Method, system and computer program for inspecting a part by X-ray radiography

Abstract

The invention relates to a method for non-destructive testing of a part, comprising: the simulation (E2), from a digital model of the part (DMOP), of projections of the part according to viewpoints, where n is a positive integer; the determination (E3), for each of the simulated projections, of an uncertainty in the determination of deformation parameters of the digital model of the part; the resolution of an optimization problem to determine (E4), among the viewpoints, a subset of viewpoints that minimizes the uncertainty in the determination of the deformation parameters, where n is a positive integer less than n; the calculation (E5) of projections of the part according to the viewpoints of said subset, from images (200) of the part acquired by an X-ray radiography device (100) according to the viewpoints of said subset; and a validation (E6) of the part based on the calculated projections. Figure for the abstract: Fig. 2

Description

Title of the invention: Method, system and computer program for inspecting a part by X-ray radiography technical field

[0001] The field of the invention is that of non-destructive testing by X-ray radiography of parts, for example of aeronautical parts such as turbine blades. STATE OF THE ART

[0002] Non-destructive testing (NDT) of aeronautical parts is an essential element of aircraft operational safety, the objective of which is to prevent any defect that could cause a failure in flight. Among NDT methods, X-ray radiography stands out for its ability to visualize the interior of the part in a minimally intrusive manner and to resolve details down to a micrometer size.

[0003] A normalized X-ray radiography is interpreted as an image of the attenuation of X-rays during passage through the part, attenuation itself related to the thickness traversed by a law which is often approximated by an exponential function as for the Beer-Lambert law.

[0004] Tomography consists of acquiring one or more thousand radiographs during a rotation, most often a complete rotation, of a part in order to calculate a complete three-dimensional image of the part. The significant time required to acquire these tomographic images, as well as the associated cost, leads manufacturers to consider only a limited number of radiographic images for performing material and dimensional non-destructive testing.

[0005] The certification of a part by X-ray radiography from a small number of views is usually carried out by inspectors, specialist technicians trained for this task, who analyze the images looking for any abnormal variation.

[0006] However, the meticulous analysis of images by controllers is a laborious and tiring task. Finally, inter- and intra-controller variability exists, reducing the reliability of the sanction.

[0007] Cédric Fragnaud et al. CAD-based X-ray CT calibration and error compensation, Measurement Science and Technology, IOP Publishing, vol. 33, no. 6, 065024, (2022) https: / / iopscience.iop.org / article / 10.1088 / 1361-6501 / ac5133 proposes an NDT solution that integrates a method for identifying the geometry of the part-cabinet system and for quantifying and reproducing the physical phenomena responsible for the formation of images observed on radiographs of an inspected part. This identification allows for the realistic and numerical simulation of images observed from a computer-aided design (CAD) model providing prior knowledge. By comparing the observed images with the simulated images, this method makes it possible to identify the geometric indications or defects of the inspected part.

[0008] To better represent the actual 3D geometry of an inspected part, the previously described solution can be supplemented by determining a parametric deformation transformation that allows the CAD model of the inspected part to be converted to the geometry of the actual part as observed in the radiographs. Once identified, this parametric transformation makes it possible to generate a 3D simulation of the inspected part and, by comparison with the observed images, to quantify any geometric indications or defects.

[0009] However, these methods require a fixed, suboptimal number of radiographs, resulting in wasted time and unnecessary costs. Indeed, given the required production rate, it would be preferable to acquire only a small number (on the order of ten) of radiographs, typically about one hundred times fewer than the radiographs needed for tomographic reconstruction. Description of the invention

[0010] One aim of the present application is to remedy the aforementioned drawbacks by proposing an NDT solution that allows, with a reduced and optimized number of radiographic images and minimal inspection time, the 3D geometry of a part and its dimensional conformity to be characterized with a high level of reliability.

[0011] To this end, the invention proposes a method for non-destructive testing of a part, comprising: • the simulation, from a digital model of the part, of M projections of the part according to M viewpoints, M being a positive integer; • the determination, for each of the M simulated projections, of an uncertainty relating to the determination of deformation parameters of the digital model of the part; • the resolution of an optimization problem to determine, among the M viewpoints, a subset of N viewpoints that minimizes the uncertainty in determining the deformation parameters, X being a positive integer less than M; • the calculation of N projections of the part according to the N viewpoints of said sub-assembly, from images of the part acquired by an X-ray radiography device according to the N viewpoints of said sub-assembly; • a sanction of the piece based on the N calculated projections.

