METHOD, DEVICE AND COMPUTER PROGRAM FOR MONITORING A PART BY X-RAY

Abstract

The invention relates to a method for non-destructively testing a part by means of transmission radiography, which method comprises the following steps of acquiring N projections (P.sup.(n)) of the part, generating calculated N images (P.sup.(n)) of the part, estimating, by successive iterations, the vector p from an initial vector p=p.sub.ini and the vector c from an initial vector c=c.sub.ini and/or the parameter vector a from an initial vector =.sub.ini, by minimising the sum of the squared differences between the projections (P.sup.(n)) and the images (P.sup.(n)), processing the projections (P.sup.(n)) and/or the images (P.sup.(n)), identifying defects in the part by comparing the processed projections (P.sup.(n)) and the processed images (P.sup.(n)).

Claims

1-8. (canceled)

9. A method of non-destructive testing of a part by transmission radiography, comprising the following steps, executed by a calculator: acquiring of N projections of the part using a transmission radiography device from N different and predetermined angles of view of the part, where N is a given natural integer, generating of N computed images of the part from a reference model of the part corresponding to the N angles of view and from a vector p of parameters characterizing a projection geometry of acquisition for the N angles of view at each of several successive iterations, estimating, by the successive iterations, of the vector p from an initial vector p=p.sub.ini and of at least one of a vector c of parameters from an initial vector c=c.sub.ini and of a vector of parameters from an initial vector =.sub.ini, where the vector c of the parameters accounts for beam hardening of radiation in the part and the vector of the parameters characterizes Compton scattering of the radiation in the part, by minimizing the sum of the norms of the squared differences between the N projections having been acquired and the N computed images, processing of the N projections and/or of the N computed images comprising a first processing and/or a second processing, the first processing comprising a correction of the beam hardening over the N projections from the vector c having been estimated or a generation of the beam hardening over the N computed images from the vector c having been estimated, the second processing comprising a correction of the Compton scattering over the N projections from the vector having been estimated or a generation of the Compton scattering over the N computed images from the vector having been estimated, identifying of defects of the part by comparison of the N projections having been processed with the N computed images having been processed.

10. The method as claimed in claim 9, comprising at each iteration estimating of the vector p of the parameters p.sub.i of the projection geometry of the acquisition for the N angles of view by the calculator, the estimating comprising: computing of projection residuals .sub.p.sup.(n)=P.sup.(n){circumflex over (P)}.sup.(n) from the initial vector p=p.sub.ini of initial values, computing of sensitivity fields s.sub.p.sub.i.sup.(n) according to $s_{p_i}^{\left(n\right)}=\frac{{\stackrel{}{P}}^{\left(n\right)}}{p_i}{.Math.}_p$ from the initial vector p=p.sub.ini of the initial values, where {circumflex over (P)}.sup.(n) are the N computed images for the N angles of view, P.sup.(n) are the N projections having been acquired of the part, p is a column vector of the parameters p.sub.i of the projection geometry, p.sub.ini is a column vector of the initial values of the parameters p.sub.i of the projection geometry, n is a natural integer denoting the number of the angle of view and ranging from 1 to N, p is updated according to p*=p+p*, where p* is a column vector of the parameters p.sub.i of the projection geometry having been updated, p* is a column vector of variation of the parameters p.sub.i of the projection geometry and is calculated as a p minimizing the sum of the norms of the squared differences between the projection residuals .sub.p.sup.(n) and the product of p by s.sub.p.sup.(n) for n ranging from 1 to N, $p^{*}=\arg \underset{p}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_p^{\left(n\right)}-s_p^{\left(n\right)}p.Math.}^2$ where p is a column vector, s.sub.p.sup.(n) is a matrix of the sensitivity fields s.sub.pi.sup.(n).

11. The method as claimed in claim 9, comprising at each iteration estimating of the vector c of the parameters c.sub.k of calibration of the beam hardening of the radiation in the part by the calculator, the estimating comprising: computing of projection residuals .sub.c.sup.(n)=P.sup.(n)P.sup.(n) from the initial vector c=c.sub.ini of initial values, computing of sensitivity fields s.sub.c.sub.k.sup.(n) according to $s_{c_k}^{\left(n\right)}=\frac{{\stackrel{}{P}}^{\left(n\right)}}{c_k}{.Math.}_c$ from the initial vector c=c.sub.ini of the initial values, where P.sup.(n)(x)=u({circumflex over (P)}.sup.(n)(x)) is an image obtained by applying a function u(y) to an intensity y of each of pixels x of {circumflex over (P)}.sup.(n), {circumflex over (P)}.sup.(n) are the N computed images for the N angles of view, P.sup.(n) are the N projections having been acquired of the part, c is a column vector of the parameters c.sub.k of calibration of the beam hardening, c.sub.ini is a column vector of the initial values of the parameters c.sub.k of calibration of the beam hardening of the radiation in the part, .sub.k(y) is a base of given form functions, $u\left(y\right)=\underset{k=1}{\overset{K_3}{.Math.}}{c}_k{}_k\left(y\right)$ K.sub.3 is a given natural integer greater than or equal to 1, k is a natural integer ranging from 1 to K.sub.3, c is updated according to c*=c+c*, where c* is a column vector of the parameters c.sub.k of calibration of the beam hardening of the radiation in the part, having been updated, c* is a column vector of variation of the parameters c.sub.k of calibration of the beam hardening of the radiation in the part and is computed as a c minimizing the sum of the norms of the squared differences between the projection residuals .sub.c.sup.(n) and the product of c by s.sub.c.sup.(n) for n ranging from 1 to N, $c^{*}=\arg \underset{c}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_c^{\left(n\right)}-s_c^{\left(n\right)}c.Math.}^2$ where c is a column vector, s.sub.c.sup.(n) is a matrix of the sensitivity fields s.sub.c.sub.k.sup.(n).

