Unsupervised calibration method for a detector

Abstract

A method for determining parameters governing the charge carrier transport properties in an ionizing radiation detector. The method involves irradiating the detector and simulating the propagation of charge carrier clouds to detector electrodes, so as to compare the simulated signal with the measured signal. This allows the definition of an error, which is backpropagated to the interaction site in the detector. During backpropagation, the charge carrier transport parameters are updated to minimize the error. The backpropagation operation is iterative and continues until a stopping criterion is met.

Description

Title of the invention: Unsupervised calibration method for a detector technical field

[0001] The technical field of the invention is the calibration of ionizing radiation detectors, for example X-ray or gamma radiation, and in particular for spectrometric imaging. EARLIER ART

[0002] Gamma cameras are devices that allow the formation of an image to establish a map of irradiating sources in a given environment, and in particular in nuclear facilities, for medical diagnostic applications, or for non-destructive testing applications, for example baggage screening.

[0003] Some gamma cameras consist of a two-dimensional pixel array connected to a detector material. The detector material is generally a semiconductor material, for example CdTe or CdZnTe. Under the effect of an interaction of ionizing radiation in the detector material, one or more pixels generate an electrical pulse, the amplitude of which is correlated with the energy released by the radiation during the interaction. Each pixel is connected to an electronic pulse processing circuit.

[0004] Each pixel is formed by an electrode, which usually acts as an anode. When incident radiation interacts with the detector material, electrons are released into the detector material. These electrons are collected by an anode. The anode generates a pulse whose amplitude depends on the number of electrons collected by the anode, this number generally being proportional to the energy lost by the ionizing radiation in the detector material.

[0005] Each detector extends over a few hundred millimeters on each side. For compactness reasons, the pixel matrix generally has one hundred, or a few hundred, pixels per row and per column.

[0006] CdZnTe, or CZT, is often used as a detection material in gamma cameras. This material combines many advantages for use in high-energy X-ray or gamma-ray imaging: - a high atomic number Z and density for a significant absorption capacity of incident photons. - a band gap width, wide enough to be used at room temperature, while being limited to keep the electron-hole pair creation energy Epaire low enough. - interesting load transport properties (lifetime r and carrier mobility p) in order to have the highest possible mean free path pr.

[0007] Much work has been carried out over the years to improve the composition, crystal growth, geometry and processing electronics around these detectors in order to optimize their performance.

[0008] However, the use of CZT presents a major difficulty: the presence of various types of defects in the crystal lattice, which modify the material's properties and thus impact the detector's output signals. These defects can be of different natures and sizes: Cd vacancies at the atomic scale, or sub-grain boundaries that can measure several centimeters in length. These defects, which appear during the growth of CZT crystals, hinder the development of larger-volume detectors and generally degrade their performance, producing a non-uniform response that varies from one pixel to another.

[0009] These defects create intermediate energy states within the band gap. When a carrier finds itself in one of these new energy states, it can either recombine or return to its initial state. The trapped carriers create space charges, which modify the electric field inside the detector. The trapping is non-uniform and affects the uniformity of the electric field.

[0010] Indeed, these defects cause an error on the signals induced by the charge carriers and therefore on the signals collected on the electrodes which disturb the current measured at the output which is used as data allowing the source to be located with the most precise resolutions possible.

[0011] To date, detector responses can be learned using supervised algorithms, such as maximum likelihood or neural networks. However, this requires precise data for each detector individually. Indeed, defects affect each detector differently. Thus, implementing a supervised algorithm requires reliable data regarding the position of interactions within the detector and the energy deposited during each interaction. This implies, for example, scanning a detector with a narrow beam of photons, preferably monoenergetic. Such a scan is hardly feasible for systematically characterizing detectors during their fabrication.

[0012] The inventors propose a solution for performing unsupervised learning, so as to learn the response function of a detector, from The detector can be irradiated using a low-constraint irradiation technique. By low-constraint irradiation, we mean irradiation that is not necessarily collimated. The entire detector can be irradiated in order to learn the response of the combined detector material and electrodes. This allows for accurate simulations of signals detected by the detector, taking into account the inherent limitations of using a CdZnTe type detection material. Description of the invention

[0013] A first object of the invention is a method for estimating a signal measured in a pixel of a detector, the detector comprising several electrodes, forming pixels, connected to a detector material, the detector material being a semiconductor, each electrode being configured to collect charge carriers moving through the detector material under the effect of an electric field, following an interaction of an X-ray or gamma photon in the detector material, the method comprising: - irradiation of the detector with X or gamma photons, so as to generate interactions in the detector material, each interaction forming a detection signal measured by at least one electrode; - for different detected interactions, association of a state vector, comprising at least one position, one charge, and each detection signal, the state vector passing from an initial state when the interaction occurs, to a final state, when the electrons, generated by the interaction, are collected by at least one electrode;

