Method for reconstructing object by X-ray tomography, comprising calculation of regularization term

Abstract

The invention relates to a method for reconstructing an object by X-ray tomography by means of an X-ray source and an X-ray detector. The method comprises the following steps: obtaining (E0) the positions of the X-ray source and the X-ray detector; pre-calculating (E1) a direction set; calculating (E2) a vector imperfect graph; calculating (E3) a metric imperfect graph IM (x); calculating (E4) a regularization term from the vector imperfect graph and the metric imperfect graph; and iteratively solving (E5) the regularization tomography reconstruction problem by means of the calculated regularization term.

Description

Methods for reconstructing objects using X-ray tomography, including the calculation of regularization terms. Technical Field

[0001] This invention relates to the field of X-ray-based imaging, and more specifically to the field of computed tomography. The invention relates to a method for reconstructing an object using X-ray tomography, including the calculation of a regularization term.

[0002] This invention is applicable to the security field, and is particularly useful for searching baggage in sensitive locations (airports, train stations, etc.) during incidents or when baggage checks are required. It is also applicable to the medical field, for example, in medical X-ray imaging equipment. More generally, this invention is suitable when it is desired to reconstruct an object using computed tomography. Background Technology

[0003] X-ray computed tomography (CT) provides two-dimensional or three-dimensional images from a series of tomographic projections using reconstruction algorithms. When the measurement data has high noise and / or contains blanks, the reconstruction algorithms used in CT become inadequate. The reconstruction algorithms then encounter data bottlenecks and may therefore generate highly inconsistent solutions.

[0004] In order to reconstruct an object, in addition to the input data, the reconstruction algorithm also needs to make pre-assumptions about the object to be reconstructed. These assumptions about the object are made through regularization, thus giving rise to regularized reconstruction algorithms.

[0005] Therefore, regularized reconstruction algorithms solve the inverse problem based on projection data. These algorithms require tuning regularization terms. The purpose of these regularization terms is to strike a trade-off between the information contained in the data and the assumptions made about the object to be reconstructed. Regularization allows for pre-assumptions about the statistical distribution of the imaged object and / or consideration of the acquisition geometry. This regularization depends on parameters, referred to here as hyperparameters. These hyperparameters are computed before the object reconstruction process, as described below.

[0006] Most of the time, regularization terms are constructed to limit the number of hyperparameters that need to be tuned. Specifically, the more hyperparameters there are, the more difficult it is to tune them manually.

[0007] To automatically adjust hyperparameters, some methods rely on cross-validation [Kohavi, 1995], or more recently, on machine learning methods [Shen, 2018], [Ding, 2021]. In 2021, [Zhang, 2021] showed that regularization of the total variation type is highly suitable for finite-angle reconstruction problems. However, this regularization requires more hyperparameters than regular total variation adjustment [Rudin, 1992].

[0008] Current methods for adjusting regularization terms require reference images and / or large amounts of diverse data. While reference images can be obtained through simulation, this approach remains difficult to implement under experimental conditions.

[0009] In addition, ideally, the choice of regularization term should take into account the geometry of the scanner, i.e. the location of the source and detector.

[0010] The 3D theory of computer-aided tomography provides tools that allow for the determination of local information about an object to be reconstructed given the geometry of a scanner. Assuming the projection is non-truncated, [Tuy, 1983] provides a condition that can be used to verify whether a continuous X-ray source path is sufficient to reconstruct the object. This condition can be written as: "Every plane intersecting the imaged object must intersect the source path at least once." In practice, each scanner is limited because only a finite number of source locations are possible; therefore, Tuy's condition is never satisfied in practice. When Tuy's condition is not satisfied, the term "incompleteness" is used.

[0011] Figure 1 illustrates an example of a source-detector path T for a scanner used to reconstruct an object. Example a) shows a circular source-detector path. This path does not allow for accurate reconstruction of the object outside the path plane. Specifically, a plane that intersects the object and, for example, is parallel to the circle of path T does not intersect the source-detector path. As shown in example b), a portion of path T with helical geometry allows for accurate reconstruction. Specifically, regardless of which plane is considered, it must intersect the helix of path T.

