Patent Application
20180088232
The present invention relates to methods and devices for probing by wave propagation.
The method for probing by wave propagation uses an assembly of transducers and allows for example measuring a characteristic parameter of the medium, and/or detecting a singular point of the medium, and/or creating an image of the medium.
Methods of this type are used in particular in detection and imaging systems, for example such as sonar, radar, ultrasound, etc.
In known methods of this type and in particular in ultrasound or radar imaging methods, a simple-scattering component of the captured signals is used: if each scatterer only interacts once with the wave, there is effectively a direct equivalence between the time of arrival of each echo and the distance separating the transducer and the scatterer that generated this echo. Detection of an echo at a given moment is indicative of the presence of a scatterer at the distance corresponding to the time of arrival of the echo. An image of the reflectivity of the medium, meaning the position of the various scatterers within the medium, can then be constructed from the received signals if necessary. Multiple scattering is not much used in ultrasound or radar imaging methods. These imaging methods are based on the assumption that there is negligible multiple scattering.
In the presence of a significant multiple-scattering component, particularly when scatterers contained in the medium have a high scattering power and/or are very dense within the medium, conventional imaging methods are greatly disrupted and are no longer reliable. Indeed, in this case, there is no equivalence between the time of arrival of an echo and the distance separating a transducer and a scatterer of the medium, which does not enable constructing an image of the medium.
Patent application WO-2010/001027 proposes a probing method capable of separating the multiple-scattering component from the simple-scattering component. More particularly, the method of that document comprises the following steps:
(a) an emitting step during which an assembly of transducers (meaning some or all of the transducers of the assembly) emits an incident wave in a medium which scatters said wave,
(b) a measuring step during which said assembly of transducers (meaning some or all of the transducers of the assembly) captures signals representative of a reflected wave reverberated by the medium from the incident wave, said captured signals comprising:
(c) a processing step during which said captured signals are processed in order to determine characteristics of the medium (the characteristics in question may consist of an image of the medium, and/or a value of a parameter of the medium, and/or the presence or absence of a singular point such as heterogeneity, etc.), at least one component selected among the multiple-scattering component and the simple-scattering component is extracted by filtering at least one frequency transfer matrix representative of responses between transducers of the assembly of transducers, and which comprises at least the following substeps:
(c1) a windowed transfer matrix determination substep during which is determined at least one windowed frequency transfer matrix K(T,f) corresponding to a windowed temporal matrix of inter-element response K(T,t)=[kij(T,t)] (or response between transducers of the assembly), said windowed temporal matrix of inter-element response corresponding, over a time window close to a time T and of duration Δt, to the temporal responses hij(t) between transducers of the assembly of transducers, f being the frequency,
(c2) a data rotation substep during which two matrices A1(T,f) and A2(t,f) are calculated from the windowed frequency transfer matrix K(T,f), by rotations in a first direction and extraction of components from the windowed frequency transfer matrix,
(c3) a filtering substep during which the multiple-scattering component is separated from the simple-scattering component in each of matrices A1, A2, thus obtaining at least two filtered matrices A1F, A2F respectively corresponding to matrices A1, A2 and each representative of either the simple-scattering component or the multiple-scattering component,
(c4) a reverse data rotation substep during which a filtered windowed transfer matrix KF(T,f) is calculated from the two filtered matrices A1F, A2F by rotations in a second direction opposite the first direction and extraction of components from the filtered matrices A1F, A2F.
The present invention is intended to refine and improve the probing methods as defined above.
To this end, the processing step of the method according to the invention is such that:
With these features of the method according to the invention, a greater amount of information from the medium can be taken into account in the processing step. The probing of the medium by wave propagation is therefore more effective and allows probing the medium to a greater depth.
In addition, the images of the medium that can be produced will have better spatial resolution. It is possible to use a transducer assembly having a smaller number of transducers, and therefore the method will be less costly.
In addition, separation of the components allows for example:
In various embodiments of the method according to the invention, one or more of the following arrangements may possibly be used.
