CN112529980A - Multi-target finite angle CT image reconstruction method based on maximum minimization - Google Patents

Multi-target finite angle CT image reconstruction method based on maximum minimization Download PDF

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CN112529980A
CN112529980A CN202011474025.7A CN202011474025A CN112529980A CN 112529980 A CN112529980 A CN 112529980A CN 202011474025 A CN202011474025 A CN 202011474025A CN 112529980 A CN112529980 A CN 112529980A
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CN112529980B (en
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王成祥
夏远梅
赵克全
李国权
王艳
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Chongqing Normal University
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Abstract

The invention relates to a multi-target finite angle CT image reconstruction method based on maximum minimization, and belongs to the field of image reconstruction. The method comprises the following steps: s1: acquiring projection data: s2: establishing a multi-target finite angle CT image reconstruction model; s3: finite angle CT iterative reconstruction; s4: and outputting the reconstructed image. By utilizing the characteristics of multi-scale and multi-resolution decomposition and artifact directivity of non-downsampling contourlet transformation, 2 optimization targets of L0 norm minimization and L0 norm of image gradient of the non-downsampling contourlet transformation are set, and a multi-objective optimization reconstruction model is established. By adopting the method, the artifacts and the noise of the reconstructed image can be effectively inhibited, the image boundary is protected, and the quality of the CT reconstructed image is improved.

Description

Multi-target finite angle CT image reconstruction method based on maximum minimization
Technical Field
The invention belongs to the field of image reconstruction, and relates to a multi-target finite angle CT image reconstruction method based on maximum minimization.
Background
In some specific applications or scanning scenarios, for example: the reconstruction problem of the scanning mode is called as finite angle CT reconstruction problem. Because the acquired data does not meet the condition of accurate reconstruction, the traditional filtering back projection reconstruction algorithm causes a plurality of artifacts to appear in the reconstructed image, so that some important structural information is lost or covered, and the accuracy of nondestructive detection or the diagnosis of a doctor is greatly influenced. Therefore, the reconstruction of high-quality images meeting the diagnosis requirements of doctors or the nondestructive testing standards has great practical significance. The traditional algebraic reconstruction algorithm causes a great deal of artifacts and distortion phenomena to the reconstructed image. The image reconstruction algorithm based on tv (total variation) can effectively deal with the problem of sparse angle CT reconstruction, but when applied to scanning of a smaller finite angle, the reconstructed image also has certain artifacts and distortion. In order to inhibit artifacts and protect boundaries, the residual dimension introduces image gradient L0 regularization, and the method can inhibit the artifacts to a certain extent and further improve the quality of a reconstructed image. The Wangchengxi et al uses the multi-scale and multi-resolution characteristics of wavelet transformation to process the finite angle CT reconstruction problem by taking the L0 quasi-norm of image wavelet transformation as regularization, and the method can improve the quality of reconstructed images to a certain extent.
Patent application publication No. CN 107978005a discloses "a finite angle CT image reconstruction algorithm based on boundary diffusion and smoothing". The method mainly carries out gradient L0 boundary diffusion correction on the reconstructed image in the x direction and the y direction respectively to further improve the quality of the reconstructed image. However, the following disadvantages still exist: (1) in the method of the patent application, gradient L0 boundary diffusion correction is carried out only in consideration of x and y axes of the image, regularization constraint is carried out on the gradient L0 of the x and y axes of the image after data fidelity constraint, and image details are lost due to the fact that the image is too smooth and lack of scanning angles correspondingly in order to inhibit artifacts; (2) the gradient transformation of the image of the method of the patent application only has high-frequency information of a high-frequency part of the image, lacks regularization constraint on a low-frequency part and does not consider the multi-directional characteristic of an artifact; (3) from the optimization point of view, the method described in the above patent application is a single-objective optimization method, and multiple indexes are not considered and optimized from the multi-objective optimization point of view.
