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 (MODP), of projections of the part according to viewpoints, being a positive integer; the determination (E3), for each of the 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 (E4), among the viewpoints, a subset of viewpoints which minimizes the uncertainty relating to the determination of the deformation parameters, being a positive integer less than; the calculation (E5) of projections of the part according to the viewpoints of said subset, from images () of the part (200) acquired by an X-ray radiography device (100) according to the viewpoints of said subset; a sanction (E6) of the part from 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 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 avoid any defect 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 non-intrusive manner and to resolve details down to micrometric size.

[0003] A standardized X-ray radiograph is interpreted as an image of the attenuation of the X-rays when passing through the part, attenuation itself linked to the thickness passed through 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 of the time complete, of a part in order to calculate a complete three-dimensional image of the part. The significant acquisition time of these tomographic images, as well as the cost incurred, leads manufacturers to consider only a limited number of radiographic images to carry out material and dimensional health NDT.

[0005] The sanctioning 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 in search of any abnormal variation.

[0006] But the careful analysis of images by the controllers constitutes a laborious and tiring task. Finally, there is inter and intra controller variability, 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, n°6, 065024, (2022) https: / / iopscience.iop.org / article / 10.1088 / 136l-6501 / ac5133 proposes a NDT solution that integrates a method for identifying the geometry of the part-cab system as well as quantifying and reproducing the physical phenomena responsible for the formation of images observed on radiographs of a controlled part. This identification makes it possible to digitally and realistically simulate the images observed from a computer-aided design (CAD) model providing a priori knowledge. By comparing the observed images with the simulated images, this method makes it possible to identify geometric indications or defects in the part being inspected.

[0008] In order to better account for the real 3D geometry of an inspected part, the previously described solution can be completed by determining a parametric deformation transformation making it possible to move from the CAD model of the inspected part to the geometry of the real part as observed on the radiographs. Once identified, this parametric transformation therefore makes it possible to generate a 3D simulation of the inspected part and, by comparison with the observed images, to quantify the geometric indications or defects.

[0009] However, these methods impose a fixed number of non-optimal radiographs, which causes a loss of time and unnecessary costs. Indeed, given the necessary production rate, it would be preferable to acquire only a small number (of the order of ten) of radiographs, typically a hundred times fewer than the radiographs necessary for tomographic reconstruction. Presentation of the invention

[0010] An aim of the present application is to remedy the aforementioned drawbacks, by proposing a NDT solution making it possible, with a reduced and optimized number of radiographic images and minimal control time, to characterize with a high level of reliability the 3D geometry of a part and its dimensional conformity.

[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 points of view, 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; • solving an optimization problem to determine, among the M viewpoints, a subset of N viewpoints which minimizes the uncertainty relating to the determination of the deformation parameters, N being a positive integer less than M; • the calculation of N projections of the part according to the N points of view of said subset, from images of the part acquired by an X-ray radiography device according to the N points of view of said subset; • a sanction of the part from the N calculated projections.

[0012] Some preferred but non-limiting aspects of this method are as follows: the determination, for each of the M simulated projections, of an uncertainty relating to 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 N viewpoints includes: • the calculation of weighting coefficients of a weighted sum of the Hessian matrices determined for each of the simulated projections, said calculation comprising the iterative optimization of the result of a cost function applied to said weighted sum; • the selection of the N points of view based on the calculated weighting coefficients. The calculation of the weighting coefficients includes the following steps: • assignment to N of the 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 zero weighting coefficient, the N points of view associated with the Hessian matrices of weighting coefficient equal to 1 / N forming an observation configuration of the first part, • at each of several iterations, and as long as a stopping criterion is not respected: • determination of an optimization pair comprising a first and a second 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 zero weighting coefficient, 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 inverted; • replacement of the first point of view by the second point of view in the observation configuration, and inversion of the weighting coefficients of the two Hessian matrices associated with said points of view. the cost function for an input matrix maps to the output the determinant of the input matrix or the smallest eigenvalue of the input matrix that is non-zero or less than a threshold; the determination of the weighting coefficients includes the following steps: • assignment to each of the Hessian matrices of a weighting coefficient, the weighting coefficients associated with each of the Hessian matrices being stored in a vector of weighting coefficients in which the index term corresponds to the weighting coefficient associated with the Hessian matrix of the projection simulated for the point of view n, n being an integer between 1 and M •> • at each of several iterations, and as long as a stopping criterion is not respected: • 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 the weighting coefficients, so that a term of the optimization vector of index fl is an optimization term intended to be added to the weighting coefficient of index n of the vector of the weighting coefficients, • where applicable, when a weighting coefficient is zero and the term of the optimization vector with the same index is negative, assignment (E422b) to said term of a value equal to 0; • modification of each of the terms of the vector of weighting coefficients by adding a quantity proportional to the term of the optimization vector of the same index; • assignment of a value equal to 0 to each of the negative terms of the vector of weighting coefficients. • normalization of the vector of weighting coefficients* 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 transformation of 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 points of view and the comparison of the N simulated projections with the N calculated projections. DESCRIPTION OF FIGURES

