CN114820387B - Image recovery method and terminal based on probability induction nuclear norm minimization - Google Patents

Image recovery method and terminal based on probability induction nuclear norm minimization Download PDF

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CN114820387B
CN114820387B CN202210588134.4A CN202210588134A CN114820387B CN 114820387 B CN114820387 B CN 114820387B CN 202210588134 A CN202210588134 A CN 202210588134A CN 114820387 B CN114820387 B CN 114820387B
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刘慧�
张中兴
王欣雨
郭强
范琳伟
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Shandong University of Finance and Economics
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Abstract

The invention provides an image recovery method and a terminal based on probability induction nuclear norm minimization, wherein the method adopts the statistical characteristic of maximum posterior modeling singular values so as to estimate the singular values of a low-rank matrix to be reconstructed. And adopting low-rank truncation to ignore singular values which have smaller values and are irrelevant to the remarkable information of the image, and inhibiting noise introduced in image recovery. Then, to further fill in the lost image details when singular values are processed, the quality of the resulting image is improved by using the intermediate restored image through a simple and efficient residual cascade mechanism. Finally, the method is applied to an image recovery task to realize image super-resolution and image denoising. The method of the present invention is superior to many methods of the prior art in both numerical index and visual effect.

Description

Image recovery method and terminal based on probability induction nuclear norm minimization
Technical Field
The invention relates to the technical field of medical image processing, in particular to an image recovery method and a terminal based on probability induction nuclear norm minimization.
Background
Image recovery (image restoration, IR) has significant utility in the fields of medical imaging, remote sensing, video surveillance, and the like, and has thus been a critical issue until now. Typically, the IR target is from an observed degraded image
Figure BDA0003666655280000011
Is to restore its potentially high quality image +.>
Figure BDA0003666655280000012
Wherein->
Figure BDA0003666655280000013
Figure BDA0003666655280000014
Is additive white gaussian noise with an average value of 0; />
Figure BDA0003666655280000015
Is a degradation matrix, which can be defined as an identity matrix for denoising analog images or a fuzzy downsampling complex for super-resolution reconstructionA matrix. Based on Bayes' theorem, obtaining an estimated solution of the original sharp image x by solving the maximum posterior probability (maximum a posterior, MAP) problem>
Figure BDA0003666655280000016
Figure BDA0003666655280000017
Wherein,,
Figure BDA0003666655280000018
is a log-likelihood term of the known image y whose probability distribution depends on the a priori distribution of additive noise. log P (x) is a priori of an unknown high quality image x and is used to regularize the image restoration process independent of y. In general, equation (1) can be converted into an objective function,
Figure BDA0003666655280000019
wherein,,
Figure BDA00036666552800000110
is the image prior, η is the regularized term coefficient.
Since restoring the original image x from the observed image y is an inverse problem, the restoration process is typically constrained using image priors as regularization terms. Currently, various regularization methods proposed for a priori are widely used in the field of image processing, such as Tikhonov and Total Variation. Recently, low rank priors of natural images are widely used in various image restoration tasks. The basic principle of low-rank regularization terms is that the matrix formed by image similarity blocks is low-rank, however, minimizing the rank of the matrix is an NP-hard problem, which most algorithms solve by kernel norm minimization (nuclear norm minimization, NNM).
In particular, the nuclear norm of the image matrix y is known to be the sum of its singular values, i.e. |y| * ＝∑ i |σ i (y)| 1 ，σ i (y) is the ith singular value of y. By minimizing the kernel norms of y * NNM may approximate x using y. However, NNM ignores differences between singular values and thus has limited flexibility. A truncated kernel norm minimization (truncated nuclear norm minimization, TNNM) algorithm is proposed in the prior art, which regularizes only part of the singular values and is therefore also limited by its flexibility.
Also proposed in the prior art is a weighted kernel-norm minimization (weighted nuclear norm minimization, WNNM) model that re-weights the singular values differently and is therefore more flexible than NNM and TNNM. However, the above algorithms all process singular values discretely, without considering the statistical relevance of the singular values.