Patent application publication No. CN 110717959a discloses "x-ray finite angle CT image reconstruction method and apparatus based on curvature constraint". The method mainly carries out image gradient L0 regularization sparse constraint on an image which is reconstructed firstly, and then carries out curvature constraint on a sparse constraint result. Although the method described in the above patent application considers 2 optimization indexes, the problem of boundary blurring or step effect in the existing finite angle CT reconstruction algorithm can be further overcome. However, the following disadvantages still exist: (1) gradient transformation and curvature constraint of the image of the method disclosed in the patent application only have high-frequency information of a high-frequency part of the image, and do not have regularization constraint on a low-frequency part and do not consider multi-directional characteristics of artifacts; (2) from the optimization point of view, although the method described in the above patent application considers 2 optimization indexes (image gradient L0 constraint and curvature constraint), the 2 optimization indexes are respectively processed by a single-target optimization method, and the 2 optimization indexes are not directly considered and optimized from the multi-target optimization point of view; (3) this single-target approach may result in an image that is too smooth due to image gradient L0 constraints, and image details corresponding to a lack of scan angles are lost, so that the latter curvature constraint indicators cannot recover the lost details.
Patent application publication No. CN 109697691a discloses "a dual regularization term optimized finite angle projection reconstruction method based on L0 norm and singular value threshold decomposition". The method comprises the steps of firstly carrying out image gradient L0 regularization sparse constraint on a reconstructed image, and then carrying out nuclear norm constraint on the image. Although the method described in the above-mentioned patent application enables the CT image contour to be restored, the limited angle artifacts are reduced. However, the following disadvantages still exist: (1) gradient transformation and curvature constraint of the image of the method disclosed in the patent application only have high-frequency information of a high-frequency part of the image, and do not have regularization constraint on a low-frequency part and do not consider multi-directional characteristics of artifacts; (2) from the optimization point of view, the method is a double-regularization single-target optimization method, and the 2 indexes of the optimized image gradient L0 constraint and the image nuclear norm are not directly considered from the multi-target optimization point of view. (3) This single-target approach may result in an image that is too smooth due to image gradient L0 constraints, and image details corresponding to a lack of scan angles are lost, so that the subsequent image kernel norm indicator cannot recover the lost details.
Most existing finite angle CT optimization reconstruction methods do not consider the multi-directional characteristic of artifacts, and adopt a single-target optimization method to optimize multiple indexes, and also solve a multi-regularization optimization model by the single-target method. The invention considers the sparsity of the image under the non-down sampling contourlet transformation and the image gradient transformation, sets 2 optimization targets of L0 norm minimization of the non-down sampling contourlet transformation and L0 norm of the image gradient by utilizing the characteristics of multi-scale, multi-resolution decomposition and artifact directivity of the non-down sampling contourlet transformation, and optimizes the two targets by adopting a maximum minimization method in multi-target optimization, thereby improving the quality of CT reconstructed images.
Disclosure of Invention
In view of the above, the present invention provides a multi-target finite angle CT image reconstruction method based on the maximum minimization.
In order to achieve the purpose, the invention provides the following technical scheme:
a multi-target finite angle CT image reconstruction method based on maximum minimization comprises the following steps:
s1: acquiring projection data:
s2: establishing a multi-target finite angle CT image reconstruction model;
s3: finite angle CT iterative reconstruction;
s4: and outputting the reconstructed image.
Optionally, the S1 specifically includes: incomplete projection data is obtained by rotating the source about a center of rotation along the scan trajectory through a limited angle.
Optionally, the S2 specifically includes: when the discrete model is used for reconstruction, f (x, y) corresponding to coordinates (x, y) needs to be rearranged into a 1-dimensional vector f according to the dimension of y, wherein the dimension of the vector f is N × 1, and N is N ═ N1×n2，n1Is the dimension of f (x, y) in the x direction, n2Is the dimension of f (x, y) in the y direction; then, all coordinates (a, s) are associated with projection data gδ(a, s) are rearranged into a 1-dimensional vector g according to the dimension of sδColumn vector gδHas a dimension of M × 1, wherein M ═ M1×m2，m1Is gδ(a, s) dimension in the a-direction, m2Is gδ(a, s) a dimension in the s direction; according to the invention, the reconstructed image is decomposed into a low-frequency part and a high-frequency part by adopting non-subsampled contourlet transform, so that the image is decomposed in multiple scales and multiple resolutions;
in order to suppress noise and artifact and avoid partial detail loss caused by over-smoothness, L0 sparse regularization constraint is carried out on coefficients of non-downsampling contourlet transform; for image smoothness, L0 regularization of image gradients is used for constraint;
the model built from the multi-target perspective is as follows:
Figure BDA0002834470720000031
wherein A ∈ RM×NIs a finite angle CT system matrix, f belongs to RN×1Is the image to be reconstructed, gδ∈RM×1Is finite angle CT projection data, and Ω is a convex set (Ω: { f ≧ 0 }); λ is the relaxation parameter, W is the non-downsampled contourlet transform, i is the index of the subband; | beta | | non-conducting phosphor0Is the number of non-0 elements of beta +(v +xβ,▽yβ)，▽xβ,▽yThe component of β is in the form of +(vxβ)i′,j′＝βi′,j′-βi′-1,j′,(▽yβ)i′，j′＝βi′,j′-βi′,j′-1(ii) a When non-downsampling contourlet transformation is carried out, f is rearranged into 1 2-dimensional matrix f (x, y), then the 2-dimensional matrix is subjected to the non-downsampling contourlet transformation, and after the non-downsampling contourlet transformation is finished, f (x, y) is rearranged into a 1-dimensional vector f according to the dimension of y.
