CN106960420B - Image reconstruction method of segmented iterative matching tracking algorithm - Google Patents

Image reconstruction method of segmented iterative matching tracking algorithm Download PDF

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CN106960420B
CN106960420B CN201710088200.0A CN201710088200A CN106960420B CN 106960420 B CN106960420 B CN 106960420B CN 201710088200 A CN201710088200 A CN 201710088200A CN 106960420 B CN106960420 B CN 106960420B
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CN106960420A (en
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石曼曼
李雷
杨真真
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses an image reconstruction method of a segmented iterative matching tracking algorithm, which is characterized in that an observation matrix is generalized and inverted to reduce atom correlation before iteration, the iteration combines the accuracy of atom selection of an OMP algorithm and the backtracking of a CoSaMP algorithm, the iteration is divided into two stages, the first stage iterates K/2 times by using the OMP algorithm (assuming that the signal sparsity K is an even number, if the K is an odd number, the K/2 is rounded downwards), the obtained residual error and atoms are used as initial input of the second stage, then the CoSaMP is used for continuing the iteration, and the atom selection criterion is changed, so that sparse signals are accurately and quickly reconstructed. The method can simultaneously give consideration to the image reconstruction time and the reconstruction precision, and has the advantages of high precision and short reconstruction time compared with the existing algorithm.

Description

Image reconstruction method of segmented iterative matching tracking algorithm
Technical Field
The invention relates to the technical field of image reconstruction based on compressed sensing, in particular to an image reconstruction method based on a segmented iterative matching tracking algorithm.
Background
With the increasing degree of social informatization, data acquisition equipment is continuously developed in technology, and information needing to be processed at a later stage is increased day by day, especially multimedia information such as voice, images and videos, which undoubtedly increases the workload of signal sampling, transmission and storage. Compared with the traditional Nyquist sampling theorem, the compressed sensing has the advantages that the signal can be acquired at a sampling rate far lower than Nyquist, the accurate reconstruction of the observed signal or image can be realized, the sampling and the compression are carried out simultaneously, the storage space is saved, and the waste of resources is greatly reduced.
The image reconstruction technology based on compressed sensing comprises three parts: sparse representation of image signals, selection of an observation matrix and reconstruction of the image signals. The most critical step is to select a reliable algorithm for image reconstruction, and the selection of the algorithm is directly related to the quality of a reconstruction result.
The indexes for evaluating the image reconstruction effect are generally image reconstruction accuracy and reconstruction speed. The image reconstruction precision and the reconstruction speed are respectively expressed by the reconstructed peak signal-to-noise ratio (PSNR) value and the reconstruction time. Wherein a higher PSNR value represents a better reconstruction accuracy; a shorter reconstruction time represents a faster reconstruction speed. However, the greedy tracking type image reconstruction algorithm in the current compressed sensing cannot simultaneously give consideration to the image reconstruction precision and the reconstruction speed.
Disclosure of Invention
The technical problem to be solved by the invention is to overcome the defects of the prior art, and provide an image reconstruction method of a segmented iterative matching tracking algorithm, which can simultaneously ensure the accuracy and the speed of image reconstruction.
