CN112070704B - Dual regularization limited angle CT image reconstruction method based on tight wavelet frame - Google Patents
Dual regularization limited angle CT image reconstruction method based on tight wavelet frame Download PDFInfo
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Abstract
The invention relates to a double regularization limited angle CT image reconstruction method based on a tight wavelet frame, and belongs to the field of image processing. The method comprises the following steps: s1: collecting projection data; s2: establishing a double regularization limited angle CT reconstruction model; s3: limited angle CT iterative reconstruction; s4: outputting a reconstructed image; when the iterative reconstruction algorithm in step S3 converges, a reconstructed image is output. The double regularization limited angle CT image reconstruction method based on the tight wavelet frame disclosed by the invention comprises the steps of carrying out hard threshold processing on a high-frequency part and carrying out image TV (television) minimum smoothing processing on a low-frequency part, and after the double regularization mechanism processing, not only can the limited angle artifact and noise of a CT image be effectively inhibited, but also the boundary can be effectively protected, so that the quality of the CT reconstructed image is improved.
Description
Technical Field
The invention belongs to the field of image processing, and relates to a double regularization limited angle CT image reconstruction method based on a tight wavelet frame.
Background
The method is limited by factors such as a scanning environment, the structure of a scanned target and the like, and sometimes the target to be detected can be scanned only in a certain angle range, and the acquisition of projection data is incomplete. For example: in-service pipeline imaging, large-size object detection, C-arm CT (Computed Tomography), and the like. This scanning situation is called limited angle CT scanning. Aiming at the reconstruction problem of limited angle CT scanning, the adoption of a traditional filtered back projection reconstruction algorithm can lead to the occurrence of a plurality of artifacts on the reconstructed image, so that some important structural information is lost or covered, and the accuracy of nondestructive detection or the diagnosis of symptoms by doctors are seriously affected. Therefore, aiming at the problem of limited angle CT image reconstruction, how to reconstruct a high-quality CT image which meets the nondestructive testing standard or the doctor diagnosis requirement has great practical significance.
In the prior art, the conventional algebraic reconstruction algorithms ART (Algebraic Reconstruction Technique), SART (Simultaneous Algebraic Reconstruction Technique), SIRT (Simultaneous Iterative Reconstruction Technique) and the like cause obvious artifacts to appear in the reconstructed image. The image reconstruction algorithm based on TV (Total Variation) has a better effect on sparse angle CT reconstruction, but can not effectively inhibit artifacts and better protect boundary structures for smaller limited angle scanning. Yu Wei introduces image gradient L0 regularization, effectively suppressing artifacts to some extent and avoiding excessive smoothness. Wang Chengxiang et al use the L0 pseudo-norm of the image wavelet transform as regularization, and use the multi-scale, multi-resolution characteristics of the wavelet transform to address the problem of limited angle CT reconstruction, which can suppress artifacts to some extent and protect boundaries.
Patent application publication No. CN 107978005a discloses "a limited angle CT image reconstruction algorithm based on guaranteed boundary diffusion and smoothing". Firstly, reconstructing by using projection data, then respectively carrying out gradient L0 boundary-preserving diffusion correction on the reconstructed image in the x and y directions, and finally repeating iterative correction until the shutdown standard is met. Although the method described in the above patent application is capable of preserving boundary smoothness and eliminating linear artifacts that may be introduced by diffusion. But still have the following drawbacks: the method described in the above patent application only considers the image x, y axis to perform gradient L0 boundary-preserving diffusion correction, but does not consider the characteristic of multi-directionality of the artifacts of the limited angle CT image. The gradient transformation of the image only has high frequency information of the high frequency part of the image, the image with limited angle artifacts has artifacts in the high frequency part and distortion in the low frequency part, and the method described in the patent application does not consider the characteristics of multi-scale and multi-resolution of the image.