[0014] the process being characterized in that it comprises, for each detected interaction: - (i) initialization of the initial state vector; - (ii) from the initial state vector, or from an initial state resulting from a previous iteration, implementation of a charge carrier propagation model in the semiconductor material, to estimate the final state vector, the propagation model being based on a parameter vector, parameterizing at least one transport property of charge carriers through the detector material, towards the electrodes; - (iii) calculation of an error, representative of a difference between the detection signal estimated in the final state vector and the measured detection signal; - (iv) backpropagation of an error gradient from the final state to the initial state, the error gradient being calculated with respect to at least one term of the state vector, and with respect to several parameters of the parameter vector; - (v) update of the state vector, at the initial time, and of the vector parameter, and repetition of steps (ii) to (v) until a stopping criterion for the iterations is reached;

[0015] the process being such that steps (i) to (v) are implemented by a processing unit connected to the detector.

[0016] According to one possibility, the detector material is discretized into voxels, and the parameter vector includes, for each voxel, a charge carrier transport property in the detector.

[0017] According to one possibility, the load-carrier transport property comprises: - a value of the electric field; - and / or a gradient of the electric field along at least one direction; - and / or a value of a divergence of the electric field; - and / or a probability of trapping in the voxel.

[0018] According to one possibility, - steps (i) to (v) are implemented for different interactions, so as to update the parameter vector at each implementation of steps (i) to (v); - the parameter vector is then updated by combining the parameter vectors updated during each interaction.

[0019] According to one possibility, - For each interaction, the time between the initial state and the final state is discretized into time steps. - step (ii) involves a numerical integration of an evolution function (fg), the evolution function reflecting a time evolution of the position and charge of the charge carriers, as well as an evolution of each detection signal, the numerical integration being carried out successively between each time step, between the initial state and the final state.

[0020] Step (iv) may include a numerical integration of an adjoint propagation equation, reflecting a time evolution of the error gradient, the numerical integration being carried out successively between each time step, between the final state and the initial state.

[0021] A second object of the invention is a method for learning a supervised artificial intelligence algorithm, intended to simulate a response from a detector, the detector comprising several electrodes, forming pixels, connected to a semiconductor material, each pixel being configured to collect charge carriers moving through the semiconductor material under the effect of an electric field, following an interaction of an X-ray or gamma photon in the semiconductor material, the method comprising the following steps: - a) definition of a position of an interaction in the detector and of an energy released during said interaction; - b) estimation of a detection signal measured by at least one electrode of the detector, by implementing the steps of a method according to any one of the preceding claims; - c) repeating steps a) and b) so as to form a database linking, for each defined interaction, the signal measured in at least one pixel; - d) use of the database to perform supervised learning of the artificial intelligence algorithm.

[0022] The artificial intelligence algorithm may be of the multilayer perceptron type.

[0023] A third object of the invention is a detector, comprising several electrodes, forming pixels, connected to a semiconductor material, each pixel being configured to collect charge carriers moving, through the semiconductor material, under the effect of an electric field, following an interaction of an X or gamma photon in the semiconductor material, the detector being connected to a processing unit, configured to implement steps (i) to (v) of a process according to the first object of the invention.

[0024] A fourth object of the invention is a detector, comprising several electrodes, forming pixels, connected to a semiconductor material, each pixel being configured to collect charge carriers moving, through the semiconductor material, under the effect of an electric field, following an interaction of an X or gamma photon in the semiconductor material, the detector being connected to a processing unit, configured to estimate an energy released by the interaction and / or a position of the interaction, the processing unit implementing a supervised artificial intelligence algorithm whose learning is carried out according to the second object of the invention.

[0025] The invention will be better understood upon reading the description of the exemplary embodiments presented later in this description, in connection with the figures listed below. FIGURES

[0026] Fig. 1 schematically represents a detector.

[0027] Fig. 2A shows the main steps of a method for determining a detector response.

[0028] Fig. 2B illustrates the steps discussed in relation to Fig. 2A.

[0029] Figures 2C and 2D show consideration of five or nine anodes in determining a collected charge.

[0030] Fig. 3 details certain operations described in relation to Figure 2.

[0031] Figures 4A to 4H show the performance of estimating the depth of interaction by implementing the invention, during successive iterations.