[0012] Some preferred but not limiting aspects of this process are as follows: The determination, for each of the M simulated projections, of an uncertainty in the determination of the deformation parameters includes: • the calculation, for each of the deformation parameters, of a sensitivity field of the simulated projection to a variation of said parameter; • the determination of a Hessian matrix from the sensitivity fields calculated for each of the deformation parameters. Determine, among the M viewpoints, the subset of A7 viewpoints that includes: • the calculation of weighting coefficients of a weighted sum of the Hessian matrices determined for each of the simulated projections, said calculation including the iterative optimization of the result of a cost function applied to said weighted sum; • the selection of A7 viewpoints based on the calculated weighting coefficients. The calculation of the weighting coefficients includes the following steps: • Assignment to Al of the Hessian matrices the weighted sum of a weighting coefficient equal to 1 / N, and assignment to the remaining MN Hessian matrices of a weighting coefficient of zero, the A' viewpoints associated with the Hessian matrices with weighting coefficients equal to 1 / N forming an observation configuration of the first room, • at each of several iterations, and as long as a stopping criterion is not met: • determination of an optimization pair comprising a first and a second viewpoint, the first viewpoint being associated with a Hessian matrix with a weighting coefficient equal to 1 / Al, and the second viewpoint being associated with a Hessian matrix with a weighting coefficient of zero, the optimization pair being determined so as to maximize the result of applying the cost function to the weighted sum of the Hessian matrices in which the weighting coefficients of the Hessian matrices associated with the two viewpoints of the pair are interchanged; • replacing the first viewpoint with the second viewpoint in the observation configuration, and interchange of the weighting coefficients of the two Hessian matrices associated with said viewpoints. the cost function for an input matrix matches in output the determinant of the input matrix or the smallest non-zero eigenvalue of the input matrix or less than a threshold; The determination of the weighting coefficients includes the following steps: • Assigning a weighting coefficient to each of the Hessian matrices, the weighting coefficients associated with each of the Hessian matrices being stored in a weighting coefficient vector in which the term with index n corresponds to the weighting coefficient associated with the Hessian matrix of the simulated projection for viewpoint n, n being an integer between 1 and M, • at each of several iterations, and as long as a stopping criterion is not met: • determination of an optimization vector of the weighting coefficients, the optimization vector being equal to the derivative of the cost function with respect to the vector of weighting coefficients, such that a term of the optimization vector of index n is an optimization term intended to be added to the weighting coefficient of index n of the vector of weighting coefficients; • where appropriate, when a weighting coefficient is zero and the term of the optimization vector of the same index is negative, assigning said term a value equal to 0; • modification of each of the terms of the weighting coefficient vector by adding a quantity proportional to the term of the optimization vector with the same index; • assigning a value equal to 0 to each of the negative terms of the weighting coefficient vector; • normalization of the weighting coefficient vector, the cost function is the determinant of the weighted sum of the Hessian matrices to which a Lagrange multiplier is added; the sanction of the part includes the determination of the deformation parameters of the digital model of the part from the N calculated projections; - the sanction of the part includes the determination of a corrected model of the part by transforming the digital model of the part using the deformation parameters; - the sanction also includes the simulation, from the corrected model of the part, of N projections of the part according to the N viewpoints and the comparison of the N simulated projections with the N calculated projections. DESCRIPTION OF THE FIGURES

[0013] Other features, objectives and advantages of the invention will become apparent from the following description, which is purely illustrative and not limiting, and which should be read in conjunction with the accompanying drawings on which:

[0014] - [Fig. 1] is a diagram of a volumetric modeling system for a part according to a possible embodiment of the invention;

[0015] - [Fig. 2] is a diagram representing different stages of a process of volumetric modeling of a part from a batch of parts according to a possible embodiment of the invention;

[0016] - Figure 3 illustrates an example of an initial observation configuration consisting from a set of viewpoints of an X-ray source arranged along a circular trajectory around a room;

[0017] - Figure 4 illustrates an optimal observation configuration for a resulting part of the implementation of a first possible method of determining the optimal subset of N viewpoints;

[0018] - Figure 5 illustrates an optimal observation configuration for a part resulting of the implementation of a second possible embodiment of the determination of the optimal subset of N viewpoints;

[0019] - [Fig. 6] is a diagram representing different stages of the first mode of possible realization of the determination of the optimal subset of N viewpoints;

[0020] - [Fig.7] is an example of pseudocode implementing the different steps of the first possible embodiment of determining the optimal subset of N viewpoints;

[0021] - [Fig.8] is a diagram representing different stages of the second mode of possible realization of the determination of the optimal subset of N viewpoints;

[0022] - [Fig.9] is a first example of pseudocode implementing the different stages of the second possible implementation of determining the optimal subset of N viewpoints;

[0023] - [Fig. 10] is a second example of pseudocode implementing the different stages of the second possible implementation of determining the optimal subset of N viewpoints; and

[0024] - [Fig. 1 1] is a diagram representing the different sub-steps of the step of sanction of the play. DETAILED DESCRIPTION OF THE INVENTION

[0025] The invention relates to a method and system for the non-destructive testing of parts that can be manufactured using various processes, for example, lost-wax casting or additive manufacturing, and that can be made of a single material. The invention is applicable to the non-destructive testing of aeronautical parts, typically turbine blades, after their manufacture or during maintenance operations in order to detect any defects that could, for example, cause a failure in flight.

[0026] With reference to Figures 1 and 2, the non-destructive testing system 1 includes an X-ray radiography device 100 for acquiring images of a real part 200 from different viewpoints. These images are denoted / «) where n denotes the number of one of the viewpoints of the part.