According to one aspect of the method, the windowed transfer matrix K(T,f) is a matrix of size N*N where N is odd, and during the data rotation step (c2) the two matrices A1(T,f)=[a1uv(T,f)] and A2=[a2uv(T,f)] are calculated from the windowed frequency transfer matrix K(T,f)=[kij(T,f)], by the following relations:
if u+v−1−(N−1)/2 and v−u+(N+3)/2−1 are both between 1 and N, inclusive,
if u+v−(N−1)/2 and v−u+(N+3)/2−1 are both between 1 and N−1, inclusive,
According to one aspect of the method, the data rotation substep (c2) comprises:
According to one aspect of the method, the windowed transfer matrix K(T,f) is a matrix of size N*N where N is odd, and:
if i, [1+(N−1)/2; N+(N−1)/2], and
then keij(T,f)=ki−(N−1)/2, j−(N−1)/2(T,f),
else keij(T,f)=0, and
a1uv(T,f)=keu+v−1, v−u+(N+3)/2−1+(N−1)/2(T,f), and
a2uv(T,f)=keu+v, v−u+(N+3)/2−1+(N−1)/2(T,f).
According to one aspect of the method, the windowed transfer matrix K(T,f) is a matrix of size N*N where N is odd, and during the reverse data rotation step (c4) the filtered windowed transfer matrix KF(T,f)=[kFij(T,f)] is calculated from the filtered matrices A1F(T,f)=[a1Fuv(T,f)], A2F(T,f)=[a2Fuv(T,f)] by the following relations:
if i−j is even,
then kFij(T,f)=a1F(i−j−1)/2+(N+1)/2, (i+j)/2,
else kFij(T,f)=a2F(i−j−1)/2+(N+1)/2, (i+j−1)/2.
According to one aspect of the method, during the windowed transfer matrix determination substep (c1), each windowed frequency transfer matrix K(T,f) is determined by wavelet transform of the windowed temporal matrix K(T,t) corresponding, over said time window close to time T and of duration Δt, to the temporal responses hij(t) between transducers of the assembly of transducers.
According to one aspect of the method, the time windows of the various windowed transfer matrices K(T,f) overlap at least pairwise.
According to one aspect of the method, during the filtering substep (c3):
According to one aspect of the method, during the emitting step the incident wave emitted in a medium is inclined by a predetermined angle to an outer surface of said medium.
The invention also relates to a device for the implementation of a probing method as defined above, comprising an assembly of transducers adapted to emit an incident wave in a scattering medium and to capture signals representative of a reflected wave reverberated by the medium from the incident wave, said captured signals comprising:
(c1) a windowed transfer matrix determination substep during which is determined at least one windowed frequency transfer matrix K(T,f) corresponding to a windowed temporal matrix of inter-element response K(T,t)=[kij(T,t)], said windowed temporal matrix of inter-element response corresponding, over a time window close to a time T and of duration Δt, to the temporal responses hij (t) between transducers of the assembly of transducers, f being the frequency,
(c2) a data rotation substep during which two matrices A1(T,f) and A2(T,f) are calculated from the windowed frequency transfer matrix K(T,f), by rotations in a first direction and extraction of components from the windowed frequency transfer matrix,
(c3) a filtering substep during which the multiple-scattering component is separated from the simple-scattering component in each of the matrices A1, A2, thus obtaining at least two filtered matrices A1F, A2F respectively corresponding to matrices A1, A2 and each representative of either the simple-scattering component or the multiple-scattering component,
(c4) a reverse data rotation substep during which a filtered windowed transfer matrix KF(T,f) is calculated from the two filtered matrices A1F, A2F by rotations in a second direction opposite the first direction and extraction of components from the filtered matrices A1F, A2F.
The device of the invention is such that the processing means are adapted for the following:
Other features and advantages of the invention will be apparent from the following description of several of its embodiments, given by way of non-limiting examples, with reference to the accompanying drawings.
In the drawings:
The medium 1 is a scattering medium for the waves in question, meaning it is heterogeneous and/or contains scatterers 2 randomly distributed and capable of reflecting the emitted waves in the medium 1.
The medium 1 in question may for example be part of the human body, and the scatterers 2 may be small particles, unresolved, contained in the medium 1 (in ultrasound such scatterers generate images of “speckles”). Of course, the medium 1 to be probed could be otherwise, for example part of an industrial object of which the structure is to be tested in a context of nondestructive testing.
The probing device represented in
Each transducer 4 of the assembly 3 can be controlled individually by a central processing unit 5 (UC), for example comprising digital signal processing means, this central processing unit 5 able for example to present an image of the medium 1 on a screen 6.
To probe the medium 1, the central processing unit 5 sends to the transducers 4 electrical signals which are converted by said transducers into waves emitted in the medium 1, in this case ultrasonic compression waves, and these waves are partially reflected by the scatterers 2 contained in the medium. A portion of the scattered waves (or echoes) thus returns to the transducers 4 which capture and convert them into electrical reception signals subsequently processed by the central processing unit 5.