Patent application publication No. CN 110717959a discloses "limited angle x-ray CT image reconstruction method and apparatus based on curvature constraints". Firstly, reconstructing by using projection data, secondly, performing image gradient L0 regularized sparse constraint on the reconstructed image, then performing curvature constraint on the result of the sparse constraint, and finally repeating iteration until the shutdown standard is met. Although the method described in the above patent application can overcome the problem of boundary blurring or the existence of a step effect in the existing limited angle CT reconstruction algorithm. But still have the following drawbacks: (1) The method described in the above patent application only considers the gradient transformation of the image, only the high frequency information of the high frequency part of the image; (2) The method described in the above patent application takes into account curvature constraints, only the variations in the high frequency profile. The method described in the above patent application does not take into account the multi-scale and multi-resolution properties of the image, not only the high frequency part of the image with limited angular artefacts has artefacts and the low frequency part has distortions.
Publication number CN 109697691a discloses "a limited angle projection reconstruction method based on a bilinear term optimization based on L0 norm and singular value thresholding. Firstly, correcting by utilizing an SART algorithm and errors to reconstruct, secondly, regularizing the image gradient L0 of the reconstructed image to sparse constraint, and then, carrying out singular value decomposition and soft threshold further processing on the image after sparse constraint. And finally, repeating the iteration until the shutdown standard is met. Although the method described in the above patent application enables restoration of CT image contours, the reduction of limited angle artifacts. But still have the following drawbacks: the method described in the above patent application only considers x, y-axis artifacts, but does not consider the multi-directional nature of the artifacts of the limited angle CT images. The gradient transformation of the image only has high frequency information of the high frequency part of the image, the image with limited angle artifacts has artifacts in the high frequency part and distortion in the low frequency part, and the method described in the patent application does not consider the characteristics of multi-scale and multi-resolution of the image.
Most of the existing limited angle CT optimal reconstruction methods either only consider the high frequency part of the image, or do not consider the artifacts of the reconstructed image of the limited angle CT to be effectively processed under the multi-scale and multi-resolution decomposition, and not only exist in the high frequency part but also exist in the low frequency part. The invention considers the sparsity of the image under the transformation of the tight wavelet frame, utilizes the B-spline tight wavelet frame to carry out multi-scale and multi-resolution decomposition on the image, adopts a double regularization constraint mechanism to carry out L0 sparse constraint on the high-frequency part, and adopts TV regularization to carry out smooth and sparse constraint on the low-frequency part, thereby improving the quality of CT reconstructed images.
Disclosure of Invention
In view of the above, the present invention aims to provide a dual regularization limited angle CT image reconstruction method based on a tight wavelet frame, which can effectively inhibit artifacts of a reconstructed image and protect image boundaries, thereby improving the quality of a CT reconstructed image.
In order to achieve the above purpose, the present invention provides the following technical solutions:
a double regularization limited angle CT image reconstruction method based on a tight wavelet frame comprises the following steps:
s1: collecting projection data;
s2: establishing a double regularization limited angle CT reconstruction model;
s3: limited angle CT iterative reconstruction;
s4: outputting a reconstructed image; when the iterative reconstruction algorithm in step S3 converges, a reconstructed image is output.
Optionally, the S1 specifically is:
under the control of the control and image processing system, the ray source and the linear array detector are rotated around the center to be detected along the track by a limited angle to obtain incomplete projection data, and then the incomplete projection data are transmitted to the control and image processing system for storage.