[0027] The non-destructive testing system 1 comprises, and the non-destructive testing method for the batch uses, one or more CAL computers. The CAL computer may be or include one or more computers, one or more servers, one or more machines, one or more processors, one or more microprocessors, one or more permanent memory (PMM) units, or one or more random access memory (RAM). The CAL computer may include one or more physical data input interfaces INT1 and one or more physical data output interfaces INT2. This or these physical data input interfaces INT1 may be or include one or more computer keyboards, one or more physical data communication ports, one or more touchscreens, or other devices. This or these physical data output interfaces INT2 may be or include one or more physical data communication ports, one or more screens, or other devices.A computer program can be recorded and executed on the CAL computer and include code instructions which, when executed on it, implement all or part of the non-destructive testing method according to the invention, including image reception / »).

[0028] 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 moving and / or rotating the support 102 and the source 101 relative to each other about an axis 103 of rotation, which may be, for example, vertical, and an X-ray detector 105 for the X-rays passing through the part 200, the part 200 thus being located in the path of the X-rays between the source 101 and the detector 105. For example, the source 101 is fixed and the support 102 is rotated about the axis 103. Alternatively, the support 102 is fixed, and the source 101 follows a movement around the support 103 and therefore around the part 200. This movement can be circular or follow any trajectory in space. In what follows, to simplify the description and to ensure compatibility with most existing tomography devices without having to modify the arrangement of the source 101 and the detector 105 during acquisitions, we will consider the movement of the first part 200 to be circular around the axis 103, which is perpendicular to the support 102 and passes through the first part 200 (in other words, the support 102 rotates on its own axis, and the source 101 and the detector 105 are stationary). Such an example is shown in Figures 3, 4, and 5, which illustrate a schematic top view including the axis 103, as well as different possible viewpoints (denoted g / n), where n denotes the viewpoint number. Source 101, the support

[0029]

[0030]

[0031] 102 and detector 105 are arranged in a high-power X-ray cabinet. The control mechanism 104 is configured for the acquisition by detector 105 of images, each associated with one of the viewpoints gjn). In the example described, where the support 102 (and therefore the part 200) rotates around the axis 103, the viewpoints g / n are defined by only one parameter, such as an angle. In the case where the movement of the source 101 and / or the support 102 relative to each other is more complex, the viewpoints can be defined by several parameters, such as different angles or distances. Referring to Figure 1, a calibration step El can be implemented by the CAL computer to identify the parameters of a parametric model describing image formation and accounting for phenomena occurring during acquisition, such as Compton scattering and beam hardening. This El step aims more specifically to estimate parameters representative of the projective geometry of the radiography device 100 and to estimate parameters of a model of expected image artifacts for the material constituting the first part and the power of the X-ray beam used. The determination of these various parameters can be carried out by following the procedure detailed in the aforementioned article. With further reference to Figure 1, the process includes steps E2 to E4 which allow the selection of / V viewpoints from a set of M possible viewpoints, M being greater than N. As will be described later, this selection is made so as to retain the viewpoints which minimize the measurement uncertainties of the shape of the part. Following these steps E2 to E4, the process according to the invention includes a step E5 of calculation, from images / «) of part 200 acquired by the X-ray radiography device 100 from the N selected viewpoints, of / V projections p*"* of the part from the N viewpoints. An implementation of isP-n>= -log^ / / 0) where denotes an intensity image of the X-rays that have passed through the part for the viewpoint n and Iq the image of white (i.e. the image captured by the detector in the absence of a part).

[0032] The process further includes a step E6 for validating part 200 (described in more detail later) which uses the N projections p[n]. In one possible embodiment, this validation includes determining, from the N projections p{n}, the deformation parameters of a digital model of the part MODP. This digital model, which can be stored in advance in the MEM memory of the CAL computer, is a geometric reference of the part, for example, a CAD model, reproducing an ideal part 200. This MODP model takes into account the composition of the material constituting the part. The deformation parameters allow the transition from the ideal geometry of the part (as represented by its digital model) to the actual geometry of the part (as observed in the images).

[0033] Starting from this MODP digital model, the CAL computer can simulate expected radiographs of the part. The computer can thus simulate projections from different viewpoints. The parameters representing the projective geometry of the radiography device can be taken into account during this simulation. Similarly, the parameters of the image artifact model can also be used to reproduce the artifacts in the simulated projections.

[0034] Steps E2 to E4 of the method according to the invention, which will now be described, aim to assist in the selection of N optimal viewpoints from among M possible viewpoints (where AT is a positive integer less than 1), that is, in the selection of a subset of N viewpoints that forms an optimal observation configuration, thereby reducing the uncertainties related to the determination of the deformation parameters. These uncertainties are linked, among other things, to the noise present in the images, which is generally white and Gaussian. The deformation parameters can be represented as a vector d. A covariance matrix associated with this vector d allows these uncertainties to be quantified.