[0012] Several measures that allow for the quantification of the effects of incompleteness have been investigated. [Metzler, Bowsher, and Jaszczak, 2003], [Lin and Meikle, 2010], and [Liu et al., 2012] numerically estimated the percentage of planes intersecting the source path by sampling a unit sphere. This type of measurement is only suitable for continuous source paths, but in practice, sampling is performed at discrete points along the source path. Therefore, the plane must pass between sources. Furthermore, the authors of these papers show that it is difficult to predict reconstruction quality based on this metric.

[0013] [Stopp, Winne, Jank, and Keeve, 2010] have proposed a local measurement of the average of the minimum angles, defined by the plane passing through the examined voxel and the associated optimal source. [Clackdoyle and Noo, 2020] have also quantified incompleteness, aiming to measure how far a voxel is from satisfying the Tuy condition in a given direction. These methods can be applied to discrete source paths.

[0014] Existing methods do not utilize the theoretical conditions for local reconstruction in tomography. Regardless of local sampling differences, the resulting adjustments are identical for any object to be reconstructed. Specifically, if local adjustment regularization is used, the number of parameters or terms that need to be adjusted becomes excessive. For example, nonlocal regularization of the total orientation variation type requires adjusting three hyperparameters, while applying local total orientation variation to a volume of one million voxels would require adjusting three million hyperparameters. This makes it impossible to select hyperparameters using current methods.

[0015] It should be noted that in this document, the term "voxel" is understood to mean a pixel or voxel, depending on whether it is a two-dimensional or three-dimensional reconstruction. Summary of the Invention

[0016] To reconstruct objects via X-ray tomography, this invention aims to provide voxel-by-voxel regularization that can be customized for any acquisition system or scanner used in its application, while maintaining low data costs.

[0017] Therefore, the present invention relates to a method for reconstructing an object by means of an X-ray source and an X-ray detector via X-ray tomography, the object belonging to an imaging space defined by a set of voxels. The method includes the following steps: obtaining the positions of the X-ray source and the X-ray detector; pre-compiling a set of directions; calculating a vector incompleteness graph, the vector incompleteness graph being for each voxel including a vector in a direction ν* in the set of directions; calculating a metric incompleteness graph, the metric incompleteness graph defining an information loss in the direction ν* for each voxel; calculating a regularization term based on the vector incompleteness graph and the metric incompleteness graph; and iteratively solving a regularized tomography reconstruction problem using the calculated regularization term.

[0018] Therefore, the solution provided by this invention enables the use of incomplete information to adjust regularization terms, such as hyperparameters of local total variation type regularization. The goal is to introduce specific voxel-by-voxel regularization that will utilize data based on incomplete estimates, i.e., directional information and the strength of the regularization (i.e., assigning weights to the regularization).

[0019] Because the incomplete information is local and specific to each acquisition system or scanner, the result is a regularization that can be tailored for any acquisition system. The advantages of this invention are as follows: it takes into account the geometry of the scanner; the introduction of local regularization allows for more accurate reconstruction; higher quality images can be reconstructed, especially when the acquisition geometry is missing data leading to image reconstruction problems (finite angle problems, ill-posed or pathological inverse problems); using analytical criteria that reflect the geometry of the acquisition system overcomes the barriers of hyperparameter tuning in local total variation regularization without the need for generating experimental data, simulations, or reconstructions, or human or artificial intelligence intervention; this invention is applicable to both known and novel geometries (static multi-source, etc.); this invention is applicable to any X-ray tomography system in the fields of medical imaging, non-destructive testing, or even airport baggage screening; this invention creates new opportunities for developing novel X-ray tomography reconstruction methods (with dynamic reconstruction of 3D total variation + local time).

[0020] The following are some practical and specific preferred features of the reconstruction method according to the present invention.

[0021] Calculating the vector incompleteness graph includes the following sub-steps for each voxel in the imaging space: selecting a source from the sources for each direction in the direction set; and selecting a direction ν* for each selected source, wherein the information loss defined by the metric incompleteness graph corresponds to the information loss of the detector associated with the selected source in the selected direction ν*.

[0022] The steps of computing the vector incompleteness graph include a preparatory sub-step of selecting a source from which voxels of an object are projected onto their associated detectors, the source being selected from sources that project voxels of an object onto their associated detectors.

[0023] The source is obtained by using an expression Calculate local orientation incompleteness I(x,ν)∈ To choose, among them It is the source The projection onto the plane defined by the voxel's point x and the normal direction ν.

[0024] The selected direction is chosen based on the local directional incompleteness I(x, ν).