These waves return to the transducers 4:
The overall wave scattered by the medium 1 and returned to the transducers 4 therefore includes two contributions:
This method allows separating these two contributions by filtering, so that only one of them is used or so that they are processed separately. For example:
In order to separate the simple-scattering and multiple-scattering contributions, first the inter-element responses of each pair of transducers 4 of the assembly 3 are recorded.
For this purpose, as represented in
The set of N2 responses forms a temporal matrix of inter-element response H(t)=[hij(t)], a square matrix of size N*N, which is the overall response of the medium 1. One will note that the temporal matrix of inter-element response H(t) may possibly be acquired more rapidly, without each transducer i of the assembly 3 successively emitting a pulse signal, by proceeding for example as disclosed in document WO-A-2004/086557.
The signals hij(t) are digitized (for example in 9 bits, or similar), sampled (for example with sampling at 20 MHz for ultrasound waves), and stored by the central processing unit 5.
The central processing unit 5 then carries out a processing step (c), comprising an initial substep (c1) of windowed transfer matrix determination.
During this substep (c1), each impulse response hij(t) is first truncated (windowed) into successive time windows of duration Δt.
A series of windowed temporal matrices K(T,t)=[kij(T,t)] is thus obtained of size N*N, where kij(T,t) is the contribution at hij(t) corresponding to the time window [T−Δt/2;T+Δt/2], which is:
k
ij(T,t)=hij(t)·WR(t−T) (1)
with:
During substep (c1), a discrete Fourier transform of the coefficients of matrix K(T,t) is performed in order to obtain for each value of T a transfer matrix of size N*N which we will call a windowed frequency transfer matrix K(T,f)=[kij(T,f)], where kij(T,f) is the discrete Fourier transform of kij(T,t) and f is the frequency.
Based on these windowed frequency transfer matrices K(T,f), the simple- and multiple-scattering contributions can be separated by filtering during a subsequent filtering substep (c3) which is part of the processing step (c).
In particular, during this subsequent filtering substep (c3), the multiple-scattering component can be separated from the simple-scattering component in each windowed frequency transfer matrix K(T,f) on the basis of the coherence of coefficients kij(T,f) of the windowed frequency domain transfer matrix K(T,f) on each antidiagonal of said windowed frequency transfer matrix K(T,f) (antidiagonal being used to refer to an alignment of coefficients kij(T,f) of said matrix, such that i+j is constant).
In effect, simply scattered waves have a particular coherence along the antidiagonals of the matrix K(T,f) while multiply scattered waves have a random aspect and do not exhibit a preferred coherence direction in said matrix K(T,f). By carefully filtering these antidiagonals, the two contributions can thus be separated.
This property can be explained as follows.
Each of the impulse responses hij(t) can be decomposed into the following form:
h
ij(t)=hijS(t)+hijM(t) (2)
where hijS(t) and hijM(t) respectively correspond to the signals originating from simple scattering (S) and multiple scattering (M).
Similarly, the coefficients kij(T,f) of the windowed frequency transfer matrix K(T,f) can each be decomposed into the form kij(T,f)=kijS(T,f)+kijM(T,f), where kijS(T,f) is the simple-scattering contribution and kijM(T,f) is the multiple-scattering contribution.
Each contribution kijS(T,f) can be considered as the sum of partial waves associated with several simple-scattering paths of which two examples (paths d1 and d2) are shown in
We use the term “isochronous volume” to denote the set of points which, at a given moment T, contribute to the signal captured in the array. In reality, the isochronous volume is not exactly a slice parallel to the array surface but results from the superposition of ellipses whose foci are the emitter element (i) and the receiver element (j). In far field, meaning when R is sufficiently large, the isochronous volume is similar to a slice of thickness DR parallel to the array and at a distance therefrom of R=cT/2.
The response kijS between elements i and j can be broken down into a sum of partial waves resulting from reflection on the Nd scatterers of the isochronous volume. In two dimensions, in a context of paraxial approximation, we can write kijS(T,f) as follows:
where the integer d denotes the dth simple-scattering path contributing to the signal received at time T, Xd is the transverse position of scatterer d (along axis X), k is the wavenumber in the surrounding medium (k=2π/λ, λ being the wavelength), and Ad is an amplitude characterizing the reflectivity of scatterer d.