Therefore, there is still room for further improvement in solving the low rank matrix approximation problem, and how to model the statistical correlation between singular values to improve the performance of the kernel norm minimization algorithm is a technical problem that is currently in need of solution.
Disclosure of Invention
The invention provides an image recovery method based on probability induction kernel norm minimization, which solves the problem of how to discover potential statistical properties of singular values to better solve the low-rank matrix approximation.
The method comprises the following steps: recovering a high-definition image x from the degraded image y based on a low-rank model of the block, and utilizing an image non-local self-similarity constraint reconstruction process through a low-rank regularization term;
if the similar blocks extracted from the image are remodeled into vector form, then a low-rank matrix based on the similar blocks can be formed by collecting block vectors;
constructing an augmented lagrangian equation by adopting an alternate direction multiplier method;
prior modeling using each singular value in equation (11), and then summing the estimates of all singular values as σ i (L j ) Is determined by the final estimate of (a);
Figure BDA0003666655280000031
by combining the mth component of formulas (10) and (11)
Figure BDA0003666655280000032
Substituted into the formula (9),
Figure BDA0003666655280000033
Figure BDA0003666655280000034
obtaining:
Figure BDA0003666655280000035
let (12) relate to sigma i (L j ) The partial derivative of (c) is 0,
Figure BDA0003666655280000036
by taking each component in equation (11) as an a priori model, σ is obtained i (L j ) And then adding and averaging the estimated values to obtain:
Figure BDA0003666655280000037
wherein, |Ω i I represents the neighborhood Ω i Number of internal singular values. First, for matrix I j The singular value decomposition is carried out and,
Figure BDA0003666655280000041
then, the first s singular value estimated values are selected to form L j Is a singular value matrix Σ of (2) s ；
Wherein sigma 1 (L j )≥…≥σ s (L j )≥σ s+1 (L j )≥…σ n (L j )，n＝min(d,p)；
By truncating the singular value sigma s+1 (L j ),…,σ n (L j ) Compression matrix L j The method comprises the steps of carrying out a first treatment on the surface of the Solving to obtain L j Is estimated by (a):
Figure BDA0003666655280000042
wherein,,
Figure BDA0003666655280000043
define weight +.>
Figure BDA0003666655280000044
Figure BDA0003666655280000045
Is the s singular value estimated in the kth iteration, epsilon is a positive number of small value.
And (3) performing enhancement processing on the intermediate reconstructed image by using a residual cascade mechanism, and increasing the quality of the resulting image.
It should be further noted that, the step of recovering the required image x from the degraded image y based on the low-rank model of the block, and using the non-local self-similarity constraint reconstruction process of the image through the low-rank regularization term includes:
is provided with
Figure BDA0003666655280000046
Representing image block->
Figure BDA0003666655280000047
Is included in the group of similar blocks to sample block->
Figure BDA0003666655280000048
The top p most similar blocks; wherein (1)>
Figure BDA0003666655280000049
Is selected to be centered on the pixel position j and of a size of
Figure BDA00036666552800000410
Is a similar block of (2);
to construct image similarity blocks
Figure BDA00036666552800000411
Mapping between the original image x, defining +.>
Figure BDA00036666552800000412
Wherein,,
Figure BDA0003666655280000051
is a binary index matrix if the image block +.>
Figure BDA0003666655280000052
Corresponds to pixel b of the original image x, then
Figure BDA0003666655280000053
Otherwise->
Figure BDA0003666655280000054
Using low rank matrix I j To approximate a high quality image x, will I j As image priors;
Figure BDA0003666655280000055
introducing an auxiliary variable L j Characterizing the low rank matrix of image x and re-estimating the block matrix I by probability induction j Is a singular value of (2);
wherein, define
Figure BDA0003666655280000056
Is I j Is estimated by (a):
Figure BDA0003666655280000057
in the method, if s is present<r＝rank(x j ) Then there is a real matrix L j ，
Figure BDA0003666655280000058
Figure BDA0003666655280000059
Wherein,,
Figure BDA00036666552800000510
is L j Is a singular value of (2); s is defined as:
Figure BDA00036666552800000511
by (18) pair of augmented Lagrangian multipliers U j The updating is performed such that,
Figure BDA00036666552800000512
it should be further noted that, the step of enhancing the intermediate reconstructed image by using a residual cascade mechanism, the enhancing the quality of the resulting image further includes:
recovering image x from the middle C The residual characteristics are extracted from the obtained product,
x R ＝x C -x S (19)
wherein x is R Is an intermediate restored image x C And corresponding source image x S Residual errors between; c is the number of cascaded layers, and when c=1,
Figure BDA0003666655280000061
x S =y; when c=c, ">
Figure BDA0003666655280000062
