CN114786018A - Image reconstruction method based on greedy random sparse Kaczmarz - Google Patents

Image reconstruction method based on greedy random sparse Kaczmarz Download PDF

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CN114786018A
CN114786018A CN202210682958.8A CN202210682958A CN114786018A CN 114786018 A CN114786018 A CN 114786018A CN 202210682958 A CN202210682958 A CN 202210682958A CN 114786018 A CN114786018 A CN 114786018A
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sparse
kaczmarz
vector
iteration
image reconstruction
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朱德梁
陈娜
彭江涛
廖殷娜
谢尔心
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Hubei University
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Hubei University
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    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/44—Decoders specially adapted therefor, e.g. video decoders which are asymmetric with respect to the encoder
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M7/00—Conversion of a code where information is represented by a given sequence or number of digits to a code where the same, similar or subset of information is represented by a different sequence or number of digits
    • H03M7/30—Compression; Expansion; Suppression of unnecessary data, e.g. redundancy reduction
    • H03M7/3059—Digital compression and data reduction techniques where the original information is represented by a subset or similar information, e.g. lossy compression
    • H03M7/3062—Compressive sampling or sensing
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/70—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals characterised by syntax aspects related to video coding, e.g. related to compression standards

Abstract

The invention relates to an image reconstruction method based on greedy random sparse Kaczmarz, which comprises the following steps of 1, carrying out image reconstruction on one image to be compressed
Figure 568726DEST_PATH_IMAGE001
Using random Gaussian matrix
Figure 629086DEST_PATH_IMAGE002
And sparse basis matrices
Figure 588951DEST_PATH_IMAGE003
To pair
Figure 240512DEST_PATH_IMAGE001
Sampling to obtain corresponding compression observed value
Figure 20250DEST_PATH_IMAGE004
Wherein, in the process,
Figure 8934DEST_PATH_IMAGE005
(ii) a 2. For compressed observed value
Figure 823307DEST_PATH_IMAGE006
Is divided into blocks according to columns
Figure 645769DEST_PATH_IMAGE007
(ii) a 3. Solving sparse solution vector by greedy random sparse Kaczmarz algorithm
Figure 381644DEST_PATH_IMAGE008
(ii) a 4. Inverse sparse transformation of sparse original vectors
Figure 580544DEST_PATH_IMAGE009
Obtain the final solution vector
Figure 249423DEST_PATH_IMAGE010
To is aligned with
Figure 836262DEST_PATH_IMAGE011
Are combined into
Figure 590591DEST_PATH_IMAGE012
，
Figure 593182DEST_PATH_IMAGE012
I.e. the reconstructed image signal. The invention not only provides the Kaczmarz iteration method for image compressed sensing reconstruction without the prior known signal compressed sparsity, but also improves the efficiency of image compressed sensing reconstruction and reduces the times and the calculation amount required by iteration.

Description

Image reconstruction method based on greedy random sparse Kaczmarz
Technical Field
The invention relates to a signal compression sampling and reconstruction method in the technical field of signal processing, in particular to an image reconstruction method based on greedy random sparse Kaczmarz.
Background
Compressed Sensing (CS) is an emerging theory of information acquisition and transmission processing, and the CS indicates that sparsity prior information in a signal can be fully utilized, and an original high-dimensional signal is projected to a low-dimensional space by using an uncorrelated measurement matrix under a condition far lower than a Nyquist sampling frequency, so as to obtain a low-dimensional observed value, and the original signal can be reconstructed without distortion. And the signal sampling and the compression coding are completed in one step in the compressed sensing, which has great convenience and advantages for the acquisition and transmission of the signal. The signal reconstruction algorithm is the core of the compressed sensing theory, and refers to the process of reconstructing a sparse signal from a measured value.
The traditional compressed sensing image reconstruction method is mainly based on optimization iteration for reconstruction, inevitably brings higher calculation cost, and has undesirable recovery effect when the measurement rate is very low. The problem of recovering the sparse signal can be regarded as a solution problem of the linear system Ax = b. The iterative projection-based Kaczmarz method is concerned by people because of small storage capacity and high operation speed.
The Kaczmarz algorithm is an important method for solving a linear system Ax = b, the traditional Kaczmarz method is a method for solving a linear compatible system, and initial values are projected on a hyperplane which is determined by each row vector of a matrix and a corresponding observation value in sequence according to the sequence of rows. It is clear that the convergence properties of Kaczmarz depend on the row order, and that the convergence results of Kaczmarz are very slow if the row order of the coefficient matrix is not good, and it is difficult to analyze the convergence results for this method.