[0032] Fig. 5A shows a comparison between an estimate of interaction positions by implementing the invention, and the measured interaction positions.

[0033] Fig. 5B shows a comparison between an estimate of interaction depths by implementing the invention, and the measured interaction depths.

[0034] Figures 5C and 5D correspond respectively to figures 5A and 5B, taking into account noise.

[0035] Fig. 6A represents a measured map of a quantity "CIE", which corresponds to a ratio between the collected charge and the deposited charge.

[0036] Fig. 6B shows an estimated mapping of the CIE by implementing the process. PRESENTATION OF SPECIFIC IMPLEMENTATION METHODS

[0037] Fig. 1 schematically illustrates the structure of a detector 1. The detector comprises a detector material 2, polarized between a cathode 3 and various anodes 4. The anodes are connected to a readout circuit 5, enabling the analysis of each detection signal formed at each anode.

[0038] Through interaction within the detector material, charge carriers are created: electrons migrate towards the anodes, while holes migrate towards the cathode. In a semiconductor such as CZT, the differences in transport properties between electrons and holes make the use of the electron signal much more advantageous from a detector performance standpoint. Hereafter, only the transport of electrons through the detector is considered, although the invention can also be applied to other types of charge carriers, such as holes. Each interaction results in the formation of an electron cloud, which migrates towards one or more anodes.

[0039] Each anode forms a pixel, allowing the measurement of a detection signal following each interaction. During an interaction, one or more anodes detect a pulse. From each pulse, a detection signal Sa is formed, which allows the interaction to be localized within the detector material. The detection signal may include one or more characteristics of the detected pulse, as described in US9322937. The subscript a designates each anode that has received a usable detection signal. The number of anodes that have received a usable detection signal can vary between 1 and 9, or even more, as described below. For each interaction, an initial instant t0, corresponding to the occurrence of the interaction, and an instant tb, corresponding to the collection of electrons by one or more anodes, are distinguished.

[0040] The collected charge Qb reaching the anodes allows us to estimate the charge Qo deposited in the detector material during the interaction. Each interaction can be assigned a state X of the charge carriers generated by the interaction, which varies between an initial state Xo, at the time of the interaction, and a final state Xb corresponding to the collection of charges by the anodes. The initial state Xo is characterized by the quantities (Qo, x0, y0, z0, Saq), corresponding to the charge released by the photon in the detector, the position of the interaction, and the signal induced at the electrodes during the interaction. The final state Xi is characterized by the quantities (Qi, Zi, xB, yi, Saj), corresponding to the charge collected at the anodes and the position of a centroid of the cloud of charge carriers reaching the electrodes, as well as the signal induced at each electrode, the latter being measurable.

[0041] Figure 1 shows field lines along which the charge carriers generated during the interaction propagate between the initial state Xo, an intermediate state X(t), and the final state Xb. For each interaction, one or more signals Sa are detected. Each charge carrier propagates through the detector according to transport properties p(r) that characterize the detector. In this example, the detector is made of CdZnTe: the charge carriers are therefore electrons. Alternatively, the charge carriers could be holes.

[0042] Each voxel is assigned a coordinate (X / y' z). Each voxel can result from a discretization of the volume of the detector material at a spatial step of 100 pm in each direction. Each voxel is associated with at least one transport property p(r), for example, a value of the electric field y(f), and / or a trapping probability f(r), or a characteristic representing a spatial variation of the electric field, for example, gradient or divergence. For example, each voxel r is assigned a property p(r), corresponding to the pair (u(r),T(r)).

[0043] The spatial distribution of the transport properties p(r), in each voxel, is parameterized by parameters 0n, forming a parameter vector 0. According to a first approach, each voxel r is associated with parameters = p(r) =(u(r), T(r))> defined for each voxel. This implies defining a parameter value for each voxel, which is constraining from a memory size perspective: there are as many parameters as voxels. n is an integer between 1 and N, where N corresponds to the number of parameters determined.

[0044] According to another approach, the spatial distribution of the parameters p(r) is parameterized by 3n parameters applied to a spatial function F^r) or to a combination of spatial functions. For example, p(p)=^ 0 F

[0045] Each spatial function Fn(r) can be, for example, a sinusoidal or polynomial function. The use of spatial functions makes it possible to reduce the number of parameters 0U of the parameter vector 0.

[0046] To summarize, each voxel is assigned at least one transport property of a charge carrier p(r)- The parameters 0U allow the spatial distribution of transport properties in the voxels of the detector to be defined.