[0025] The selected direction ν* is chosen based on the norms of multiple directions, and the local directional incompleteness I(x, ν) is calculated for the multiple directions.

[0026] The norm is the Lp norm of p∈[1;+∞], where p is preferably equal to +∞.

[0027] The selected direction ν* is chosen based on the weighted average of multiple directions, and the local directional incompleteness I(x, ν) is calculated for the multiple directions.

[0028] The regularization term refers to the regularization function. The calculated regularization function is as follows: in For each finite difference, j, k, l correspond to the index of each voxel, and d corresponds to the direction of the finite difference.

[0029] Regularization function This is the type of total directional variation. Attached Figure Description

[0030] Other features and advantages of the invention will become more apparent from the following description given with reference to the accompanying drawings, which are given by way of non-limiting example: Figure 1 shows an example of a source-detector path for a scanner used to reconstruct an object; Figure 2 shows a flowchart of a method for reconstructing an object by X-ray tomography according to an embodiment of the invention; Figure 3 shows an example of calculating a set of orientations; Figure 4 shows an example of calculating local orientation incompleteness; and Figure 5 shows an example of a 3D incompleteness map. Detailed Implementation

[0031] Figure 2 shows a flowchart of a method for reconstructing an object imaged by X-ray tomography according to one embodiment. The reconstruction algorithm used to reconstruct the object in this invention is a regularized reconstruction algorithm. The method first involves calculating a regularization term, and according to an example of an embodiment, the steps required to do so are shown in Figure 2.

[0032] X-ray computed tomography (CT) is performed using a scanner. The scanner consists of an X-ray source and an X-ray detector. The set of source-detector pairs representing the scanner geometry is denoted as {s1, s2, ..., s...}. N}∈ and {d1, d2, ..., d N}∈ There are N source-detector pairs. Each source is associated with one detector.

[0033] The calculation of regularization terms and the reconstruction of objects are performed by computers.

[0034] The reconstruction method includes steps E0 of obtaining the positions of the X-ray source and X-ray detector and defining an imaging space Ω containing the object to be reconstructed. The object to be reconstructed is independent of this method. In particular, the shape of the object has no effect on the calculation of incompleteness, and therefore no effect on regularization. Only the size of the imaging space Ω containing the object to be reconstructed affects the method.

[0035] The imaging space Ω is meshed into multiple voxels. The imaging space Ω can be of any shape and size. For example, the imaging space Ω is a cube with dimensions of 256mm × 256mm × 256mm.

[0036] The reconstruction method also includes a step of pre-compiling a set of directions. This pre-computation step allows for the discretization of directions in space. Therefore, a finite number of directions are predefined in the space. The more directions pre-computed or predefined, the more accurate the calculated incomplete directions will be. However, the more directions pre-computed, the more computations are required. The number of directions should be chosen to satisfactorily describe the entire space while limiting the computation to a reasonable number.

[0037] For example, the pre-computation step includes placing a set of unit vectors uniformly on a sphere S. 2 As shown in Figure 3. For example, using a method based on the Fibonacci lattice [Gonzalez, 2009], these points are uniformly arranged along a spiral of the sphere. Iteratively, each new point is placed between the previous points that are furthest apart.

[0038] Then, the reconstruction method includes steps E2 and E3 for calculating the incomplete graph. Specifically, the method includes step E2 for calculating the vector incomplete graph and step E3 for calculating the metric incomplete graph.

[0039] According to one example of an embodiment, step E2 of computing the incomplete vector graph includes a first sub-step E20, for each voxel in the imaging space or at a point x ∈ Ω, selecting a source that projects the voxel in the imaging space onto its associated detector. In other words, selecting a source such that the direction (s i x) and detector d i Intersecting sources. A set of sources that project voxels of the imaging space Ω onto their associated detectors is represented as { , … }⊆{s1、s2、…、s N}

[0040] Therefore, the number of sources is reduced. For each voxel in the imaging space Ω, only sources capable of viewing that voxel are selected.

[0041] Step E2, which computes the vector incompleteness graph, includes a second sub-step E21, for each voxel in the imaging space or at a point x∈Ω, selecting the source (called the best source) from which the voxel can be best viewed.

[0042] The optimal source is obtained by calculating the point x∈Ω and pre-computing the direction ν∈S. 2 Local directional incompleteness I(x, ν)∈ The selection is based on the expression. Local orientation incompleteness is determined by... The calculation yielded the result. It is the source In the voxel's point x and normal direction = The projection onto the defined plane. This expression is defined by [Clackdoyle & Noo, 2020].