[0035] In general, a covariance matrix is ​​(convergently) inversely proportional to a Hessian matrix, so that studying such a Hessian matrix (via a cost function to be defined later) allows us to determine the covariance matrix that minimizes the uncertainties in determining the deformation parameters. In one possible embodiment, the invention proposes defining such a Hessian matrix as a weighted sum of Hessian matrices, each associated with one of the M possible viewpoints.

[0036] Step E2 includes the selection of M possible viewpoints and the simulation, from the MODP digital model of the part, of M p11' projections of the part according to

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044]

[0045] The M possible viewpoints. As shown in Figures 3 and 4, which depict a circular path of the source 101 around the part, the M possible viewpoints can be equally distributed along this circular path. In one embodiment, N is chosen to have a value of 12, while M is 180, even though the 180 viewpoints are not each represented separately. Step E3 includes the determination, for each of the M simulated jpv projections, of an uncertainty relating to the determination of the deformation parameters. This step E3 may include performing the following operations for each of the M simulated projections (x denoting a pixel of the X-ray detector): • the calculation, for each of the deformation parameters di forming the vector d, of a sensitivity field of the simulated projection pn' x ) to a variation of said parameter d, ; and • the determination of a Hessian matrix from the sensitivity fields ( x ) calculated for each of the deformation parameters d; forming the vector d. The sensitivity field x) of a simulated projection pin^xj to a variation of the l-th parameter of the vector d can be expressed as follows: ^w)(x) Such a sensitivity field can be calculated by finite difference. The Hessian matrix / T^ determined from the sensitivity fields is the matrix consisting of terms (in one row * and one column *) such that: x) where ^(x) and ^(x) correspond respectively to the sensitivity fields of the simulated projection pW to a variation of the i-th deformation parameter d} and the same deformation parameter d^ of the vector d. This Hessian matrix H^ associated with a projection p(H) is a square matrix of dimension equal to the dimension of the vector d. As seen previously, it characterizes the uncertainty in the determination of the deformation parameters. Step E4 involves solving an optimization problem to determine, from among M viewpoints, the subset of N viewpoints that minimizes (in the sense that it satisfies a stopping criterion) the uncertainty in determining the deformation parameters. This step E4 may include performing the following operations: the calculation of weighting coefficients for a weighted sum H of the Hessian matrices determined in step E3 for each of the simulated projections, said calculation including the iterative optimization of the result of a cost function f applied to said weighted sum; and the selection of N viewpoints based on the calculated weighting coefficients.

[0046] The weighted sum H of the Hessian matrices can thus be expressed as follows:

[0047]

[0048] where is a weighting coefficient associated with the Hessian matrix associated with the viewpoint ^l«). The set of weighting coefficients forms a weighting vector.

[0049] The weighting coefficients are all positive and of unit sum and characterize a probability of the associated viewpoint of belonging to an optimal configuration CONFopt comprising a predetermined number N of viewpoints allowing to reduce to the maximum the uncertainty related to the determination of the transformation parameters forming the vector d.

[0050] The weighting coefficients ^«) are useful in selecting the N viewpoints from among the possible viewpoints. The selected N viewpoints, denoted ..., ^f, ... ■>, form an optimal configuration CONFopt observation of part 200. These N viewpoints are "optimal" in the sense that they minimize the uncertainty in determining the transformation parameters forming the vector d, and therefore are the viewpoints of the set of M possible viewpoints that allow obtaining precise values ​​of said parameters with a reduced number of acquisitions.

[0051] The determination of the weighting coefficients by iterative optimization of the result of a cost function applied to the weighted sum H can be carried out according to different embodiments. First implementation of step E4:

[0052] In this first embodiment illustrated in Figures 6 and 7, the weighting coefficients are binary, that is to say they can only take two values, for example 0 or 1 / N, one of the possible values ​​(for example 0) of a weighting coefficient at the last iteration meaning that the viewpoint associated with this coefficient is not part of the optimal configuration CONFopt, and the other possible value (for example VN) of a weighting coefficient at the last iteration meaning that the viewpoint associated with this coefficient is part of the optimal configuration CONFopt.

[0053] In this first embodiment, at each of the iterations of determining the optimal configuration, only 'V weighting coefficients (associated with N viewpoints) have the value 1 / N and the M - N other coefficients (associated with the MN other viewpoints) have the value 0. [Fig.6] illustrates the different sub-steps of step E4 according to the first embodiment, and [Fig.7] is an example of pseudocode implementing said first embodiment of step E4.

[0054] In this first embodiment, step E4 includes a first substep E41a in which an initial configuration CONFini is determined by assigning one of the possible values, for example the value 1 / A\, to the N weighting coefficients associated with N of the Hessian matrices of the weighted sum H (and therefore to Ar of the points of view), and by assigning the other possible value, for example the value 0, to the MN other weighting coefficients associated with the MN other Hessian matrices of the weighted sum H.