[0013] Other characteristics, aims and advantages of the invention will emerge from the following description, which is purely illustrative and non-limiting, and which must be read in conjunction with the appended drawings in which:

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

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

[0016] - [Fig.3] illustrates an example of an initial observation configuration consisting of a set of viewpoints of an X-ray source arranged in a circular path around a room;

[0017] - [Fig.4] illustrates an optimal configuration for observing a part resulting from the implementation of a first possible embodiment for determining the optimal subset of N viewpoints;

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

[0019] - [Fig.6] is a diagram representing different steps of the first possible em bodiment for determining the optimal subset of N viewpoints;

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

[0021] - [Fig.8] is a diagram representing different steps of the second possible em possible implementation of the determination of the optimal subset of N points of view;

[0022] - [Fig.9] is a first example of a pseudo-code implementing the dif different steps of the second possible embodiment of the determination of the optimal subset of N points of view;

[0023] - [Fig. 10] is a second example of a pseudo-code implementing the dif different steps of the second possible embodiment of the determination of the optimal subset of N points of view; and

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

[0025] The invention relates to a method and a system for non-destructive testing of parts which can be produced using different manufacturing processes, for example by lost wax casting or by additive manufacturing, and which can be composed of a single material. The invention finds application in the non-destructive testing of aeronautical parts, typically turbine blades, at the end of their manufacture or during maintenance operations in order to detect possible defects which could, for example, cause a failure in flight.

[0026] With reference to Figures 1 and 2, the non-destructive testing system 1 comprises an X-ray radiography device 100 making it possible to acquire images of a real part 200 from different viewpoints. These images are noted where n designates the number of one of the viewpoints of the part.

[0027] The non-destructive testing system 1 comprises, and the non-destructive testing method of the batch uses, one or more CAL computers. The CAL computer 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 computer 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 can be recorded and executed on the CAL computer and include code instructions, which when executed thereon, implement all or part of the non-destructive testing method according to the invention, including the reception of images.

[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 around an axis 103 of rotation, which may be vertical for example, 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. For example, the source 101 is fixed and the support 102 is rotated around the axis 103. Alternatively, the support 102 is fixed and the source 101 follows a movement around the support 103 and therefore the part200, this movement being able to be circular, or following any trajectory in space. In what follows, it will be considered, to simplify the description, but also to be compatible with most existing tomography devices without having to modify the arrangement of the source 101 and the detector 105 during the acquisitions, that the movement of the first part 200 is circular around the axis 103 perpendicular to the support 102 and passing through the first part 200 (in other words, the support 102 rotates on itself 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 view from above including the axis 103, as well as different possible viewpoints, noted where n denotes the number of the viewpoint. The source 101, the support 102 and the detector 105 are arranged in a high-power X-ray cabin.The control mechanism 104 is configured for the acquisition by the detector 105 of images j(«) each associated with one of the viewpoints (^n). In the example described in which the support 102 (and therefore the part 200) rotates around the axis 103, the viewpoints ¢)00 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 ¢)00 can be defined by several parameters such as different angles or distances.

[0029] With reference to FIG. 1, a calibration step E1 can be implemented by the computer CAL in order to identify the parameters of a parametric model describing the formation of the images and accounting for phenomena occurring during the acquisition, such as Compton scattering and beam hardening. This step E1 aims more particularly to estimate parameters representative of the projective geometry of the radiography device 100 and to estimate parameters of a model of image artifacts expected for the material constituting the first part and the power of the X-ray beam used. The determination of these different parameters can be carried out by following the procedure detailed in the aforementioned article.