[0043] The above calculation of local directional incompleteness I(x, ν) quantifies the minimum tangent of the angle defined by the plane (x, ν) and the straight line traced by the X-ray (i.e., the straight line passing through the source point and x along the path). Therefore, local directional incompleteness I(x, ν) quantifies the degree to which a voxel or point x can be reconstructed in the direction ν.

[0044] The higher the value of I(x,ν), the less locally point x satisfies Tuy's condition, and the greater the loss of tomographic information in the direction ν. Specifically, the ratio of I(x,ν)... The lower the value of I(x,ν), the closer the source is to the normal plane. When the source is located in the normal plane, I(x,ν) equals 0. Conversely, the closer the direction ν is to any line passing through the source that intersects point x, the higher the ratio of I(x,ν). The larger it gets, the closer it tends to infinity. Therefore, the ratio of I(x,ν) The higher the value, the more difficult it is to locally satisfy the conditions of Tuy, and the less likely it is to see point x in a plane (or normal plane) defined by the same direction ν.

[0045] By calculating the ratio Find the minimum value of the vector and the direction most orthogonal to the source vector. Therefore, the optimal source is obtained.

[0046] Here, the optimal source is the source that projects voxels of the imaging space Ω onto their associated detectors. , … The first sub-step E20, which involves selecting the source to project the voxels of the imaging space Ω onto their associated detector, is optional. According to an example of an embodiment, consider the source set {s1, s2, ..., s}. N From the source set {s1, s2, ..., s} N Choose the best source from the list.

[0047] Figure 4 illustrates the calculation of the local orientation incompleteness I(x,ν), with the view showing the selection of the optimal source to be used for reconstructing point x. For simplicity, only two sources, s1 and s2, are shown. The angle between the line passing through source s1 and point x and the plane perpendicular to ν is denoted as ψ. i The local directional incompleteness I(x,ν) is equal to tan(ψ). iThe tangent of source s1, tan(ψ) i The calculated value of ) is 0.28, while the calculated value of the tangent tan(ψ2) of source s2 is 0.65. Therefore, the local directional incompleteness I(x,ν) is equal to 0.28 here.

[0048] According to another example of the embodiment, the source selected in step E21 can be a source other than the optimal source. For example, a second optimal source can be selected.

[0049] Step E2, which computes the incomplete vector graph, includes a third sub-step E22, for each voxel in the imaging space or for a point x∈Ω, selecting the direction ν* for the selected optimal source.

[0050] The selected direction ν* can be chosen or calculated in various ways. The selected direction ν* is chosen or calculated based on the local direction incompleteness I(x,ν).

[0051] The selected direction ν* can be chosen based on the calculation of the norm. The selected direction ν* is, for example, chosen based on the calculation of the norm of multiple directions (v1, ..., vn), which correspond to a pre-computed set of directions or a subset thereof. The norm is, for example, the Lp norm of the considered direction, where p∈[1; +∞].

[0052] A preferred example is to select the worst-case direction ν*. The worst-case direction ν* corresponds to the direction in which the detector associated with the selected best source finds each voxel the most difficult to see.

[0053] via expression: Calculate the worst direction ν*∈S at point x∈Ω. 2 In other words, the worst-case direction ν* corresponds to the Lp norm (of the considered direction) that p tends towards +∞.

[0054] Therefore, the worst-case direction ν* is the direction that maximizes the local directional incompleteness I(x,ν). This is a direction problem in which the worst possible outcome will be obtained even if the optimal source is chosen, i.e., point x appears to be the worst.

[0055] In the simplified example, for illustrative purposes, the direction set contains three directions {ν1, ν2, ν3}, and a local orientation incompleteness I(x,ν) is computed for each of the three directions. Thus, three values ​​are obtained: I(x,ν1), I(x,ν2), and I(x,ν3). The worst direction ν* corresponds to the direction with the highest local orientation incompleteness among the three directions {ν1, ν2, ν3}.

[0056] According to another example of the embodiment, a selected direction ν* can be selected by calculating a weighted average of the considered directions. Local directional incompleteness is calculated for each considered direction. A weight is associated with each considered direction, and this weight depends on the local incompleteness in the considered direction. The higher the local incompleteness in the considered direction, the greater the weight.