One will note that in the above equation (3) and in the other equations of the present patent application, j is the imaginary number such that j2=−1 when it is not a subscript, but denotes the position of a matrix element when it is a subscript.
The multiple-scattering contribution can also be broken down into partial waves corresponding to multiple-scattering paths of a length comprised within the interval [R−DR/2; R+DR/2], as represented in
We can write KijM(T,f) as follows:
where the integer p denotes the index of the multiple-scattering path concerned. The pairs (X1(p),Z1(p)) and (X2(p),Z2(p)) respectively denote the coordinates of the first and last scatterer of path p in the example represented in
Although the distribution of the scatterers 2 in the medium 1 is completely random and without correlation between scatterers, the signal associated with simple scattering kijS has a particular coherence, unlike the multiple-scattering contribution. In effect, equation (3) can be rewritten as follows:
The term appearing in front of the sum of equation (5) is independent of the exact distribution of the scatterers, and is therefore a deterministic contribution, which characterizes simple scattering. The right term is random, because it is explicitly dependent on the position of the scatterers.
In contrast, the signal related to multiple scattering (Equation 4) cannot be written this way.
This property of signals originating from simple scattering is expressed as a particular coherence along the antidiagonals of each matrix K(T,f), as illustrated in
However, in multiple scattering, this property is no longer satisfied and matrix K(T,f) has no particular coherence: elements kijM are independent of one another.
The present method exploits this property in order to isolate the simple- and multiple-scattering contributions by filtering based on experimentally measured signals, making use of the particular symmetry of the simple-scattering contribution within each matrix K(T,f). One can thus filter out:
Two example filtering techniques which can be used to separate the two contributions are presented below.
In these two techniques, the processing step (c) comprises the following two substeps, after the windowed transfer matrix determination substep (c1):
These substeps (c2) to (c4) are detailed below.
In this substep (c2), the central processing unit 5 calculates two matrices A1(T,f)=[a1uv(T,f)] and A2=[a2uv(T,f)] from each matrix K(T,f), where:
a1uv(T,f)=ku+v−1, v−u+2M−1(T,f),
a2uv(T,f)=ku+v, v−u+2M−1(T,f),
N is selected so that M is an integer: for example, N=125 and M=32. If the total number of transducers 3 is such that M is not an integer, we work with a reduced number N of transducers, such that M=(N+3)/4 is an integer (in the specific example considered here, we can for example use an array of 128 transducers and work with only N=125 of them).
Matrices A1 and A2 are square matrices composed of subsets of matrix K(T,f) which are rotated 45° in the counterclockwise direction. These matrices A1 and A2 are respectively illustrated in
In the following, we will use Ar=[arij] to refer to matrices A1 and A2 (r=1 or 2) and we will use L to refer to the size of matrix Ar (for matrix A1, we therefore have L=2M−1, and for matrix A2, L=2M−2).
Due to spatial reciprocity, matrix K is symmetric with respect to its main diagonal D (kij=kji). Matrix Ar therefore also has symmetry: each row of its upper portion is identical to a row in the lower portion, symmetric with respect to a horizontal center line corresponding to the main diagonal D. The upper portion of matrix Ar can therefore be directly deduced from the lower portion. Thus, each column of matrix A1 only contains M independent coefficients although it is of size L>M, and each column of matrix A2 contains M−1 independent coefficients. More generally, the number of independent coefficients of matrix Ar is therefore a number Mr such that Mr=M if r=l, and Mr=M−1 if r=2.
During the filtering substep (c3), the central processing unit 5 separates the multiple-scattering component from the simple-scattering component in each of the matrices Ar, r being an index equal to 1 or 2, and we thus obtain at least two filtered matrices ArF respectively corresponding to the two matrices Ar and each representative of either the simple-scattering component or the multiple-scattering component.
This filtering may be performed in particular according to the abovementioned technique 1 or according to technique 2.
In this first technique, during the filtering substep (c3), the central processing unit 5 calculates two filtered matrices ArF which are representative of simple scattering.
Each of these filtered matrices is calculated by the formula:
Ar
F
=S
t
S*Ar, where:
S=[su] is a column vector, tS* being the transpose of the conjugate vector of vector S,
the components su of vector S are complex numbers equal to:
s is a constant, in practice equal to 1 (and therefore not mentioned below),
yu=(xi−xj)/√{square root over (2)}, where u=(i−j)/2+M if r=1 and u=(i−j−1)/2+M if r=2,
xi and xj are the abscissas of the transducers of indices i and j along an axis X, the assembly of transducers extending at least along said axis X,
L=2M−1 for r=1 and L=2M−2 for r=2.