12. The method as claimed in claim 9, comprising at each iteration estimating of the vector of the parameters .sub.j of Compton scattering of the radiation in the part by the calculator, the estimating comprising: computing of projection residuals .sub..sup.(n)=P.sup.(n){tilde over (P)}.sup.(n) from the initial vector =.sub.ini of the initial values, computing of sensitivity fields s.sub..sub.j.sup.(n) according to $s_{{}_j}^{\left(n\right)}=\frac{{\stackrel{}{P}}^{\left(n\right)}}{{}_j}{.Math.}_{}$ from the initial vector =.sub.ini of the initial values, where P.sup.(n) are the N projections having been acquired of the part, {circumflex over (P)}.sup.(n) are the N computed images for the N angles of view, {tilde over (P)}.sup.(n)=P.sup.(n)+P.sup.(n)*K is an image obtained by convolution of simulated images P.sup.(n), having been obtained from at least the N computed images {circumflex over (P)}.sup.(n), with a kernel +K, is a column vector of the parameters .sub.j of Compton scattering of the radiation in the part, .sub.ini is a column vector of the initial values of the parameters .sub.j of Compton scattering of the radiation in the part, K is a convolution kernel defined by $K\left(x\right)=\underset{j=1}{\overset{K_2}{.Math.}}{}_j\left(g_{{}_j}\left(x\right)-\left(x\right)\right)$ K.sub.2 is a prescribed natural integer greater than or equal to 1, j is a natural integer ranging from 1 to K.sub.2, g.sub..sub.j a two-dimensional Gaussian kernel of prescribed standard deviation j, (x) is a Dirac function at the pixel x, is updated according to *=+*, where * is a column vector of the parameters .sub.j of Compton scattering of the radiation in the part, having been updated, * is a column vector of variation of the parameters .sub.j of Compton scattering of the radiation in the part and is calculated as a minimizing the sum of the norms of the squared differences between the projection residuals .sub..sup.(n) and the product of by s.sub..sup.(n) for n ranging from 1 to N, ${}^{*}=\arg \underset{}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_{}^{\left(n\right)}-s_{}^{\left(n\right)}.Math.}^2$ where is a column vector, s.sub..sup.(n) is a matrix of the sensitivity fields s.sub..sub.j.sup.(n).

13. The method as claimed in claim 11, comprising at each iteration estimating of the vector of the parameters .sub.j of Compton scattering of the radiation in the part by the calculator, the estimating comprising: computing of projection residuals .sub..sup.(n)=P.sup.(n){tilde over (P)}.sup.(n) from the initial vector =.sub.ini of the initial values, computing of sensitivity fields s.sub..sub.j.sup.(n) according to $s_{{}_j}^{\left(n\right)}=\frac{{\stackrel{}{P}}^{\left(n\right)}}{{}_j}{.Math.}_{}$ from the initial vector =.sub.ini of the initial values, where P.sup.(n) are the N projections having been acquired of the part, {circumflex over (P)}.sup.(n) are the N computed images for the N angles of view, {tilde over (P)}.sup.(n)=P.sup.(n)+P.sup.(n)*K is an image obtained by convolution of simulated images P.sup.(n), having been obtained from at least the N computed images {circumflex over (P)}.sup.(n), with a kernel +K, is a column vector of the parameters .sub.j of Compton scattering of the radiation in the part, .sub.ini is a column vector of the initial values of the parameters .sub.j of Compton scattering of the radiation in the part, K is a convolution kernel defined by $K\left(x\right)=\underset{j=1}{\overset{K_2}{.Math.}}{}_j\left(g_{{}_j}\left(x\right)-\left(x\right)\right)$ K.sub.2 is a prescribed natural integer greater than or equal to 1, j is a natural integer ranging from 1 to K.sub.2, g.sub..sub.j a two-dimensional Gaussian kernel of prescribed standard deviation j, (x) is a Dirac function at the pixel x, is updated according to *=+*, where * is a column vector of the parameters .sub.j of Compton scattering of the radiation in the part, having been updated, * is a column vector of variation of the parameters .sub.j of Compton scattering of the radiation in the part and is calculated as a minimizing the sum of the norms of the squared differences between the projection residuals .sub..sup.(n) and the product of by s.sub..sup.(n) for n ranging from 1 to N, ${}^{*}=\arg \underset{}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_{}^{\left(n\right)}-s_{}^{\left(n\right)}.Math.}^2$ where is a column vector, s.sub..sup.(n) is a matrix of the sensitivity fields s.sub..sub.j.sup.(n), wherein P.sup.(n)(x)=u({circumflex over (P)}.sup.(n)(x)) is the simulated image, obtained by applying the function u(y) to the intensity of each of the pixels x of the computed image {circumflex over (P)}.sup.(n).

14. The method as claimed in claim 9, characterized in that N is less than or equal to 1000.

15. A computer program, comprising code instructions for implementing the following steps of a method of non-destructive testing of a part by transmission radiography, when it is executed by a calculator: receiving of N projections of the part from a transmission radiography device from N different and predetermined angles of view of the part, where N is a given natural integer, generating of N computed images of the part from a reference model of the part corresponding to the N angles of view and from a vector p of parameters characterizing the projection geometry of the acquisition for the N angles of view at each of several successive iterations, estimating, by the successive iterations, of the vector p from an initial vector p=p.sub.ini and of at least one of a vector c of parameters from an initial vector c=c.sub.ini and of a vector of parameters from an initial vector =.sub.ini, where the vector c of the parameters accounts for the beam hardening of the radiation in the part and the vector of the parameters characterizes the Compton scattering of the radiation in the part, by minimizing the sum of the norms of the squared differences between the N projections having been acquired and the N computed images, processing of the N projections and/or of the N computed images comprising a first processing and/or a second processing, the first processing comprising a correction of the beam hardening over the N projections from the vector c having been estimated or a generation of the beam hardening over the N computed images from the vector c having been estimated, the second processing comprising a correction of the Compton scattering over the N projections from the vector having been estimated or a generation of the Compton scattering over the N computed images from the vector having been estimated, identifying of defects of the part by comparison of the N projections having been processed with the N computed images having been processed.

16. A device for non-destructive testing of a part by transmission radiography, comprising: a transmission radiography device, for acquiring of N projections of the part along N different and predetermined angles of view of the part, where N is a given natural integer, a calculator configured to carry out the following steps: generating of N computed images of the part from a reference model of the part corresponding to the N angles of view and from a vector p of parameters characterizing the projection geometry of the acquisition for the N angles of view at each of several successive iterations, estimating, by the successive iterations, of the vector p from an initial vector p=p.sub.ini and of at least one of a vector c of parameters from an initial vector c=c.sub.ini and of a vector of parameters from an initial vector =.sub.ini, where the vector c of the parameters accounts for the beam hardening of the radiation in the part and the vector of the parameters characterizes the Compton scattering of the radiation in the part, by minimizing the sum of the norms of the squared differences between the N acquired projections and the N computed images, processing of the N projections and/or of the N computed images comprising a first processing and/or a second processing, the first processing comprising a correction of the beam hardening over the N projections from the vector c having been estimated or a generation of the beam hardening over the N computed images from the vector c having been estimated, the second processing comprising a correction of the Compton scattering over the N projections from the vector having been estimated or a generation of the Compton scattering over the N computed images from the vector having been estimated, identifying of defects of the part by comparison of the N projections having been processed with the N computed images having been processed.