Optionally, the S2 specifically is:
when reconstructing using the discrete model, all (x, y) corresponding reconstructed pixels f (x, y) are first transformed into a long column vector f according to the dimension of y, where n=n×1 1 ×n 2 ，n 1 Is the dimension of f (x, y) in the x direction, n 2 Is the dimension of f (x, y) in the y direction;
then, the projection data g of all projection view angle indexes s corresponding to the linear array detector position a are obtained δ (a, s) converting it into a long column vector g in accordance with the dimension of s δ Column vector g δ Is M x 1, where m=m 1 ×m 2 ，m 1 G is g δ (a, s) dimension in the a direction, m 2 G is g δ (a, s) dimension in the s-direction, i.e., total projection view angle number;
b spline tight wavelet frame transformation is adopted to decompose the reconstructed image into a low-frequency part and a high-frequency part, so that the image is subjected to multi-scale and multi-resolution decomposition;
performing L0 sparse regularization constraint on the high-frequency part of the B spline tight wavelet frame transformation to suppress noise and artifacts in high frequency, and performing TV regularization constraint on the low-frequency part of the B spline tight wavelet frame transformation to smooth a reconstructed image and suppress artifacts in the low-frequency part;
the model built is as follows:
wherein A is E R M×N Is a finite angle CT system matrix, f epsilon R N×1 Is the image to be reconstructed g δ ∈R M×1 Is limited angle CT projection data, Ω is a convex set (Ω: = { f|f. Gtoreq.0 }), |x| D ＝<Dx,x>The method comprises the steps of carrying out a first treatment on the surface of the D is a diagonal matrix with diagonal elements ofAnd for all i' =1, 2,..m,/v>λ i Is a regularization parameter, W is a B-spline tight wavelet framework; omega shape 1 Is an index set of high frequency sub-bands, Ω 2 Is the index set of the low frequency sub-band, Ω 1 ∪Ω 2 An index set representing all B-spline tight wavelet frame transform subbands; beta 0 The number of non-0 elements of beta is counted,
and (3) during B spline tight wavelet frame transformation, arranging the y dimension of (x, y) in f into 1 2-dimensional matrix f (x, y), then performing B spline tight wavelet frame transformation on the 2-dimensional matrix, and after performing B spline tight wavelet frame transformation inverse transformation, converting f (x, y) into a long column vector f according to the y dimension again.
Optionally, the S3 specifically is:
according to the established model (1), adopting an alternating direction iteration ADMM method to solve the model (1);
the specific process of the limited angle CT iterative reconstruction in the step S3 is as follows:
firstly, the model (1) is converted into the following iterative format by an iterative method of alternating directions:
k in the iteration format (2) represents the iteration times, ρ is a relaxation parameter, α is an auxiliary variable, v is a dual variable, and t is a parameter introduced when ADMM splits;
secondly, to avoid the disadvantage of solving the system matrix a in the sub-problem about the first variable f or solving the sub-problem using an iterative method, the iterative format (2) is converted into an ADMM iterative format close to the alternate linearization, specifically as follows:
in iterative format (3)μ is the relaxation parameter introduced by the proximity linearization; (3) In (a)Updating the format for ART iteration; the classical ART iterative algorithm is skillfully integrated into the method through adjacent alternate linearization;
then, to solve the sub-problem α, α is decomposed into high and low frequenciesSolving separately, the iterative format becomes the following equivalent:
finally, the optimal solution of the sub-problem of the iterative format (4) is obtained, and the iterative format is as follows:
wherein W is T Representing the inverse transform of the B-spline tight wavelet framework,representing an image TV minimal smoothness algorithm>The image TV minimum smoothness algorithm iteration format is as follows:
wherein the method comprises the steps ofτ is a parameter introduced by the split Bregman algorithm, G n The component forms of (a) are:
n represents the number of iterations of the TV minimum smoothing algorithm.
Optionally, the iterative format includes the following steps:
s31: ART iterative reconstruction and B spline tight wavelet frame inverse transformation are linearly combined and ensure that the combined image is non-negative, and a preliminary reconstruction result, namely a 1 st equation of a formula (5) is obtained;
s32: performing hard threshold processing on the high-frequency part in the B spline tight wavelet frame transformation domain, and inhibiting noise and directivity artifact of the high-frequency part, namely the 2 nd equation of the formula (5);
s33: performing TV minimization smoothing processing on the low-frequency part in the B spline tight wavelet frame transformation domain, and inhibiting artifacts of the low-frequency part and keeping the smoothness of the image, namely a 3 rd equation of the formula (5);
s34: the dual variable v is updated, namely the 4 th equation of the formula (5); stopping iteration when a certain number of iterations is reached, otherwise repeating steps S31-S34.