[0047] The parameter vector of a detector is unknown, particularly due to the presence of randomly distributed defects in the detector material, which influence charge transport between the interactions and the electrodes that collect the charge carriers, i.e., the anodes in the case of electrons. It constitutes a signature, conditioning the detector's response. For each detector, the values ​​of each parameter are considered stable over time.

[0048] The parameter vector 0 allows us to establish an evolution function f allowing us to follow the state of the cloud of charge carriers generated by the interaction, from the initial state to the final state, and this for each interaction.

[0049] The reading circuit 5 is connected to a processing unit 6, configured to determine the parameter set, from signals Sa respectively measured during each interaction, by implementing the steps described in connection with Figures 2A and 3. The processing unit 6 includes in particular one or more microprocessors, programmed to implement the steps described below.

[0050] Figures 2A summarize the main steps of a method for learning the parameter set governing the transport properties of the detector. The detector is exposed to an irradiation source whose emission energy or energies are known. Under the effect of the irradiation, interactions occur in the detector material. During each interaction, a detection signal Sa is measured by anodes. The subscript a denotes each anode detecting a usable signal. The signal Sa is a vector formed by each detection signal generated by a single anode or several adjacent anodes, for example: - four anodes: these are the 4 anodes closest to the position of the electron cloud when the latter reaches the anodes; - five anodes: four electrodes around anode C which has collected the highest amount of charge, the latter being the collecting anode; - nine anodes: 8 anodes around the collecting anode.

[0051] During irradiation, the detection signals Sa measured during the irradiation are collected. Each measured signal Sa corresponds to a detected interaction. Iterative steps 100 to 150 are implemented for each detected interaction. Each interaction extends between the initial time t0 (occurrence of the interaction) and the final time ti (collection of charges by the anodes) previously defined.

[0052] Step 100: Initialization: an initialized value of the initial state is estimated for each interaction. The position of the interaction can be defined randomly. The deposited energy Qo can be determined randomly, or as a function of a maximum energy defined according to the nature of the irradiation source.

[0053] Step 110: During this step, the state resulting from step 100 or a previous iteration is propagated. This step is implemented using an evolution function f, described below, and parameterized by the parameter vector 0. The evolution function is described by a differential equation of the type:

[0054] (O

[0055] Thus, 100561 c+cymmdt 121

[0057] The integral of expression (2) can be calculated numerically, step by step, for example according to a Runge-Kutta method.

[0058] Step 120: estimation of the final state Xi and calculation of an error, error calculation: during this step, an error function is determined between the estimated final state XiCt and the measured final state. In the measured final state, an estimate of each signal measured at the electrodes is available, as well as a measurement of this signal.

[0059] _ || ç ç il 2(3) : error norm. £ — he '“'u II

[0060] and [006i] AS„=â„-S„(3'): this is the deviation from the observation. ex ex a

[0062] Other expressions of the error standard may be considered.

[0063] Step 130: Backpropagation. Starting from the error AXp, a deep learning algorithm of the N-ODE type, as previously mentioned, is implemented to obtain an error AX0 with respect to the initial state considered in step 100, and to update this initial state. This phase, called backpropagation, involves calculating error gradients with respect to each component of the parameter vector, as well as with respect to the components of the state vector (spatial components x, y, and z, charges Q, signal or signals Sa). Backpropagation is performed from the final state (time ti) to the initial state (time t0), knowing that these times are known.

[0064] Step 140: Calculation of difference.

[0065] During this step, for each interaction, a deviation AX0 is calculated to be applied to the initial state as a correction term, so as to progressively minimize the error Σ. A deviation is also determined for each term of the vector of parameters. Steps 100 to 140 are then repeated until a stopping criterion for iterations is reached, for example a predetermined number of iterations or a threshold value of the error function below which the parameter vector is considered to describe the behavior of the detector sufficiently well.

[0066] Step 150: Taking into account different interactions.

[0067] During this step, the parameters 9a determined for different interactions are taken into account to estimate a mean parameter vector 0n, or another statistical function, so as to allow a correction of the parameter vector by taking a large number of interactions.

[0068] Fig. 2B illustrates the main steps of the process.

[0069] Steps 110 and 130, which are the basis of the process, are now described in detail. Step 110 is a propagation phase, aimed at estimating the state X1 from a state estimated by successive integration of the evolution function f according to predetermined time steps.

