensitivity fields s α j n according to s α j n = ∂ P ˜ n ∂ α j α from the initial vector α = αini of the initial values, where P(n) are the N projections having been acquired of the part (200), P̂(n) are the N computed images for the N angles (ANG(n)) of view, P ˜ n = P ⌣ n + P ⌣ n ∗ K is an image obtained by convolution of simulated images , having been obtained from at least the N computed images P̂(n), with a kernel δ + K, α is a column vector of the parameters αj of Compton scattering of the radiation in the part (200), αini is a column vector of the initial values of the parameters αj of Compton scattering of the radiation in the part (200), K is a convolution kernel defined by K x = ∑ j = 1 K 2 α j g σ j x − δ x K2 is a prescribed natural integer greater than or equal to 1, j is a natural integer ranging from 1 to K2, gσj, a two-dimensional Gaussian kernel of prescribed standard deviation σj, δ(x) is a Dirac function at the pixel x, α is updated according to α* = α + δα*, where α* is a column vector of the parameters αj of Compton scattering of the radiation in the part (200), having been updated, δα* is a column vector of variation of the parameters αj of Compton scattering of the radiation in the part (200) and is calculated as a δα minimizing the sum of the squared norms of the differences between the projection residuals ρ α n and the product of δα by s α n for n ranging from 1 to N, δ α * = arg min δ α ∑ n ρ α n − s α n δ α 2 where δα is a column vector, s α n is a matrix of the sensitivity fields s α j n .

5. The method as claimed in claims 3 and 4 taken together, characterized in that P ⌣ n x = u P ^ n x is the simulated image, obtained by applying the function u(y) to the intensity of each of the pixels x of the computed image P̂(n).

6. The method as claimed in any of the preceding claims, characterized in that N is less than or equal to 1000.

7. A computer program, comprising code instructions for implementing the following steps of a method of non-destructive testing of a part (200) by transmission radiography, when it is executed by a calculator (CAL): receiving (E1) of N projections (P(n) ) of the part (200) from a transmission radiography device (100) from N different and predetermined angles (ANG(n)) of view of the part (200), where N is a given natural integer, generating (E2) of N computed images (P̂(n)) of the part (200) from a reference model (MODP) of the part (200) corresponding to the N angles (ANG(n)) of view and from a vector p of parameters (pi) characterizing the projection geometry of the acquisition for the N angles (ANG(n)) of view at each of several successive iterations, estimating (E3a, E3b, E3c), by the successive iterations, of the vector p from an initial vector p = pini and of at least one of a vector c of parameters from an initial vector c = cini and of a vector α of parameters from an initial vector α = αini, where the vector c of the parameters (ck) accounts for the beam hardening of the radiation in the part (200) and the vector α of the parameters (αj) characterizes the Compton scattering of the radiation in the part (200), by minimizing the sum of the norms of the squared differences between the N projections (P(n)) having been acquired and the N computed images (P̂(n)), processing (E4) of the N projections (P(n)) and / or of the N computed images (P̂(n)) comprising a first processing and / or a second processing, the first processing comprising a correction of the beam hardening over the N projections (P(n)) from the vector c having been estimated or a generation of the beam hardening over the N computed images (P̂(n)) from the vector c having been estimated, the second processing comprising a correction of the Compton scattering over the N projections (P(n)) from the vector α having been estimated or a generation of the Compton scattering over the N computed images (P̂(n)) from the vector α having been estimated, identifying (E5) of defects of the part (200) by comparison of the N projections (P(n)) having been processed with the N computed images (P̂(n)) having been processed.

8. A device for non-destructive testing of a part (200) by transmission radiography, comprising: a transmission radiography device (100), for acquiring (E1) of N projections (P(n)) of the part (200) along N different and predetermined angles (ANG(n)) of view of the part (200), where N is a given natural integer, a calculator (CAL) configured to carry out the following steps: generating (E2) of N computed images (P̂(n)) of the part (200) from a reference model (MODP) of the part (200) corresponding to the N angles (ANG(n)) of view and from a vector p of parameters (pi) characterizing the projection geometry of the acquisition for the N angles (ANG(n)) of view at each of several successive iterations, estimating (E3a, E3b, E3c), by the successive iterations, of the vector p from an initial vector p = pini and of at least one of a vector c of parameters from an initial vector c = cini and of a vector α of parameters from an initial vector α = αini, where the vector c of the parameters (ck) accounts for the beam hardening of the radiation in the part (200) and the vector α of the parameters (αj) characterizes the Compton scattering of the radiation in the part (200), by minimizing the sum of the norms of the squared differences between the N acquired projections (P(n)) and the N computed images (P̂(n)), processing (E4) of the N projections (P(n)) and / or of the N computed images (P̂(n)) comprising a first processing and / or a second processing, the first processing comprising a correction of the beam hardening over the N projections (P(n)) from the vector c having been estimated or a generation of the beam hardening over the N computed images (P̂(n)) from the vector c having been estimated, the second processing comprising a correction of the Compton scattering over the N projections (P(n)) from the vector α having been estimated or a generation of the Compton scattering over the N computed images (P̂(n)) from the vector α having been estimated, identifying (E5) of defects of the part (200) by comparison of the N projections (P(n)) having been processed with the N computed images (P̂(n)) having been processed.