The invention has the beneficial effects that: the invention discloses a double regularization limited angle CT image reconstruction method based on a tight wavelet frame, and relates to a limited angle CT reconstruction technology. According to the invention, under the condition of multi-scale and multi-resolution decomposition, the artifacts of the limited angle CT reconstructed image are considered to be not only high-frequency parts but also low-frequency parts, and the reconstructed image is decomposed into the low-frequency parts and the high-frequency parts by adopting B-spline tight wavelet frame transformation, so that the artifacts are decomposed by multiple scales. In order to suppress noise and directional artifacts in high frequencies, an image TV regularization constraint is applied to the low frequency portion of the B-spline tight wavelet frame transform by applying an L0 sparse regularization constraint to the high frequency portion of the B-spline tight wavelet frame transform in order to smooth the reconstructed image and suppress artifacts in the low frequency portion. The double regularization limited angle CT image reconstruction method based on the tight wavelet frame disclosed by the invention comprises the steps of carrying out hard threshold processing on a high-frequency part and carrying out image TV (television) minimum smoothing processing on a low-frequency part, and after the double regularization mechanism processing, not only can the limited angle artifact and noise of a CT image be effectively inhibited, but also the boundary can be effectively protected, so that the quality of the CT reconstructed image is improved.
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 objects and other advantages of the invention may be realized and obtained by means of the instrumentalities and combinations particularly pointed out in the specification.
Drawings
For the purpose of making the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in the following preferred detail with reference to the accompanying drawings, in which:
FIG. 1 is a schematic diagram of a scanning structure of an object to be inspected according to the present invention;
FIG. 2 is a schematic diagram of the geometry of the limited angle CT reconstruction algorithm of the present invention;
FIG. 3 is a flow chart of dual regularization limited angle CT image reconstruction based on a tight wavelet framework.
Reference numerals: 1-ray source, 2-linear array detector, 3-object to be detected, 4-track and 5-image processing system.
Detailed Description
Other advantages and effects of the present invention will become apparent to those skilled in the art from the following disclosure, which describes the embodiments of the present invention with reference to specific examples. The invention may be practiced or carried out in other embodiments that depart from the specific details, and the details of the present description may be modified or varied from the spirit and scope of the present invention. It should be noted that the illustrations provided in the following embodiments merely illustrate the basic idea of the present invention by way of illustration, and the following embodiments and features in the embodiments may be combined with each other without conflict.
Wherein the drawings are for illustrative purposes only and are shown in schematic, non-physical, and not intended to limit the invention; for the purpose of better illustrating embodiments of the invention, certain elements of the drawings may be omitted, enlarged or reduced and do not represent the size of the actual product; it will be appreciated 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 numbers in the drawings of embodiments of the invention correspond to the same or similar components; in the description of the present invention, it should be understood that, if there are terms such as "upper", "lower", "left", "right", "front", "rear", etc., that indicate an azimuth or a positional relationship based on the azimuth or the positional relationship shown in the drawings, it is only for convenience of describing the present invention and simplifying the description, but not for indicating or suggesting that the referred device or element must have a specific azimuth, be constructed and operated in a specific azimuth, so that the terms describing the positional relationship in the drawings are merely for exemplary illustration and should not be construed as limiting the present invention, and that the specific meaning of the above terms may be understood by those of ordinary skill in the art according to the specific circumstances.
Fig. 1 is a schematic diagram of a scanning structure of an object to be inspected according to the present invention: before data acquisition, the ray source 1 and the linear array detector 2 are respectively arranged at two sides of the object 3 to be detected, so that a fan beam ray beam generated by the ray source 1 can cover the object 3 to be detected. When data is acquired, the ray source 1 rotates along the track 4, and meanwhile the linear array detector 2 synchronously rotates, so that the ray source 1 and the linear array detector 2 can only rotate within a limited angle to acquire the data due to the limitation of a scanning scene. The acquired data are transmitted to the control and image processing system 5 for reconstruction.