[0070] In the final state: y _ / A e

[0071] In the final state: - - - A _ A \ (5)

[0072] Among the modeling assumptions, it is considered that: The spatial distribution of electrons within the detector volume follows a spherical Gaussian distribution, the broadening of which is due to scattering and Coulomb repulsion. The temporal evolution of the variance of this distribution is known, as described in the publication by G. Montemont, S. Lux, O. Monnet, S. Stanchina, and L. Verger, "Studying Spatial Resolution of CZT Detectors Using Sub-Pixel Positioning for SPECT," IEEE Transactions on Nuclear Science, vol. 61, no. 5, pp. 2559–2566, Oct. 2014, and more specifically in relation to Figure 2 of that publication. Each primary photon generates a number of secondary electrons depending on its energy, which can themselves generate secondary photons, and so on. The interaction of a photon can then be represented schematically as a series of highly localized deposits separated by a large distance. Each of these deposition positions generates an electron cloud that expands under the influence of thermal scattering and Coulomb repulsion.

[0073] The evolution function is such that:

[0074] (6)

[0075] Where: - p is the position of the centroid of the carrier cloud. It depends on time p — r(t) ■ the different terms r(t) represent a trajectory of the charge carrier cloud. - corresponds to the speed of electrons in the detector material and is the electric field; y and are voxelized data: y = √(r) ' E = - T is the lifetime of electrons in the detector material: it is a quantity spatially distributed in the detector therefore q- = Ej'); - Q is the total charge of the electron cloud: it is the charge integral of the cloud of charge carriers at a time t, Q = Q(t)- This charge evolves according to the trapping of the carriers. - Sa is the signal from each anode of rank a; - 1 = spatial coordinate index: 1 = 1, 2 and 3 correspond respectively at spatial coordinates x> y and z "> - w is the detector weighting field, which corresponds to the gradient of the weighting potential. w is discretized at each anode, with each anode corresponding to a vector wa whose values ​​wa(r) are defined for each voxel of the detector. Figure 1 shows the weighting field of an anode. Each dashed curve corresponds to identical values ​​of the weighting field.

[0076] The weighting potential is used to calculate the signal collected by an anode under the influence of a moving charge in the detector material. The weighting potential has no physical unit and represents the influence of an anode on signal induction as a function of the distance to the moving charged particle. The weighting potential is significant near each anode: near each anode, electrons are subjected to the weighting potential, through which a detection signal is formed due to the movement of the electrons. The weighting potential is considered invariant from one anode to another: it depends on the size of each anode, the space between two adjacent anodes, and the thickness and permittivity of the detector material.

[0077] The spatial gradient of the weighting potential forms a weighting field in the vicinity of an anode. The value of the current collected at the anode depends on the scalar product of the weighting field and the electric field extending through the detector material.

[0078] Thus, at a time t, when the electron cloud extends along a coordinate r(t) = (x₁, y, z) in the detector, it induces, on each anode a, a signal Sa(t) such that: = Q(t) - ε). ε₀ dt

[0079] Equation (6) corresponds to the time transport equation, allowing the simulation of the electron cloud drift in the detector, so as to estimate from . f is the time evolution function (or propagation function). The electron cloud drift is obtained by predetermined time increments, for example 1 ns or a few ns, knowing that, given the usual dimensions of a detector, the propagation time between interaction and detection is a few tens of ps

[0080] The propagation step consists of a step-by-step numerical integration of the type:

[0081] x(T + dt)= X(T) + f™1 f ^t)dt (9)

[0082] It is thus possible to estimate the evolution of the state of the electron cloud between the initial and final times t0, given that these times are known. The integration can, for example, implement a Runge-Kutta integration method.

[0083] This allows us to obtain an estimate of f - an estimate of the position - an estimate of the load Q. - an estimate of the detected detection signal knowing that the signal of The measured detection value (Sa) is known. Backpropagation

[0084] Backpropagation is the subject of substeps 131 to 133 of step 130.

[0085] Step 130 corresponds to a backpropagation of the term x* = —, for each interaction, between the final state (X = Xj), resulting from the measurements of Sa, and the initial state (A = Xq). This involves performing backpropagation, step by step, of the errors r* = $h Q^=Set g* = relative to the position of the electron cloud and the charge respectively, between Ci, r = LQ and Q = Q^, Q = Q® respectively. The backpropagation of charge Q1 is carried out according to expression (20)

[0086] The backpropagation of the error r* is carried out according to expressions (30) to (32).

[0087] The backpropagation of the error q* with respect to the parameter vector is carried out according to expression (40).