Optionally, the S3 specifically includes:
solving the model (1) by adopting a maximum minimization method according to the established model (1);
the step S3 finite angle CT iterative reconstruction specifically includes:
firstly, converting the model (1) into a single-target optimization model by a maximum minimization method, wherein the form of the model is as follows:
Figure BDA0002834470720000032
then, in order to effectively solve the single-target optimization model, taking lambda as two special values, firstly, taking lambda as 1, and solving a corresponding optimization problem; then lambda is equal to 0, and the corresponding optimization problem is solved; repeating the value-taking process of the lambda in order to make the final solution converge; the single-objective optimization model is then simplified in the form:
Figure BDA0002834470720000041
the model (3) is equivalent to the following form:
Figure BDA0002834470720000042
3) when λ ═ 1, to solve the first constrained optimization model in (4), we convert to the following form:
Figure BDA0002834470720000043
wherein | x | purpleD＝＜Dx,x>(ii) a D is a diagonal matrix having diagonal elements of
Figure BDA0002834470720000044
And for all i' ═ 1,2, ·, M,
Figure BDA0002834470720000045
γithe regularization parameters are adopted, and then an alternating direction iteration method (ADMM) is adopted to solve the model (5), wherein the iteration format is as follows:
Figure BDA0002834470720000046
t is a parameter introduced by the ADMM algorithm;
in order to avoid the deficiency of solving the inverse of the system matrix A in the subproblems about the first variable f or solving the subproblems by adopting an iteration method, the idea of approaching alternating linearization is embedded, and the iteration format (6) is converted into an ADMM iteration format approaching alternating linearization, which specifically comprises the following steps:
Figure BDA0002834470720000047
in iterative Format (7)
Figure BDA0002834470720000048
Is a relaxation parameter introduced near linearization; (7) in
Figure BDA0002834470720000049
Iteratively updating the format for the ART; skillfully integrating a classical ART iterative algorithm into the linear;
then the sub-problem optimal solution is found for the iterative format (7), which is as follows:
Figure BDA0002834470720000051
wherein WTRepresenting the inverse non-downsampled contourlet transform,
Figure BDA0002834470720000052
4) when λ is 0, to solve the second constrained optimization model in (4), the following form is converted:
Figure BDA0002834470720000053
where D is a diagonal matrix having diagonal elements of
Figure BDA0002834470720000054
And for all i' ═ 1,2, ·, M,
Figure BDA0002834470720000055
mu is a regularization parameter, and the idea of near linearization is adopted to convert the problem into:
Figure BDA0002834470720000056
wherein
Figure BDA0002834470720000057
Equation (10) is solved by the minimization method of L0, and the iteration format is as follows:
Figure BDA0002834470720000058
Figure BDA0002834470720000059
the iterative form of (1) is as follows: for all image coordinates i ', j'
Figure BDA00028344707200000510
Wherein
Figure BDA00028344707200000511
Which represents the fourier transform of the signal,
Figure BDA00028344707200000512
which represents the inverse fourier transform, is used,
Figure BDA00028344707200000513
represents the complex conjugate of the Fourier transform +x,▽yGradient operators respectively representing x and y; beta control
Figure BDA00028344707200000514
A parameter for similarity, κ (κ > 1) denotes a parameter controlling the rate of β growth; n represents
Figure BDA00028344707200000515
The number of iterations of the algorithm is,
Figure BDA00028344707200000516
the stopping criterion of the algorithm is that beta is larger than a preset parameter beta before iterationmaxWhen is coming into contact with
Figure BDA00028344707200000517
The output image after shutdown of the algorithm is fk+1；
Setting the stop standards corresponding to the iteration formats (8) and (11) as the preset iteration times N are respectively reachedite1And Nite2Setting the total iteration number of the whole algorithm as NTotAnd tt is the index of the total iteration times, the iteration flow is as follows:
Figure BDA0002834470720000061
the finite angle CT iterative reconstruction of step S3 includes the following 2 sub-steps:
s31, reconstructing an L0 regularized image based on non-downsampling contourlet transformation;
s32, reconstructing an L0 regularization image based on gradient transformation;
wherein S31 comprises the following 3 sub-steps:
s311, combining ART iterative reconstruction and non-subsampled contourlet inverse transformation to obtain a preliminary reconstruction result;
s312, carrying out hard threshold processing on the high-frequency part and the low-frequency part in a non-subsampled contourlet transform domain according to the obtained preliminary result to inhibit artifacts and noises and simultaneously carrying out boundary protection; s313, updating the dual variable v;
wherein S32 comprises the following 2 sub-steps:
s321, ART iterative reconstruction, wherein an initial image is an output result of S31;
s322, ART result is processed
Figure BDA0002834470720000062
The smoothing processing is minimized to further inhibit artifacts and noise, and the smoothing processing is carried out; when a certain number of iterations N is reachedTotStopping the iteration, otherwise, repeating the steps S31-S32.
Optionally, the S4 specifically includes: when the iterative algorithm in step S3 stops iterating, a reconstructed image is output.
The invention has the beneficial effects that: according to the method, the sparsity of the image under the non-downsampling contourlet transformation and the image gradient transformation is considered, the L0 norm minimization of the non-downsampling contourlet transformation and the L0 norm 2 optimization targets of the image gradient are set by utilizing the characteristics of multi-scale, multi-resolution decomposition and artifact directivity of the non-downsampling contourlet transformation, and a multi-objective optimization reconstruction model is established. In order to suppress noise, directional artifacts and protect boundaries, the method carries out hard threshold processing on high-frequency and low-frequency parts of a non-downsampling contourlet transform domain on the basis of an ART algorithm; in order to further suppress artifacts and noise and smooth the reconstructed image, an image gradient L0 minimization process is performed on the basis of the ART algorithm. The invention discloses a maximum minimized multi-target finite angle CT image reconstruction method, which is characterized in that a multi-target optimization model is established from a multi-target angle, and the multi-target model is solved by utilizing the maximum minimized method, wherein the method comprises ART iteration, hard threshold processing of a non-downsampling contourlet transform domain and L0 minimization processing of image gradient. By the extremely-minimized multi-target method, the boundary of the CT image is protected, and meanwhile, limited angle artifacts and noise are effectively inhibited, so that the quality of the CT reconstructed image is improved to a great extent.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the means of the instrumentalities and combinations particularly pointed out hereinafter.
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For the purposes of promoting a better understanding of the objects, aspects and advantages of the invention, reference will now be made to the following detailed description taken in conjunction with the accompanying drawings in which:
FIG. 1 is a schematic diagram of a geometric structure of a multi-target finite angle CT image reconstruction method based on maximum minimization;
FIG. 2 is a flow chart of a multi-target finite angle CT image reconstruction method based on maximum minimization.
Detailed Description
The embodiments of the present invention are described below with reference to specific embodiments, and other advantages and effects of the present invention will be easily understood by those skilled in the art from the disclosure of the present specification. The invention is capable of other and different embodiments and of being practiced or of being carried out in various ways, and its several details are capable of modification in various respects, all without departing from the spirit and scope of the present invention. It should be noted that the drawings provided in the following embodiments are only for illustrating the basic idea of the present invention in a schematic way, and the features in the following embodiments and examples may be combined with each other without conflict.