[0055] The N viewpoints forming the initial CONFini configuration may be chosen randomly or not from among the M viewpoints (end). For example, the initial CONFini configuration may include A' predetermined viewpoints corresponding to positions more likely a priori to provide accurate information on the deformation parameters contained in the vector a. Alternatively, the N viewpoints of the initial CONFini configuration may correspond to a regular sampling of the trajectory followed by the support 102 or the source 101, as illustrated in Figure 3, which shows, for a circular trajectory 5 of the source 101 around the part 200, all the viewpoints (end), among which those in black form an initial CONFini configuration in which the different viewpoints are equally distributed along the circular trajectory.

[0056] The MN viewpoints whose weighting coefficient is equal to 0 form a set of candidate viewpoints, denoted |... ç / MA)|, for a current iteration. The N viewpoints whose weighting coefficient is equal to 1 / N during the current iteration form a configuration called the current configuration CONF of the current iteration. Thus, the current configuration CONF at the end of the last iteration is the optimal configuration CONFopt. And, during the first iteration, the set of candidate viewpoints includes the viewpoints not part of the initial configuration CONFini.

[0057] In a step E42a of the first embodiment of step E6, the invention determines the optimal configuration CONFopt.

[0058] This determination is carried out iteratively by exploiting, at each iteration, and starting from a current configuration CONF, a weighted sum H associated with said current configuration CONF and a set of viewpoints candidates at the beginning of said iteration, a cost function f is applied to the weighted sum H in which one viewpoint from the current configuration CONF and one viewpoint from the set of candidate viewpoints of said iteration have been swapped. In other words, the cost function f is applied to a second weighted sum H' of the Hessian matrices constructed similarly to the first weighted sum H, but from a second configuration. This second configuration includes the same viewpoints as the current configuration CONF in the current iteration, except for one viewpoint, which is replaced by one of the candidate viewpoints.

[0059] The viewpoints swapped between the current configuration CONF and the candidate configuration of the current iteration form a viewpoint optimization pair and are determined via a loop (lines 5 to 9 of the pseudocode in Figure 7) on each of the viewpoints among the current configuration CONF and the set of candidate viewpoints ^(0, .......}, and by the evaluation of the cost function f applied to the second weighted sum H* of the Hessian matrices.

[0060] Step E42a can thus follow an iterative process comprising several iterations at each stage, and as long as a convergence criterion is not met: • the determination (first sub-step E421a of step E42a) of the optimization pair ¢(¢)} comprising a first and a second viewpoint çjW and chosen from the set of viewpoints ...^), the first striking viewpoint associated with a The first viewpoint is a Hessian matrix with a weighting coefficient of 1 / N during the current iteration, and the second viewpoint is associated with a Hessian matrix with a weighting coefficient of 0 during the current iteration. More specifically, the optimization pair | is determined such that the result of applying a cost function f to a second weighted sum A) ■opt "c is maximized. • the replacement (second substep E422a of step E42a) of the first viewpoint Ap) of the couple by the second viewpoint (Aq) of the couple r opt " c in the current CONF configuration and the interchange of the weighting coefficients and the two associated Hessian matrices to said viewpoints. Thus, the weighted sums H and H' are also interchanged.

[0061] Detailed description of the determination of the optimization torque during a given iteration (step E 4 21a):

[0062] The determination of the pair is carried out via a loop in which the set of possible pairs of a first viewpoint corresponding to a viewpoint from the current configuration CONF and a second viewpoint corresponding to a viewpoint from the set of candidate viewpoints of the current iteration is tested (lines 5 to 9 of the pseudocode in Figure 7). More precisely, for each of the possible pairs, a second weighted sum g' is calculated, and the result of applying the cost function / to said second weighted sum H' is recorded, for example in a cost matrix F. Each term of the cost matrix F is thus associated with a possible optimization pair.

[0063] The cost function f used and applied to determine the optimization pair is defined in the first embodiment as a function that, for the input matrix H or H, maps the determinant of the input matrix ff to its output. Alternatively, the cost function f can be defined as a function that, for the input matrix H or H, maps the minimum eigenvalue 2lmin? to lmin' (non-zero or less than a user-defined threshold) of the input matrix H, H •

[0064] When the loop during which the set of possible pairs are tested is finished, the maximum term of the matrix F completed during the loop is determined (line 10 of the pseudo-code in Figure 7), and the viewpoints | g / jd, qfô) | forming the optimization pair are deduced from it, for example via the indices of the row and column of said maximum term.

[0065] The first sub-step E421a is broken down as follows: • determination of a cost matrix F in which each term Fîj of row i and column j is associated with a possible optimization pair, said term Fig corresponding to the result of the cost function f applied to the second weighted sum H in which the weighting coefficients associated with the two points of view of said possible optimization pair are interchanged, • determination of the optimal optimization pair as being the possible optimization pair associated with the maximum term of the cost matrix F.

[0066] When the first substep E421a is carried out, the second substep E422a of step E62a can then be implemented by modifying the current configuration by replacing the first viewpoint of the optimal optimization pair with the second viewpoint of said pair, and by interchanging the coefficients of weightings associated with the Hessian matrices corresponding to said viewpoints (lines 10 to 13 and 3 of the pseudo-code of [Fig.7]).