The Kaczmarz algorithm solves for a least squares solution, which is usually dense. The sparse solution for solving the linear system has wide application in the fields of image reconstruction, signal processing, machine learning and the like. By introducing the l1 norm, a sparse solution to solve a linear system can be converted to a regularized least squares problem. The subsequent random sparse Kaczmarz algorithm is characterized in that when the row norm of the coefficient matrix is not large, the random Kaczmarz and the random sparse Kaczmarz select the row of the coefficient matrix at equal probability, and the convergence rate of the random Kaczmarz algorithm is slow, so that a new probability criterion is adopted for selecting a working row by the aid of the later greedy random Kaczmarz.
Disclosure of Invention
The invention aims to provide an image reconstruction method based on greedy random sparse Kaczmarz. The invention adopts the following technical scheme:
an image reconstruction method based on greedy random sparse Kaczmarz comprises the following steps:
step 1, one image to be compressed
Figure 381161DEST_PATH_IMAGE001
Using random Gaussian matrix
Figure 32722DEST_PATH_IMAGE002
And sparse basis matrices
Figure 671514DEST_PATH_IMAGE003
To pair
Figure 801144DEST_PATH_IMAGE001
Sampling to obtain corresponding compressed observed value
Figure 349937DEST_PATH_IMAGE004
Wherein, in the step (A),
Figure 906820DEST_PATH_IMAGE005
，
Figure 173853DEST_PATH_IMAGE006
for the M-dimensional euclidean space,
Figure 231808DEST_PATH_IMAGE007
is composed ofBThe number of rows of (a) to (b),nis composed ofBThe number of columns;
step 2, compressing the observed value
Figure 635108DEST_PATH_IMAGE008
Blocking into compressed observation value blocks according to columns
Figure 362892DEST_PATH_IMAGE009
；
Step 3, utilizing greedy random sparse Kaczmarz algorithmRespectively calculate
Figure 117222DEST_PATH_IMAGE009
Corresponding sparse solution vector
Figure 588654DEST_PATH_IMAGE010
；
Step 4, calculating
Figure 971094DEST_PATH_IMAGE010
Corresponding final solution vector
Figure 135359DEST_PATH_IMAGE011
The calculation method is as follows,
for the computed sparse solution vector
Figure 376985DEST_PATH_IMAGE012
Performing an anti-sparseness transformation
Figure 386529DEST_PATH_IMAGE013
Obtain the final solution vector
Figure 764421DEST_PATH_IMAGE014
，
p∈[1,2,……，n]Will be
Figure 99587DEST_PATH_IMAGE011
Are combined into
Figure 687563DEST_PATH_IMAGE015
，
Figure 500798DEST_PATH_IMAGE015
I.e. the reconstructed image signal.
Further, in the step 3, calculation is performed
Figure 998776DEST_PATH_IMAGE016
Corresponding sparse solution vector
Figure 504844DEST_PATH_IMAGE017
The method comprises the following steps:
step 3.1, use
Figure 189903DEST_PATH_IMAGE018
Computing a quantity for determining a probability criterion in a greedy stochastic Kaczmarz method
Figure 72408DEST_PATH_IMAGE019
，
Figure 283947DEST_PATH_IMAGE018
Representing sparse solution vectors obtained from the last iteration, at the first iteration
Figure 960916DEST_PATH_IMAGE018
For n-dimensional all-0 vectors
Figure 398850DEST_PATH_IMAGE020
Wherein n is
Figure 819467DEST_PATH_IMAGE017
The number of rows ofBThe number of columns;
step 3.2, defining a positive integer index set
Figure 760879DEST_PATH_IMAGE021
：
Figure 874328DEST_PATH_IMAGE022
Wherein
Figure 799559DEST_PATH_IMAGE023
Show to obtain
Figure 617342DEST_PATH_IMAGE024
The number of the 2-norm,
Figure 678839DEST_PATH_IMAGE025
show to obtain
Figure 432031DEST_PATH_IMAGE024
2 modelThe square of the number of the bits is,
Figure 844558DEST_PATH_IMAGE026
representing a vector
Figure 872557DEST_PATH_IMAGE016
To (1)
Figure 913194DEST_PATH_IMAGE027
The value of the one or more parameters,
Figure 102867DEST_PATH_IMAGE028
representing a matrix of coefficients
Figure 2690DEST_PATH_IMAGE029
To (1) a
Figure 772063DEST_PATH_IMAGE027
A row vector formed by the rows of the image,
Figure 73731DEST_PATH_IMAGE030
；
step 3.3, calculating residual error vector
Figure 434305DEST_PATH_IMAGE031
，
Figure 414900DEST_PATH_IMAGE031
To (1)
Figure 784701DEST_PATH_IMAGE027
Each component is
Figure 675297DEST_PATH_IMAGE032
And according to probability
Figure 206772DEST_PATH_IMAGE033
Selecting
Figure 815608DEST_PATH_IMAGE029
Work line of (1)
Figure 989100DEST_PATH_IMAGE034
In which
Figure 327678DEST_PATH_IMAGE035
Represents the selected coefficient matrix of
Figure 295634DEST_PATH_IMAGE034
The line is used as the probability of the iterative work line;
step 3.4, calculating by using a soft threshold shrinkage operator to obtain sparse solution vectors of the iteration
Figure 126187DEST_PATH_IMAGE036
；
And 3.5, judging whether the iteration termination condition is met, if so, turning to the next step, otherwise, turning to the step 3.1, adding 1 to the iteration times, and adding the iteration times to the number of times
Figure 837791DEST_PATH_IMAGE018
Is updated to
Figure 702979DEST_PATH_IMAGE036
；
Step 3,6, taking the sparse solution vector of the iteration as a sparse solution vector
Figure 181451DEST_PATH_IMAGE016
Corresponding sparse solution vector
Figure 764879DEST_PATH_IMAGE017
And (6) outputting.