[0057] Therefore, in the same simplified example considering a set of directions {ν1, ν2, ν3}, the local directional incompleteness I(x,ν) is calculated for each of the three directions. Thus, three values ​​I(x,ν1), I(x,ν2), and I(x,ν3) are obtained. Depending on the obtained values ​​I(x,ν1), I(x,ν2), and I(x,ν3), the corresponding weights p1, p2, and p3 are associated with each of the three directions {ν1, ν2, ν3} (weight p1 is associated with I(x,ν1), weight p2 with I(x,ν2), and weight p3 with I(x,ν3)). The selected direction is calculated by taking the weighted average of the three directions {ν1, ν2, ν3}.

[0058] The following are unrestricted examples of calculations for a selected direction ν*, some of which are as described above: Weighted mean: , where N is the number of pre-calculated directions; Weighted average of the worst directions: ,in It is the i-th highest incompleteness (or lowest incompleteness), and Selected as less than or equal to N pre-calculated directions; mean: Optimal direction: .

[0059] Regardless of how the direction ν* is calculated or chosen, at each point x in the imaging space Ω, a vector giving the selected direction ν* will be obtained. Therefore, the direction ν* is obtained as follows: The incomplete 3D vector map is obtained, that is, the incomplete 3D spatial distribution in the imaging space Ω is obtained.

[0060] After selecting the direction ν*, step E3 is performed to compute the metric incompleteness graph. The metric incompleteness depends on the local directional incompleteness I(x,ν). In other words, the 3D metric incompleteness graph IM: Defined by IM(x)=h(I(x,ν)), where h is a function.

[0061] 3D Measurement Incompleteness Map IM: For example, it is defined by: IM(x) = I(x, ν*(x)). In other words, the metric incompleteness IM for a given voxel corresponds to the incompleteness function applied to the selected direction ν*.

[0062] The metric incompleteness map defines, for each voxel, the information loss of the detector associated with the selected optimal source in the chosen direction ν*. Metric incompleteness provides information about the degree to which a point x in the imaging space Ω is not well visible in the chosen direction ν*. In the sense of tomographic reconstruction conditions, the higher the point x, the better.

[0063] The 3D metric incompleteness graph IM can be defined by another function. For example, the 3D metric incompleteness graph IM: It can be defined as follows: IM(x) = I 2 (x,ν*(x)), or IM(x) = tanI(x,ν*(x)).

[0064] Once calculated as described above, the 3D metric incompleteness graph IM can be further reduced to have a and b∈ The segment [a, b]. Therefore, we obtain the following: IM(x) = a + .

[0065] Therefore, two incomplete 3D graphs are obtained that are directly related to the inherent geometry of the scanner: a vector incomplete graph indicating the direction of missing information ν*(x) at each point x∈Ω (e.g., the worst direction ν*(x) with the least information at that point in the projection terms); and a metric incomplete graph that quantifies this missing information using IM(x).

[0066] Figure 5 shows an example of a 3D incompleteness map when the imaging space Ω is a cube. For each point x∈Ω, the figure shows a cone with a selected orientation ν*(x) and a value corresponding to the missing information IM(x). For readability, the grid of the imaging space Ω used here is 3×3 (i.e., 27 cones). A finer grid, such as 9×9, would be preferred.

[0067] The variation of the missing information IM(x) has been represented by a point cloud, with the points arranged on a cone. The higher the IM(x), the greater the density of points.

[0068] Part a) of Figure 5 illustrates the case where the source-detector path T is circular. This figure shows that only points within the circular plane of the path satisfy Tuy's condition. Incompleteness increases with distance from the circular plane. In other words, voxels become increasingly difficult to reconstruct using tomography as the distance from the circular plane increases.

[0069] Part b) of Figure 5 illustrates the case where the source-detector path T is spiral. This figure shows that the incompleteness is more uniform and very low throughout. From the perspective of tomographic reconstruction, every direction (even the worst direction) is satisfactory.

[0070] The reconstruction method includes step E4, which computes regularization terms based on the vector incompleteness graph and the metric incompleteness graph. Information ν*(x) and IM(x) are used to adjust the regularization terms of the regularization function.

[0071] Regularization allows for statistical pre-assumptions about the imaging object and / or takes into account the scanner's acquisition geometry.