FIG. 2 is a schematic diagram of the geometry of the limited angle CT reconstruction algorithm of the present invention: a space right-hand rectangular coordinate system O-xy is established by taking a vertical foot from the ray source 1 to a central axis of the object 3 to be detected as a coordinate origin O, a y-axis is a connecting line of the origin and the ray source 1, the X-axis points to the ray source 1 from the origin to be in a positive direction, and the x-axis is an axis vertical to the y-axis and is in a positive direction to the right. With the origin of coordinates O as the center of rotation, (x, y) represents the coordinates of the reconstructed pixel J, the source 1 is located at S, SL represents the center ray of the fan beam ray, and SF represents one ray passing through the reconstructed point J. The a-axis represents the position of the detection unit in the linear array detector and the positive direction is consistent with the x-axis of the spatial coordinate system, and (a, s) represents the position a of the linear array detector corresponding to the projection view angle index s.
FIG. 3 is a flow chart of dual regularization limited angle CT image reconstruction based on a tight wavelet framework: the double regularization limited angle CT image reconstruction method based on the tight wavelet frame comprises the following steps:
s1, collecting projection data: under the control of the control and image processing system 5, firstly, the radiation source 1 and the linear array detector 2 are rotated around the center to be detected along the track 4 by a limited angle to obtain incomplete projection data, and then the incomplete projection data are transmitted to the control and image processing system 5 for storage;
s2, establishing a double regularization limited angle CT reconstruction model: when reconstructing using the discrete model, all (x, y) corresponding reconstructed pixels f (x, y) as shown in fig. 2 need to be first converted into a long column vector f according to the dimension of y, where the dimension of the column vector f is n×1, where n=n 1 ×n 2 ，n 1 Is the dimension of f (x, y) in the x direction, n 2 Is the dimension of f (x, y) in the y direction. Then, as shown in FIG. 2Projection data g of all projection view angle indexes s corresponding to the linear array detector position a δ (a, s) converting it into a long column vector g in accordance with the dimension of s δ Column vector g δ Is M x 1, where m=m 1 ×m 2 ，m 1 G is g δ (a, s) dimension in the a direction, m 2 G is g δ (a, s) dimension in the s-direction, i.e. the total projection view angle number. According to the invention, the reconstructed image is decomposed into a low-frequency part and a high-frequency part by adopting B-spline tight wavelet frame transformation, so that the image is decomposed in a multi-scale and multi-resolution mode. In order to suppress noise and artifacts in high frequencies, the low frequency portion of the B-spline tight wavelet frame transform is subjected to TV regularization constraints by subjecting the high frequency portion of the B-spline tight wavelet frame transform to L0 sparse regularization constraints in order to smooth the reconstructed image and suppress artifacts in the low frequency portion. The model established by the invention is as follows:
wherein A is E R M×N Is a finite angle CT system matrix, f epsilon R N×1 Is the image to be reconstructed g δ ∈R M×1 Is limited angle CT projection data, Ω is a convex set (Ω: = { f|f. Gtoreq.0 }), |x| D ＝<Dx,x>. D is a diagonal matrix with diagonal elements ofAnd for all i' =1, 2,..m,/v>λ i Is a regularization parameter, W is a B-spline tight wavelet framework. Omega shape 1 Is an index set of high frequency sub-bands, Ω 2 Is the index set of the low frequency sub-band, Ω 1 ∪Ω 2 An index set representing all B-spline tight wavelet frame transform subbands. Beta 0 The number of non-0 elements of beta is counted, when we do B-spline tight wavelet frame transform, we rank f into 1 2-dimensional matrix according to y-dimension of (x, y) in fig. 2, then do B-spline tight wavelet frame transform to the 2-dimensional matrix, when we do B-spline tight wavelet frame transform inverse transform, we convert f (x, y) into a long column vector f according to y-dimension again.
S3, limited angle CT iterative reconstruction: solving the model (1) by adopting an alternating direction iteration (ADMM, alternating direction method of multipliers) method according to the established model (1);
the specific process of the limited angle CT iterative reconstruction in the step S3 is as follows:
firstly, the model (1) is converted into the following iterative format by an iterative method of alternating directions:
k in the iterative format (2) represents the iteration number, ρ is a relaxation parameter, α is an auxiliary variable, v is a dual variable, and t is a parameter introduced when the ADMM splits.