[0030] Still with reference to Figure 1, the method comprises steps E2 to E4 which make it possible to select A / viewpoints from a set of M possible viewpoints, M being greater than N. As will be described later, this selection is carried out so as to retain the viewpoints which minimize the uncertainties in measuring the shape of the part.

[0031] Following these steps E2 to E4, the method according to the invention comprises a step E5 of calculating, from images of the part 200 acquired by the X-ray radiography device 100 according to the selected A- points of view, A / projections p(«) of the part according to the A points of view. A realization of p(«) is pW _ _]og / / (") / / 1 where denotes an image of the intensity of the X-rays having passed through the part for point of view 11 and Iq the white image (i.e. the image captured by the detector in the absence of a part).

[0032] The method further comprises a step E6 of sanctioning the part 200 (described in more detail below) which uses the N projections p(«). In a possible embodiment, this sanction comprises the determination, from the N projections pi»), of deformation parameters of a digital model of the part MODP. This digital model, which can be recorded in advance in the memory MEM of the computer CAL, 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 make it possible to move from the ideal geometry of the part (as represented by its digital model) to the real geometry of the part (as observed in the images / "1).

[0033] Based on 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 representative of the projective geometry of the radiography device can be taken into consideration 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, have the objective of assisting in the selection of N optimal viewpoints from among M possible viewpoints (Ai being a positive integer less than i.e. the selection of a subset of N viewpoints which forms an optimal observation configuration making it possible to reduce the uncertainties relating to the determination of the deformation parameters. These uncertainties are linked, among other things, to the noise present in the images, generally white and Gaussian. The deformation parameters can be represented in the form of a vector d. A covariance matrix associated with this vector d makes it possible to quantify these uncertainties.

[0035] Generally speaking, a covariance matrix is ​​(at convergence) proportional to the inverse of a Hessian matrix, so that the study of such a Hessian matrix (by means of a cost function which will be defined later) makes it possible to determine the covariance matrix minimizing the uncertainties relating to the determination of the deformation parameters. In a possible embodiment, the invention proposes to define such a Hessian matrix as a weighted sum of Hessian matrices each associated with one of the M possible points of view.

[0036] Step E2 includes the selection of the M possible viewpoints and the simulation, from the MODP digital model of the part, of M pW projections of the part according to the M possible viewpoints. As shown in Figures 3 and 4 which represent a circular trajectory of the source 101 around the part, the M possible viewpoints can be equally distributed on this circular trajectory. In an exemplary embodiment, N is chosen as having a value equal to 12, while M is worth 180, even if the 180 viewpoints are not each represented distinctly.

[0037] Step E3 comprises the determination, for each of the M simulated projections p00, of an uncertainty relating to the determination of the deformation parameters.

[0038] This step E3 may comprise carrying out the following operations for each of the M simulated projections (x designating a pixel of the X-ray detector): • the calculation, for each of the deformation parameters dt forming the vector d, of a sensitivity field of the simulated projection p(n) to a variation of said parameter; and • the determination of a Hessian matrix Hp» from the sensitivity fields (x) calculated for each of the deformation parameters d, forming the vector d.

[0039] The sensitivity field of a simulated projection p(n) to a variation of the i-th parameter dt of the vector d can be expressed as follows:

[0040] . 4) s^Hx) =—A—-iv ' tod;

[0041] Such a sensitivity field can be calculated by finite difference.

[0042] The Hessian matrix / / ^determined from the sensitivity fields is the matrix made up of the terms ( H ;«. ) (with a row i and a column k) such that: ■ ' ik

[0043] (x) s<”>(x)

[0044] where s(") and s(n) (x) correspond respectively to the sensitivity fields of the simulated projection pM to a variation of the f-th deformation parameter dt and the k-th deformation parameter dk respectively of the vector d. This Hessian matrix H^ associated with a projection p(n) is a square matrix of dimension equal to the dimension of the vector d. It characterizes, as we saw previously, the uncertainty relating to the determination of the deformation parameters.