The invention adopts the following technical scheme for solving the technical problems:
the invention provides an image reconstruction method of a segmented iterative matching tracking algorithm, which comprises the following steps:
step 1, inputting an observation matrix phi, an observation vector y, a signal sparsity K and a threshold value delta; first stage initialization residual
Figure BDA0001228531990000011
Index set
Figure BDA0001228531990000012
Original subset
Figure BDA0001228531990000013
The iteration time t is 1; wherein the content of the first and second substances,
Figure BDA0001228531990000014
is an empty set;
step 2, generalized inversion is carried out on the atom set phi to obtain A ═ phi (phi)T)-1Phi, wherein A is a generalized inverse matrix of phi, and superscript T represents transposition operation;
step 3, finding the index
Figure BDA0001228531990000015
So that
Figure BDA0001228531990000016
Wherein the content of the first and second substances,
Figure BDA0001228531990000017
denotes the residual at the t-1 th iteration
Figure BDA0001228531990000018
Column j a of AjPerforming inner product operation;
step 4, updating the index set and the original subset:
Figure BDA0001228531990000021
wherein
Figure BDA0001228531990000022
For the index set updated for the t-th time,
Figure BDA0001228531990000023
for the atomic set updated for the t-th time,
Figure BDA0001228531990000024
is the first in the corresponding matrix A
Figure BDA0001228531990000025
A column vector;
step 5, solving the t iteration
Figure BDA0001228531990000026
Least squares solution of
Figure BDA0001228531990000027
Figure BDA0001228531990000028
Wherein the content of the first and second substances,
Figure BDA0001228531990000029
a column vector representing t × 1;
step 6, updating residual error in the t iteration
Figure BDA00012285319900000210
Figure BDA00012285319900000211
Step 7, if t is equal to t +1 and is equal to or less than K/2, returning to the step 3; otherwise, executing step 8;
step 8, initialization of the second stage: let residual error
Figure BDA00012285319900000212
Index set
Figure BDA00012285319900000213
Original subset
Figure BDA00012285319900000214
The iteration number s is 1;
step 9, calculating ∞<rs-1,aj>1,2, …, N will not smoke<rs-1,aj>Marking | as u, calculating floor (3K/2), marking floor (3K/2) as q, arranging elements in u in descending order, selecting former q item values, and forming a set J by the column number J of the values corresponding to phi0(ii) a Wherein the content of the first and second substances,
Figure BDA00012285319900000215
denotes the residual r at the s-1 th iterations-1And j column a in AjPerforming inner product operation;
step 10, updating index set Λs＝Λs-1∪J0Atomic set of phis＝Φs-1∪{φj},j∈J0(ii) a Wherein, ΛsFor the s-th updated index set, the number of elements is Ls，ΦsFor the atomic set updated s, phijIs the jth column vector in the corresponding matrix A;
step 11, solving y ═ Φ during the s-th iterationsxsLeast squares solution of
Figure BDA00012285319900000216
Figure BDA00012285319900000217
Wherein x issIs LsA column vector of x 1;
step 12, from
Figure BDA00012285319900000218
The top K term with the largest absolute value is recorded as
Figure BDA00012285319900000219
Where the K term element corresponds to phisK in (1) is represented by phisK，ΦsKThe column number corresponding to the middle K column in phi is marked as LambdasKUpdating the set Λs＝ΛsK；
Step 13, updating residual error r in the s-th iterations：
Figure BDA00012285319900000220
Step 14, making s equal to s +1, if s is less than or equal to K/2, returning to step 9, if s is less than or equal to K/2>K/2, judgment of rsWhether the difference is less than or equal to the threshold value delta or not, if not, returning to the step 9; if s is>K/2 and rsIf the delta is less than or equal to delta, the iteration is terminated and output
Figure BDA00012285319900000221
The observation matrix phi is obtained by firstly sampling image signals, secondly thinning by utilizing wavelet transformation and then observing the thinned matrix projection by utilizing a Gaussian matrix.
As a further optimization scheme of the image reconstruction method of the segmentation iteration matching tracking algorithm, the wavelet transform is a single-scale two-dimensional discrete wavelet transform method.
As a further optimization scheme of the image reconstruction method of the segmented iterative matching tracking algorithm, the steps 1 to 7 are the calculation process of the OMP algorithm, and the steps 8 to 14 are the calculation process of the CoSaMP algorithm.
As a further optimization scheme of the image reconstruction method of the segmented iterative matching pursuit algorithm, the threshold value delta is 10-5*‖y‖2。
Compared with the prior art, the invention adopting the technical scheme has the following technical effects: the method can give consideration to both the image reconstruction time and the reconstruction precision, and has the advantages of high precision and short reconstruction time compared with the existing algorithm.
Drawings
FIG. 1 is a flowchart of the image reconstruction method of a segmented iterative matching pursuit algorithm designed by the present invention.
Fig. 2a is a diagram of a one-dimensional gaussian random signal, fig. 2b is a diagram of a reconstruction result of the one-dimensional gaussian random signal by the reconstruction method designed by the present invention, and fig. 2c is a diagram of an error between the one-dimensional gaussian random signal and the reconstruction result of the one-dimensional gaussian random signal by the reconstruction method designed by the present invention.
