CN115937351B - Greedy random Kaczmarz image reconstruction method based on accurate step length - Google Patents
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Abstract
The invention relates to a greedy random Kaczmarz image reconstruction method based on accurate step length, which comprises the following steps of 1, splitting images to be reconstructed according to columns; 2. setting an iteration termination condition, a soft threshold function coefficient and an initial iteration vector of a solving process; 3. selecting iterative row vectors to determine a projection direction in a greedy random mode; 4. obtaining an accurate step length by using a linear search method, updating a solution vector, adding 1 to the iteration times, judging whether an iteration termination condition is met, if so, outputting a sparse solution vector obtained by the last iteration, otherwise, turning to a step 3; 5. and (3) respectively carrying out anti-sparse transformation on all the final sparse solution vectors obtained through calculation to obtain final solution vectors, and merging according to columns to obtain a reconstructed image. The invention uses the linear search method to obtain the accurate step length of each iteration on the basis of the Kaczmarz method, reduces the times and calculated amount required by iteration, and improves the reconstruction efficiency and accuracy.
Description
Technical Field
The invention relates to a signal compression sampling and reconstruction method in the technical field of signal processing, in particular to a greedy random Kaczmarz image reconstruction method based on accurate step length.
Background
Compressed sensing (Compressed Sensing, CS) is a theory and method of information acquisition, information transmission and information recovery that has emerged in recent years, the basic idea of which is similar to the principle of the human visual system, and uses the sensitivity of the human eye to specific features of an image to compress the image, for example, the human eye is generally sensitive to changes in details in the image and is less sensitive to changes in smooth portions of the image, so that the CS retains details in the image and compresses the smooth portions. Compressed sensing algorithms typically implement compression using sparse representations, which are feature-based decomposition of an image, retaining only a few important features, which may be ignored. The compressed image is therefore typically highly sparse and requires reconstruction of the image by recovering these important features at the time of reconstruction.
The compressed sensing image reconstruction algorithm has the advantage of realizing higher compression ratio on the premise of ensuring the image quality. The method has good application prospect in the fields of image storage, image transmission and the like. However, the conventional compressed sensing image reconstruction method is mainly based on the convex optimization theory for reconstruction, and has high computational complexity, so that improvement is required in practical application to improve efficiency. The problem of restoration of sparse images can be seen as a solution to the linear system ax=b. The Kaczmarz method based on iterative projection has the advantages of small information storage amount, high operation speed and the like, and is attracting attention in the field of compressed sensing image reconstruction.
The Kaczmarz algorithm is an important method for solving the linear system ax=b, and the traditional Kaczmarz method is a method for solving a compatible linear system, and initial solution vectors are projected onto a hyperplane determined by each row vector of a coefficient matrix and a corresponding observation value in sequence according to a row sequence. It is clear that the convergence property of Kaczmarz depends on the line order, and if the line order of the coefficient matrix a is not good, the convergence result of Kaczmarz is very slow, and the convergence speed of the conventional Kaczmarz method is difficult to estimate.
On the other hand, the Kaczmarz algorithm solves for a least squares solution, which is typically dense. Whereas the solution vectors required in the compressed perceived image reconstruction method are sparse, for this purpose it is necessary to introduce l 1 The norm, and thus the sparse solution solving the linear system, can be converted into a regularized least squares problem, so the Kaczmarz method can be extended to sparse Kaczmarz algorithms. The improvement of a series of subsequent Kaczmarz algorithms mainly comprises that the convergence rate is improved by selecting proper rows, for example, the random sparse Kaczmarz algorithm is characterized in that when the row norms of coefficient matrixes are not different, the random Kaczmarz selects the rows of the coefficient matrixes with equal probability, and the convergence rate of the random Kaczmarz algorithm is improved compared with that of the traditional Kaczmarz algorithm but is still slower; for example, the greedy random Kaczmarz algorithm proposes to select a working line by adopting a new probability criterion, and the probability criterion combines the greedy thought and a random strategy, so that the convergence speed of the greedy random Kaczmarz algorithm is improved compared with that of the greedy random Kaczmarz algorithm. But for the need to obtain sparsityThe solution vector sparse Kaczmarz algorithm is still not efficient enough, mainly because it uses orthogonal projection, the step size of which is fixed, but not necessarily optimal, which makes the iteration not efficient enough. The method is an effective improvement method for greedy random sparse Kaczmarz by determining the optimal step length of each projection.