Wherein the showings are for the purpose of illustrating the invention only and not for the purpose of limiting the same, and in which there is shown by way of illustration only and not in the drawings in which there is no intention to limit the invention thereto; to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product; it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The same or similar reference numerals in the drawings of the embodiments of the present invention correspond to the same or similar components; in the description of the present invention, it should be understood that if there is an orientation or positional relationship indicated by terms such as "upper", "lower", "left", "right", "front", "rear", etc., based on the orientation or positional relationship shown in the drawings, it is only for convenience of description and simplification of description, but it is not an indication or suggestion that the referred device or element must have a specific orientation, be constructed in a specific orientation, and be operated, and therefore, the terms describing the positional relationship in the drawings are only used for illustrative purposes, and are not to be construed as limiting the present invention, and the specific meaning of the terms may be understood by those skilled in the art according to specific situations.
Fig. 1 is a schematic geometric structure diagram of a multi-target finite angle CT image reconstruction method based on maximum minimization. As shown in the figure: and establishing a right-hand rectangular coordinate system O-xy by taking the ray source S to the rotation center O, wherein the y axis passes through the straight line between the rotation center O and the ray source S, the direction from the rotation center O to the ray source S is the positive direction, and the x axis is vertical to the y axis and is the positive direction to the right. (x, y) are the coordinates of the reconstructed pixel J and SL represents a ray passing through the reconstructed point J. The axis a represents the position corresponding to the detection unit of the detector, and the positive direction is consistent with the axis x, and the (a, s) represents the coordinate formed by the projection visual angle index s and the detector position a.
Fig. 2 is a flowchart of a multi-target finite angle CT image reconstruction method based on maximum minimization, as shown in the figure: the multi-target finite angle CT image reconstruction method based on the maximum minimization comprises the following steps:
s1, acquiring projection data: rotating the ray source around the rotation center along the scanning track for a limited angle to obtain incomplete projection data;
s2, establishing a multi-target finite angle CT image reconstruction model: when the discrete model is used for reconstruction, f (x, y) corresponding to all coordinates (x, y) as shown in fig. 1 needs to be rearranged into a 1-dimensional vector f according to the dimension of y, wherein the dimension of f is N × 1, where N is N1×n2，n1Is the dimension of f (x, y) in the x direction, n2Is the dimension of f (x, y) in the y direction. Then, all coordinates (a, s) are associated with projection data gδ(a, s) are rearranged into a 1-dimensional vector g according to the dimension of sδColumn vector gδHas a dimension of M × 1, wherein M ═ M1×m2，m1Is gδ(a, s) dimension in the a-direction, m2Is gδ(a, s) dimension in the s direction. In accordance with the present invention, it is contemplated to use a non-downsampling contourlet transform to decompose a reconstructed image into a low frequency portion and a high frequency portion, resulting in a multi-scale, multi-resolution decomposition of the image. Is composed ofNoise and artifacts are suppressed and partial detail loss due to over-smoothing is avoided by applying L0 sparse regularization constraints on the coefficients of the non-downsampled contourlet transform. Since the reconstructed image may be less smooth to preserve details, L0 regularization of the image gradients is used for the constraint to make the image smooth. The model established from the multi-target angle is as follows:
Figure BDA0002834470720000081
wherein A ∈ RM×NIs a finite angle CT system matrix, f belongs to RN×1Is the image to be reconstructed, gδ∈RM×1Is finite angle CT projection data and Ω is the convex set (Ω: { f ≧ 0 }). λ is the relaxation parameter, W is the non-downsampled contourlet transform, and i is the index of the subband. | beta | | non-conducting phosphor0Is the number of non-0 elements of beta +(v +xβ,▽yβ)，▽xβ,▽yThe component of β is in the form of +(vxβ)i′,j′＝βi′,j′-βi′-1,j′,(▽yβ)i′，j′＝βi′,j′-βi′,j′-1. When non-downsampling contourlet transformation is carried out, f is rearranged into 1 2-dimensional matrix f (x, y), then the 2-dimensional matrix is subjected to the non-downsampling contourlet transformation, and after the non-downsampling contourlet transformation is finished, f (x, y) is rearranged into a 1-dimensional vector f according to the dimension of y.