[0088] We pose ^(T dt)= X{T) - f ^t)dt 1 ~Cl L

[0089]

[0090] with e = || AX||2 Generally speaking, if = ' dQ [Q* = di dt' de _ dr de । ad d dt % * __ ™ 1 - 1 T c 0 0 U.W1 0 d»,- dUy dX dX dX Q dx 0 dUx dlly dy dy dUz dy 0fu.IV1) 0~7ÿ~ 0 QdT duz duy ~dZ~ dZ dUy dUy dUz ~dz dllz afïiw1) Q~dT~ . QdT 1 dux dlly 90^ d0;y dUz ôë~ V dy ^ 4 i l of 0Q of dX of dy of dz of I (10)

[0091] Expression (10) is a propagation function called adjoint of the function of evolution. Backpropagation consists of propagating errors back, respectively explained according to the following expressions:

[0092] Evolution of the charge of the electron cloud: (substep 131)

[0093] _d ( of \ 1 of , y t-'~a) (20) dt\dQ ) ~ r 3q TZ,adSa\uw > Position of the electron cloud (sub-step 132)

[0094] j (- 1 f. 2£ i c3e). z-pç ~ dt \ dX ) ~ \ dX ' dX ' dX ' dy ' dX ' dZ / ' dX ' 3Sa (30)

[0095] j { \ _ ( dUx de । dUy de । de ​​) । ^-wt^ of dt \ dy / ~ l ay * dx + dy ' dy + dy ' dz ) + dy ' asa (31) [°096] d ( of \ / of, of auy ae . duz of \ d(T^) " dt V dZ J [ dZ ' dx ' dZ ' dy ' dZ ' dZ / ' dZ •-^-(32) Parameter vector (substep 133)

[0097] dt J. 3¾ ) ~ / 3¾ of । of , d»z of , riX 3e: of the —?Ta \ , of Qd? \ ' dx + den ' dy + deB ' dz + aen ' w / +ôq T^en (40)

[0098] Substeps 131 to 133 are implemented by successive integration, according to time steps dt as described in connection with the propagation step, according to the expressions: 100991 %(T-dt) = %(T) -dt (21)

[0100] — ( T-dt} — ~ ( T~} - ( — • — + — • — 4- — • — ) + OT —— dt dx \ 1 ~ dx \ 1 JJ T-rft [ dx dx dx dy dx dz ) dSa u L (33)

[0101] of (T J_ âe (T} cT dy^1 UL)-Sy{l) JT.dt\Sy dx^dy Sy^dy dz)^^Zja dy dSaUL (34)

[0102] ae, „,. _âÊrmi / 3¾ de, duy de, 3¾ 3e ], z-v'r de j j. dz v 1 ~alf ~ dZ 1 1 / y.dt\ dz ' dX dZ ' dy + dZ ' dZ J + dZ dSaal (35)

[0103] — fr (. il£ -U 22 . de . 2k . de , nx? de du .-» / . ae Qdr ] -Ul) d0n y1 l~ J T.dt\d6a dX^ddB dy^d0a dz T VZaaS“ d0B W ^dQT^0,JaL (41)

[0104] We know △ Sa = g =J£.. eï tx ôSq.

[0105] Expressions (21), (33), (34), (35), and (41) allow backpropagation between times ti and t0, which are known because they are defined during propagation. These expressions can be combined with:

[0106] of _ x' of dSa = yç. ç dSa (22) 0Q ~ £adSa dQ a dQ

[0107] 22 —V de d$a =V a ç (36) dx ~ ZjadSa dx dx

[0108] 22 _ V of dSa ~ dSa ^1) dy “ ^aasn ây Za n dy

[0109] de _ \ de dSa = _ dSa (3g) dz ~ ZjadSa dz Zj^^^adz

[0110] of _ a ç = ac (42) d9n ~ aden n ^den

[0111] This yields expressions allowing backpropagation to be performed, expressing the dependence of the signals Sa on the different parameters: those of the event Q,x,y,z and those of detector 0:

[0112] Evolution of the charge of the electron cloud:

[0113] ssa _ iasQ "dQ ~ t ao + uw a (23) Position of the electron cloud

[0114] dSa _ / 3¾ as”. 3¾ dX \ dX ' dX Sx dSa, 3¾ dy + dx dSa 1 dZ ) (39)

[0115] dSa _ / 3¾ . aS” 3¾ dy ~ \ dy ' dx ■*" dy dS* , 3¾ dS^ ) , dy + dy ' dz} + dx (39')

[0116] as« / 3¾ asg, diiy ~ dz ~ \ dz ' dx ' dz dSa, 3¾. as” j , n3(”.wg) dy ' dz ' dz / ' V dx (39”) Parameter vector

[0117] _ss^_ oe n / 3¾ aces”, 3¾ aces^ 3¾ bones”. n of —as” Qar ( 9¾ ' Sx "*■ ddn ' dy sen ' dz ■+■ ' W dQ T^d&n (43) Update step

[0118] During step 140, an update of the initial state is performed Xq. This step consists of determining gaps △ Q (substep 141), &r(Ar=(Ax, Ay, Az)) (substep 142), A 6 (formed of correction terms A 6^) (substep 143) and this for each interaction.