Disclosure of Invention
The invention aims to provide a greedy random Kaczmarz image reconstruction method based on accurate step length. In order to solve the technical problems, the invention adopts the following technical scheme:
a greedy random Kaczmarz image reconstruction method based on accurate step size, the method comprising the steps of:
step 1, splitting an image to be reconstructed according to columns;
step 2, setting an iteration termination condition of a solving process, soft threshold function coefficients and initial iteration vectors;
step 3, selecting iterative row vectors to determine a projection direction in a greedy random mode;
step 4, obtaining an accurate step length by using a linear search method, updating a solution vector, adding 1 to the iteration times, judging whether an iteration termination condition is met, if so, outputting a sparse solution vector obtained by the last iteration, otherwise, turning to step 3;
and 5, performing anti-sparse transformation on all the calculated final sparse solution vectors to obtain final solution vectors, and merging according to columns to obtain a reconstructed image.
Further, the step 4 specifically includes the following steps:
the problem of determining the optimal step size in the step 4.1 is essentially to solve the following optimization problem:
wherein the method comprises the steps ofIs the solution vector before the soft threshold function is acted in the last iteration, t k For the optimal step length of the iteration to be solved, a is the selected working line vector +.>Beta is the observation value corresponding to the selected working line vector, S λ (. Cndot.) is a soft threshold function, I.I 2 Is l 2 A norm;
solving the optimization problem by using a linear search method to obtain an accurate step length t k ；
Step 4.2, updating the iterative solution vector x according to the following formula k ：
Wherein a is the projection direction, t k In steps.
Further, in the step 4.1, the method for obtaining the accurate step by using the linear search method includes the following steps:
step 4.11, initializing an estimated value of the intercept b by the following formula:
b＝a T x-β
wherein a is T Representing the transpose of the vector, s=sign (b), where sign (·) represents the sign function, i.e. when b>At 0 sign (b) =1; when b=0, sign (b) =0; when b<At 0 sign (b) = -1.
If the value of s is equal to 0, the search is aborted, the final step t=0; if the value of s is-1, then a and b are taken the opposite number, i.e., a= -a, b= -b;
in step 4.11, the purpose of initializing the intercept b is to let b= -g ′ (0) I.e. let the initial value of b be the inverse of the function value of the first derivative of the objective function of the problem to be optimized at t=0, is used in a subsequent step to determine whether t crosses the break point. At the same time, if s has a value of-1, the opposite numbers of a and b are taken to ensure the mostStep t of final finding>0。
Step 4.12, recording an index of the non-zero item in a, and using the index to reserve non-zero elements in a and z, and removing the elements with zero;
in step 4.12, only non-zero elements in a and z are reserved for subsequent calculation, so that unnecessary searching for the element with 0 in a can be avoided, and the searching efficiency is improved.
Step 4.13, calculating all break points of the objective function g (t), and dividing all the break points into a left break point l and a right break point r according to the following calculation formula:
subsequently, combining l and r, marking as king, and arranging elements in the king order;
in step 4.13, due to the soft threshold operator S λ The presence of (-), the derivative function g of the objective function g (t) ′ (t) cannot be directly expressed, but g ′ (t) is a monotonically increasing function and has the property of a piecewise linear function, i.e. the slope m and intercept-b vary in different intervals, so if the derivative is denoted as g ′ (t) =m t-b, each break point needs to be found, and searching is performed in a section formed by every two break points, so that the calculated left break point and right break point are all break points, and on the other hand, in the subsequent searching process, the formulas for updating the slopes are different according to the difference of the left break point and the right break point.