Secondly, in order to avoid the defect that the inverse of the system matrix A is solved or the sub-problem is solved by adopting an iteration method in the sub-problem about the first variable f, the invention embeds the idea of adjacent alternate linearization, and converts the iteration format (2) into an ADMM iteration format of adjacent alternate linearization, which is specifically as follows:
in iterative format (3)μ is the relaxation parameter introduced by the proximity linearization. (3) In (a)The format is iteratively updated for ART. The classical ART iterative algorithm is smartly incorporated into it by close-by alternate linearization.
Then, to solve the sub-problem α, we decompose α into high and low frequenciesSolving separately, the iterative format becomes the following equivalent:
finally, the optimal solution of the sub-problem of the iterative format (4) is obtained, and the iterative format is as follows:
wherein (W) T Representing a B-spline tight wavelet frame inverse transform),representing an image TV minimal smoothness algorithm>The image TV minimum smoothness algorithm iteration format is as follows:
wherein the method comprises the steps ofτ is a parameter introduced by the split Bregman algorithm, G n The component forms of (a) are:
n represents the number of iterations of the TV minimum smoothing algorithm.
According to equation (5), step S4 of limited angle CT iterative reconstruction comprises the following 4 sub-steps: s31.ART iterative reconstruction and B spline tight wavelet frame inverse transformation are linearly combined and ensure that the combined image is non-negative, so as to obtain a preliminary reconstruction result (1 st equation of formula (5)); s32, performing hard threshold processing on the high-frequency part in the B spline tight wavelet frame transformation domain, and inhibiting noise and directivity artifact of the high-frequency part (equation 2 of the formula (5)); s33, performing TV minimum smoothing processing on the low-frequency part in a B spline tight wavelet frame transformation domain, and inhibiting artifacts of the low-frequency part and keeping the smoothness of an image (the 3 rd equation of the formula (5)); s34, updating the dual variable v (the 4 th equation of the formula (5)). Stopping iteration when a certain number of iterations is reached, otherwise repeating steps S31-S34.
S4, outputting a reconstructed image. When the iterative reconstruction algorithm in step S3 converges, a reconstructed image is output.
Finally, it is noted that the above embodiments are only for illustrating the technical solution of the present invention and not for limiting the same, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications and equivalents may be made thereto without departing from the spirit and scope of the present invention, which is intended to be covered by the claims of the present invention.
Claims (2)
1.A double regularization limited angle CT image reconstruction method based on a tight wavelet frame is characterized in that: the method comprises the following steps:
s1: collecting projection data;
s2: establishing a double regularization limited angle CT reconstruction model;
s3: limited angle CT iterative reconstruction;
s4: outputting a reconstructed image; outputting a reconstructed image when the iterative reconstruction algorithm in the step S3 converges;
the S1 specifically comprises the following steps:
under the control of the control and image processing system, firstly, a ray source and a linear array detector rotate around the center to be detected along a track for a limited angle to obtain incomplete projection data, and then the incomplete projection data are transmitted to the control and image processing system for storage;
the step S2 is specifically as follows:
when reconstructing using the discrete model, all (x, y) corresponding reconstructed pixels f (x, y) need to be first converted into a column vector f according to the dimension of y, the dimension of the column vector f is n×1, where n=n 1 ×n 2 ，n 1 Is the dimension of f (x, y) in the x direction, n 2 Is the dimension of f (x, y) in the y direction;
then, the projection data g of all projection view angle indexes s corresponding to the linear array detector position a are obtained δ (a, s) converting it into a column vector g in accordance with the dimension of s δ Column vector g δ Is M x 1, where m=m 1 ×m 2 ，m 1 G is g δ (a, s) dimension in the a direction, m 2 G is g δ (a, s) dimension in the s-direction, i.e., total projection view angle number;
b spline tight wavelet frame transformation is adopted to decompose the reconstructed image into a low-frequency part and a high-frequency part, so that the image is subjected to multi-scale and multi-resolution decomposition;