Description

[0083] The invention will be better understood on reading the following description, given solely by way of non-limiting example with reference to the figures below of the appended drawings.

[0084] FIG. 1 schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0085] FIG. 2 schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0086] FIG. 3 schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0087] FIG. 4 schematically represents a modular block diagram of a device for non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0088] FIG. 5 schematically represents an organizational chart of a method for non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0089] FIG. 6 schematically represents a curve of the intensity of the pixels of a projection P(x) in the ideal scenario and in a practical scenario altered by beam hardening artefacts of a projection obtained by a device of the prior art.

[0090] FIG. 7 represents an explanatory diagram of Compton scattering during the acquisition of an X-ray projection.

[0091] FIG. 8 represents the difference between, firstly, the inverse of the exponential of the intensity of an acquired projection, and secondly, the inverse of the exponential of the intensity of a simulated projection, before correction on a part of a blade airfoil of a turbomachine turbine, obtained by a device of the prior art.

[0092] FIG. 9 represents the difference between, firstly, the inverse of the exponential of the intensity of an acquired projection, and secondly, the inverse of the exponential of the intensity of a simulated projection, before correction on a part of a blade root of a turbomachine turbine, obtained by a device of the prior art.

[0093] FIG. 10 represents the difference between, firstly, the inverse of the exponential of the intensity of an acquired projection, and secondly, the inverse of the exponential of the intensity of a simulated projection, after correction on a part of the same blade airfoil of a turbomachine turbine as in FIG. 8, obtained by the method, a device and a computer program for the non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0094] FIG. 11 represents the difference between, firstly, the inverse of the exponential of the intensity of an acquired projection, and secondly, the inverse of the exponential of the intensity of a simulated projection, after correction on a part of the same blade root of a turbomachine turbine as in FIG. 9, obtained by the method, a device and a computer program for the non-destructive testing of a part by X-ray radiography according to an embodiment of the invention.

[0095] As illustrated in FIGS. 1 to 5, the device 1 for non-destructive testing of an actual part 200 comprises a transmission radiography device 100 used to acquire, during the step E1 of the method, a number of projections P.sup.(n), also known as raw acquired projections, of the part 200. In the following text, the radiation is X-rays. The radiography device 100 makes it possible to inspect the part 200 under controlled acquisition conditions with a known acquisition geometry and makes it possible to perform transmission imaging of the internal structure of the part 200. In the remainder of the text, the projections P.sup.(n) and the images mentioned below are defined by their gray levels (or intensity) for their different pixels x.

[0096] In the embodiments described below, the projections P.sup.(n) are X-ray images P.sup.(n) in X-ray transmission mode, in the method of non-destructive testing of a part by X-ray radiography, the device 1 for non-destructive testing of a part by X-ray radiography and the computer program for non-destructive testing of a part by X-ray radiography.

[0097] The part 200 is a mechanical part, and can be without Limitation an aeronautical part, particularly a turbomachine blade (for example of a turbojet engine), for example one of their turbine blades (turbine airfoil), or can be of another kind. The part 200 is made of a material semi-transparent to radiation, here to X-rays, and can be composed of a single material. The part 200 can be a metallic part or a ceramic matrix composite part, or another kind. The method, the device 1 and the computer program for the non-destructive testing of the part by X-ray radiography are used to test the part 200 during its fabrication or during maintenance operations, in order to detect defects in this part, which can for example cause an in-flight failure in the case of an aeronautical part.

[0098] The device 1 for non-destructive testing of the part 200 comprises, and the method of non-destructive testing of the part 200 uses, one (or more) calculators CAL. The calculator CAL 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, or one or more random-access memories MEM. The calculator CAL may comprise 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 comprise one or more computer keyboard, one or more physical data communication ports, one or more touch-sensitive screens, or other kinds. This or these physical data output interfaces INT2 may be or comprise one or more physical data communication ports, one or more touch-sensitive screens, or other kinds. A computer program may be recorded and executed on the calculator CAL of the device 1 for non-destructive testing of the part 200 and comprise code instructions, which when they are executed on it, implement all or part of the method of non-destructive testing of the part 200 according to the invention (comprising the reception of the N projections during step E1).

[0099] The X-ray radiography device 100 comprises a source 101 of X-rays, a support 102 on which the part 200 is located, a control mechanism 104 to turn the support 102 and the source 101 with respect to one another 101 about an axis 103 of rotation, which can for example be vertical (for example the source 101 is fixed and the support 102 is rotated about the axis 103), a detector 105 of the X-rays crossing the part 200, the part 200 being therefore located on the path of the X-rays between the source 101 and the detector 105. The source 101, the support 102 and the detector 105 are disposed in a high-power X-ray cabinet. The detector 105 supplies the projections P.sup.(n) of the part 200 during the first step E1. The control mechanism 104 is controlled for the acquisition at the N angles ANG.sup.(n) of view, different from one another, of the part 200 as regards the X-rays, by the detector 105, of N projections P.sup.(n). N is a given natural integer, greater than or equal to 1. The natural integer n ranges from 1 to N and denotes the number of the respective angle ANG.sup.(n) of view and therefore the number of the acquired projection P.sup.(n). The radiography device 100 thus makes it possible to acquire N projections (i.e. N projections P.sup.(n)) of the volume of the part 200 from the N angles ANG.sup.(n) of view respectively. The N angles ANG.sup.(n) of view have been predetermined to be used for the later step E5 of analysis of the part. The long acquisition time of the projections acquired by X-rays leads to the consideration of only a limited number N of acquired projections P.sup.(n), less than 1000 or even less than 100, or otherwise. The X-ray radiography device 100 thus supplies all the projections P.sup.(n) acquired during step E1.

[0100] During a second step E2, after the first step E1, the calculator CAL generates N computed images {circumflex over (P)}.sup.(n) of the part 200 based on a reference digital model MODP of the part 200, as illustrated in FIGS. 1 to 5. These N computed images {circumflex over (P)}.sup.(n) correspond to the N angles ANG.sup.(n) of view, i.e. adopt the same angle ANG.sup.(n) of view as the N projections P.sup.(n) respectively. This model MODP is a geometrical reference of the part 200, is recorded in advance in a memory of the calculator CAL and can be, for example, a computer-assisted design (or CAD) model, of the part 200, replicating an ideal defect-free part 200, having an external shape of defined three-dimensional coordinates. This model MODP can take into account the composition of the material of the part 200. At each step E2, the calculator CAL generates N computed images {circumflex over (P)}.sup.(n) of the part 200 based on the reference model MODP of the part 200 corresponding to the N angles ANG.sup.(n) of view and based on a vector p of parameters pi characterizing the geometry of projection of the acquisition for the N angles ANG.sup.(n) of view at each of several successive iterations.