Step 4.14, initializing the slope m of g (t), wherein the initial value of m is as follows: the sum of the squares of all elements in a that satisfy l >0 and r < 0;
and 4.15, carrying out N times of searching, wherein N is equal to the number of folding points, and firstly checking whether t crosses one folding point in the ith searching, wherein the judgment basis is as follows:
m*kink(i)<b
wherein king (i) represents the ith value in king; if the conditions are met, the condition represents that the folding point is crossed, otherwise, the next search is carried out; if t crosses a left break point, then b and m are updated by the following formula:
b＝b-kink(i)*a 2 (i)
m＝m-a 2 (i)
wherein a is 2 (i) An ith value representing the square of array a; if the step size crosses a right break point, then b and m are updated by the following formula:
b＝b+kink(i)*a 2 (i)
m＝m+a 2 (i)
after N times of searching, the final step length t can be determined k If the updated value of m is 0, the step size is:
t k ＝s*kink(N)
i.e. assigning t to the largest break point in the array king k The method comprises the steps of carrying out a first treatment on the surface of the If the updated value of m is not 0, the step size is:
when t passes through the inflection point, m and b need to be updated to determine a new interval, so that different intervals can be traversed and the optimal solution can be found. The final accurate step length after traversing all the intervals is as follows:let t=s×king (N) be the approximate optimal step size if m=0.
Further, the termination condition is: the change amount of the solution vector corresponding to the previous iteration and the next iteration is sufficiently small or the maximum iteration number is reached.
After the technical scheme is adopted, compared with the prior art, the invention has the following advantages:
the method is suitable for image reconstruction of compressed measurement images of any known measurement matrix and sparse matrix, performs image reconstruction by using a greedy random sparse Kaczmarz image reconstruction method based on accurate step length, obtains accurate step length of each iteration by using a linear search method on the basis of the greedy random sparse Kaczmarz method, reduces the times and calculated amount required by iteration, and improves reconstruction efficiency and reconstruction precision.
The invention will now be described in detail with reference to the drawings and examples.
Drawings
FIG. 1 is a schematic flow diagram of the method of the present invention;
FIG. 2 is a flow chart of a greedy random sparse Kaczmarz algorithm with accurate step size;
FIG. 3 is a schematic diagram of the convergence of a greedy random sparse Kaczmarz algorithm of the precise step size used in the present invention; where the coefficient matrix is m=1000, n=400, λ=3. The horizontal axis represents iteration times, and the vertical axis represents reconstruction errors of the solution obtained by each step of iteration of the algorithm and the real solution;
FIG. 4 is a graph of experimental results of the present invention for image reconstruction in which the image to be reconstructed is a gray scale image; wherein (a) is an original image having a resolution of 256×256; (b) The image to be reconstructed is also the image of the input algorithm; (c) The graph is the result of reconstructing the image matrix of (b) graph with the algorithm proposed by the present invention (psnr=26.73); (d) The graph is the result of reconstructing the image matrix of (b) graph with the GRSK algorithm (psnr=20.46); (e) The graph is the result of reconstructing the image matrix of (b) graph with PGD algorithm (psnr=22.95); (f) The figure is the result of reconstructing the image matrix of (b) figure with ADMM algorithm (psnr= 22.68)
Detailed Description
The principles and features of the present invention are described below with reference to the drawings, the examples are illustrated for the purpose of illustrating the invention and are not to be construed as limiting the scope of the invention.
1-2, FIG. 1 is a flow chart of a greedy random sparse Kaczmarz image reconstruction method based on accurate step length, specifically comprising the following steps:
step 1, dividing the obtained sampling observation value B into blocks B according to columns 1 ,b 2 ,...,b n Wherein B is E R M×n M is the number of rows of B, n is the number of columns of B；
The reconstruction method is a greedy random sparse Kaczmarz method based on accurate step length, so that the sampled image signals need to be reconstructed in a column-by-column block mode and then combined in a column-by-column mode.