[0119] The update can be carried out according to the diagonal Gauss Newton method: cf expressions (50) to (54).

[0120] ^(t0) (50)

[0121] Jf(tO) (51) v éAi2

[0122] ^(to) (52)

[0123] ff(tO) (53) ^Z = ——to

[0124] (54)

[0125] During the update,

[0126] XQ = Xo + AXo(6O)

[0127] Thus, in this example: x0+ Ax; yQ«- yQ + AyQ; zQ^ z0+ to z0; 6n^ 0a+ A 9n (61)

[0128] During a step 150, the correction terms A 6n defined during the implementation of the process are taken into account for several interactions, typically several hundred, to form an average correction vector 0. The average correction vector is formed from an average of the parameters 0tl defined for the interactions.

[0129] During this step, other criteria than a mean of the correction terms 6U can be used to form the correction vector 0, for example a median.

[0130] The previously described procedure was implemented. Figures 4A to 4H represent the estimation of the interaction depth in the detector, for different modeled interactions, for eight successive iterations. Each point corresponds to one interaction. The x-axis represents the modeled depths, while the y-axis represents the depths estimated by the algorithm. The initial depths are chosen randomly (see [Fig. 4A]). It can be observed that after eight iterations, the estimated depths are consistent with the modeled depths.

[0131] Figures 5A and 5B respectively represent an estimation of the interaction positions along an X-axis, parallel to the anodes, and along the depth (Z-axis). These figures were obtained by taking into account 32768 interactions of photons with an energy of 122 keV on a detector comprising 2x2 anodes.

[0132] Figures 5C and 5D are equivalent to Figures 5A and 5B, but with consideration of realistic noise at the energy of 122 keV.

[0133] In another series of tests, the charge induction efficiency (CIE), which represents the ratio, normalized to 1, between the charge collected and the charge deposited by a photon during an interaction, was simulated. A detector consisting of 2 x 2 anodes was used. Figure 6A shows a CIE measured by scanning the detector with a collimated beam of 122 keV energy. Figure 6B shows a simulated CIE implemented by the invention. The simulation is consistent with the measurements.

[0134] The method thus allows for an accurate estimation of the response of a detector.

[0135] It is also possible to estimate the response of a detector using a supervised learning neural network algorithm. Using such an algorithm requires less computing power than the method of the invention. However, a neural network involves a supervised learning phase. The invention referred to in the subject matter can be used to perform such learning. supervised by a neural network, replacing tedious experimental trials, which assume control of the position of interactions in the detector material.

[0136] The invention makes it possible to generate interactions at any location within the detector and to estimate the detection signal on one or more anodes. It can therefore be used to define training sets, associating data related to an interaction in the detector (position in the detector and energy) and the signals induced at different anodes.

[0137] The neural network can be of the multilayer perceptron type, comprising: - as an input layer: at least one characteristic of the signal from the anode that collected the maximum signal (collecting anode) and of the anodes neighboring the latter; For example, one can take into account the maximum signal from the collecting anode and the maximum signal from anodes neighboring the collecting anode; - as output layer: the position of the interaction in the detector, in two or three dimensions, as well as the charge released during the interaction: thus, the output layer is formed of 3 or 4 nodes.

[0138] Between the input layer and the output layer, the neural network can, for example, comprise 3 interconnected layers, with 16 nodes per layer. Such a neural network allows for accurate modeling of a detector

[0139] The input layer advantageously includes, for each anode, a normalized characteristic based on the same characteristic of the anode that collected the maximum signal. The characteristic can be a maximum amplitude or the amplitude of a transient signal. In this case, the output layer is multiplied by the normalization term to estimate the energy. An example of a neural network is described, for instance, in Yang et al., "Joint extimation of interaction position and energy deposition in semiconductor SPECT imaging sensors using fully connected neural network." In this publication, the neural network undergoes supervised learning using experimental data.

[0140] Such a neural network can easily be coded onto a compact electronic board, such as a Field-Programmed Gated Array (FPGA). The advantage is the use of a relatively simple neural network to model the detector's response, the implementation of which does not require any complex components. Training the neural network is carried out simply, particularly through simulations, using the method described above.

[0141] Although described in connection with an X or gamma photon detector, the invention can be applied to other types of ionizing radiation for example a, [3- or neutrons.