Further, in the step 3.1,
Figure 14595DEST_PATH_IMAGE019
the determination rule of (2) is:
Figure 468710DEST_PATH_IMAGE037
wherein
Figure 778468DEST_PATH_IMAGE038
Shows FrobenThe norm of the ius is obtained by calculating the sum of the signals,
Figure 849192DEST_PATH_IMAGE039
the square of Frobenius norm is calculated,
Figure 761654DEST_PATH_IMAGE023
show to obtain
Figure 601434DEST_PATH_IMAGE024
The number of the 2 norm is the same as the standard,
Figure 82094DEST_PATH_IMAGE025
expression solution
Figure 577797DEST_PATH_IMAGE024
2 norm square.
Further, in the step 3.2, a positive integer index set is defined
Figure 434895DEST_PATH_IMAGE021
By satisfying the inequality
Figure 129181DEST_PATH_IMAGE040
Index of (2)
Figure 639797DEST_PATH_IMAGE027
Put into an index set
Figure 685113DEST_PATH_IMAGE021
In (1).
Further, in said step 3.3, calculating
Figure 80323DEST_PATH_IMAGE032
The method comprises the following steps:
Figure 97957DEST_PATH_IMAGE041
。
further, the detailed steps of step 3.4 are as follows:
step 3.41, the work line selected according to the step 3.3
Figure 185999DEST_PATH_IMAGE034
Calculating a solution vector
Figure 780928DEST_PATH_IMAGE042
；
Step 3.42, given the hyperparameter in the soft threshold shrink operator
Figure 979828DEST_PATH_IMAGE043
Will solve the vector
Figure 383128DEST_PATH_IMAGE042
Introducing a soft threshold shrinking operator to obtain a sparse solution vector obtained by the iteration
Figure 376492DEST_PATH_IMAGE036
。
Further, in said step 3.41, the vector is solved
Figure 130821DEST_PATH_IMAGE044
Wherein
Figure 867833DEST_PATH_IMAGE045
Indicating a to-be-processed line vector
Figure 719114DEST_PATH_IMAGE046
The translation is a column vector.
Further, in step 3.42, the soft threshold shrinking operator is:
Figure 148959DEST_PATH_IMAGE047
wherein
Figure 125005DEST_PATH_IMAGE048
Show to get
Figure 400128DEST_PATH_IMAGE049
And the maximum value of the sum of 0,
Figure 43599DEST_PATH_IMAGE050
representing a symbolic function, i.e. when
Figure 972241DEST_PATH_IMAGE051
When the utility model is used, the water is discharged,
Figure 701163DEST_PATH_IMAGE052
(ii) a When the temperature is higher than the set temperature
Figure 779977DEST_PATH_IMAGE053
When the temperature of the water is higher than the set temperature,
Figure 481217DEST_PATH_IMAGE054
(ii) a When in use
Figure 987285DEST_PATH_IMAGE055
When the temperature of the water is higher than the set temperature,
Figure 203502DEST_PATH_IMAGE056
therefore, the soft threshold shrink operator can also be expressed as:
Figure 945062DEST_PATH_IMAGE057
wherein
Figure 297546DEST_PATH_IMAGE058
To represent
Figure 240094DEST_PATH_IMAGE018
To (1)
Figure 146870DEST_PATH_IMAGE027
And (4) a component.