Step 2, selecting iterative row vectors to determine a projection direction in a greedy random mode;
when the Kaczmarz method is iterated, the projection direction and the step length are required to be obtained at the same time, a more accurate projection direction can be obtained by using a greedy random rule for selecting row vectors, and meanwhile, the calculated amount is not very large.
Step 3, determining an optimization problem of the optimal step length of the iteration:
wherein x is * Is the solution vector before the soft threshold function is not acted in the last iteration, t is the optimal step length to be solved, and a is the selected working line vectorBeta is the observation value corresponding to the selected working line vector, S λ (. Cndot.) is a soft threshold function, I.I 2 Is l 2 Norms.
The essence of the optimization problem is to solveAnd x k-1 In the super plane H k (a,β)：＝{x∈R n |<a,x>Bregman projection on =β }. Wherein the method comprises the steps of<·,·>Representing the euclidean inner product. By doing so, the t value obtained by solving the optimization problem later can be ensured to be the optimal step length.
Step 4, initializing the estimated value of the intercept b by the following formula
b＝a T x-β
Wherein a is T Representing the transpose of the vector. The notation s=sign (b), where sign (·) represents the sign function, i.e. when b>At 0 sign (b) =1; when (when)b=0, sign (b) =0; when b<At 0 sign (b) = -1.
If the value of s is equal to 0, the search is aborted, the final step t=0; if the value of s is-1, then a and b are taken the opposite number, i.e., a= -a, b= -b.
The purpose of initializing the intercept b is to let b= -g ′ (0) I.e. let the initial value of b be the inverse of the function value of the first derivative of the objective function of the problem to be optimized at t=0, is used in a subsequent step to determine whether t crosses the break point. Meanwhile, if the value of s is-1, the opposite numbers of a and b are taken, so as to ensure the finally found step length t>0。
Step 5, record the index of the non-zero entry in a, and use this index to hold the non-zero elements in a and z, the zero elements being removed.
The method has the advantages that in the subsequent searching and calculating, unnecessary searching of the element with 0 in a can be avoided, the searching efficiency is improved, and the calculating time and the calculating cost are saved.
Step 6, calculating all the break points of the objective function g (t) (objective function derivative function g ′ Break points of (t), and dividing all break points into a left break point l and a right break point r according to the following calculation formula:
subsequently, l and r are combined and denoted as king, and the elements therein are arranged in ascending order.
This is done because of the soft threshold operator S λ The presence of (-), the derivative function g of the objective function g (t) ′ (t) cannot be directly expressed, but g ′ (t) is a monotonically increasing function and has the property of a piecewise linear function, i.e. the slope m and intercept-b vary in different intervals, so if the derivative is denoted as g ′ (t) =m×t-b, and each space needs to be foundThe break points are searched in the interval formed by every two break points, so that the left break point and the right break point are all calculated, and on the other hand, the update slope and the solving formula are different according to the difference of the left break point and the right break point in the subsequent searching process.
Step 7, initializing the slope m of g (t), wherein the initial value of m is as follows: the sum of the squares of all elements in a that satisfy l >0 and r < 0.
And 8, carrying out N times of searching, wherein N is equal to the number of folding points. In the ith search, checking whether t crosses a break point or not, and judging according to the following criteria:
m*kink(i)<b
where king (i) represents the ith value in king. Satisfying the above condition represents crossing the break point, otherwise, the next search is performed. If t crosses a left break point, updates b and m pass the following formula:
b＝b-kink(i)*a 2 (i)
m＝m-a 2 (i)
wherein a is 2 (i) Representing the i-th value of the element squared in array a. If t crosses a right break point, then updates b and m pass the following formula:
b＝b+kink(i)*a 2 (i)
m＝m+a 2 (i)
after k times of searching are completed, the final step length t can be determined, and if the updated value of m is 0, the step length t is:
t＝s*kink(N)
namely, assigning the maximum folding point in the array king to t; if the updated value of m is not 0, the step size is:
when t passes through the inflection point, m and b need to be updated to determine a new interval, so that different intervals can be traversed and the optimal solution can be found. The exact optimal step size is determined after traversing all the intervals,let t=s×king (N) be the approximate optimal step size if m=0.