S3, finite angle CT iterative reconstruction: solving the model (1) by adopting a maximum minimization method according to the established model (1);
the step S3 finite angle CT iterative reconstruction specifically includes:
firstly, converting the model (1) into a single-target optimization model by a maximum minimization method, wherein the form of the model is as follows:
Figure BDA0002834470720000091
then, in order to effectively solve the single-target optimization model, taking lambda as two special values, firstly, taking lambda as 1, and solving a corresponding optimization problem; then lambda is equal to 0, and the corresponding optimization problem is solved. The above-described process of taking the value of λ is repeated in order to make the final solution converge. The single-objective optimization model is then simplified in the form:
Figure BDA0002834470720000092
the model (3) is equivalent to the following form:
Figure BDA0002834470720000093
5) when λ ═ 1, to solve the first constrained optimization model in (4), we convert to the following form:
Figure BDA0002834470720000094
wherein | x | purpleD＝<Dx,x>. D is a diagonal matrix having diagonal elements of
Figure BDA0002834470720000095
And for all i' ═ 1,2, ·, M,
Figure BDA0002834470720000096
γithe regularization parameters are adopted, and then an alternating direction iteration method (ADMM) is adopted to solve the model (5), wherein the iteration format is as follows:
Figure BDA0002834470720000097
t is a parameter introduced by the ADMM algorithm.
In order to avoid the defect of solving the inverse of the system matrix A in the subproblem about the first variable f or solving the subproblem by adopting an iteration method, the invention embeds the idea of adjacent alternate linearization, and converts the iteration format (6) into an ADMM iteration format of adjacent alternate linearization, and the specific steps are as follows:
Figure BDA0002834470720000101
in iterative Format (7)
Figure BDA0002834470720000102
Is the relaxation parameter introduced adjacent to the linearization. (7) In
Figure BDA0002834470720000103
The format is iteratively updated for ART. The classical ART iterative algorithm is ingeniously incorporated by approaching alternating linearization.
Then the sub-problem optimal solution is found for the iterative format (7), which is as follows:
Figure BDA0002834470720000104
wherein (W)TRepresenting an inverse non-downsampled contourlet transform),
Figure BDA0002834470720000105
6) when λ is 0, to solve the second constrained optimization model in (4), the following form is converted:
Figure BDA0002834470720000106
where D is a diagonal matrix having diagonal elements of
Figure BDA0002834470720000107
And for all i' ═ 1,2, ·, M,
Figure BDA0002834470720000108
mu is a regularization parameter, and the idea of near linearization is adopted to convert the problem into:
Figure BDA0002834470720000109
wherein
Figure BDA00028344707200001010
Equation (10) is solved by the minimization method of L0, and the iteration format is as follows:
Figure BDA00028344707200001011
Figure BDA00028344707200001012
the iterative form of (1) is as follows: for all image coordinates i ', j'
Figure BDA0002834470720000111
Wherein
Figure BDA0002834470720000112
Which represents the fourier transform of the signal,
Figure BDA0002834470720000113
which represents the inverse fourier transform, is used,
Figure BDA0002834470720000114
represents the complex conjugate of the Fourier transform +x,▽yGradient operators respectively representing x and y; beta control
Figure BDA0002834470720000115
A parameter for similarity, κ (κ > 1) denotes a parameter controlling the rate of β growth; n represents
Figure BDA0002834470720000116
The number of iterations of the algorithm is,
Figure BDA0002834470720000117
the stopping criterion of the algorithm is that beta is larger than a preset parameter beta before iterationmaxWhen is coming into contact with
Figure BDA0002834470720000118
The output image after shutdown of the algorithm is fk+1。
Setting the stop standards corresponding to the iteration formats (8) and (11) as the preset iteration times N are respectively reachedite1And Nite2Setting the total iteration number of the whole algorithm as NTot(tt is the index of the total number of iterations), the iteration flow is as follows:
Figure BDA0002834470720000119
according to the above method, the finite angle CT iterative reconstruction of step S3 includes the following 2 sub-steps: s31, reconstructing an L0 regularized image based on non-downsampling contourlet transformation; s32, L0 regularization image reconstruction based on gradient transformation. Wherein S31 comprises the following 3 sub-steps: s311, combining ART iterative reconstruction and non-subsampled contourlet inverse transformation to obtain a preliminary reconstruction result; s312, carrying out hard threshold processing on the high-frequency part and the low-frequency part in a non-subsampled contourlet transform domain according to the obtained preliminary result to inhibit artifacts and noises and simultaneously carrying out boundary protection; s313, updating the dual variable v. Wherein S32 comprises the following 2 sub-steps: s321, ART iterative reconstruction, wherein an initial image is an output result of S31; s322, ART result is processed
Figure BDA00028344707200001110
Smoothing is minimized to further suppress artifacts and noise and is performed. When a certain number of iterations N is reachedTotStopping the iteration, otherwise, repeating the steps S31-S32.