[0067] The iterations of the first embodiment are stopped when a stopping criterion is met, for example when a maximum number of iterations is reached or, as illustrated in the pseudo-code of Figure 6 on line 2, when a difference between the results of the application of the cost function / to the weighted sum of the Hessian matrices determined at the end of the previous iteration on the one hand and to the weighted sum of the Hessian matrices determined at the end of the current iteration on the other hand becomes less than a predetermined gap toi. Second embodiment of step E4:

[0068] An example of the second embodiment is illustrated in Figures 8, 9, 10. [Fig.8] illustrates the different sub-steps of step E4 according to the second embodiment, and Figures 9 and 10 are two examples of pseudo-code implementing said second embodiment of step E4, these examples differing by a stopping criterion.

[0069] In this second embodiment, the weighting coefficients can be (positive) decimal numbers between 0 and 1. Furthermore, in the second embodiment, the sum of the values ​​of all the weighting coefficients (for any ” between 1 and Af) can be equal to 1.

[0070] Step E4 includes a first substep E41b in which a predetermined value is assigned to each of the Af weighting coefficients associated with the M Hessian matrices of the weighted sum H. The values ​​assigned to each of the M weighting coefficients are stored in a weighting coefficient vector denoted by . More particularly, the weighting coefficient vector is ordered such that the term with index n of the vector is the weighting coefficient associated with the nth Hessian matrix of the weighted sum H, i.e. associated with the Hessian matrix itself associated with the nth viewpoint g / n).

[0071] For example, a uniform value il M is assigned in the first step E41b to each of the Af weighting coefficients of the vector ? in order to account for a uniform initial probability for each of the viewpoints to minimize the uncertainties relating to the estimation of the parameters of the vector d. Alternatively, different values ​​can be assigned to the M different weighting coefficients depending on whether they are associated with a viewpoint a priori more likely to be part of the optimal configuration CONFopt.

[0072] In a step E42b, the invention optimizes the distribution of the values ​​of the Af weighting coefficients. This optimization is carried out iteratively.

[0073] More specifically, step E42b includes, at each of several iterations, and as long as a stopping criterion is not met: • The determination (substep E421b) of an optimization vector of the weighting coefficients denoted *, the vector * being equal to the derivative of the cost function / with respect to the weighting coefficient vector P, that is to say that a term of index n denoted hn of the vector h is equal to the partial derivative of the cost function f with respect to the term of index n of the weighting coefficient vector f, in other words with respect to the weighting coefficient associated with the Hessian matrix associated with the R-th viewpoint • where applicable, when a weighting coefficient of the weighting coefficient vector Y is zero and the associated optimization vector term A, i.e., with the same index as this weighting coefficient, is negative, the assignment (substep E422b) of said optimization vector term * to a value equal to 0, • the modification (substep E423b) of each of the terms of the weighting coefficient vector, i.e., each of the M weighting coefficients, by adding a quantity proportional to the associated optimization vector term *, and • the assignment (substep E424b) of a value equal to 0 to each of the negative terms of the weighting coefficient vector Y, and • the normalization (substep E425b) of the weighting coefficient vector

[0074] Each of the terms of the optimization vector * is intended to be added to the associated weighting coefficient of the same index of the weighting coefficient vector Y.

[0075] As described previously, the weighted sum H of the Hessian matrices associated with each of the viewpoints can be expressed as follows:

[0076] H = yr

[0077] where Y^ is a weighting coefficient associated with the corresponding Hessian matrix from the point of view ^n). Considering a vector H with index term n corresponding to the Hessian matrix, the weighted sum H can thus be written:

[0078]

[0079] Moreover, the cost function f considered in this second embodiment E6, which is to be maximized for the same reasons as in the first embodiment, is defined as a determining function of H to which is added a Lagrange multiplier to force normalization. The cost function f under consideration can thus be expressed as: f(y) = det(H(y))-2(1)

[0080] Thus, the optimization vector * calculated during step E421b can be expressed, using Jacobi's formula, as follows: [00811 h = ^detWrntrUHCrB-'ff;)-^!)

[0082] Substeps E422b (lines 5 to 7 of the pseudo-codes in Figures 9 and 10) and E424b (lines 9 to 11 of the pseudo-codes in Figures 9 and 10) prevent terms in the weighting coefficient vector from being negative.

[0083] The modification E423b of each term of the weighting coefficient vector P is achieved by adding a quantity proportional to the term of the associated optimization vector. Thus, substep E423b assigns, as described in line 8 of the pseudocode in Figures 9 and 10, to the vector P the new value y + wh, in which w is chosen, for example, such that w11 fj| < 0.11| y -

[0084] Thus, over the course of iterations, modification E423b (line 8 of the pseudo-codes in Figures 9 and 10) allows the weighting coefficients to be increased or decreased individually in order to determine the distribution P maximizing the cost function f-

[0085] In order to comply with the normalization of the weighting coefficients, an additional step E425b of normalization of the vector of weighting coefficients can be executed at the end of step E424b, as illustrated in lines 12 of the two pseudo-codes of figures 9 and 10.