And 9, the projection direction is a, the step length is t, and the updated solution vector is calculated by the following formula:
step 10, iteratively operating the steps 2-9 until a termination condition is met; and (5) outputting lean fluffing.
The termination conditions are: the change amount of the sparse solution vector corresponding to the current iteration and the subsequent iteration is sufficiently small or reaches the maximum iteration number, and the algorithm is terminated. At this time, a sparse solution vector x may be output l 。
Step 11, solving the obtained sparse solution vector x i Performing anti-sparseness transformation X i ＝Ψ·x i Obtaining a solution vector X i For X 1 ,X 2 ,...,X n And combining the two images into X according to columns, namely, reconstructing the image signals.
The solution vector obtained by solving the greedy random sparse kaczmarz algorithm with accurate step length is sparse, the solution vector is true solution vector after inverse sparse transformation, and the reconstructed image is obtained after column combination.
Effect verification
To verify the effectiveness of the present invention, we have experimentally verified on a randomly generated 1000×400 gaussian matrix and a pepers picture matrix with a resolution of 256×256.
In experiment 1, we first randomly generate a 1000×400 gaussian matrix as coefficient matrix a, and regenerate a sparse 400×1 random vector as a true solution of the linear systemUse->And obtaining a vector b at the right end of the linear system. Comparing greedy based on accurate step lengthRandom sparse Kaczmarz algorithm (Exact-Step Greedy Randomized Sparse Kaczmarz, denoted EGRSK) and greedy random sparse Kaczmarz algorithm (Greedy Randomized Sparse Kaczmarz, denoted GRSK), near-end gradient descent algorithm (Proximal Gradient Descent, denoted PGD), cross direction multiplier algorithm (Alternating Direction Method of Multipliers, denoted ADMM) are used. Fig. 3 records the convergence curves of the four algorithms, wherein the RSK algorithm is a dotted line, the GRSK algorithm is a solid line, and the PGD and ADMM algorithms are a dotted line and a virtual point line, respectively. The calculation formula of the reconstruction error in the experiment is as follows:
it can be seen that the algorithm provided by the invention has better result.
In experiment 2, the results of reconstructing an image to be reconstructed based on a greedy random sparse Kaczmarz algorithm (corresponding to (c) diagram in fig. 4), a GRSK algorithm (corresponding to (d) diagram in fig. 4), a PGD algorithm (corresponding to (e) diagram in fig. 4) and an ADMM algorithm (corresponding to (f) diagram in fig. 4) proposed by the present invention are compared, wherein (a) diagram in fig. 4 is an original image, and (b) diagram in fig. 4 is an image to be reconstructed. The decision criteria for the reconstruction results of the various algorithms uses the peak signal-to-noise ratio (Peak Signal to Noise Ratio, noted PSNR), an engineering term that represents the ratio of the maximum possible power of the signal to the destructive noise power affecting its accuracy of representation, which is used in this experiment to represent the accuracy of the image reconstruction. The peak signal-to-noise ratios are respectively:
PSNR (EGRSK) =26.73 (corresponding to (c) in fig. 4), PSNR (GRSK) =20.46 (corresponding to (d) in fig. 4), PSNR (PGD) =22.95 (corresponding to (e) in fig. 4), PSNR (ADMM) = 22.68 (corresponding to (f) in fig. 4). It can be seen that the algorithm proposed by the present invention is superior to some existing algorithms in terms of the accuracy of image reconstruction.
The foregoing is illustrative of the best mode of carrying out the invention, and is not presented in any detail as is known to those of ordinary skill in the art. The protection scope of the invention is defined by the claims, and any equivalent transformation based on the technical teaching of the invention is also within the protection scope of the invention.