[0101] The calculator CAL carries out one or more of the steps E3a, E3b, E3c of estimation of multi-view acquired projection artefacts, which will be described below. According to embodiments, the determination of the parameters described below is done by an optimization procedure which makes use of the sensitivity fields to minimize projection residuals.

[0102] During the third step E3a, after the first step E1, the calculator CAL estimates the vector p of the parameters p.sub.i of the projection geometry of the acquisition of step E1 for the N angles ANG.sup.(n) of view, by minimizing the sum of the quadratic norms of the squared differences, computed by the calculator CAL, between the N acquired projections P.sup.(n) and the N computed images {circumflex over (P)}.sup.(n). During the third step E3a the calculator CAL estimates the vector p describing the geometry of the X-ray radiography device 100 with respect to the model MODP of the ideal part 200 (for example its CAD model). The third step E3 makes it possible to accurately determine by the calculator CAL the vector p of the projection geometry used during the acquisition of the projections P.sup.(n) by the X-ray radiography device 100. This makes it possible to ensure a more reliable simulation of the images. The third step E3a makes it possible to carry out, during the next iteration of step E2, a geometric registration of the N images {circumflex over (P)}.sup.(n) computed for the N angles ANG.sup.(n) of view by computing the N computed images {circumflex over (P)}.sup.(n) by projection of the model MODP onto the plane of the detector 105 according to the geometry determined by the vector p of the N acquired projections P.sup.(n). The step E3a can be implemented by a simulator SIMX, as represented in FIG. 4. During the third step E3a, the calculator CAL estimates the vector p of the parameters p.sub.i of the projection geometry of the acquisition by the successive iterations based on an initial vector p=p.sub.ini.

[0103] The vector p of parameters of the projection geometry is used to simulate the computed images {circumflex over (P)}.sup.(n). Thus, during step E2, use is already being made of the vector p.sub.ini of the parameters of the given initial values of the parameters of the projection geometry during the first iteration; then during the following iterations of step E2 use is made of the estimated values of the vector p. The computed images {circumflex over (P)}.sup.(n) are subsequently used to estimate the vector c and/or a in step E3a and step E3b and/or E3c. The computed images {circumflex over (P)}.sup.(n) are not corrected solely in step E4, but also in step E2. Thus, step E2 comprises the regeneration of the computed image {circumflex over (P)}.sup.(n) at each new iteration with the new vector p*, which has been estimated during the step E3a. The estimate of the vector p during step E3a is an iterative procedure, and therefore at each iteration one uses the best possible estimate (up to convergence) to generate the computed images {circumflex over (P)}.sup.(n). The vector p is used in step E2 and at each of the iterations of step E3a.

[0104] The reproduction or correction of the artefacts is based on the digital simulation of the acquired projections P.sup.(n) by an X-ray system. The artefacts are then estimated based on a parametric model (steps E3b, E3c) before being either reproduced (step E4), or corrected (step E4).

[0105] The projection parameters p.sub.i (such as the angles ANG.sup.(n) of projection, distance from the source 101 to the detector 105) make it possible to link the ideal model MODP of the part 200 to the acquired projections P.sup.(n) of the part 200 and are thus registration parameters of the ideal model MODP of the part 200 with the acquired projections P.sup.(n).

[0106] According to an embodiment of the invention, during step E3a, at each iteration the calculator CAL computes (for example via the residuals computer CRES in FIG. 4) the projection residuals .sub.p.sup.(n), these residuals being the differences between the observed projections P.sup.(n) and the simulated projections {circumflex over (P)}.sup.(n) according to the equation .sub.p.sup.(n)=P.sup.(n){circumflex over (P)}.sup.(n). Based on the first given approximation, written p.sub.ini, of these parameters p.sub.i, the calculator CAL estimates (for example by the estimator ESTV in FIG. 4) the vector p* describing the system observed as well as possible, i.e. minimizing the projection residuals .sub.p.sup.(n). The sensitivity fields s.sub.p.sub.i.sup.(n) are computed (for example by finite differences, or otherwise) by the calculator CAL according to the following equation:

[00009]${s_{p_i}^{\left(n\right)}=\frac{{\stackrel{}{P}}^{\left(n\right)}}{p_i}.Math.}_p$ [0107] and express the effect of the variation of each of the parameters p.sub.i on the computed image. The parameters p.sub.i of the projection geometry are computed by the calculator CAL by optimization based on the projection residuals .sub.p.sup.(n), on the value p.sub.ini, and on the sensitivity fields s.sub.p.sub.i.sup.(n). The calculator CAL computes the optimal variation p* of the vector p of parameters and updates the vector p according to the following equation:

[00010]$p^{*}=p+p^{*}$ [0108] where p* is the updated column vector of the parameters p.sub.i of the projection geometry, p* is the column vector of variation of the parameters p.sub.i of the projection geometry and is computed by the calculator CAL as the p minimizing the sum of the norms of the squared differences between the projection residuals .sub.p.sup.(n) and the product of p by s.sub.p.sup.(n) for n ranging from 1 to N,

[00011]$p^{*}=\arg \underset{p}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_p^{\left(n\right)}-s_p^{\left(n\right)}p.Math.}^2$ [0109] where p is a column vector, [0110] s.sub.p.sup.(n) is the matrix of sensitivity fields s.sub.p.sub.i.sup.(n).
The number of rows of the matrix s.sub.p.sup.(n) is the number of pixels exploited in the projections P.sup.(n), in the computed images {circumflex over (P)}.sup.(n) and in the residual .sub.p.sup.(n). The number of columns of the matrix s.sub.p.sup.(n) is the number of parameters p.sub.i.
The calculator CAL then computes during the step E3a the set of sensitivity fields s.sub.p.sub.i.sup.(n) for n ranging from 1 to N and i ranging from 1 to K.sub.1.
The estimation of the vector p of parameters p.sub.i of the projection geometry for the N angles ANG.sup.(n) of view by the calculator CAL is carried out by successive iterations over p which becomes p.sub.ini, to: [0111] compute at each current iteration, based on the vector p of the preceding iteration taking the place of the vector p.sub.ini mentioned above, the sensitivity fields s.sub.p.sub.i.sup.(n) according to the equation mentioned above, [0112] for each current iteration: compute a new p* and update p according to p*=p+p*, with p* computed as p minimizing the sum of the norms of the squared differences between the projection residuals .sub.p.sup.(n) and the product of p by s.sub.p.sup.(n) for n ranging from 1 to N, according to

[00012]$p^{*}=\arg \underset{p}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_p^{\left(n\right)}-s_p^{\left(n\right)}p.Math.}^2$ [0113] until p converges on a better estimate (determined for example by the fact that the norm of the computed difference between the p of the current iteration and the p of the preceding iteration becomes less than a prescribed threshold).