And S4, outputting a reconstructed image. When the iterative algorithm in step S3 stops iterating, a reconstructed image is output.
Finally, the above embodiments are only intended to illustrate the technical solutions of the present invention and not to limit the present invention, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that modifications or equivalent substitutions may be made on the technical solutions of the present invention without departing from the spirit and scope of the technical solutions, and all of them should be covered by the claims of the present invention.

Claims (5)

1.A multi-target finite angle CT image reconstruction method based on maximum minimization is characterized in that: the method comprises the following steps:
s1: acquiring projection data:
s2: establishing a multi-target finite angle CT image reconstruction model;
s3: finite angle CT iterative reconstruction;
s4: and outputting the reconstructed image.
2. The maximum minimization-based multi-target finite angle CT image reconstruction method according to claim 1, characterized in that: the S1 specifically includes: incomplete projection data is obtained by rotating the source about a center of rotation along the scan trajectory through a limited angle.
3. The maximum minimization-based multi-target finite angle CT image reconstruction method according to claim 1, characterized in that: the S2 specifically includes: when the discrete model is used for reconstruction, f (x, y) corresponding to coordinates (x, y) needs to be rearranged into a 1-dimensional vector f according to the dimension of y, wherein the dimension of the vector f is N × 1, and N is N ═ N1×n2，n1Is the dimension of f (x, y) in the x direction, n2Is the dimension of f (x, y) in the y direction; then, all coordinates (a, s) are associated with projection data gδ(a, s) are rearranged into a 1-dimensional vector g according to the dimension of sδColumn vector gδHas a dimension of M × 1, wherein M ═ M1×m2，m1Is gδ(a, s) dimension in the a-direction, m2Is gδ(a, s) a dimension in the s direction; according to the invention, the reconstructed image is decomposed into a low-frequency part and a high-frequency part by adopting non-subsampled contourlet transform, so that the image is decomposed in multiple scales and multiple resolutions;
in order to suppress noise and artifact and avoid partial detail loss caused by over-smoothness, L0 sparse regularization constraint is carried out on coefficients of non-downsampling contourlet transform; for image smoothness, L0 regularization of image gradients is used for constraint;
the model built from the multi-target perspective is as follows:
Figure FDA0002834470710000011
wherein A ∈ RM×NIs a finite angle CT system matrix, f belongs to RN×1Is the image to be reconstructed, gδ∈RM×1Is finite angle CT projection data, and Ω is a convex set (Ω: { f ≧ 0 }); λ is the relaxation parameter, W is the non-downsampled contourlet transform, i is the index of the subband; | beta | | non-conducting phosphor0The number of non-0 elements of beta is counted,
Figure FDA0002834470710000012
Figure FDA0002834470710000013
is in the form of a component of
Figure FDA0002834470710000014
When non-downsampling contourlet transformation is carried out, f is rearranged into 1 2-dimensional matrix f (x, y), then the 2-dimensional matrix is subjected to the non-downsampling contourlet transformation, and after the non-downsampling contourlet transformation is finished, f (x, y) is rearranged into a 1-dimensional vector f according to the dimension of y.