[0086] The iterations of step E42b are stopped when a stopping criterion is met, for example, when a maximum number of iterations is reached or, as illustrated in the pseudocode in Figure 9, when a difference between the results of applying the cost function f to the weighted sum H of the Hessian matrices determined in the previous iteration and to the weighted sum H of the Hessian matrices determined in the current iteration becomes less than a predetermined deviation toi. Alternatively, another stopping criterion, which is shown in Figure 10 illustrating a second example of pseudocode for the second embodiment of step E6, can be met when a norm of the difference between the vector of weighting coefficients determined in the current iteration and that determined in the previous iteration becomes less than a predetermined tolerance deviation toi.

[0087] Thus, at the end of step E6, a value for each of the M weighting coefficients is determined.

[0088] As described previously, step E6 then includes a selection of the N optimal viewpoints forming an optimal configuration CONFopt = | ^pj. ..., (p^c ... observation of room 200 among the M viewpoints q / n) associated with the different simulated projections. This selection is made based on the weighting coefficients associated with said viewpoints, specifically by selecting the N viewpoints with the highest associated weighting coefficients. Thus, in the first embodiment described above, the A selected viewpoints are those whose weighting coefficient has a value equal to 1 / A, the others having a value equal to 0. In the second embodiment described, the selection of the N optimal viewpoints involves studying the weighting coefficients forming the weighting coefficient vector and selecting the N viewpoints associated with the N highest weighting coefficients.

[0089] Once the N viewpoints have been selected, steps E5 and E6 can be implemented. As previously stated, step E5 involves calculating, from images of part 200 acquired from the N selected viewpoints, N projections pw of the part from the N viewpoints. Step E6 involves a calibration of part 200 during which the deformation parameters are determined in a substep E61 from the N projections p1. It is understood that for the calibration, the number of images acquired is limited to N rather than M, which considerably reduces the acquisition time, and thus the inspection time, without sacrificing reliability.

[0090] In one possible embodiment, the determination of the deformation parameters forming the vector d can be carried out in accordance with the procedure presented and described in detail in French patent application no. 2204535 and a summary of which is described below.

[0091] This determination exploits a difference or residual p^ between p^n- the projection calculated for viewpoint number n (for all n between 1 and N) and a simulated projection for the same viewpoint which is expressed for example in the following form:

[0092] ^x . ; )

[0093] This determination can follow an iterative process comprising at each iteration: • the simulation of projections p^Çx • dj corresponding to the projections calculated from the acquired images, from the digital model of the part MODP; • for each viewpoint number, the determination of a difference pW(x ; d) between the calculated projections p^x • and the simulated projections P^ix ; d ) from the acquired images; • the modification of the vector d in order to reduce said gap.

[0094] Moreover, at each of the iterations, the modification of the vector d may include: • for each viewpoint, the calculation of sensitivity fields of the projection simulated for that viewpoint to a variation of the parameters contained in the vector d; • the calculation of a correction vector as being the vector bd minimizing a difference between the projection residuals and the product of bd by the sensitivity fields; • Updating the vector d using the correction vector

[0095] The sensitivity field of a simulated projection for viewpoint number n to a variation of a parameter dk of the vector d is expressed by definition as follows:

[0096] / \ ........has;........

[0097] The calculation of the correction vector may include minimizing the sum over the viewpoints of the squares of the differences between, for each viewpoint, the projection residual calculated for that viewpoint and the product of bd by the sensitivity fields calculated for that viewpoint. Thus, the update of the vector d using the correction vector after an iteration can be denoted by For example, the correction vector is given by:

[0098] Ôd* = æ'gmin]L(AJ|M^^ ; d) -^(x ; d) Ôrf)|| 2'

[0099] where ; d) cst 'a sensitivity field matrix ^(x ; dk) for the viewpoint and is a weighting term which can be used to take into account local uncertainties for example due to noise or dead pixels.

[0100] The iterations are stopped when a criterion is met, for example when a maximum number of iterations is reached, when the residuals calculated at the end of an iteration are less than a given threshold or when the decrease in the value of the residuals between two successive iterations is less than a given threshold.

[0101] Once the deformation parameters forming the vector d have been determined by substep E61, the sanction may include the determination, in a substep E62, of a corrected model of the part by transformation of the MODP digital model of the part using the deformation parameters.

[0102] The sanction may further include in a substep E63 the simulation, from the corrected model of the part, of N projections of the part according to the N viewpoints and the comparison of the N simulated projections with the N projections calculated from the N acquired images.

[0103] The sanction can be carried out in two different ways, on the one hand from information directly from the corrected model, and on the other hand from information from the differences between projections simulated using the corrected model and projections calculated from the acquired images.

[0104] In particular, a control procedure can be applied to the corrected model. The values ​​obtained are compared with the values ​​of the control procedure applied to the numerical model, taking into account the tolerances.