This formula is justified as follows.
Each matrix Ar is the sum of two terms, ArS and ArM, respectively denoting the contributions due to simple scattering and multiple scattering:
Ar=Ar
S
+Ar
M (6)
The data rotation, in other words the transition from K(T,f) to Ar, is mathematically expressed by the change of coordinates (xi,xj) fi (yu,yv):
y
u=(xi−xj)/√{square root over (2)} and yv=(xi+xj)/√{square root over (2)}
Equation (5) is then rewritten on this new basis:
Thus for a given medium 1, each column of matrix ArS has a fully determined dependency on the index of the rows (u).
However, the multiple-scattering contribution (Equation 4) cannot be factored so simply. Even after rotation of the matrix, the randomness of the position of the scatterers persists in the columns as well as in the rows of matrix ArM.
Filtering of simply scattered signals can therefore be done by projecting the columns of the complete matrix Ar into the “characteristic space of simple scattering” generated by vector S of coordinates:
The presence of √{square root over (L)} in the denominator ensures normalization of vector S. The row vector P resulting from this projection is written:
P=
t
S*Ar (9),
where tS* is the conjugate transpose of vector S.
The coordinates of vector P are given by:
The residual term
corresponds to the projection of multiply scattered signals on vector S.
Next, the filtered matrix ArF is obtained by multiplying the column vector S by the row vector P:
Ar
F
=SP=S·
t
S*·A (11)
The coordinates of matrix ArF are then written:
The first term is strictly equal to the simply scattered component (Equation 7). We therefore have:
In terms of matrices, equation (13) is rewritten as follows:
Matrix ArF does indeed contain the contribution related to simple scattering (AS), but it also contains a residual term related to the presence of multiple scattering (StS*AM). The persistence of this term is due to the fact that the multiple-scattering signals are not strictly orthogonal to the characteristic space of simple scattering, generated by vector S.
The filtering achieved is therefore not perfect; the extent of the residual noise can be assessed, however.
Indeed, as noted in the paragraph concerning data rotation, each column in matrix A1 only has M independent coefficients and matrix A2 only M−1 independent coefficients; the contribution from multiple scattering is therefore decreased by a factor √{square root over (Mr)} after filtering. As the simple-scattering contribution remains unchanged, the signal-to-noise ratio or more specifically the “simple scattering/multiple scattering” ratio is therefore on the order of √{square root over (Mr)}.
The filtering technique described above (technique 1) is to be used in particular when one wishes to extract a simple-scattering contribution obscured by the multiple scattering, meaning in the case of media where the simply scattered signals are of very low amplitude compared to the signals from multiple scattering. This particularly applies to the case of detecting targets buried in a scattering medium.
This second technique consists of separating simple scattering and multiple scattering by performing singular value decomposition (SVD) of the matrices A1 and A2 obtained after rotation. SVD has the property of factoring a matrix into two subspaces: a “signal space” (matrix characterized by a significant correlation between rows and/or columns of the matrix) and a “noise space” (random matrix without correlation between elements). By applying SVD to the matrices Ar obtained after rotation, the signal space corresponds to matrix ArS (simple-scattering contribution, characterized by a high correlation along its columns) and the noise space is associated with matrix Arm (multiple-scattering contribution), where Ar=ArS+ArM (Equation 6 already described in the paragraph concerning technique 1).
The SVD of the matrices Ar is written as follows:
where U and V are unitary square matrices of size L. Their respective columns Ui and Vi correspond to the eigenvectors associated with the singular value li,r. L is a square diagonal matrix of dimension L whose diagonal elements correspond to the singular values li,r arranged in descending order. In the paragraph concerning data rotation, a particular symmetry of matrix Ar was highlighted: this matrix only comprises Mr independent rows, and is therefore of rank Mr<L. Matrix Ar therefore only has Mr non-zero singular values and equation (15) is rewritten as:
As simple scattering is characterized, after data rotation, by great coherence along the columns of the matrices Ar, SVD reveals this contribution in the signal space (the simple-scattering contribution will be associated with the highest singular values) while the multiple-scattering contribution will be associated with the lowest singular values. Here, unlike in the first filtering technique, there is no a priori hypothesis on the form of the existing coherence on the antidiagonals of matrix K(T,f) in the case of simple scattering: it is simply assumed that this coherence exists.