4. The maximum minimization-based multi-target finite angle CT image reconstruction method according to claim 3, characterized in that: the S3 specifically includes:
solving the model (1) by adopting a maximum minimization method according to the established model (1);
the step S3 finite angle CT iterative reconstruction specifically includes:
firstly, converting the model (1) into a single-target optimization model by a maximum minimization method, wherein the form of the model is as follows:
Figure FDA0002834470710000021
then, in order to effectively solve the single-target optimization model, taking lambda as two special values, firstly, taking lambda as 1, and solving a corresponding optimization problem; then lambda is equal to 0, and the corresponding optimization problem is solved; repeating the value-taking process of the lambda in order to make the final solution converge; the single-objective optimization model is then simplified in the form:
Figure FDA0002834470710000022
the model (3) is equivalent to the following form:
Figure FDA0002834470710000023
1) when λ ═ 1, to solve the first constrained optimization model in (4), we convert to the following form:
Figure FDA0002834470710000024
wherein | x | purpleD＝<Dx,x>(ii) a D is a diagonal matrix having diagonal elements of
Figure FDA0002834470710000025
And for all i' ═ 1,2, ·, M,
Figure FDA0002834470710000026
γithe regularization parameters are adopted, and then an alternating direction iteration method ADMM is adopted to solve the model (5), and the iteration format is as follows:
Figure FDA0002834470710000027
t is a parameter introduced by the ADMM algorithm;
in order to avoid the deficiency of solving the inverse of the system matrix A in the subproblems about the first variable f or solving the subproblems by adopting an iteration method, the idea of approaching alternating linearization is embedded, and the iteration format (6) is converted into an ADMM iteration format approaching alternating linearization, which specifically comprises the following steps:
Figure FDA0002834470710000031
in iterative Format (7)
Figure FDA0002834470710000032
Is a relaxation parameter introduced near linearization; (7) in
Figure FDA0002834470710000033
Iteratively updating the format for the ART; skillfully integrating a classical ART iterative algorithm into the linear;
then the sub-problem optimal solution is found for the iterative format (7), which is as follows:
Figure FDA0002834470710000034
wherein WTRepresenting the inverse non-downsampled contourlet transform,
Figure FDA0002834470710000035
2) when λ is 0, to solve the second constrained optimization model in (4), the following form is converted:
Figure FDA0002834470710000036
whereinDIs aA diagonal matrix having diagonal elements of
Figure FDA0002834470710000037
And for all i' ═ 1,2, ·, M,
Figure FDA0002834470710000038
mu is a regularization parameter, and the idea of near linearization is adopted to convert the problem into:
Figure FDA0002834470710000039
wherein
Figure FDA00028344707100000310
Equation (10) is solved by the minimization method of L0, and the iteration format is as follows:
Figure FDA00028344707100000311
Figure FDA00028344707100000312
the iterative form of (1) is as follows: for all image coordinates i ', j'
Figure FDA0002834470710000041
Wherein
Figure FDA0002834470710000042
Which represents the fourier transform of the signal,
Figure FDA0002834470710000043
which represents the inverse fourier transform, is used,
Figure FDA0002834470710000044
the complex conjugate of the fourier transform is represented,
Figure FDA0002834470710000045
gradient operators respectively representing x and y; beta control
Figure FDA0002834470710000046
A parameter for similarity, κ (κ > 1) denotes a parameter controlling the rate of β growth; n represents
Figure FDA0002834470710000047
The number of iterations of the algorithm is,
Figure FDA0002834470710000048
the stopping criterion of the algorithm is that beta is larger than a preset parameter beta before iterationmaxWhen is coming into contact with
Figure FDA0002834470710000049
The output image after shutdown of the algorithm is fk+1；
Setting the stop standards corresponding to the iteration formats (8) and (11) as the preset iteration times N are respectively reachedite1And Nite2Setting the total iteration number of the whole algorithm as NTotAnd tt is the index of the total iteration times, the iteration flow is as follows:
For tt＝1:NTot
For k＝1:Nite1+Nite2
If k≤Nite1
Figure FDA00028344707100000410
else
Figure FDA00028344707100000411
End
End
End
the finite angle CT iterative reconstruction of step S3 includes the following 2 sub-steps:
s31, reconstructing an L0 regularized image based on non-downsampling contourlet transformation;
s32, reconstructing an L0 regularization image based on gradient transformation;
wherein S31 comprises the following 3 sub-steps:
s311, combining ART iterative reconstruction and non-subsampled contourlet inverse transformation to obtain a preliminary reconstruction result;
s312, carrying out hard threshold processing on the high-frequency part and the low-frequency part in a non-subsampled contourlet transform domain according to the obtained preliminary result to inhibit artifacts and noises and simultaneously carrying out boundary protection; s313, updating the dual variable v;
wherein S32 comprises the following 2 sub-steps:
s321, ART iterative reconstruction, wherein an initial image is an output result of S31;
s322, ART result is processed
Figure FDA0002834470710000051
The smoothing processing is minimized to further inhibit artifacts and noise, and the smoothing processing is carried out; when a certain number of iterations N is reachedTotStopping the iteration, otherwise, repeating the steps S31-S32.
5. The maximum minimization-based multi-target finite angle CT image reconstruction method according to claim 4, characterized in that: the S4 specifically includes: when the iterative algorithm in step S3 stops iterating, a reconstructed image is output.
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