[0114] During the fourth step E3b, after the first step E1, the calculator CAL estimates the vector c of the parameters c.sub.k of calibration of the beam hardening of the X-rays in the part 200 by minimizing the sum of the norms of the squared differences between the N acquired projections P.sup.(n) and the N computed images {circumflex over (P)}.sup.(n), computed by the calculator CAL. Beam hardening refers to the fact that the absorption of a photon by the constituent material of the analyzed part depends on its energy. Low-energy photons are absorbed preferably, compared to those of high energy. So-called beam hardening artefacts emerge from the gap between the actual curve CRI given by the gray level P(x) of the pixels x of a measured projection P.sup.(n) and the theoretical curve CTI resulting from the modeling used in the analyses of the projections (Beer-Lambert law), illustrated in FIG. 6. Beam hardening gives rise to a non-linear relationship between the measured intensity P(x) of a projection and the thickness (x) of material crossed (deviation from the Beer-Lambert law) in the case of a single-material part. The thickness (x) is the thickness crossed by the ray before reaching the pixel x. In the case of single-material parts 200, beam hardening artefacts can be exactly corrected by calibration of the beam hardening attenuation. The calculator CAL makes an identification of the calibration function connecting the effective attenuation of the X-rays crossing the part 200 and the length of material crossed in the part 200. The step E3b can be implemented by an X-ray beam hardening simulator SIMBH, as shown in FIG. 4. In step E3b, the calculator CAL estimates, by the successive iterations, the vector c based on an initial vector c=c.sub.ini.

[0115] Below is a description of the embodiments of the fourth step E3b. What is described below is carried out at each of the iterations of step E3b.

[0116] The calculator CAL calibrates the beam hardening by identifying the function u making the change from {circumflex over (P)}{umlaut over (()}n)(x) to P.sup.(n)(x), namely P.sup.(n)(x)=u({circumflex over (P)}.sup.(n)(x)).

[0117] According to an embodiment of the invention, the beam hardening calibration is done by application of a piecewise linear function u connecting P.sup.(n)(x) and {circumflex over (P)}.sup.(n)(x).

[0118] According to an embodiment of the invention, the function u is discretized over a base of given form functions .sub.k(y) (for example one-dimensional finite elements, or spline functions, or polynomial functions, or others):

[00013]$u\left(y\right)=\underset{k=1}{\overset{K_3}{.Math.}}{c}_k{}_k\left(y\right)$

where K.sub.3 is a given natural integer greater than or equal to 1, k is a natural integer ranging from 1 to K.sub.3, y is a mute variable denoting the gray level (intensity) of a pixel of a reference image {circumflex over (P)}.sup.(n). The column vector c containing the parameters c.sub.k quantifies the X-ray beam hardening calibration function for k ranging from 1 to K.sub.3. Based on a given initial estimation c.sub.ini of these parameters, the calculator CAL computes the vector c* describing the observed effect as well as possible.

[0119] According to an embodiment of the invention, the calculator CAL computes (for example by finite differences) the sensitivity fields s.sub.c.sub.k.sup.(n) associated with each degree of freedom c.sub.k:

[00014]${s_{c_k}^{\left(n\right)}=\frac{{\stackrel{}{P}}^{\left(n\right)}}{c_k}.Math.}_c$

based on the vector c=c.sub.ini of the initial given values of the parameters c.sub.k of calibration of the beam hardening of the X-rays in the part 200, with P.sup.(n) the simulated image, obtained by applying the function u (with the parameters contained in the vector c.sub.ini) at each of the pixels of {circumflex over (P)}.sup.(n). The vector c.sub.ini of the initial values is such that u(y)=y. One thus has:

[00015]${\stackrel{}{P}}^{\left(n\right)}\left(x\right)=\underset{k=1}{\overset{K_3}{.Math.}}{c}_k{}_k\left({\stackrel{}{P}}^{\left(n\right)}\left(x\right)\right)$

The sensitivity fields s.sub.c.sub.k.sup.(n) express the effect of the variation of each of the parameters c.sub.k on the computed image.

[0120] According to an embodiment of the invention, during step E3b, the calculator CAL computes (for example by the residuals computer CRES in FIG. 4) the projection residuals .sub.c.sup.(n)=P.sup.(n)P.sup.(n), these residuals being the differences between the observed projections P.sup.(n) and the simulated projections P.sup.(n). Based on a first given approximation c.sub.ini of the column vector c of parameters c.sub.k, the calculator CAL estimates (for example by the estimator ESTV in FIG. 4) the vector c* describing the observed system as well as possible, i.e. minimizing the projection residuals .sub.c.sup.(n). The vector c is computed by the calculator CAL by optimization based on the projection residuals .sub.c.sup.(n), on the value c.sub.ini, and on the sensitivity fields s.sub.c.sub.k.sup.(n). The calculator CAL computes the optimal variation c* of the vector c of parameters and updates the vector c according to the following equations:

[00016]$c^{*}=c+c^{*}$ [0121] where c* is the updated column vector of the parameters c.sub.k of X-ray beam hardening in the part 200, [0122] c* is the column vector of variation of the parameters c.sub.k of calibration of the beam hardening of the X-rays in the part 200 and is computed as the c minimizing the sum of the norms of the squared differences between the projection residuals .sub.c.sup.(n) and the product of c by s.sub.c.sup.(n) for n ranging from 1 to N,

[00017]$c^{*}=\arg \underset{c}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_c^{\left(n\right)}-s_c^{\left(n\right)}c.Math.}^2$ [0123] where c is a column vector, [0124] s.sub.c.sup.(n) is the matrix of sensitivity fields s.sub.c.sub.k.sup.(n).
The number of rows of the matrix s.sub.c.sup.(n) is the number of pixels exploited in the projections P.sup.(n), in the computed images {circumflex over (P)}.sup.(n) and in the residual .sub.c.sup.(n). The number of columns of the matrix s.sub.c.sup.(n) is the number of parameters c.sub.k.
The calculator CAL then computes during step E3b the set of sensitivity fields s.sub.c.sub.k.sup.(n) for n ranging from 1 to N and k ranging from 1 to K.sub.3.
The estimation of the vector c of the parameters c.sub.k of calibration of the beam hardening by the calculator CAL is carried out by successive iterations over c which becomes c.sub.ini, to: [0125] compute at each current iteration, based on the vector c of the preceding iteration taking the place of the vector c.sub.ini mentioned above, the sensitivity fields s.sub.c.sub.k.sup.(n) according to the equation mentioned above, [0126] for each current iteration: compute a new c* and update c according to c*=c+c*, with c* computed as the c minimizing the sum of the norms of the squared differences between the projection residuals .sub.c.sup.(n) and the product of c by s.sub.c.sup.(n) for n ranging from 1 to N, according to

[00018]$c^{*}=\arg \underset{c}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_c^{\left(n\right)}-s_c^{\left(n\right)}c.Math.}^2$ [0127] until c converges on a better estimate (determined for example by the fact that the norm of the difference computed between the c of the current iteration and the c of the preceding iteration becomes less than a given threshold).