x S =y, reconstruct x by solving the objective function in equation (19) R The method comprises the following steps:
Figure BDA0003666655280000063
wherein U is R Is an introduced lagrangian multiplier; λ and γ are coefficients of the data fidelity term and regularization term, respectively, in equation (19);
by x R The singular value decomposition is carried out and,
Figure BDA0003666655280000064
solution is obtained through PINNM algorithm:
Figure BDA0003666655280000065
wherein,,
Figure BDA0003666655280000066
is->
Figure BDA0003666655280000067
Is the i-th singular value estimate of (a); residual error to be reconstructed->
Figure BDA0003666655280000068
Adding back the intermediate recovery image to obtain a result image;
Figure BDA0003666655280000069
updating U through ADMM R ，
Figure BDA0003666655280000071
The invention also provides a terminal machine for realizing the image recovery method based on probability induction kernel norm minimization, which comprises the following steps:
the memory is used for storing a terminal program and an image recovery method based on probability induction kernel norm minimization;
and the processor is used for executing the terminal program and the image restoration method based on the probability induction nuclear norm minimization so as to realize the image restoration method based on the probability induction nuclear norm minimization.
From the above technical scheme, the invention has the following advantages:
the image recovery method based on probability induction nuclear norm minimization solves the problem of low-rank matrix approximation through the statistical characteristic of maximum posterior modeling singular values, adopts probability induction singular value estimation to estimate the singular values of a potential low-rank matrix better, and suppresses the interference of factors such as noise through low-rank truncation. In order to further compensate for the lost image details in low-rank truncation, a residual cascade scheme is designed, and the quality of a result image is gradually improved by utilizing an intermediate recovery image.
The method provided by the invention can recover clearer textures and improve the visual quality of the recovered images. Compared with the existing method, the method can realize superior image recovery performance in quantitative and qualitative comparison.
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In order to more clearly illustrate the technical solutions of the present invention, the drawings that are needed in the description will be briefly introduced below, it being obvious that the drawings in the following description are only some embodiments of the present invention, and that other drawings can be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flowchart of a PINNM algorithm;
FIG. 2 is a graph of a singular value statistical probability distribution;
FIG. 3 is a view of 10 test images selected for use in the experiments of the present invention;
FIG. 4 is a graph showing the super-resolution effect of different algorithms on the natural images of the plants;
fig. 5 is a comparison chart of denoising effects of different algorithms on a camera natural image.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The present invention aims at solving the technical problem of how to model the statistical correlation between singular values to improve the performance of the kernel norm minimization algorithm. I.e., how to explore the potential statistical properties of singular values to better solve the low rank matrix approximation problem. A new probability-induced kernel-norm minimization (PINNM) algorithm is proposed and applied to image super-resolution and denoising.
It should be noted that, the units and algorithm steps of each example described in the embodiments disclosed in the image restoration method based on the probability-induced kernel norm minimization provided in the present invention can be implemented in electronic hardware, terminal software, or a combination of both, and in order to clearly illustrate the interchangeability of hardware and software, the components and steps of each example have been generally described in terms of functions in the foregoing description. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the solution. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention.
In the image restoration method based on probability-induced kernel norm minimization provided by the present invention, it should be understood that the disclosed system, apparatus and method may be implemented in other manners. For example, the apparatus embodiments described above are merely illustrative, e.g., the division of the units is merely a logical function division, and there may be additional divisions when actually implemented, e.g., multiple units or components may be combined or integrated into another system, or some features may be omitted or not performed. In addition, the coupling or direct coupling or communication connection shown or discussed with each other may be an indirect coupling or communication connection via some interfaces, devices, or elements, or may be an electrical, mechanical, or other form of connection.