performing L0 sparse regularization constraint on the high-frequency part of the B spline tight wavelet frame transformation to suppress noise and artifacts in high frequency, and performing TV regularization constraint on the low-frequency part of the B spline tight wavelet frame transformation to smooth a reconstructed image and suppress artifacts in the low-frequency part;
the model built is as follows:
wherein A is E R M×N Is a finite angle CT system matrix, f epsilon R N×1 Is the image to be reconstructed g δ ∈R M×1 Is limited angle CT projection data, Ω is a convex set, Ω: = { f|f is not less than 0}, |x| D ＝<Dx,x>The method comprises the steps of carrying out a first treatment on the surface of the D is a diagonal matrix with diagonal elements ofAnd for all i' =1, 2,..m,/v>λ i Is a regularization parameter, W is a B-spline tight wavelet framework; omega shape 1 Is an index set of high frequency sub-bands, Ω 2 Is the index set of the low frequency sub-band, Ω 1 ∪Ω 2 An index set representing all B-spline tight wavelet frame transform subbands; beta 0 The number of non-0 elements of beta is counted,(▽ x β) i′,j′ ＝β i′,j′ -β i′-1,j′ ,(▽ y β) i′，j′ ＝β i′,j′ -β i′,j′-1 ；
when B spline tight wavelet frame transformation is performed, the y dimension of (x, y) in f is arranged into 1 2-dimensional matrix f (x, y), then the 2-dimensional matrix is subjected to B spline tight wavelet frame transformation, and after B spline tight wavelet frame transformation inverse transformation is performed, f (x, y) is converted into a column vector f again according to the y dimension;
the step S3 is specifically as follows:
according to the established model (1), adopting an alternating direction iteration ADMM method to solve the model (1);
the specific process of the limited angle CT iterative reconstruction in the step S3 is as follows:
firstly, the model (1) is converted into the following iterative format by an iterative method of alternating directions:
k in the iteration format (2) represents the iteration times, alpha is an auxiliary variable, v is a dual variable, and t is a parameter introduced when the ADMM splits the variable;
secondly, to avoid the disadvantage of solving the system matrix a in the sub-problem about the first variable f or solving the sub-problem using an iterative method, the iterative format (2) is converted into an ADMM iterative format close to the alternate linearization, specifically as follows:
in the iterative format (3) gamma is calculated,μ is the relaxation parameter introduced by the proximity linearization; (3) Middle->Updating the format for ART iteration; the classical ART iterative algorithm is skillfully integrated into the method through adjacent alternate linearization;
then, to solve the sub-problem α, α is decomposed into high and low frequenciesSolving separately, the iterative format becomes the following equivalent:
finally, the optimal solution of the sub-problem of the iterative format (4) is obtained, and the iterative format is as follows:
wherein W is T Representing the inverse transform of the B-spline tight wavelet framework, representing an image TV minimal smoothness algorithm>The image TV minimum smoothness algorithm iteration format is as follows:
wherein the method comprises the steps ofτ is a parameter introduced by the Split Bregman algorithm, G n The component forms of (a) are:
n represents the number of iterations of the TV minimum smoothing algorithm.
2. The dual regularization limited angle CT image reconstruction method based on a tight wavelet framework of claim 1, wherein: the iterative format comprises the following steps:
s31: ART iterative reconstruction and B spline tight wavelet frame inverse transformation are linearly combined and ensure that the combined image is non-negative, and a preliminary reconstruction result, namely a 1 st equation of a formula (5) is obtained;
s32: performing hard threshold processing on the high-frequency part in the B spline tight wavelet frame transformation domain, and inhibiting noise and directivity artifact of the high-frequency part, namely the 2 nd equation of the formula (5);
s33: performing TV minimization smoothing processing on the low-frequency part in the B spline tight wavelet frame transformation domain, and inhibiting artifacts of the low-frequency part and keeping the smoothness of the image, namely a 3 rd equation of the formula (5);
s34: the dual variable v is updated, namely the 4 th equation of the formula (5); stopping iteration when a certain number of iterations is reached, otherwise repeating steps S31-S34.
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