[0045] Step E4 comprises solving an optimization problem to determine, among the M viewpoints, the subset of N viewpoints which minimizes (in the sense that it respects a stopping criterion) the uncertainty relating to the determination of the deformation parameters. This step E4 may comprise carrying out the following operations: • the calculation of weighting coefficients of a weighted sum H of the Hessian matrices determined in step E3 for each of the simulated projections, said calculation comprising the iterative optimization of the result of a cost function f applied to said weighted sum; and • the selection of the N points of view based on the calculated weighting coefficients.

[0046] The weighted sum H of the Hessian matrices can thus be expressed according to: 100471 h =

[0048] where y^ is a weighting coefficient associated with the Hessian matrix associated with the point of view (pn>. The set of weighting coefficients 1^«) forms a weighting vector y.

[0049] The weighting coefficients y^m are all positive and have a unit sum and characterize a probability of the associated point of view of belonging to an optimal configuration CONFopt comprising a predetermined number N of points of view making it possible to reduce as much as possible the uncertainty linked to the determination of the transformation parameters forming the vector d.

[0050] The weighting coefficients find their use in the selection of the N points of view among the M possible points of view. The N selected points of view, noted [ 1, forming an optimal configuration CONFopt Top? ^bpt] observation of the part 200. These N points of view are “optimal” in the sense that they make it possible to minimize the uncertainty in the determination of the transformation parameters forming the vector d, and therefore are the points of view of the set of M possible points of view which make it possible to obtain 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 embodiment of step E4:

[0052] In this first embodiment illustrated in Figures 6 and 7, the weighting coefficients y^ are binary, that is to say that they can take only 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 point of view associated with this coefficient is not part of the optimal configuration CONFopt, and the other possible value (for example 1 / N) of a weighting coefficient at the last iteration meaning that the point of view 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, N weighting coefficients (associated with N viewpoints) only 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 comprises a first sub-step E41a in which an initial configuration C ON Fini is determined by assigning one of the possible values, for example the value 1 / N, to the weighting coefficients associated with N of the Hessian matrices of the weighted sum H (and therefore to N of the viewpoints), 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 configuration CONFini may be chosen randomly or not from the M viewpoints ^(«). For example, the initial configuration C ON Fini may comprise N predetermined viewpoints corresponding to positions more likely a priori to give precise information on the deformation parameters contained in the vector d. Alternatively, the Ai viewpoints of the initial configuration C ON Fini may correspond to a regular sampling of the trajectory followed by the support 102 or the source 101, as illustrated in FIG. 3, which presents for a circular trajectory 5 of the source 101 around the part 200 the set of viewpoints among which those blackened form an initial configuration C ON Fini in which the different viewpoints are equally distributed on the circular trajectory.

[0056] The MN viewpoints whose weighting coefficient is equal to 0 form a set of candidate viewpoints rated ... of an iteration in current. 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, during each iteration, and starting from a current configuration CONF, a weighted sum H associated with said current configuration CONF and a set of candidate viewpoints at the start of said iteration, a cost function / applied to the sum weighted H in which one viewpoint of the current configuration CONF and one viewpoint of the candidate viewpoint set of said iteration have been swapped. In other words, the cost function / is applied to a second H' weighted sum of Hessian matrices constructed similarly to the first H weighted sum, but from a second configuration. This second configuration includes the same viewpoints as the current configuration CONF in the current iteration, except that only one of these viewpoints is replaced by one of the candidate points.

[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 by means of a loop (lines 5 to 9 of the pseudo-code of figure 7) on each of the viewpoints among the current configuration CONF and the set of candidate viewpoints |¢)(1( ... cl by the evaluation of the cost function f applied to the second weighted sum H' of the Hessian matrices.

[0060]

[0061] Step E42a can thus follow an iterative process comprising at each of several iterations, and as long as a convergence criterion is not respected: • the determination (first sub-step E421a of step E42a) of the optimization pair ^)| comprising a first and a second viewpoint and elected from among the set of viewpoints ¢ / 0 ..., (ff n\ ... the first point of view çibh being associated with a Hessian matrix with a weighting coefficient equal to 1 / N during the current iteration, and the second point of view being associated with a Hessian matrix with a weighting coefficient equal to 0 during the current iteration. More particularly, the optimization pair f ^p) ^)1 is determined so that the result of the application of a cost function f to a second weighted sum H ~ H is maximized. the replacement (second sub-step E422a of step E42a) of the first point of view çjûd of the couple by the second point of view qFd of the couple in the current configuration CONF, and the inversion of the weighting coefficients and of the two Hessian matrices Has associated with said points of view. Thus, the weighted sums H and H' are also inverted. Detailed description of the determination of the optimization torque during a given iteration (step E421a):