Claims (2)
1. The greedy random Kaczmarz image reconstruction method based on the accurate step length is characterized by comprising the following steps:
step 1, splitting an image to be reconstructed according to columns;
step 2, setting an iteration termination condition of a solving process, soft threshold function coefficients and initial iteration vectors;
step 3, selecting iterative row vectors to determine a projection direction in a greedy random mode;
step 4, obtaining an accurate step length by using a linear search method, updating a solution vector, adding 1 to the iteration times, judging whether an iteration termination condition is met, if so, outputting a sparse solution vector obtained by the last iteration, otherwise, turning to step 3;
the step 4 specifically comprises the following steps:
the problem of determining the optimal step size in the step 4.1 is essentially to solve the following optimization problem:
wherein the method comprises the steps ofIs the solution vector before the soft threshold function is acted in the last iteration, t k For the optimal step length of the iteration to be solved, a is the selected working line vector +.>Beta is the observation value corresponding to the selected working line vector, S λ (. Cndot.) is a soft threshold function, I.I 2 Is l 2 A norm;
solving the above-mentioned optimization problem by using linear search methodTo an accurate step length t k ；
In the step 4.1, the method for obtaining the accurate step length by using the linear search method comprises the following steps:
step 4.11, initializing an estimated value of the intercept b by the following formula:
b＝a T x-β
wherein a is T Representing the transpose of the vector, the intermediate variable s=sign (b), sign (·) representing the sign function, i.e. when b>At 0, s=sign (b) =1; when b=0, s=sign (b) =0; when b<At 0, s=sign (b) = -1; if the value of s is equal to 0, the search is aborted, the final step t=0; if the value of s is-1, then a and b are taken the opposite number, i.e., a= -a, b= -b;
step 4.12, recording an index of the non-zero item in a, and utilizing the index to reserve the non-zero element in a and remove the element with zero;
step 4.13, calculating all break points of the objective function g (t), and dividing all the break points into a left break point l and a right break point r according to the following calculation formula:
then combining l and r, marking as king, arranging elements in ascending order, and lambda is a preset super parameter;
step 4.14, initializing the slope m of g (t), wherein the initial value of m is as follows: the sum of the squares of all elements in a that satisfy l >0 and r < 0;
and 4.15, carrying out N times of searching, wherein N is equal to the number of folding points, and firstly checking whether t crosses one folding point in the ith searching, wherein the judgment basis is as follows:
m*kink(i)<b
wherein king (i) represents the ith value in king; if the conditions are met, the condition represents that the folding point is crossed, otherwise, the next search is carried out; if t crosses a left break point, then b and m are updated by the following formula:
b＝b-kink(i)*a 2 (i)
m＝m-a 2 (i)
wherein a is 2 (i) An ith value representing the square of array a; if the step size crosses a right break point, then b and m are updated by the following formula:
b＝b+kink(i)*a 2 (i)
m＝m+a 2 (i)
after N times of searching, the final step length t can be determined k If the updated value of m is 0, the step size is:
t k ＝s*kink(N)
i.e. assigning t to the largest break point in the array king k The method comprises the steps of carrying out a first treatment on the surface of the If the updated value of m is not 0, the step size is:
step 4.2, updating the iterative solution vector x according to the following formula k ：
Wherein a is the projection direction, t k Is the step length;
and 5, performing anti-sparse transformation on all the calculated final sparse solution vectors to obtain final solution vectors, and merging according to columns to obtain a reconstructed image.
2. The greedy random Kaczmarz image reconstruction method based on accurate step size of claim 1, wherein the termination condition is: the change amount of the solution vector corresponding to the previous iteration and the next iteration is sufficiently small or the maximum iteration number is reached.
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| CN114398589A (en) * | 2021-12-31 | 2022-04-26 | 西安交通大学 | Method and system for accelerating solving of accumulated residual error Kaczmarz of regularized linear system |
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