In the image recovery method based on probability induction nuclear norm minimization provided by the invention, a nuclear norm minimization algorithm based on maximum posterior is provided by analyzing the problem of probability induction nuclear norm minimization. The probability induction singular value estimator is introduced to estimate singular values of a required low-rank matrix, and low-rank truncation is adopted to ignore small singular values, so that noise is suppressed.
Then, to compensate for the lost image details when dealing with singular values, a simple and efficient residual cascade mechanism is designed that gradually enhances the resulting image quality with intermediate reconstructed images.
Finally, the effectiveness of the algorithm of the invention is verified through experiments. Experimental results show that the algorithm of the invention is superior to most of the latest algorithms in terms of objective evaluation indexes and subjective visual effects.
The PINNM image recovery model is processed, specifically, firstly, a low-rank matrix reconstruction model based on image blocks is introduced, and a required low-rank matrix is constructed by utilizing non-local self-similarity of natural images; then, a probability-induced kernel-norm minimization algorithm is used for processing. Finally, it is briefly described how the residual cascade mechanism is utilized to enhance the image quality. The model reconstruction process is shown in fig. 1.
The invention recovers the required image x from the degraded image y by adopting a low-rank model based on blocks, and utilizes the non-local self-similarity constraint reconstruction process of the image by using a low-rank regularization term. If the similar blocks extracted from the image are reshaped into vector form, then a low rank matrix based on the similar blocks can be formed by collecting the block vectors. Based on the above principle, the inventionThe model introduces euclidean distance to quantify the similarity of image blocks and constructs a low rank matrix of similar blocks accordingly. Assume that
Figure BDA0003666655280000091
Representing image block->
Figure BDA0003666655280000092
Is included in the group of similar blocks to sample block->
Figure BDA0003666655280000093
The top p most similar blocks. Wherein (1)>
Figure BDA0003666655280000094
Is selected to be centered on the pixel position j and of size +.>
Figure BDA0003666655280000101
Is a similar block of (c). For constructing image similarity block->
Figure BDA0003666655280000102
Mapping between the original image x, definition +.in the invention>
Figure BDA0003666655280000103
Wherein (1)>
Figure BDA0003666655280000104
Is a binary index matrix if the image block +.>
Figure BDA0003666655280000105
Pixel a of (2) corresponds to pixel b of the original image x, then +.>
Figure BDA0003666655280000106
Otherwise->
Figure BDA0003666655280000107
Then, utilize low rank matrix I j To approximate a high quality image x, will I j Low rank as image firstAnd (5) checking.
Figure BDA0003666655280000108
For convenience of optimization, an auxiliary variable L is introduced j Characterizing the low rank matrix of image x and re-estimating the block matrix I by probability induction j And the approximation accuracy is improved. Wherein, define
Figure BDA0003666655280000109
Is I j Is a function of the estimate of (2).
Figure BDA00036666552800001010
The PINNM model adopts an alternate direction multiplier method (Alternating Direction Method of Multipliers, ADMM) to construct an augmented Lagrange equation, the constraint optimization problem is converted into an unconstrained optimization problem, and an objective function in a formula (4) is solved, namely:
Figure BDA00036666552800001011
wherein U is j For Lagrangian multiplier matrix, μ is U j Is a parameter of (a). The objective function in equation (5) is convex for each parameter, so that optimization can be carried out by iteratively updating each parameter until convergence to the setpoint co occurs. Since other parameters need to be fixed for each parameter update, solution (5) can be divided into the following alternately iterative process:
Figure BDA0003666655280000111
solving x in the formula (6) by the formula (7), that is:
Figure BDA0003666655280000112
as shown in fig. 2 (a), the singular values of the natural image satisfy the heavy tail distribution, i.e., a few larger singular values implies most of the information in the image. As can be seen from fig. 2 (b), the singular value distribution of the logarithmic domain has a gaussian-like distribution, which can be characterized by a gaussian distribution. Based on the observation, the invention provides the PINNM algorithm to solve the low-rank matrix approximation problem through the statistical correlation among MAP modeling singular values, and meanwhile, no additional parameters are introduced. Known I j Is estimated by maximum a posteriori probability j Is a singular value of (c).