[0128] During the fifth step E3c, after the first step E1, the calculator CAL estimates the vector of the parameters .sub.j of Compton scattering of the X-rays in the part 200 by minimizing the sum of the norms of the squared differences, computed by the calculator CAL, between the N acquired projections P.sup.(n) and the N simulated images P.sup.(n), having been computed based on the N computed images {circumflex over (P)}.sup.(n). The Compton scattering parameters .sub.j represent the effect of the Compton scattering on the acquired projections P.sup.(n). As illustrated in FIG. 7, Compton scattering consists in the interaction of a photon of the X-ray with a free electron (or an electron weakly attached to its atom) of the inspected part 200, causing the deviation of the photon from its initial trajectory. Reaching the detector 105, the photon thus scattered and deviated PDEV gives rise to an increase in the measured intensity (known as secondary signal P.sub.sec.sup.(n) in the remainder of the text) in the area DEV located around the point of direct and central impact CENT of the primary ray PCENT, and a reduction of this intensity (called primary signal P.sub.prim.sup.(n) in the remainder of the text) at this point of impact CENT, creating in the projections a blur degrading the metrological analysis. Unlike the known methods using a direct correction of the Compton effect by a deconvolution operation, which is numerically unstable, rendering these reprocessed radiographies unreliable, the step E3c makes provision for reproducing this Compton scattering effect on simulated images, to allow them to be compared with the acquired projections. The step E3c can be implemented by a Compton scattering estimator ESDC, as represented in FIG. 4. During step E3c, the calculator CAL estimates, by successive iterations, the vector based on an initial vector =.sub.ini. What is described below is carried out for each iteration of step E3c.

[0129] According to an embodiment of the invention, each acquired projection P.sup.(n) is modelled by the calculator CAL as the sum of two components: the primary signal P.sub.prim.sup.(n) coming from photons crossing the object without Compton scattering but undergoing absorption by the constituent atoms of the material (photoelectric effect); and the secondary signal P.sub.sec.sup.(n) which corresponds to the contribution of the photons scattered by Compton scattering from the part 200, according to the following equation:

[00019]$P^{\left(n\right)}=P_{prim}^{\left(n\right)}+P_{\sec }^{\left(n\right)}$

The secondary, or scattered, signal, P.sub.sec.sup.(n) is computed by the calculator CAL as the result of the convolution of the primary signal P.sub.prim.sup.(n) with a parametric kernel K:

[00020]$P^{\left(n\right)}=P_{prim}^{\left(n\right)}+P_{prim}^{\left(n\right)}*K$

The calculator CAL computes the parametric kernel K as a weighted sum of two-dimensional Gaussians g.sub. of different widths to one another (different standard deviations to one another):

[00021]$K\left(x\right)=\underset{j=1}{\overset{K_2}{.Math.}}{}_j\left({}_{{}_j}\left(x\right)-\left(x\right)\right)$ [0130] where K.sub.2 is a given natural integer greater than or equal to 1, [0131] j is a natural integer ranging from 1 to K.sub.2, [0132] g.sub. a two-dimensional Gaussian kernel of given standard deviation (following a sequence .sub.j), and (x) is the Dirac function at the pixel x.
In a variant, the parameters .sub.j of the Gaussian kernel g.sub..sub.j may be identified experimentally or via Monte-Carlo simulations.

[0133] According to an embodiment of the invention, by identifying the primary signal P.sub.prim.sup.(n) on the digitally simulated images P.sup.(n), the calculator CAL computes the vector * that best approximates the observed Compton scattering. * is the column vector of the parameters .sub.j of Compton scattering of the X-rays in the part 200, for j ranging from 1 to K.sub.2.

[0134] Based on a vector .sub.ini of the given initial values of the parameters .sub.j of Compton scattering of the X-rays in the part 200, the calculator CAL computes images {tilde over (P)}.sup.(n)=P.sup.(n)+P.sup.(n)*K based on the simulated images P.sup.(n), obtained based on the N computed images {circumflex over (P)}.sup.(n), according to the equation:

[00022]${\stackrel{}{P}}^{\left(n\right)}={\stackrel{}{P}}^{\left(n\right)}+{\stackrel{}{P}}^{\left(n\right)}*K$

where the scattered signal P.sup.(n)*K is simulated and added to the images P.sup.(n) such as to generate the images {tilde over (P)}.sup.(n). The calculator CAL thus computes the set of simulated N images P.sup.(n) during step E3c.

[0135] According to an embodiment of the invention, P.sup.(n)(x)=u({circumflex over (P)}.sup.(n)(x)) is the image obtained by applying the function u to the intensity (gray level) of each of the pixels x of the computed image {circumflex over (P)}.sup.(n). Of course, the simulated images P.sup.(n) may be obtained in another way based on the N computed images {circumflex over (P)}.sup.(n), in the case where step E3c is implemented without step E3b.

The vector .sub.ini of the given initial values of the parameters .sub.j of Compton scattering is such that {tilde over (P)}.sup.(n)=P.sup.(n) for .sub.ini.

[0136] According to an embodiment of the invention, during step E3c, the calculator CAL computes (for example by the residuals calculator CRES in FIG. 4) the projection residuals .sub..sup.(n), these projection residuals .sub..sup.(n) being the differences between the acquired projections P.sup.(n) and the simulated projections {tilde over (P)}.sup.(n) according to the equation:

[00023]$p_{}^{\left(n\right)}=P^{\left(n\right)}-{\stackrel{}{P}}^{\left(n\right)}$

[0137] Based on the first given approximation of the vector of parameters, written .sub.ini, the calculator CAL estimates (for example by the estimator ESTV in FIG. 4) the vector * describing the observed system as well as possible, i.e. minimizing the projection residuals .sub..sup.(n).