Further, in step 3.42:
Figure 567487DEST_PATH_IMAGE059
Figure 633532DEST_PATH_IMAGE036
indicating the dilution obtained in this iterationAnd (5) thinning the vector.
Further, in step 3.5, the iteration termination condition is: the change of the sparse solution vector corresponding to the two iterations is minimized or reaches the maximum iteration number.
After the technical scheme is adopted, compared with the prior art, the invention has the following advantages:
the image reconstruction method based on greedy random sparse Kaczmarz is suitable for signal reconstruction of compressed measurement signals of any known measurement matrix and sparse basis matrix.
The invention is described in detail below with reference to the figures and examples.
Drawings
FIG. 1 is a schematic flow diagram of the process of the present invention;
FIG. 2 is a schematic flow diagram of a greedy stochastic sparse Kaczmarz algorithm;
FIG. 3 is a schematic diagram of the convergence of the greedy stochastic sparse kaczmarz algorithm used by the present invention; wherein the coefficient matrix is m =600, n =200,
Figure 481403DEST_PATH_IMAGE043
and = 1. The horizontal axis represents iteration times, and the vertical axis represents reconstruction errors of a solution obtained by each step of iteration of the algorithm and a real solution;
FIG. 4 is a graph of experimental results of compressed sensing image reconstruction for grayscale images in accordance with the present invention; wherein (a) is an original image with a resolution of 256 × 256; (b) the image is an image after low-rank adjustment and is also an image of an input algorithm; (c) the image is an image after being sampled by a sensing matrix; (d) the figure is the result of recovering the image matrix of the (c) figure with the proposed algorithm of the invention (PSNR = 22.4503); (e) the graph is the result of recovering the image matrix of (c) graph with orthogonal matching pursuit algorithm (PSNR = 14.4409); (f) the figure is the result of recovering the image matrix of (c) the figure with the iterative shrinkage thresholding algorithm (PSNR = 20.5339).
Detailed Description
The principles and features of this invention are described below in conjunction with the following drawings, which are set forth to illustrate, but are not to be construed to limit the scope of the invention.
As shown in fig. 1-2, fig. 1 is a flowchart of an image reconstruction method based on greedy random sparse Kaczmarz according to the present invention, which specifically includes the following steps:
step 1, for an image to be compressed
Figure 875475DEST_PATH_IMAGE001
Using random Gaussian matrix
Figure 99783DEST_PATH_IMAGE002
And sparse basis matrices
Figure 426859DEST_PATH_IMAGE003
For is to
Figure 39106DEST_PATH_IMAGE001
Sampling to obtain corresponding compression observed value
Figure 451633DEST_PATH_IMAGE004
Wherein, in the process,
Figure 214052DEST_PATH_IMAGE005
，
Figure 130056DEST_PATH_IMAGE006
for the M-dimensional euclidean space,
Figure 585308DEST_PATH_IMAGE007
is composed ofBThe number of rows of (a) to (b),nis composed ofBNumber of columns of (a) and
Figure 485131DEST_PATH_IMAGE007
much less than
Figure 644717DEST_PATH_IMAGE060
；
Compressed sensing is a process of recovering an original signal from a compressed observation value, so that the compressed observation value, and a corresponding measurement matrix and a sparse basis matrix need to be obtained first.
Step 2, compressing the observed value
Figure 680806DEST_PATH_IMAGE008
Is divided into blocks according to columns
Figure 775801DEST_PATH_IMAGE009
；
The reconstruction method of the present invention is an image reconstruction method based on one-dimensional signals, and therefore, the image signals need to be restored in blocks according to columns and then merged according to columns.
Step 3, inputting the data into a greedy random sparse Kaczmarz algorithm, wherein
Figure 162920DEST_PATH_IMAGE030
，
Figure 267142DEST_PATH_IMAGE016
Is a compression observation of blocking by column.
Determining quantities of probability criterion in greedy stochastic Kaczmarz method
Figure 16792DEST_PATH_IMAGE019
The rule of (1) is:
Figure 813847DEST_PATH_IMAGE037
wherein
Figure 422683DEST_PATH_IMAGE038
The Frobenius norm is calculated,
Figure 65017DEST_PATH_IMAGE039
the square of Frobenius norm is calculated,
Figure 75698DEST_PATH_IMAGE023
expression solution
Figure 43654DEST_PATH_IMAGE024
The number of the 2-norm,
Figure 733261DEST_PATH_IMAGE025
show to obtain
Figure 179286DEST_PATH_IMAGE024
2 norm square.