[0105] Furthermore, the differences between the projections simulated from the corrected model and the projections calculated from the acquired images can be compared to a noise level present in a projection. This measure has the advantage of being pixel-dependent. If the differences are less than the noise level, then they are considered insignificant. Otherwise, it means that the corrected model failed to capture the shape variability necessary for this comparison.

[0106] The invention is not limited to the method described above, but also extends to a non-destructive testing system for parts which includes a processor configured to implement this method and to a computer program product including instructions which, when the program is executed by a computer, cause the computer to implement the method.

Claims

1. Demands A non-destructive testing method for a part, comprising: • the simulation (E2), from a digital model of the part, of M projections of the part according to M viewpoints, M being a positive integer; • the determination (E3), for each of the M simulated projections, of an uncertainty relating to the determination of deformation parameters of the digital model of the part; said determination comprising: • the calculation, for each of the deformation parameters, of a sensitivity field of the simulated projection to a variation of said parameter; and • the determination of a Hessian matrix (g- based on the sensitivity fields calculated for each of the deformation parameters; • the resolution of an optimization problem to determine (E4), among the M viewpoints, a subset of V viewpoints which minimizes the uncertainty in determining the deformation parameters, where E4 is a positive integer less than M; • the calculation (E5) of V projections of the part according to the V viewpoints of said sub-assembly, from images ( / »)) of the part (200) acquired by an X-ray radiography device (100) according to the V viewpoints of said sub-assembly; • a sanction (E6) of the piece based on the N calculated projections, in which to determine (E4), among the M viewpoints, the subset of V viewpoints comprises: • the calculation of weighting coefficients of a weighted sum (H) of the Hessian matrices determined for each of the simulated projections, said calculation including the iterative optimization of the result of a cost function ( / ) applied to said weighted sum;

2. • the selection of N viewpoints based on the calculated weighting coefficients. A method according to claim 1, wherein the calculation of the weighting coefficients comprises the following steps:

3.

4. • Assignment (E41a) to Hessian matrices of the weighted sum of a weighting coefficient of a value equal to 1 / N, and assignment to the remaining MN Hessian matrices of a weighting coefficient of zero, the N viewpoints associated with the Hessian matrices of weighting coefficients equal to UN forming an observation configuration of the first piece (CONF\ • at each of several iterations, and as long as a stopping criterion is not met: • determination (E421a) of an optimization pair comprising a first and a second from a point of view, the first point of view being associated with a Hessian matrix with a weighting coefficient equal to 1 / N, and the second point of view being associated with a Hessian matrix with a weighting coefficient of zero, the optimization pair being determined so as to maximize the result of the application of the cost function to the weighted sum of the Hessian matrices in which the weighting coefficients of the Hessian matrices associated with the two points of view of the pair are interchanged; • replacement (E422a) of the first viewpoint by the second viewpoint in the observation configuration, and interchange of the weighting coefficients of the two Hessian matrices associated with said viewpoints. A method according to any one of claims 1 and 2, wherein the cost function for an input matrix matches in output the determinant of the input matrix or the smallest non-zero eigenvalue of the input matrix or less than a threshold. A method according to claim 1, wherein the determination (E4) of the weighting coefficients comprises the following steps: • Assignment (E41b) to each of the Hessian matrices a weighting coefficient, the weighting coefficients associated with each of the Hessian matrices being stored in a weighting coefficient vector (?) in which the index term with index n / j corresponds to the weighting coefficient associated with the Hessian matrix of the simulated projection for viewpoint n, n being an integer between 1 and M, • at each of several iterations, and as long as a stopping criterion is not met: • determination (E421b) of an optimization vector of the weighting coefficients (A), the optimization vector being equal to the derivative of the cost function with respect to the weighting coefficient vector, such that a term of the optimization vector of index n is an optimization term intended to be added to the weighting coefficient of index n of the weighting coefficient vector; • where appropriate, when a weighting coefficient is zero and the term of the optimization vector of the same index is negative, assign (E422b) said term a value equal to 0; • modification (E423b) of each of the terms of the weighting coefficient vector by adding a quantity proportional to the term of the optimization vector of the same index; • assignment (E424b) of a value equal to 0 to each of the negative terms of the weighting coefficient vector; • normalization (E425b) of the weighting coefficient vector. A method according to claim 4, wherein the cost function is the determinant of the weighted sum of the Hessian matrices to which a Lagrange multiplier is added.

6. A method according to any one of claims 1 to 5, wherein the sanction (E6) of the part includes the determination of the deformation parameters of the digital model of the part from the N calculated projections.

7. A method according to claim 6, wherein the sanction (E6) of the part includes the determination of a corrected model of the part by transforming the digital model of the part using the deformation parameters.

8. A method according to claim 7, wherein the sanction further comprises the simulation, from the corrected model of the part, of N projections of the part according to the N viewpoints and the comparison of the N simulated projections with the N calculated projections.

9. Non-destructive testing system for parts, comprising a processor configured to implement the method according to any one of claims 1 to 8.

10. Product computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method according to any one of claims 1 to 8.

Images