Fig. 3 is a schematic diagram showing the change of the influence of the sparsity of the one-dimensional gaussian random signal on the reconstruction success rate of the OMP algorithm, the CoSaMP algorithm and the reconstruction method provided by the present invention at the same sampling rate.
Fig. 4 is a schematic diagram of changes in the influence of the observation value of the one-dimensional gaussian random signal on the reconstruction success rate of the OMP algorithm, the CoSaMP algorithm and the reconstruction method provided by the present invention under the same sparsity.
Fig. 5 is a schematic diagram of changes in the influence of the sparsity of the one-dimensional gaussian random signal on the operation time of the OMP algorithm, the CoSaMP algorithm, and the reconstruction method proposed by the present invention at the same sampling rate.
Detailed Description
In order that the objects, aspects and advantages of the present invention will become more apparent, the present invention will be further described in detail with reference to the accompanying drawings in which specific embodiments are shown. The following description of the embodiments of the present invention with reference to the accompanying drawings is intended to explain the general inventive concept of the present invention and should not be construed as limiting the invention.
As shown in fig. 1, an image reconstruction method of a segmented iterative matching pursuit algorithm includes two stages, and the specific calculation process is as follows:
inputting: the observation matrix phi is used to observe the matrix,
the observation vector y is then used to determine,
signal sparsity K, threshold δ;
and (3) outputting: reconstructed signal estimation
Figure BDA0001228531990000031
Step 1, initialization residue of the first stageDifference (D)
Figure BDA0001228531990000041
Index set
Figure BDA0001228531990000042
Original subset
Figure BDA0001228531990000043
The iteration time t is 1; wherein the content of the first and second substances,
Figure BDA0001228531990000044
is an empty set;
step 2, generalized inversion is carried out on the atom set phi to obtain A ═ phi (phi)T)-1Phi, wherein A is a generalized inverse matrix of phi, and superscript T represents transposition operation;
step 3, finding the index
Figure BDA0001228531990000045
So that
Figure BDA0001228531990000046
Wherein the content of the first and second substances,
Figure BDA0001228531990000047
denotes the residual at the t-1 th iteration
Figure BDA0001228531990000048
Column j a of AjPerforming inner product operation;
step 4, updating the index set and the original subset:
Figure BDA0001228531990000049
wherein
Figure BDA00012285319900000410
For the index set updated for the t-th time,
Figure BDA00012285319900000411
for the atomic set updated for the t-th time,
Figure BDA00012285319900000412
is the first in the corresponding matrix A
Figure BDA00012285319900000413
A column vector;
step 5, solving the t iteration
Figure BDA00012285319900000414
Least squares solution of
Figure BDA00012285319900000415
Figure BDA00012285319900000416
Wherein
Figure BDA00012285319900000417
Representing a t x 1 column vector.
Step 6, updating residual error in the t iteration
Figure BDA00012285319900000418
Figure BDA00012285319900000419
Step 7, if t is equal to t +1 and is equal to or less than K/2, returning to the step 3; otherwise, executing step 8;
step 8, initialization of the second stage: let residual error
Figure BDA00012285319900000420
Index set
Figure BDA00012285319900000421
Original subset
Figure BDA00012285319900000422
The iteration number s is 1;
step 9, calculating ∞<rs-1,aj>1,2, …, N will not smoke<rs-1,aj>Marking | as u, and calculating floor: (3K/2), recording floor (3K/2) as q, arranging elements in u in descending order, selecting the first q items, and forming a set J by the column serial numbers J of the values corresponding to phi0(ii) a Wherein the content of the first and second substances,
Figure BDA00012285319900000423
denotes the residual r at the s-1 th iterations-1And j column a in AjPerforming inner product operation;
step 10, updating index set Λs＝Λs-1∪J0Atomic set of phis＝Φs-1∪{φj},j∈J0(ii) a Wherein, ΛsFor the s-th updated index set, the number of elements is Ls，ΦsFor the atomic set updated s, phijIs the jth column vector in the corresponding matrix A;
step 11, solving y ═ Φ during the s-th iterationsxsLeast squares solution of
Figure BDA00012285319900000424
Figure BDA00012285319900000425
Wherein x issIs LsA column vector of x 1.