Quantity determined according to the above rules
Figure 310053DEST_PATH_IMAGE019
Has the following properties:
Figure 917752DEST_PATH_IMAGE062
and for the case = 0:
Figure 501180DEST_PATH_IMAGE063
the above property can be such that the set of positive integer indices defined in said step 3
Figure 750896DEST_PATH_IMAGE021
Is non-empty, so that iteration can be performed, and the efficiency of iteration is improved relative to a method for randomly selecting a working line.
Step 4, defining a positive integer index set
Figure 329645DEST_PATH_IMAGE021
And find all elements that fit the index set
Figure 639403DEST_PATH_IMAGE027
；
Figure 710128DEST_PATH_IMAGE064
Wherein
Figure 232376DEST_PATH_IMAGE023
Expression solution
Figure 72156DEST_PATH_IMAGE024
The number of the 2 norm is the same as the standard,
Figure 552816DEST_PATH_IMAGE025
expression solution
Figure 438732DEST_PATH_IMAGE024
The square of the 2 norm is calculated,
Figure 295830DEST_PATH_IMAGE026
representing right-hand vectors
Figure 458958DEST_PATH_IMAGE016
To (1)
Figure 110519DEST_PATH_IMAGE027
The value of the one or more of the one,
Figure 155835DEST_PATH_IMAGE028
representing a matrix of coefficients
Figure 144520DEST_PATH_IMAGE029
To (1)
Figure 693313DEST_PATH_IMAGE027
A row vector formed by the rows of the image,
Figure 781355DEST_PATH_IMAGE018
represent iteration proceeds to
Figure 517230DEST_PATH_IMAGE065
The resulting solution vector.
Defining a set of positive integer indices that will satisfy the inequality
Figure 450551DEST_PATH_IMAGE040
Index of (2)
Figure 119429DEST_PATH_IMAGE027
Put into an index set
Figure 971848DEST_PATH_IMAGE021
In (1). The advantage of this is that it is possible to do so,
Figure 460598DEST_PATH_IMAGE066
representing the current solution
Figure 463189DEST_PATH_IMAGE018
To coefficient matrix of
Figure 455416DEST_PATH_IMAGE027
The distance of the hyperplane formed by the rows is such that the set is made
Figure 619681DEST_PATH_IMAGE021
In which the current solution is included
Figure 861306DEST_PATH_IMAGE018
Indices having greater distance to respective hyperplane
Figure 995484DEST_PATH_IMAGE027
。
Step 5, calculating residual error vector
Figure 373376DEST_PATH_IMAGE067
；
Computing residual vectors
Figure 442963DEST_PATH_IMAGE068
To (1)
Figure 906306DEST_PATH_IMAGE027
The number of the components is such that,
Figure 250699DEST_PATH_IMAGE041
is ready for
Figure 342152DEST_PATH_IMAGE027
In an index set
Figure 848220DEST_PATH_IMAGE021
In this case, the residual vector is directly calculated, and is 0 if not.
Step 6, selecting working lines according to the probability
Figure 64437DEST_PATH_IMAGE034
；
According to probability
Figure 415784DEST_PATH_IMAGE069
Select the line of work
Figure 768268DEST_PATH_IMAGE034
To select the index
Figure 445237DEST_PATH_IMAGE034
Is defined as a set of metrics proportional to the square of each residual component
Figure 476647DEST_PATH_IMAGE021
Index corresponding to larger residual vector
Figure 162843DEST_PATH_IMAGE034
Can be taken with a greater probability.
Step 7, updating the solution vector
Figure 838675DEST_PATH_IMAGE042
；
Updating solution vector passes
Figure 952125DEST_PATH_IMAGE044
Is obtained wherein
Figure 877356DEST_PATH_IMAGE045
Representing a row vector
Figure 695139DEST_PATH_IMAGE046
The translation is to be a column vector that,
Figure 22215DEST_PATH_IMAGE042
show that
Figure 306566DEST_PATH_IMAGE018
Orthogonal projection to
Figure 922355DEST_PATH_IMAGE046
The solution vector obtained by the constructed hyperplane.
Step 8, defining a soft threshold shrinkage operator, and giving a hyperparameter in the soft threshold shrinkage operator
Figure 950354DEST_PATH_IMAGE043
；
The soft threshold shrink operator is
Figure 131936DEST_PATH_IMAGE047
Wherein
Figure 180664DEST_PATH_IMAGE048
Show to get
Figure 80487DEST_PATH_IMAGE070
And the maximum value of the sum of 0,
Figure 646597DEST_PATH_IMAGE050
representing a symbolic function, i.e. when
Figure 151528DEST_PATH_IMAGE051
When the utility model is used, the water is discharged,
Figure 777681DEST_PATH_IMAGE052
(ii) a When in use
Figure 899221DEST_PATH_IMAGE053
When the utility model is used, the water is discharged,
Figure 862498DEST_PATH_IMAGE054
(ii) a When in use
Figure 18673DEST_PATH_IMAGE055
When the utility model is used, the water is discharged,
Figure 284569DEST_PATH_IMAGE056
. The soft threshold shrink operator can therefore also be expressed as:
Figure 893405DEST_PATH_IMAGE071
wherein
Figure 801318DEST_PATH_IMAGE058
To represent
Figure 405475DEST_PATH_IMAGE018
To (1)
Figure 373431DEST_PATH_IMAGE027
And (4) a component.