[0062] The determination of the pair is carried out by means of a loop in which the set of possible pairs of a first point of view corresponding to a point of view from the current configuration CONF and of a second point of view corresponding to a point of view from the set of candidate points of view of the current iteration is tested (lines 5 to 9 of the pseudo-code of figure 7). More precisely, for each of the possible pairs, a second weighted sum H is calculated, and the result of the application of 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 for the determination of the optimization couple is defined in the first embodiment as a function which, to the matrix H or / / as input, makes the determinant of the input matrix ff fj correspond as output. Alternatively, the cost function / can be defined as a function which, to the matrix H or H as input, makes the minimum eigenvalue &lmin' (non-zero or less than a threshold defined by the user) of the input matrix H, h' correspond as output.

[0064] When the loop during which all 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 of figure 7), and the viewpoints | j forming the optimization torque 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 F of row i and column j is associated with a possible optimization pair, said term Fj j 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 reversed, • 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 sub-step E421a is carried out, the second sub-step E422a of step E62a can then be implemented by modifying the current configuration via the replacement of the first point of view of the optimal optimization pair by the second point of view of said pair, and by the inversion of the weighting coefficients associated with the Hessian matrices corresponding to said points of view (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 verified, 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 applying 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 difference 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 Y(,p> can be decimal numbers (positive) between 0 and 1. Furthermore, in the second embodiment, the sum of the values ​​of all the weighting coefficients (for all n between 1 and M) can be equal to 1.

[0070] Step E4 comprises a first sub-step E41b in which a predetermined value is assigned to each of the M weighting coefficients associated with the M Hessian matrices of the weighted sum H. The values ​​assigned to each of the M weighting coefficients are recorded in a vector of weighting coefficients denoted y. More particularly, the vector of weighting coefficients F is ordered such that the index term 11 of the vector F is the weighting coefficient associated with the ^-th Hessian matrix of the weighted sum H, i.e. associated with the Hessian matrix itself associated with the ^-th point of view (fFF

[0071] For example, a uniform value 1 IM is assigned in the first step E41b to each of the M weighting coefficients of the vector F in order to account for an initial uniform probability for each of the viewpoints of minimizing 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 M weighting coefficients. This optimization is carried out iteratively.

[0073] More precisely, step E42b comprises, at each of several iterations, and as long as a stopping criterion is not respected: • The determination (sub-step E421b) of an optimization vector of the weighting coefficients noted h, the vector h being equal to the derivative of the cost function f with respect to the vector of weighting coefficients V, that is to say that a term of index n noted 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 vector of weighting coefficients V, in other words with respect to the weighting coefficient associated with the Hessian matrix H associated with the Zî-th point of view ¢)00, • where applicable, when a weighting coefficient of the vector of weighting coefficients Y is zero and the term of the associated optimization vector h, i.e. with the same index as this weighting coefficient, is negative, the assignment (sub-step E422b) to said term of the optimization vector h of a value equal to 0, • the modification (sub-step E423b) of each of the terms of the vector of weighting coefficients F, that is to say of each of the M weighting coefficients, by adding a quantity proportional to the term of the associated optimization vector h, and • the assignment (sub-step E424b) of a value equal to 0 to each of the negative terms of the vector of weighting coefficients F, and • the normalization (sub-step E425b) of the vector of weighting coefficients h

[0074] Each of the terms of the optimization vector h is intended to be added to the associated weighting coefficient of the same index of the vector of weighting coefficients 1'.