Step 12, from
Figure BDA00012285319900000426
The top K term with the largest absolute value is recorded as
Figure BDA00012285319900000427
Where the K term element corresponds to phisK in (1) is represented by phisK，ΦsKThe column number corresponding to the middle K column in phi is marked as LambdasKUpdating the set Λs＝ΛsK；
Step 13, updating residual error r in the s-th iterations：
Figure BDA0001228531990000051
Step 14, let s equal to s +1, if s is less than or equal to K/2,returning to step 9, if s>K/2, judgment of rsWhether the difference is less than or equal to the threshold value delta or not, if not, returning to the step 9; if s is>K/2 and rsIf the delta is less than or equal to delta, the iteration is terminated and output
Figure BDA0001228531990000052
The invention relates to an image reconstruction method of a segmented iterative matching tracking algorithm, wherein an atom selection criterion is that in a first stage, an OMP algorithm is adopted, atoms (a column vector) with the largest inner product with the current residual error are selected from an observation matrix, namely an atom library, in each iteration and are added into a support set, and the residual error and the support set are updated by solving the current least square solution; in the second stage, a CoSaMP algorithm is adopted, the atom set and the residual error selected in the first stage are used as initial input, in each iteration, atom selection comprises two parts of atom selection and atom elimination, the first 3K/2 item with the largest inner product of an observation matrix and the current residual error is selected in the atom selection stage, after the index is updated, the least square solution is solved, the first K item with the largest absolute value is selected from the least square solution, K/2 atoms are eliminated, the residual error is updated, and the iteration is continued until the residual error is smaller than the threshold value 10-5*‖y‖2And reconstruction precision is achieved.
The above-mentioned matching pursuit algorithms, OMP and CoSaMP algorithms are well known.
Examples
The reconstruction effect of the algorithm is verified under the experimental simulation environment of MATLAB 7.9, Intercore (TM) CPU, main frequency 2.30GHZ and memory 8G for the one-dimensional Gaussian random signal with the signal length N of 256 and the sparsity K of 26 and the Lena standard gray image of 256 multiplied by 256.
And analyzing results of the technical scheme. The results of the analysis and comparison in the case of a one-dimensional gaussian random signal are shown in fig. 2a, 2b, 2c, 3, 4 and 5.
In fig. 2a, 2b, and 2c, fig. 2a is a one-dimensional gaussian random signal diagram, fig. 2b is a diagram of a reconstruction result of the one-dimensional gaussian random signal by the reconstruction method of the present invention, and fig. 2c is a diagram of an error between the one-dimensional gaussian random signal and a reconstruction result of the one-dimensional gaussian random signal by the reconstruction method of the present invention. Wherein the content of the first and second substances,the length N of the one-dimensional gaussian random signal is 256, the sparsity K is 26, and the one-dimensional gaussian random signal is reconstructed by an observation matrix Φ: m × N is a Gaussian random matrix, the sampling rate is M/N is 0.5, and the stop threshold value delta is 10-5*‖y‖2Defining reconstruction error
Figure BDA0001228531990000053
The criterion for judging the success of signal reconstruction is epsilonr|≤1e-5. It can be seen that the error epsilon of the reconstruction of the one-dimensional Gaussian random signalr∈(-1e-14,1e-14) Far less than 1e-5The reconstructed signal is very close to the original signal, so that the reconstruction method provided by the invention can accurately reconstruct the original signal.
In fig. 3, the length N of the one-dimensional gaussian random signal is 256, and the observation matrix Φ: and M × N selects a Gaussian random matrix, and reconstruction is realized by using an OMP algorithm, a CoSaMP algorithm and the reconstruction method provided by the invention respectively, wherein the sampling rates are all 0.5. The reconstruction success rate is obtained by averaging each algorithm by running 200 times. As can be seen from fig. 3, the reconstruction success rate of the reconstruction method provided by the present invention is higher than that of the other two algorithms, and the reconstruction success rates of the three algorithms are all above 90% under the condition of small sparsity, but the reconstruction performance of the reconstruction method provided by the present invention is more obvious along with the increase of sparsity, and when the sparsity K is 50, the reconstruction success rates of the OMP and CoSaMP algorithms are sharply reduced and are both less than 15%, and the reconstruction success rate of the reconstruction method provided by the present invention is still above 90%.