Step 9, updating sparse solution vector
Figure 469563DEST_PATH_IMAGE036
；
Figure 650008DEST_PATH_IMAGE072
Figure 780775DEST_PATH_IMAGE036
Representing the sparse solution vector resulting from this iteration.
Step 10, iteratively operating the steps 2-8 until a termination condition is met; and outputting the sparse solution.
The termination conditions were: and when the change quantity of the sparse solution vector corresponding to the two iterations is sufficiently small or reaches the maximum iteration number, the algorithm is terminated. At this time, sparse solution vectors may be output
Figure 919633DEST_PATH_IMAGE036
。
Step (ii) of11, for the computed sparse solution vector
Figure 830957DEST_PATH_IMAGE036
Namely, it is
Figure 346252DEST_PATH_IMAGE012
Performing an inverse sparse transform
Figure 800367DEST_PATH_IMAGE013
Obtain the final solution vector
Figure 110126DEST_PATH_IMAGE014
，p∈[1,2,……，n]Will be
Figure 508746DEST_PATH_IMAGE011
Are combined into one column
Figure 562153DEST_PATH_IMAGE015
，
Figure 401933DEST_PATH_IMAGE015
I.e. the reconstructed image signal.
The solution vectors obtained by solving the greedy random sparse kaczmarz algorithm are sparse, real solution vectors are obtained after inverse sparse transformation, and reconstructed signals are obtained after column combination.
Effect verification
To verify the effectiveness of the present invention, we performed experimental verification on a randomly generated gaussian matrix of 600 x 200 and LENA picture matrix with a resolution of 256 x 256.
In experiment 1, we first randomly generated 600 x 200 gaussian matrices as coefficient matrices
Figure 351434DEST_PATH_IMAGE029
Regenerating a sparse 200 x 1 random vector as the true solution of the linear system
Figure 909454DEST_PATH_IMAGE001
By using
Figure 500973DEST_PATH_IMAGE073
Obtaining the vector at the right end of the linear system
Figure 54314DEST_PATH_IMAGE016
. Curves of convergence speed based on Greedy random Sparse Kaczmarz algorithm (GRSK) and random Sparse Kaczmarz algorithm (RSK), near-end Gradient Descent algorithm (PGD) and cross Direction multiplier Algorithm (ADMM) are compared. Fig. 3 records the convergence curves of the above four algorithms, where the RSK algorithm is a dashed line, the GRSK algorithm is a solid line, and the PGD and ADMM algorithms are a dashed line and a dotted line, respectively. It can be seen that the algorithm provided by the invention has better results.
In experiment 2, the results of reconstructing the LENA image matrix after sub-sampling based on the greedy stochastic sparse Kaczmarz algorithm (corresponding to the d diagram in fig. 4), the iterative algorithm based on the near-end gradient descent (corresponding to the f diagram in fig. 4), and the orthogonal matching pursuit algorithm (corresponding to the e diagram in fig. 4) proposed by the present invention are compared, and the judgment standard uses Peak Signal to Noise Ratio (PSNR), which is an engineering term representing the Ratio of the maximum possible power of a Signal and the destructive Noise power affecting its representation precision, and is used to represent the reconstruction precision in this experiment. The peak signal-to-noise ratios are:
psnr (grsk) =22.45, psnr (ista) =20.53, psnr (omp) = 14.44. It can be seen that the algorithm proposed by the present invention is also better in image reconstruction.
The foregoing is illustrative of the best mode of the invention and details not described herein are within the common general knowledge of a person of ordinary skill in the art. The scope of the present invention is defined by the appended claims, and any equivalent modifications based on the technical teaching of the present invention are also within the scope of the present invention.