[0075] As described previously, the weighted sum H of the Hessian matrices H^) associated with each of the viewpoints can be expressed as: [00761

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

[0078] H=H(y)=Hj.y

[0079] Furthermore, 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 determinant function of H to which a Lagrange multiplier is added in order to force normalization. The cost function / considered can thus be expressed such that -.f (y) = det(H( y ) ) - d ( 1 )

[0080] Thus, the optimization vector h calculated during step E421b can be expressed, by the Jacobi formula, such that: 100811 h= =det(H( r )) WH< 1)

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] Sub-steps E422b (lines 5 to 7 of the pseudo-codes of figures 9 and 10) and E424b (lines 9 to 11 of the pseudo-codes of figures 9 and 10) make it possible to avoid terms of the vector of weighting coefficients ? being negative. The modification E423b of each of the terms of the vector of the weighting coefficients y is done by adding a quantity proportional to the term of the associated optimization vector. Thus the sub-step E423b assigns, as described in line 8 of the pseudo-code of figures 9 and 10, to the vector F the new value y + wh , in which w is chosen for example such that vt' II f II S0.1 II F II • Thus, over the course of the iterations, modification E423b (line 8 of the pseudo-codes in figures 9 and 10) makes it possible to individually increase or decrease the weighting coefficients in order to determine the distribution V maximizing the cost function f. In order to respect the normalization of the weighting coefficients, an additional step E425b of normalization of the vector of the weighting coefficients can be executed at the end of step E424b, as illustrated in lines 12 of the two pseudocodes of figures 9 and 10. The iterations of step E42b are stopped when a stopping criterion is verified, for example when a maximum number of iterations is reached or, as illustrated in the pseudo-code of figure 9, when a difference between the results of the application of the cost function f to the weighted sum H of the Hessian matrices determined at the previous iteration on the one hand and to the weighted sum H of the Hessian matrices determined at the current iteration on the other hand becomes less than a predetermined deviation toi. Alternatively, another stopping criterion, which is transcribed in figure 10 which illustrates a second example of pseudo-code of the second embodiment of step E6, can be verified when a norm of the difference between the vector of the weighting coefficients V determined at the current iteration and that determined at the previous iteration becomes less than a predetermined tolerance deviation toi. Thus, at the end of step E6, a value of each of the M weighting coefficients y (pi») is determined. As described previously, step E6 then comprises a selection of the N optimal viewpoints forming an optimal configuration CONFopt^ j of observation of the part 200 among the M viewpoints qF) associated with the different simulated projections. This selection is carried out according to the weighting coefficients associated with said viewpoints, in particular by selecting the N viewpoints whose associated weighting coefficients are the highest. Thus, in the first embodiment previously described, the N viewpoints selected are the viewpoints whose co efficient weighting coefficient has a value equal to 1 / N, the others having a value equal to 0. In the second embodiment described, the selection of the N optimal viewpoints involves the study of the weighting coefficients forming the vector of weighting coefficients V, and by the selection of 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 indicated previously, step E5 comprises the calculation, from images of the part 200 acquired according to the N selected viewpoints, of N projections pW of the part according to the N viewpoints. And step E6 comprises a sanction of the part 200 during which the deformation parameters are determined during a sub-step E61 from the N projections p(n). It is understood that for the sanction, the number of images acquired is limited to N rather than M, which considerably reduces the acquisition time, and thus the control time without losing 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, a summary of which is described below.

[0091] This determination exploits a difference or residue p(n) between pi") the projection calculated for the point of view number n (for all n between 1 and N) and pW a projection simulated for the same point of view which is expressed for example in the following form:

[0092] p(n)Çx. d)=P{n\x)-P{"^ d)

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

[0094] Furthermore, at each of the iterations, the modification of the vector d can comprise: • for each point of view, the calculation of sensitivity fields of the projection simulated for this point of view to a variation of the parameters contained in the vector d; • the calculation of a corrective vector fid* as being the 8d vector minimizing a gap between the projection residues and the product of ôd by the sensitivity fields; • updating the vector d using the corrective vector fyf.

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

[0096] / x A; 4 =—è-— 2 k \ H ddk

[0097] The calculation of the corrective vector may comprise the minimization of the sum over the viewpoints of the squares of the differences between, for each viewpoint, the projection residual calculated for this viewpoint and the product of ôd by the sensitivity fields calculated for this viewpoint. Thus the update of the vector d using the corrective vector at the end of an iteration may be noted d *- d+ ôd*- For example, the corrective vector gd is given by:

[0098] & / '* = argmin]T || H'(«)(xXp(”\x ; d)-s^ (x ; d) ôd\ || 2 ' Od

[0099] where ; d) is the matrix of sensitivity fields $(«) ) for the point of view and yyln)(x) is a weighting term that can be used to account for local uncertainties due to, for example, noise or dead pixels.