In fig. 4, the length N of the one-dimensional gaussian random signal is 256, and the observation matrix Φ: and selecting a Gaussian random matrix by M x N, and realizing reconstruction by respectively using an OMP algorithm, a CoSaMP algorithm and the reconstruction method provided by the invention, wherein the sparsity is 26. The reconstruction success rate is obtained by averaging each algorithm by running 200 times. As can be seen from fig. 4, when the observed value is small, the reconstruction success rates of the three algorithms are all low, but the reconstruction success rates of the three algorithms are increased with the gradual increase of the observed value, but the reconstruction method provided by the present invention has the fastest growth rate and the best performance, when the observed value M is 70, that is, the sampling rate is about 0.27, the reconstruction success rate of the reconstruction method provided by the present invention reaches more than 60%, and under the same observation, the CosaMP algorithm is unsuccessfully reconstructed, and the reconstruction success rate of the OMP algorithm is less than half of the reconstruction success rate of the reconstruction method provided by the present invention. Along with the increase of the observed value, the reconstruction success rate of the reconstruction method provided by the invention is rapidly increased, the robustness of a new algorithm is good, the reconstruction is more stable, and the reconstruction success rate is superior to that of the other two algorithms.
In fig. 5, the length N of the one-dimensional gaussian random signal is 256, and the observation matrix Φ: the M multiplied by N selects a Gaussian random matrix, and the reconstruction is realized by using an OMP algorithm, a CoSaMP algorithm and the reconstruction method provided by the invention respectively, wherein the sampling rate is 0.5, namely M is 128 and N is 256. Run time was averaged by running 200 runs per algorithm. As can be seen from fig. 5, for one-dimensional gaussian random signals, the running time of OMP and the algorithm herein is smoothly and slowly increased with increasing sparsity, and the running time of the new algorithm is faster than that of OMP algorithm, because iteration is continued with the CoSaMP algorithm step after the first stage iteration, so that the atom selection is more accurate, and the algorithm convergence time is shortened. When the sparsity is less than 41, the running time of the CoSaMP algorithm is the minimum, but the reconstruction time of the new algorithm is very close to that of the original algorithm, the running time of the CoSaMP algorithm is greatly increased along with the increase of the sparsity, and the new algorithm is the algorithm with the fastest reconstruction among the three algorithms and has higher robustness.
In the case of a Lena standard gray image of 256 × 256, the reconstruction is realized by using an OMP algorithm, a CoSaMP algorithm and the reconstruction method proposed by the present invention, respectively, and the analysis results are shown in tables 1 and 2.
TABLE 1 reconstruction time t (unit: s) for different algorithms at different sampling rates
Figure BDA0001228531990000061
TABLE 2 PSNR values (in dB) for different algorithms at different sampling rates
Figure BDA0001228531990000071
As can be seen from tables 1 and 2, the reconstruction time of OMP is short, but the reconstruction accuracy is poor; the reconstruction precision of the CoSaMP is high, but the reconstruction time is long; the image reconstruction method provided by the invention gives consideration to both the image reconstruction precision and the reconstruction speed, and has a better reconstruction effect.
The algorithm reduces atom correlation by generalized inversion of an observation matrix before iteration, the iteration combines the accuracy of atom selection by an OMP algorithm and the backtracking of a CoSaMP algorithm, the iteration is divided into two stages, namely, in the first stage, the OMP algorithm is used for iterating K/2 times (assuming that the signal sparsity K is an even number, and if the K is an odd number, the K/2 is rounded downwards), the obtained residual error and atoms are used as initial input in the second stage, then the CoSaMP is used for continuing iteration, and the atom selection criterion is changed, so that sparse signals are accurately and quickly reconstructed. The experimental result shows that for one-dimensional random Gaussian signals, the reconstruction success rate of the new algorithm is high, the reconstruction effect is superior to that of the OMP algorithm and the CoSaMP algorithm, the reconstruction time is superior to that of the OMP algorithm, and the new algorithm is more robust; for two-dimensional image signals, the new algorithm is fast in reconstruction time, good in reconstruction effect and high in efficiency and practicability.