Claims

1. Demands Method for estimating a signal measured in a pixel of a detector (1), the detector comprising several electrodes (4a), forming pixels, connected to a detector material (2), the detector material being a semiconductor, each electrode being configured to collect charge carriers moving through the detector material under the influence of an electric field, following an interaction of ionizing radiation in the detector material, the method comprising: - irradiation of the detector with X or gamma photons, so as to generate interactions in the detector material, each interaction forming a detection signal (Sa) measured by at least one electrode; - for different interactions detected, association of a state vector (X), comprising at least one position (X, y, Zy), a charge (Q), and the detection signal (Sa), the state vector passing from an initial state (Xq) when the interaction occurs, to a final state Qf when the electrons, generated by the interaction, are collected by at least one electrode; the process being characterized in that it comprises, for each interaction detected:

2. - (i) initialization of the initial state vector (XQ); - (ii) from the initial state vector, or from an initial state resulting from a previous iteration, implementation of a charge carrier propagation model in the semiconductor material, to estimate the final state vector (X), the propagation model being based on a parameter vector (0), parameterizing at least one transport property (pir^ ïfr)) of charge carriers through the detector material, towards the electrodes; - (iii) calculation of an error (£), representative of a difference between the detection signal estimated in the final state vector (§ ) and the measured detection signal (Sa); - (iv) backpropagation of an error gradient (2£ , ut / 7 ae ôe, je pétat final vers l'état initial, le gradient de l'erreur étant compte avec le moins un terme du vector de état, et (2£) avec le plusieurs parameters de vector de parameter ; - (v) update of the state vector (Xq)' at the initial time, and of the parameter vector, and reiteration of steps (ii) to (v) until a stopping criterion for the iterations is reached; the process being such that: - steps (i) to (v) are implemented by a processing unit connected to the detector; - steps (i) to (v) are implemented for different interactions, so as to update the parameter vector at each implementation of steps (i) to (v); - the parameter vector is then updated by combining the parameter vectors updated during each interaction. A method according to any one of the preceding claims, wherein the detector material is discretized into voxels, and wherein the parameter vector includes, for each voxel, a (p(r), tX^)' property of transporting charge carriers in the detector.

3. A method according to claim 2, wherein the charge carrier transport property comprises: - an electric field value; - and / or an electric field gradient along at least one direction; - and / or an electric field divergence value; - and / or a voxel trapping probability.

4. A method according to any one of the preceding claims, wherein - for each interaction, the time between the initial state and the final state is discretized into time steps - step (ii) comprises a numerical integration of an evolution function (fg), the evolution function representing a time evolution of the position and charge of the charge carriers, as well as an evolution of each detection signal, the numerical integration being carried out successively between each time step, between the initial state and the final state.

5. A method according to claim 4, wherein step (iv) comprises a numerical integration of an adjoint propagation equation, reflecting a time evolution of the error gradient, the numerical integration being carried out successively between each time step, between the final state and the initial state.

6. A method for learning a supervised artificial intelligence algorithm, intended to simulate a response from a detector (1), the detector comprising several electrodes, forming pixels, connected to a semiconductor material, each pixel being configured to collect charge carriers moving through the semiconductor material under the effect of an electric field, following an interaction of ionizing radiation in the semiconductor material, the method comprising the following steps: - a) defining a position of an interaction in the detector and an energy released during said interaction; - b) estimating a detection signal measured by at least one electrode of the detector, by implementing the steps of a method according to any one of the preceding claims; - c) repeating steps a) and b) so as to form a database linking, for each defined interaction, the signal measured in at least one pixel; - d) using the database to perform supervised learning of the artificial intelligence algorithm.

7. A method according to claim 6, wherein the artificial intelligence algorithm is of the multilayer perceptron type.

8. Detector (1), comprising several electrodes, forming pixels, connected to a semiconductor material (2), each pixel being configured to collect charge carriers moving, through the semiconductor material, under the effect of an electric field, following an interaction of ionizing radiation in the semiconductor material, the detector being connected to a processing unit (6), configured to carry out steps (i) to (v) of a process according to any one of claims 1 to 5.

9. Detector, comprising several electrodes, forming pixels, connected to a semiconductor material (2), each pixel being configured to collect charge carriers moving, through the semiconductor material, under the effect of an electric field, following an interaction of ionizing radiation in the semiconductor material, the detector being connected to a processing unit (6), configured to estimate an energy released by the interaction and / or a position of the interaction, the processing unit implementing a supervised artificial intelligence algorithm whose learning is carried out according to any one of claims 6 or 7.

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