Claims (10)

1. The image reconstruction method based on greedy random sparse Kaczmarz is characterized by comprising the following steps of:
step 1, one image to be compressed
Figure 118099DEST_PATH_IMAGE001
Using random Gaussian matrix
Figure 35240DEST_PATH_IMAGE002
And sparse basis matrices
Figure 814977DEST_PATH_IMAGE003
For is to
Figure 741345DEST_PATH_IMAGE001
Sampling to obtain corresponding compression observed value
Figure 555717DEST_PATH_IMAGE004
Wherein, in the step (A),
Figure 909338DEST_PATH_IMAGE005
，
Figure 176371DEST_PATH_IMAGE006
for the M-dimensional euclidean space,
Figure 906430DEST_PATH_IMAGE007
is composed ofBThe number of rows of (a) to (b),nis composed ofBNumber of columns of (a) and
Figure 575309DEST_PATH_IMAGE007
much less than
Figure 568672DEST_PATH_IMAGE008
；
Step 2, compressing the observed value
Figure 854160DEST_PATH_IMAGE009
Blocking into compressed observation value blocks according to columns
Figure 591172DEST_PATH_IMAGE010
；
Step 3, respectively calculating by utilizing greedy random sparse Kaczmarz algorithm
Figure 114557DEST_PATH_IMAGE010
Corresponding sparse solution vector
Figure 75560DEST_PATH_IMAGE011
；
Step 4, calculating
Figure 317185DEST_PATH_IMAGE011
Corresponding final solution vector
Figure 592309DEST_PATH_IMAGE012
The calculation method is as follows,
for the computed sparse solution vector
Figure 766938DEST_PATH_IMAGE013
Performing an anti-sparseness transformation
Figure 102105DEST_PATH_IMAGE014
Obtain the final solution vector
Figure 831026DEST_PATH_IMAGE015
，
p∈[1,2,……，n]Will be
Figure 440999DEST_PATH_IMAGE012
Are combined into
Figure 938977DEST_PATH_IMAGE016
，
Figure 710624DEST_PATH_IMAGE016
I.e. the reconstructed image signal.
2. The image reconstruction method based on greedy random sparse Kaczmarz as claimed in claim 1, wherein in the step 3, calculation is performed
Figure 192421DEST_PATH_IMAGE017
Corresponding sparse solution vector
Figure 74926DEST_PATH_IMAGE018
The method comprises the following steps:
step 3.1, use
Figure 427410DEST_PATH_IMAGE019
Computing quantities for determining probability criterion in greedy stochastic Kaczmarz method
Figure 901116DEST_PATH_IMAGE020
，
Figure 339051DEST_PATH_IMAGE019
Representing the sparse solution vector obtained from the last iteration, at the first iteration
Figure 759668DEST_PATH_IMAGE019
For n-dimensional all-0 vectors
Figure 763396DEST_PATH_IMAGE021
Wherein n is
Figure 876846DEST_PATH_IMAGE018
The number of rows ofBThe number of columns;
step 3.2, defining a positive integer index set
Figure 802076DEST_PATH_IMAGE022
：
Figure 557543DEST_PATH_IMAGE023
In which
Figure 619040DEST_PATH_IMAGE024
Show to obtain
Figure 903391DEST_PATH_IMAGE025
The number of the 2-norm,
Figure 847076DEST_PATH_IMAGE026
show to obtain
Figure 875075DEST_PATH_IMAGE025
The square of the 2-norm,
Figure 56657DEST_PATH_IMAGE027
representing a vector
Figure 777489DEST_PATH_IMAGE017
To (1) a
Figure 677311DEST_PATH_IMAGE028
The value of the one or more of the one,
Figure 243422DEST_PATH_IMAGE029
representing a matrix of coefficients
Figure 76249DEST_PATH_IMAGE030
To (1) a
Figure 436823DEST_PATH_IMAGE028
A row vector formed by the rows of the image,
Figure 823942DEST_PATH_IMAGE031
；
step 3.3, calculating residual error vector
Figure 724902DEST_PATH_IMAGE032
，
Figure 615498DEST_PATH_IMAGE032
To (1) a
Figure 943711DEST_PATH_IMAGE028
Each component is
Figure 552547DEST_PATH_IMAGE033
And according to probability
Figure 726039DEST_PATH_IMAGE034
Selecting
Figure 267879DEST_PATH_IMAGE030
Work line of (1)
Figure 235835DEST_PATH_IMAGE035
In which
Figure 863125DEST_PATH_IMAGE036
Represents the selected coefficient matrix
Figure 574729DEST_PATH_IMAGE035
The line is used as the probability of the iterative work line;
step 3.4, calculating by using a soft threshold shrinkage operator to obtain a sparse solution vector of the iteration
Figure 971075DEST_PATH_IMAGE037
；
And 3.5, judging whether the iteration termination condition is met, if so, turning to the next step, otherwise, turning to the step 3.1, adding 1 to the iteration times, and adding the iteration times to the number of times
Figure 109933DEST_PATH_IMAGE019
Is updated to
Figure 693361DEST_PATH_IMAGE037
；
Step 3,6, taking the sparse solution vector of the iteration as a sparse solution vector
Figure 474235DEST_PATH_IMAGE017
Corresponding sparse solution vector
Figure 459509DEST_PATH_IMAGE018
And (6) outputting.