Claims (4)

1. An image reconstruction method of a segmented iterative matching pursuit algorithm is characterized by comprising the following steps:
step 1, inputting an observation matrix phi, an observation vector y, a signal sparsity K and a threshold value delta; first stage initialization residual
Figure FDA0002129740390000011
Index set
Figure FDA0002129740390000012
Original subset
Figure FDA0002129740390000013
The iteration time t is 1; wherein the content of the first and second substances,
Figure FDA0002129740390000014
is an empty set;
step 2, generalized inversion is carried out on the atom set phi to obtain A ═ phi (phi)T)-1Phi, wherein A is a generalized inverse matrix of phi, and superscript T represents transposition operation;
step 3, finding the index
Figure FDA0002129740390000015
So that
Figure FDA0002129740390000016
Wherein the content of the first and second substances,
Figure FDA0002129740390000017
denotes the residual at the t-1 th iteration
Figure FDA0002129740390000018
Column j a of AjPerforming inner product operation;
step 4, updating the index set and the original subset:
Figure FDA0002129740390000019
wherein
Figure FDA00021297403900000110
For the index set updated for the t-th time,
Figure FDA00021297403900000111
for the atomic set updated for the t-th time,
Figure FDA00021297403900000112
is the first in the corresponding matrix A
Figure FDA00021297403900000113
A column vector;
step 5, solving the t iteration
Figure FDA00021297403900000114
Least squares solution of
Figure FDA00021297403900000115
Figure FDA00021297403900000116
Wherein the content of the first and second substances,
Figure FDA00021297403900000117
a column vector representing t × 1;
step 6, updating residual error in the t iteration
Figure FDA00021297403900000118
Figure FDA00021297403900000119
Step 7, if t is equal to t +1 and is equal to or less than K/2, returning to the step 3; otherwise, executing step 8;
step 8, initialization of the second stage: let residual error
Figure FDA00021297403900000120
Index set
Figure FDA00021297403900000121
Original subset
Figure FDA00021297403900000122
The iteration number s is 1;
step 9, calculating ∞<rs-1,aj>1,2, …, N will not smoke<rs-1,aj>Marking | as u, calculating floor (3K/2), marking floor (3K/2) as q, arranging elements in u in descending order, selecting former q item values, and forming a set J by the column number J of the values corresponding to phi0(ii) a Wherein the content of the first and second substances,
Figure FDA00021297403900000123
denotes the residual r at the s-1 th iterations-1And j column a in AjPerforming inner product operation;
step 10, updating index set Λs＝Λs-1∪J0Atomic set of phis＝Φs-1∪{φj},j∈J0(ii) a Wherein, ΛsFor the s-th updated index set, the number of elements is Ls，ΦsFor the atomic set updated s, phijIs the jth column vector in the corresponding matrix A;
step 11, solving y ═ Φ during the s-th iterationsxsLeast squares solution of
Figure FDA00021297403900000124
Figure FDA00021297403900000125
Wherein x issIs LsA column vector of x 1;
step 12, from
Figure FDA0002129740390000021
The top K term with the largest absolute value is recorded as
Figure FDA0002129740390000022
Figure FDA0002129740390000025
Where the K term element corresponds to phisK in (1) is represented by phisK，ΦsKThe column number corresponding to the middle K column in phi is marked as LambdasKUpdating the set Λs＝ΛsK；
Step 13, updating residual error r in the s-th iterations：
Figure FDA0002129740390000023
Step 14, making s equal to s +1, if s is less than or equal to K/2, returning to step 9, if s is less than or equal to K/2>K/2, judgment of rsWhether the difference is less than or equal to the threshold value delta or not, if not, returning to the step 9; if s is>K/2 and rsDelta is less than or equal toIteration ends, outputs
Figure FDA0002129740390000024
The observation matrix phi is obtained by firstly sampling image signals, secondly thinning by utilizing wavelet transformation, and then projecting and observing the thinned matrix by utilizing a Gaussian matrix.
2. The image reconstruction method of a piecewise iterative matching pursuit algorithm of claim 1, wherein said wavelet transform is a single-scale two-dimensional discrete wavelet transform method.
3. The image reconstruction method of a segmentation iterative matching pursuit algorithm according to claim 1, characterized in that the steps 1 to 7 are the calculation process of the OMP algorithm, and the steps 8 to 14 are the calculation process of the CoSaMP algorithm.
4. The image reconstruction method of a segmented iterative matching pursuit algorithm according to claim 1, characterized in that the threshold δ is 10-5*‖y‖2。
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