[0138] According to an embodiment of the invention, the calculator CAL computes (for example by finite differences) the sensitivity fields s.sub..sub.j.sup.(n), associated with each degree of freedom .sub.j:

[00024]${s_{{}_j}^{\left(n\right)}=\frac{{\stackrel{}{P}}^{\left(n\right)}}{{}_j}.Math.}_{}$

based on the column vector .sub.ini of the given initial values of the parameters .sub.j of Compton scattering of the X-rays in the part 200.
The parameters .sub.j of Compton scattering of the X-rays in the part 200 are computed by the calculator CAL by optimization based on the projection residuals .sub..sub.ini.sup.(n), on the value .sub.ini, and on the sensitivity fields s.sub..sub.j.sup.(n)).

[0139] The calculator CAL computes the optimal variation * of the vector of parameters and updates the vector according to the following equation:

[00025]${}^{*}=+{}^{*}$

where * is the column vector of variation at least of the parameters .sub.j of Compton scattering of the X-rays in the part 200 and is computed as the minimizing the sum of the norms of the squared differences between the projection residuals .sub..sup.(n) and the product of by s.sub..sup.(n) for n ranging from 1 to N,

[00026]${}^{*}=\arg \underset{}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_{}^{\left(n\right)}-s_{}^{\left(n\right)}.Math.}^2$

where is a column vector,
s.sub..sup.(n) is the matrix at least of the sensitivity fields s.sub..sub.j.sup.(n).
The number of rows of the matrix s.sub..sup.(n) is the number of pixels exploited in the projections P.sup.(n), in the computed images {circumflex over (P)}.sup.(n) and in the residual .sub..sup.(n). The number of columns of the matrix s.sub..sup.(n) is the number of parameters .sub.j.
The calculator CAL then computes during step E3c the set of sensitivity fields s.sub..sup.(n) for n ranging from 1 to N and j ranging from 1 to K.sub.1.
The estimation of the vector of the parameters .sub.j of Compton scattering of the X-rays by the calculator CAL is done by successive iterations over a which becomes .sub.ini, to: [0140] compute at each current iteration, based on the vector of the preceding iteration taking the place of the vector .sub.ini mentioned above the sensitivity fields s.sub..sub.j.sup.(n) according to the equation mentioned above, [0141] for each current equation: compute a new * and update a according to *=+*, with * computed as the minimizing the sum of the norms of the squared differences between the projection residuals .sub..sup.(n) and the product of by s.sub..sup.(n) for n ranging from 1 to N, according to

[00027]${}^{*}=\arg \underset{}{\min }\underset{\left(n\right)}{.Math.}{.Math.{}_{}^{\left(n\right)}-s_{}^{\left(n\right)}.Math.}^2$ [0142] until converges on a better estimate (determined for example by the fact that the norm of the computed difference between the of the current iteration and the of the preceding iteration becomes less than a given threshold).

[0143] According to an embodiment of the invention, N is less than or equal to 1000 or to 100 or to 50 or to 10. This is an expression of the small number of acquired projections P.sup.(n) in spite of which the invention is capable of working. This number is less than the numbers of acquired projections required in the prior art for the tomographic reconstruction of the part 200.

[0144] During the seventh step E4, after the iterations of step E3a and of step E3b and/or E3c, the calculator CAL processes the N projections P.sup.(n) obtained in step E1 and/or the N computed images {circumflex over (P)}.sup.(n), having been obtained after the iterations of step E2. The calculator CAL carries out a first processing and/or a second processing.

[0145] The first processing comprises a correction of the beam hardening over the N projections P.sup.(n) based on the vector c having been estimated or a generation of the beam hardening over the N computed images {circumflex over (P)}.sup.(n) based on the vector c having been estimated.

[0146] The second processing comprises a correction of the Compton scattering over the N projections P.sup.(n) based on the vector having been estimated or a generation of the Compton scattering over the N computed images {circumflex over (P)}.sup.(n) based on the vector having been estimated.

[0147] The calculator thus obtains N acquired projections P.sub.a.sup.(n), which are equal to or corrected based on the N acquired projections P.sup.(n) of the part 200, and N simulated images P.sub.s.sup.(n), which are equal to or generated based on the N computed images {circumflex over (P)}.sup.(n) of the part 200, these acquired projections P.sub.a.sup.(n) being directly comparable to the N simulated images P.sub.s.sup.(n), to then be able to carry out the analysis of the part 200 during step E5.

[0148] For example, during step E2 the calculator CAL first generates the N computed images {circumflex over (P)}.sup.(n) by applying to them the geometric registration for the N angles ANG.sup.(n) of view, and does so by using the projection parameters p.sub.i to simulate the N computed images {circumflex over (P)}.sup.(n). It is these computed images {circumflex over (P)}.sup.(n), thus modified (registered), which are used as computed images {circumflex over (P)}.sup.(n) in the processing below. This modification by registration using the projection parameters p.sub.i is done beforehand on the N computed images {circumflex over (P)}.sup.(n) in the first and second embodiments of step E4, described below.

[0149] According to a first embodiment of step E4, the calculator CAL corrects the acquired projections P.sup.(n) by applying to them the inverse of the function u depending on the parameters c.sub.k of calibration of the beam hardening of the X-rays at each of the pixels of P.sup.(n) (to thus make a correction of the beam hardening), to obtain the N corrected acquired projections P.sub.a.sup.(n) according to the following equation:

[00028]$P_a^{\left(n\right)}=u^{-1}\left(P^{\left(n\right)}\right)$

The calculator CAL computes the function u depending on the parameters c.sub.k of calibration of the beam hardening of the X-rays according to the following equation:

[00029]$u\left(y\right)=\underset{k=1}{\overset{K_3}{.Math.}}{c}_k^{*}{}_k\left(y\right)$

where c*k denotes the parameters c.sub.k of calibration of the beam hardening of X-rays, contained in the vector c* and having been computed.
The calculator CAL processes the N computed images {circumflex over (P)}.sup.(n) by convolving this with the kernel +K having been computed as a function of the Compton scattering parameters .sub.j (reproduction of Compton scattering over the N computed images {circumflex over (P)}.sup.(n)), to obtain the N simulated images P.sub.s.sup.(n) according to the following equations:

[00030]$P_s^{\left(n\right)}={\stackrel{}{P}}^{\left(n\right)}*\left(+K\right)$$K=\underset{j=1}{\overset{K_2}{.Math.}}{}_i^{*}\left(g_{{}_j}-\right)$

where *.sub.j denotes the Compton scattering parameters .sub.j, contained in the vector * and having been computed.