3. Greedy random sparse Kaczmarz-based image reconstruction method according to claim 2, characterized in that, in step 3.1,
Figure 769267DEST_PATH_IMAGE020
the determination rule of (2) is:
Figure 371150DEST_PATH_IMAGE038
wherein
Figure 424556DEST_PATH_IMAGE039
The Frobenius norm is calculated,
Figure 264337DEST_PATH_IMAGE040
which means squaring the Frobenius norm,
Figure 276155DEST_PATH_IMAGE024
show to obtain
Figure 568596DEST_PATH_IMAGE025
The number of the 2 norm is the same as the standard,
Figure 425694DEST_PATH_IMAGE026
expression solution
Figure 651138DEST_PATH_IMAGE025
2 norm squared.
4. The method for image reconstruction based on greedy random sparse Kaczmarz as claimed in claim 2, wherein in step 3.2, a set of positive integer indices is defined
Figure 302700DEST_PATH_IMAGE022
By satisfying the inequality
Figure 348016DEST_PATH_IMAGE041
Index of (2)
Figure 274384DEST_PATH_IMAGE028
Put into an index set
Figure 823177DEST_PATH_IMAGE022
In (1).
5. The method for image reconstruction based on greedy random sparse Kaczmarz as claimed in claim 2, wherein in said step 3.3, the calculation is performed
Figure 911219DEST_PATH_IMAGE033
The method comprises the following steps:
Figure 709410DEST_PATH_IMAGE042
。
6. the greedy random sparse Kaczmarz-based image reconstruction method according to claim 2, wherein the detailed steps of step 3.4 are as follows:
step 3.41, the work line selected according to the step 3.3
Figure 908310DEST_PATH_IMAGE035
Calculating a solution vector
Figure 311610DEST_PATH_IMAGE043
；
Step 3.42, given the hyperparameter in the soft threshold shrink operator
Figure 101711DEST_PATH_IMAGE044
Will solve the vector
Figure 856041DEST_PATH_IMAGE043
Introducing a soft threshold shrinking operator to obtain sparse solution vectors obtained by the iteration
Figure 593053DEST_PATH_IMAGE037
。
7. The image reconstruction method based on greedy random sparse Kaczmarz as claimed in claim 6, wherein in the step 3.41, the vector solution is carried out
Figure 116438DEST_PATH_IMAGE045
In which
Figure 77441DEST_PATH_IMAGE046
Representing a row vector
Figure 53487DEST_PATH_IMAGE047
The translation is a column vector.
8. The image reconstruction method based on greedy random sparse Kaczmarz as claimed in claim 6, wherein in said step 3.42, the soft threshold shrinking operator is:
Figure 125348DEST_PATH_IMAGE048
wherein
Figure 768819DEST_PATH_IMAGE049
Express get
Figure 103985DEST_PATH_IMAGE050
And the maximum value of the sum of 0,
Figure 364065DEST_PATH_IMAGE051
representing a symbolic function, i.e. when
Figure 442880DEST_PATH_IMAGE052
When the utility model is used, the water is discharged,
Figure 940857DEST_PATH_IMAGE053
(ii) a When in use
Figure 978083DEST_PATH_IMAGE054
When the utility model is used, the water is discharged,
Figure 194301DEST_PATH_IMAGE055
(ii) a When the temperature is higher than the set temperature
Figure 76807DEST_PATH_IMAGE056
When the temperature of the water is higher than the set temperature,
Figure 960449DEST_PATH_IMAGE057
；
the soft threshold shrink operator can therefore also be expressed as:
Figure 902997DEST_PATH_IMAGE058
wherein
Figure 340932DEST_PATH_IMAGE059
Represent
Figure 292707DEST_PATH_IMAGE019
To (1)
Figure 765277DEST_PATH_IMAGE028
And (4) a component.
9. The greedy random sparse Kaczmarz-based image reconstruction method according to claim 6, wherein in step 3.42:
Figure 613147DEST_PATH_IMAGE060
Figure 69536DEST_PATH_IMAGE037
the sparse solution vector resulting from this iteration is represented.
10. The image reconstruction method based on greedy random sparse Kaczmarz as claimed in claim 2, wherein in the step 3.5, the iteration termination condition is: the change amount of the sparse solution vector corresponding to the two iterations before and after is minimized or reaches the maximum iteration number.
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