[0072] Regularization is achieved through an iterative method. In this method, a large-dimensional minimization problem is iteratively solved for tomographic reconstruction. The problem here is to find an approximation of f. , so that: Where: A is the tomographic projection matrix that can be transferred from the object domain to the projection domain. In other words, matrix A is a transformation that can be transferred from the imaging space to the space to which the data is projected; b is the acquired data (also known as a sine curve); This is called the data fidelity term. The data fidelity term ensures that the solution remains close to the measurements taken; and This is called the regularization term, where α is a parameter that allows adjusting the importance of regularization relative to the data fidelity term. The regularization used in the example shown is of the local total variation type. In this case, the regularization term can be expressed as... .

[0073] The above equation sets a regularity condition, thereby allowing the reconstruction algorithm to diverge due to inconsistencies in the input data to be controlled. The regularity condition is a set of assumptions.

[0074] Data fidelity items can be, for example, This is a question that allows for data fidelity evaluation, assuming the acquisition noise is Gaussian noise. The data fidelity evaluation terms can, of course, be different.

[0075] Regularization function Specifically, it can be as follows: in: Corresponding to finite differences in each direction (e.g., three directions (x, y, z)); j, k, l are indices of each voxel; g is a function, for example, could be... For example, finite differences are calculated as follows (subtraction of two adjacent voxels): Of course, one could think of using finite differences of different numbers of adjacent voxels.

[0076] Besides the norms mentioned above, other non-restrictive examples of the function g are conceivable and are listed below: The selected weights {w1, w2, ...} have, for example, a Gaussian distribution.

[0077] item and Weights can be introduced into regularization. The weights are adjusted in the spatial direction (index d) and for each voxel (index j, k, l). In which directions is regularization expected to be more or less? Set the desired regularization force.

[0078] Consider an example of a voxel, with terms in each of the three directions x, y, z. , , There are differences. If the local orientation incompleteness I(x, ν) of the voxel is low, then despite the presence of orientation effects (terms) , , (Different), but due to the low incompleteness I(x, ν) of voxels, the overall regularization strength is reduced. Set to low. Conversely, for voxels that are fundamentally unreconstructable (i.e., voxels with high local orientation incompleteness I(x, ν)), the global regularization strength is [low]. Set to high, and item Weighting is allowed in all directions.

[0079] Therefore, the regularization performed here is not only directional but also local, that is, specific to each voxel.

[0080] The regularization function defined above The regularization function is given as a non-limiting example. The regularization function may have some other definitions. For example, the regularization function may be an adaptation of the local regularization of this invention as described in [Bayram, 2012]. Regularization function Specifically, it can be as follows: Where θ= It is the direction of regularization (e.g.) = And α is the stretching factor to be selected.

[0081] The reconstruction method according to the invention further includes step E5, which involves solving a minimization problem using a calculated regularization term.

[0082] The reconstruction method according to the present invention is implemented by a computer. The computer may include various computing, storage, and communication units configured to interact with each other. The computer may include a processor, one or more storage peripherals, input / output interfaces, and a human-machine interface. The term "computer" herein refers to any information technology device or system capable of implementing the reconstruction method according to the present invention.

[0083] This invention provides a standard for automatically or semi-automatically adjusting local regularization terms. This standard is based on computed tomographic incompleteness calculated from the scanner's geometry. The regularization according to the invention is particularly suitable for cases involving regularization of the directional total variation type. However, any reconstruction algorithm with locally adaptable regularization terms (or multiple terms) is conceivable.

[0084] The main advantage is that the image reconstruction quality is better both visually and quantitatively.

[0085] This invention also allows for the analytical construction of regularization while taking into account the scanner's geometry. Furthermore, this regularization is applicable to any scanner geometry. Finally, local adjustments were made to the presets regarding objects in the tomographic reconstruction algorithm, thereby improving its accuracy.

[0086] This invention enables the introduction of voxel-by-voxel regularization using directional information and regularization strength. The regularization force allows defining the weight of regularization relative to data fidelity guided by the data. For high incompleteness, regularization is given a higher weight, and the data itself is given a lower weight. Conversely, for low incompleteness (meaning the object can be highly reconstructed using only the acquired data), data fidelity is given a higher weight, while regularization is given a lower weight.