The problem is to determine to what rank of singular value corresponds the threshold separating the “signal” space (associated with simple scattering) from the “noise” space (associated with multiple scattering). If equation (5) were strictly accurate, only the first singular values would correspond to the signal space. When the assumptions leading to equation (5) are not strictly accurate, the simple-scattering contribution is not of rank 1 and several singular values bear traces of this contribution. A criterion must then be established for separation between the simple-scattering contributions (signal space) and multiple-scattering contributions (noise space).
To do this, we use results from random matrix theory. By convention and for simplification, the singular values λi,r are normalized by their root mean square:
For a large matrix in which the coefficients are completely random, with no correlation between them, the first singular value {tilde over (l)}1 never exceeds a value {tilde over (l)}max ({tilde over (l)}max=2 in the case of a square matrix).
Experimentally, the multiple-scattering contribution does not exactly correspond to a completely random matrix, because residual correlations exist between sensors (in particular due to mechanical or electrical coupling between adjacent transducers in the assembly 3), which modifies {tilde over (l)}max. Based on [A. M. Sengupta and P. P. Mitra, “Distributions of singular values for some random matrices,” Phys. Rev. E, Vol. 60(3), pp 3389-3392, 1999.], one can establish the new law of probability for the singular values of such a matrix and use this to determine the value of {tilde over (l)}max which will allow defining an objective criterion for separation between the signal space and noise space.
After rotation of the experimental data, the matrix Ar to be processed (see equation 6) is therefore the sum of a matrix ArS of rank p<M associated with simple scattering and a matrix ArM of rank M associated with multiple scattering which one wishes to filter.
The proposed technique is as follows: after having performed SVD, the central processing unit 5 considers the first singular value {tilde over (l)}1,r after normalization. If this value is greater than {tilde over (l)}max, this means that the first eigenspace is associated with simple scattering.
The process is then repeated at rank 2 and at higher ranks if necessary.
As represented in
(c31) q is initialized to 1,
(c32) a normalized singular value is calculated using λq,r:
(c33) if {tilde over (l)}q,r is at least equal to a predetermined threshold value {tilde over (l)}max, {tilde over (l)}q,r is attributed to simple scattering and substep (c32) is repeated after incrementing q by one,
(c34) if {tilde over (l)}q,r is less than the threshold value {tilde over (l)}max, {tilde over (l)}q,r and any subsequent singular values are attributed to multiple scattering.
If the rank for which {tilde over (l)}p+1,r<{tilde over (l)}max is referred to as p+1, one thus obtains:
Matrix ArS then contains the simple-scattering contribution (plus a residual multiple-scattering contribution) and matrix ArM is associated with multiple scattering.
Note that technique 2 assumes that the first of the normalized singular values exceeds the threshold {tilde over (l)}max. It cannot be used in highly scattering media, meaning media for which the multiple-scattering contribution is predominant relative to simple scattering. In this case, the technique of filtering by projection of antidiagonals onto the simple-scattering space is used instead (technique 1) in order to extract the simple-scattering contribution. Conversely, if the simple-scattering contribution is predominant or on the same order of magnitude as multiple scattering, one can use the technique of SVD filtering of the matrices A (technique 2) and thus extract the multiple-scattering contribution.
During the reverse data rotation substep (c4), the central processing unit 5 performs a reverse transformation of the transformation described in substep (c1), and thus calculates a filtered windowed transfer matrix KF(T,f)=[kFij(T,f)], where:
when i−j is even: kFij(T,f)=a1F(i−j)/2+M, (i+j)/2,
when i−j is odd: kFij(T,f)=a2F(i−j−1)/2+M, (i−j−1)/2.
Matrix KF(T,f) is a square matrix of size (2M−1). (2M−1), containing signals which originate either from simple scattering or multiple scattering, depending on the type of filtering applied. The reverse data rotation procedure is illustrated in
The filtered matrices KF(T,f) can then be used in various ways:
if KF(T,f) corresponds to the simple-scattering component, it can be used in particular to detect a singular point of the medium 1 or to construct an image of the medium 1. For this purpose, two imaging methods can be used for example:
if the matrices KF(T,f) correspond to the multiple-scattering component, they can be used in particular to calculate, from said multiple-scattering component, an index representative of the significance of the multiple scattering in the medium. In this case, one can for example create an image (ultrasound or other) of the medium by any known means or by the above methods, and quantify the reliability of said image as a function of said index representative of the significance of the multiple scattering. Advantageously, one can calculate this index at multiple points (in particular at various depths R) and quantify the reliability of multiple portions of the image corresponding to said multiple points, as a function of said index representative of the significance of the multiple scattering.