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

[0101] Once the deformation parameters forming the vector d have been determined by sub-step E61, the sanction may comprise the determination, during a sub-step 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 comprise, during a sub-step E63, the simulation, from the corrected model of the part, of N projections of the part according to the N points of view 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 digital 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 depending on the pixel considered. If the differences are lower than the noise level, then they are considered insignificant. Otherwise, this means that the corrected model was unable to capture the shape variability necessary for this comparison.

[0106] The invention is not limited to the method previously described, but also extends to a non-destructive part testing system which comprises a processor configured to implement this method as well as to a computer program product comprising instructions which, when the program is executed by a computer, lead the latter to implement the method.

Claims

Claims

1. A method for non-destructive testing of a part, comprising: • the simulation (E2), from a digital model of the part, of M projections of the part according to M points of view, 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; • the resolution of an optimization problem to determine (E4), among the M points of view, a subset of N points of view which minimizes the uncertainty relating to the determination of the deformation parameters, N being a positive integer less than M; • the calculation (E5) of N projections of the part according to the N points of view of said subset, from images ( / n)) of the part (200) acquired by an X-ray radiography device (100) according to the N points of view of said subset;• a sanction (E6) of the part from the N calculated projections.;

2. Method according to claim 1, in which the determination (E3), for each of the M simulated projections, of an uncertainty relating to the determination of the deformation parameters comprises: • 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.

3. Method according to claim 2, in which determining (E4), among the M viewpoints, the subset of N 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 comprising the iterative optimization of the result of a cost function ( / ) applied to said weighted sum; • the selection of the N points of view based on the calculated weighting coefficients.

4. The method of claim 3, wherein the calculation of the weighting coefficients comprises the following steps: • assignment (E41a) to N of the Hessian matrices of the weighted sum of a weighting coefficient of a value equal to 1 / N, and assignment to the remaining M - N Hessian matrices of a zero weighting coefficient, the N points of view associated with the Hessian matrices of weighting coefficient equal to 1 / N forming an observation configuration of the first part (CONF), • at each of several iterations, and as long as a stopping criterion is not respected: • determination (E421a) of an optimization pair (comprising a first and a second 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 inverted; • replacement (E422a) of the first point of view by the second point of view in the observation configuration, and inversion of the weighting coefficients of the two Hessian matrices associated with said points of view.

5. A method according to one of claims 3 and 4, wherein the cost function for an input matrix maps the output to the die ending of the input matrix or the smallest eigenvalue of the input matrix that is non-zero or less than a threshold.

6. Method according to claim 3, in which the determination (E4) of the weighting coefficients comprises the following steps: • assignment (E41b) to each of the Hessian matrices of a weighting coefficient, the weighting coefficients associated with each of the Hessian matrices being stored in a vector of weighting coefficients (X) in which the index term n / \ corresponds to the weighting coefficient deration 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 respected: • determination (E421b) of an optimization vector of the weighting coefficients (h), the optimization vector being equal to the derivative of the cost function with respect to the vector of the weighting coefficients, so 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 the weighting coefficients, • where appropriate, when a weighting coefficient is zero and the term of the optimization vector of the same index is negative, assignment (E422b) to said term of a value equal to 0; • modification (E423b) of each of the terms of the vector of weighting coefficients 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 vector of weighting coefficients. • normalization (E425b) of the vector of weighting coefficients*

7. The method of claim 6, wherein the cost function is the determinant of the weighted sum of the Hessian matrices to which a Lagrange multiplier is added.

8. Method according to one of claims 1 to 7, in which the sanction (E6) of the part comprises the determination of the deformation parameters of the digital model of the part from the N calculated projections.

9. The method of claim 8, wherein the sanctioning (E6) of the part comprises determining a corrected model of the part by transforming the digital model of the part using the deformation parameters.

10. The method of claim 9, wherein the sanction further comprises simulating, from the corrected model of the part, N projections of the part according to the N points of view and comparing the N simulated projections with the N calculated projections.

11. Non-destructive testing system for parts, comprising a processor configured to implement the method according to one of claims 1 to 10.

12. Computer program product comprising instructions which, when the program is executed by a computer, cause the latter to implement the method according to one of claims 1 to 10.