[0150] According to a second embodiment of step E4, the calculator CAL computes the acquired projections P.sub.a.sup.(n) according to the following equation:

P.sub.a.sup.(n)=P.sup.(n)

The calculator processes the N computed images {circumflex over (P)}.sup.(n), by applying to them the function u (computed using the computation described above) depending on the parameters c.sub.k of calibration of the beam hardening of the X-rays (to thus reproduce the beam hardening), then by convolving them with the kernel +K having been computed (computed using the computation described above) as a function of the parameters .sub.j of Compton scattering (reproduction of the Compton scattering on the N computed images {circumflex over (P)}.sup.(n)), to obtain the N simulated images P.sub.s.sup.(n) according to the following equations:

[00031]$P_s^{\left(n\right)}=\left(u\left({\stackrel{}{P}}^{\left(n\right)}\right)\right)*\left(+K\right)$

[0151] During the eighth step E5 of analysis of the actual part 200, after step E4, the calculator CAL identifies the defects of the part 200 by comparison of the N processed projections P.sup.(n) with the N processed computed images {circumflex over (P)}.sup.(n). The calculator CAL identifies the defects of the part 200 by comparison of the N acquired projections P.sub.a.sup.(n) with the N simulated images P.sub.s.sup.(n) according to one of the embodiments described above. The calculator CAL carries out an identification of the defects of the part by comparison of the acquired P.sup.(n) and simulated {circumflex over (P)}.sup.(n) images having taken into account the correction or processing either on some of these or on others of these.

The calculator CAL computes N projection residuals .sup.(n), equal to the difference between the acquired projections P.sub.a.sup.(n) and the simulated images P.sub.s.sup.(n) according to the equation:

[00032]${}^{\left(n\right)}=P_a^{\left(n\right)}-P_s^{\left(n\right)}$

The calculator CAL records in the memory MEM the N acquired projections P.sub.a.sup.(n) and/or the N simulated images P.sub.s.sup.(n) and/or the N projection residuals .sup.(n). The calculator CAL analyzes these N projection residuals .sup.(n) in order to identify defects inherent to the inspected part 200. The calculator CAL can supply, via the output interface INT2, these N projection residuals .sup.(n) and/or the defects identified based on these N projection residuals .sup.(n). The calculator CAL can supply, via the output interface INT2, a certificate of validity or invalidity of the part, determined by the calculator CAL based on the N projection residuals .sup.(n). The calculator CAL or the user can, on the basis of the N acquired projections P.sub.a.sup.(n) and/or of the N simulated images P.sub.s.sup.(n) and/or of the N projection residuals .sup.(n), evaluate the part 200 as non-valid since it has too many defects, or validate the part 200 as being valid since it does not have too many defects.

[0152] According to an embodiment of the invention, the vector p groups one after the other all the K.sub.1 parameters p.sub.i of the projection geometry, the K.sub.3 parameters c.sub.k of beam hardening of the radiation in the part 200 and the K.sub.2 parameters .sub.j of Compton scattering of the radiation, and thus has a dimension of K.sub.1+K.sub.2+K.sub.3. Correspondingly, the vector s.sub.p.sup.(n) groups one after the other all the K.sub.1 sensitivity fields s.sub.pi.sup.(n), the K.sub.3 sensitivity fields s.sub.c.sub.k.sup.(n) and the K.sub.2 sensitivity fields s.sub..sub.j.sup.(n) and thus has a dimension of K.sub.1+K.sub.2+K.sub.3. Correspondingly, the vector p groups one after the other all the variations of the K.sub.1 parameters p.sub.i of the projection geometry, the K.sub.3 parameters c.sub.k of beam hardening of the radiation in the part 200 and the K.sub.2 parameters .sub.j of Compton scattering of the radiation, and thus has a dimension of K.sub.1+K.sub.2+K.sub.3.

[0153] FIG. 10 shows an example of the difference between, on the one hand, the image of corrected acquired intensity I.sub.a.sup.(n)=I.sub.0 exp(P.sub.a.sup.(n)) having been obtained according to the invention based on the acquired projection P.sup.(n), where I.sub.0 is a given setting based on the acquired projections P.sup.(n), and, on the other hand, the corrected simulated image of intensity I.sub.s.sup.(n)=exp(P.sub.s.sup.(n)) having been obtained based on the simulated projection P.sup.(n) for a blade airfoil of a turbomachine turbine. FIG. 11 shows an example of the difference between, on the one hand, the corrected acquired image of intensity I.sub.a.sup.(n)=I.sub.0 exp(P.sub.a.sup.(n)) having been obtained according to the invention based on the acquired projection P.sup.(n), where I.sub.0 is a given setting based on the N acquired projections P(n), and, on the other hand, the simulated image of intensity I.sub.s.sup.(n)=exp(P.sub.s.sup.(n)) having been obtained based on the simulated projection P.sub.s.sup.(n) for a blade root of a turbomachine turbine. FIGS. 10 and 11 show that the analysis of the part 200 is simplified by the invention taking into account the artefacts. FIGS. 8 and 9 show I.sub.sI.sub.a, with no correction on I.sub.a, and when I.sub.s is generated with p.sub.ini, c.sub.ini and .sub.ini. FIGS. 10 and 11 show I.sub.sI.sub.a, without correction on I.sub.a, and when I.sub.s is generated with p*, c* and *.

[0154] In the case of the combination of steps E3a, E3b and E3c, the fact of taking into account the knowledge of the acquisition system (cabin and part, step E3a) and coupling it with a suitable image processing algorithmic method E3a, E3b, E3c allows significant gains in the quality of the information contained in the images produced by the system and in those reproduced by simulation, and therefore of their difference. This allows the estimation of the artefacts (1) of beam hardening and (2) of Compton scattering, in the context of the analysis of aeronautical parts based on a limited number of multi-view projections. Step E3a allows the estimation of geometrical parameters making it possible to connect the model MODP of the ideal part with the acquired projections of the part to be tested. This step E3a makes it possible to find the parameters with the aim, in particular, of using them to digitally simulate the projections and reproduce artefacts in the simulated images. The invention also allows in step E4, either to correct the artefacts due to beam hardening (E3b) in the acquired projections, or to reproduce them in the simulated images; and to reproduce the artefacts due to Compton scattering (E3c) in the simulated images or to correct them in the acquired projections. The image acquisition system and the algorithmic sequence improve the reliability of the validation and of the evaluation of indicators of material health and of indicators of the three-dimensional geometry of the part 200 based on a limited number N of radiographic views by X-ray.

[0155] According to an embodiment of the invention, the method, the device and the program according to the invention can be implemented over one or more regions of interest, selected by a selection module of the calculator CAL, in the projection P.sup.(n).

[0156] Of course, the embodiments, features, possibilities and examples described above may be combined with one another or be selected independently of one another.