The method of the present invention improves the method described above.
In particular, the loss of information from the matrices during the steps of data rotation (c2) and reverse data rotation (c4) is avoided.
In fact, as is represented in
According to a first embodiment of the method according to the invention, the data rotation substep (c2) comprises:
The size of the extended matrix Ke is adapted so that the elements from the windowed frequency transfer matrix are all included in a square Ker tilted by 45° and itself included in the extended matrix Ke. In particular, the size of the extended matrix Ke is greater than or equal to 2×N−1.
The extended matrix Ke then comprises corner portions Ke1, Ke2, Ke3, Ke4, side portions Ke5, Ke6, Ke7, Ke8, and a central portion Kec. The side portions are also corners of the square Ker. All elements of the extended matrix Ke aside from the central portion Kec (which contains the elements of the windowed frequency transfer matrix K(T,f)), are zero. In other words, all elements of the corner portions Ke1, Ke2, Ke3, Ke4 and side portions Ke5, Ke6, Ke7, Ke8 are zero.
The square Ker serves as a basis for forming matrices A1 and A2 of the second substep by taking every other element from the extended matrix Ke, as explained below.
For example, if N is odd, in the first substep (c21) the size of the extended matrix is 2×N—1, and the extended matrix Ke is defined by Ke(T,f)=[keij(T,f)] based on matrix K(T,f), using the following relations or any other equivalent formulas:
if i, [1+(N−1)/2; N+(N−1)/2], and
then keij(T,f)=ki−(N−1)/2, j−(N−1)/2(T,f),
else keij(T,f)=0.
Equivalent formulas can be established for N being even.
Then, during the second step (c22), the two matrices A1(T,f)=[a1uv(T,f)] and A2=[a2uv(T,f)] are calculated from the extended matrix Ke(T,f)=[keij(T,f)].
In the case presented in
In fact, matrix A1 has a size identical to that of the windowed frequency transfer matrix K(T,f), meaning a size equal to N×N, and matrix A2 has a size equal to that of the windowed frequency transfer matrix K(T,f) minus one, meaning a size equal to (N−1)×(N−1).
More specifically, if N is odd, during the second substep (c22), matrices A1(T,f) and A2(T,f))] are calculated from the extended matrix Ke(T,f)=[keij(T,f)], by the following relations or any other equivalent formulas:
a1uv(T,f)=keu+v−1, v−u+(N+3)/2−1+(N−1)/2(T,f), and
a2uv(T,f)=keu+v, v−u+(N+3)/2−1+(N−1)/2(T,f).
Equivalent formulas can be established if N is even.
According to a second embodiment of the method according to the invention, the data rotation substep (c2) is performed directly (without constructing an extended matrix Ke).
For example, if N is an odd integer, the two matrices A1(T,f)=[a1uv(T,f)] and A2=[a2uv(T,f)] are calculated from the windowed frequency transfer matrix K(T,f)=[kij(T,f)], using the following relations or any equivalent formulas:
if u+v−1−(N−1)/2 and v−u+(N+3)/2−1 are both between 1 and N, inclusive,
then a1uv(T,f)=ku+v−1−(N−1)/2, v−u+(N+3)/2−1(T,f),
else a1uv(T,f)=0, and
if u+v−(N−1)/2 and v−u+(N+3)/2−1 are both between 1 and N−1, inclusive,
Equivalent formulas can be established if N is even.
In the method of the invention, the filtering substep (c3) is for example identical to the filtering substep of the prior art. In particular, two filtered matrices A1F and A2F are calculated from matrices A1 and A2, for example either using the projection filtering technique described above, or using the singular value decomposition filtering technique described above, or by any other technique.
The filtered matrices A1F and A2F have the same size as the matrices A1 and A2 determined in the data rotation step (c2).
In particular, if N is odd, advantageously the filtered matrix A1F has a size of N×N and the filtered matrix A2F has a size of (N−1)×(N−1).
In addition, during the reverse data rotation substep (c4), a substantially inverse transformation of the transformation described for the data rotation substep (c2) is performed. In this substep, a filtered transfer matrix KF(T,f)=[kFij(T,f)] is calculated from the filtered matrices A1F(T,f)=[a1Fuv(T,f)] and A2F=[a2Fuv(T,f)] which were calculated during the preceding filtering substep (c3).