[0087] References [Bayram, 2012]: Bayram and Kamasak 2012, “Directional Total Variation”, IEEE Signal Processing Letters, Vol. 19, No. 12. [Clackdoyle & Noo, 2020]: Clackdoyle, R., & Noo, F. (January 2020). Quantification of Tomographic Incompleteness in Cone-Beam Reconstruction. 4, 63-80. doi:10.1109 / trpms.2019.2918222.

[0088] [Ding, 2021]: Ding, Q.N. (2021). Deep learning with adaptive hyper-parameters for low-dose CT image reconstruction. IEEE Transactions on Computational Imaging, 7, 648-660.

[0089] [Gonzalez, 2009]: Gonzalez, A. (November 2009). Measurement of Areas on a Sphere Using Fibonacci and Latitude–Longitude Lattices. Mathematical Geosciences, 42, 49-64. doi:10.1007 / s 11004-009-9257-x.

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[0091] [Lin&Meikle, 2010]: Lin, J, & Meikle, S.R. (October 2010). Truncated pinhole SPECT: Sufficient sampling criteria and applications. IEEE Nuclear Science Symposium: Medical Imaging Conference. IEEE. doi:10.1109 / nssmic.2010.5874140.

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[0093] [Metzler, Bowsher, & Jaszczak, 2003]: Metzler, S.D., Bowsher, J.E., & Jaszczak, R.J. (October 2003). Geometrical similarities of the Orlov and Tuy sampling criteria and a numerical algorithm for assessing sampling completeness. IEEE Transactions on Nuclear Science, 50, 1550-1555. doi: 10.1109 / tns.2003.817385.

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Claims

1. A method for using an X-ray source (s1, s2, ..., s...) N ) and X-ray detectors (d1, d2, ..., d N A method for reconstructing an object by X-ray tomography, the object belonging to an imaging space (Ω) defined by a set of voxels, the method comprising the following steps: Obtain the positions of the X-ray source and X-ray detector (E0); pre-compute the direction set (E1); compute the vector incompleteness graph (E2) for each voxel including a vector in a direction ν* in the direction set; compute the metric incompleteness graph IM(x) (E3) for each voxel, which defines the information loss in the direction ν*; compute the regularization term (E4) based on the vector incompleteness graph and the metric incompleteness graph; and iteratively solve the (E5) regularized tomographic reconstruction problem using the computed regularization term.

2. The reconstruction method according to claim 1, wherein, Calculating (E2) the vector incompleteness map includes the following sub-steps for each voxel in the imaging space (Ω): from the source (s1, s2, ..., s... N In the direction set, a source is selected (E21) for each direction; and a direction ν* is selected (E22) for each selected source, wherein the information loss defined by the metric incompleteness graph IM(x) corresponds to the information loss of the detector associated with the selected source in the selected direction ν*.

3. The reconstruction method according to claim 2, wherein, The steps for calculating the vector incompleteness graph (E2) include starting from the source (s1, s2, ..., s...). N Selecting (E20) is a preparatory sub-step of a source that projects the voxels of the object onto their associated detectors, the source being from the source that projects the voxels of the object onto their associated detectors (d1, d2, ..., d...). N The source on ) 、 、…、 Selected from ).

4. The reconstruction method according to claim 2 or 3, wherein, The source is obtained by using an expression Calculate the local orientation incompleteness I(x, ν)∈ Choose (E21), among which It is the source The projection onto the plane defined by the voxel's point x and the normal direction ν.

5. The reconstruction method according to claim 4, wherein, The selected direction is chosen based on the local directional incompleteness I(x, ν).

6. The reconstruction method according to claim 4 or 5, wherein, The selected direction ν* is chosen based on the calculation of the norm of multiple directions (v1, ..., vn), and the local directional incompleteness I(x,ν) is calculated for the multiple directions.

7. The reconstruction method according to claim 6, wherein, The norm is the Lp norm of p∈[1;+∞], where p is preferably equal to +∞.

8. The reconstruction method according to claim 4 or 5, wherein, The selected direction ν* is chosen based on the weighted average of multiple directions (v1, ..., vn), and the local directional incompleteness I(x, ν) is calculated for the multiple directions.

9. The reconstruction method according to any one of claims 1 to 8, wherein, The regularization term refers to the regularization function. The calculated regularization function is as follows: in For each finite difference, j, k, l correspond to the index of each voxel, and d corresponds to the direction of the finite difference.

10. The reconstruction method according to claim 9, wherein, The regularization function This is the type of total directional variation.

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