This substep is very similar or is identical to the reverse data rotation substep (c4) of the prior art described above. In this substep, the elements of the central portions of the filtered matrices A1F and A2F are included and arranged in the filtered transfer matrix KF by a rotation in a second direction that is the reverse of the first direction (45° rotation in the clockwise direction). The elements of the corner portions of the filtered matrices A1F, A2F are not included in the filtered transfer matrix KF. These elements are zero in any event, as are those of the corner portions of matrices A1 and A2.
For example, if N is an odd integer, the filtered transfer matrix KF(T,f)=[kFij(T,f)] is calculated based on two matrices A1F(T,f)=[a1Fuv(T,f)] and A2F=[a2Fuv(T,f)], by the following relations or any equivalent formulas:
if i−j is even,
then kFij(T,f)=a1F(i−j)/2+(N+1)/2, (i+j)/2,
else kFij(T,f)=a2F(i−j−1)/2+(N+1)/2, (i+j−1)/2.
Unlike the method of the prior art, the filtered windowed frequency transfer matrix KF(T,f) then has a size of N×N, which is a size identical to the initial windowed frequency transfer matrix K(T,f).
In the method of the prior art (see
With the method of the invention, the filtered windowed frequency transfer matrix KF(T,f) contains a large number of elements, and it is possible to take into account a larger number of multiple scatterings. Therefore the medium can be probed by wave propagation to a larger depth. In addition, the images one wishes to produce of the medium will have better spatial resolution. It is possible to use an assembly 3 of transducers having a lower number of transducers, and the method and device will then be less costly.
In addition, in an advantageous embodiment, during the windowed transfer matrix determination substep (c1), each windowed frequency transfer matrix K(T,f) is determined by wavelet transform of the windowed temporal matrix K(T,t).
The wavelet transform of a temporal function ƒ, over is defined by:
where
y is a mother wavelet function,
yu,s is a wavelet of the wavelet family,
* denotes the complex conjugate,
u is the translation factor, and
s is the expansion factor.
A wavelet family therefore corresponds to a bank of frequency filters. This filter bank must contain the frequencies to be studied. Calculation of the wavelet transform is then equivalent to convolution of the impulse response of each filter with each time signal, impulse response hij(t). The windowed temporal matrices K(T,t) are said impulse responses hij(t) between transducers of the assembly of transducers, over a time window close to time T and of duration Dt. And the windowed frequency matrices K(T,f) are calculated by wavelet transform, for example by the convolutions defined above.
The mother wavelet function y may be selected among:
and
and
The family of wavelets can thus be chosen so as to best correspond to the transducer bandwidth and to the type of pulse signals to be processed (therefore to the material of the medium).
Through the use of said wavelet transforms, the method of the invention allows probing the medium by wave propagation to an even greater depth, with greater precision and resolution.
In addition, according to an advantageous embodiment, during the emitting step (a), the assembly 3 of transducers emits an incident wave, said incident wave being inclined by a predetermined angle a to an outer surface 1a of said medium 1, as represented in
To do this, the assembly 3 of transducers is for example placed in a coupling medium 7 as shown. It is possible for this coupling medium 7 to be placed between the assembly 3 and the external surface 1a of the medium to be probed, or this coupling medium 7 is integrated with a probe incorporating at least the assembly and said coupling medium.
The coupling medium 7 has for example a refraction index n7 that is different from the refraction index n1 of the medium 1 to be probed. The propagation speeds in these media are therefore respectively V1 and V7.
Thus, in the projection filtering technique of the filtering substep (c3), the components of the column vector S=[su] are determined by the following modified formula:
where
df is the focal distance, which is the sum of a first distance d7 traveled in the coupling medium 7 corrected by the variation in index between the two media and a second distance d1 traveled in the medium to be probed; in other words:
Due to the use of the emission of inclined incident waves, the method of the invention is also improved and allows probing the medium by wave propagation to an even greater depth, with greater accuracy and resolution. In particular, it is possible to obtain more information on the geometry of any defects (planar or volumetric) in the probed medium 1.
| Number | Date | Country | Kind |
|---|---|---|---|
| 15 54258 | May 2015 | FR | national |
| Filing Document | Filing Date | Country | Kind |
|---|---|---|---|
| PCT/FR2016/051036 | 5/3/2016 | WO | 00 |