σ i (L j )＝arg max log P(σ i (L j )|σ i (Ι j )) (8)
Wherein sigma i (L j ) Is L j Is the i-th singular value of (c). According to the Bayesian theory, equation (8) can be rewritten as,
Figure BDA0003666655280000121
let f j Is a known matrix I j Standard deviation of singular values, the first term P (σ) in equation (9) i (Ι j )|σ i (L j ) 0 as usable mean and 0 as variance
Figure BDA0003666655280000122
Is a gaussian function fit to (a),
Figure BDA0003666655280000123
is the second term P (sigma) in the structural formula (9) i (L j ) The present invention utilizes a known matrix I j Singular value estimation of a potentially low rank matrix L j Is used to determine the desired singular value of (a).
Wherein the known singular value is represented by the singular value sigma i (I j ) A 4 x 4 neighborhood Ω centered i And (5) determining. In the present invention, L j Each singular value to be solved for is sigma with the mean value m (I j ) Standard deviation is h i Is characterized by a gaussian distribution. Wherein sigma m (I j ) Is a neighborhood Ω i Inner I j The mth singular value of (2); h is a i Is a neighborhood Ω i Standard deviation of all singular values covered. Then, P (sigma) i (L j ) Is modeled as:
Figure BDA0003666655280000124
typically, the PINNM comprises a series of singular values to be estimated, the invention uses each singular value in equation (11) to model a priori, and then adds the estimates of all singular values as σ i (L j ) Is used to estimate the final estimate of (a).
Specifically, by combining the mth component in the formulas (10) and (11)
Figure BDA0003666655280000125
Substituting formula (9) to obtain:
Figure BDA0003666655280000131
let (12) relate to sigma i (L j ) The partial derivative of (c) is 0,
Figure BDA0003666655280000132
then, by taking each component in equation (11) as an a priori model, σ is obtained i (L j ) And then adding and averaging the estimated values to obtain:
Figure BDA0003666655280000133
wherein, |Ω i I represents the neighborhood Ω i Number of internal singular values. First, for matrix I j The singular value decomposition is carried out and,
Figure BDA0003666655280000134
then, the first s singular value estimated values are selected to form L j Is a singular value matrix Σ of (2) s 。
Wherein sigma 1 (L j )≥…≥σ s (L j )≥σ s+1 (L j )≥…σ n (L j ) N=min (d, p). By truncating the singular value sigma s+1 (L j ),…,σ n (L j ) Compression matrix L j The small singular values and the related singular vectors which do not carry significant information are ignored, and meanwhile, the interference of tiny factors such as noise and the like is restrained.
Finally, solve to L j Is estimated by (a):
Figure BDA0003666655280000141
wherein,,
Figure BDA0003666655280000142
to ensure that larger singular values are contracted less to effectively preserve image information, weights are defined +.>
Figure BDA0003666655280000143
Figure BDA0003666655280000144
Is the s singular value estimated in the kth iteration, epsilon is a positive number with a smaller value, and the divisor is avoided to be 0.
By introducing the Eckhart-Young-Mirsky theorem, prove L j Is the original image block x j Is a good approximation of the optimal approximation of (a).
Theorem 1 (The Eckhart-Young-Mirsky Theore)
If s is present<r＝rank(x j ) Then there is a real matrix L j ，
Figure BDA0003666655280000145
Figure BDA0003666655280000146
Wherein,,
Figure BDA0003666655280000147
is L j Is a singular value of (c).
As shown in fig. 2 (a), most of the information of the image is concentrated on a few larger singular values. Thus, the present invention determines the threshold δ for s by experimental selection. Wherein s is defined as:
Figure BDA0003666655280000148
finally, the augmented Lagrangian multiplier U is multiplied by (18) j The updating is performed such that,
Figure BDA0003666655280000151
in order to further improve the image quality and refine texture details, the image restoration method based on probability induction kernel norm minimization provided by the invention also designs an effective residual cascade mechanism, and the mechanism gradually improves the quality of a result image by utilizing an intermediate restoration image.
First, image x is restored from the middle C The residual characteristics are extracted from the obtained product,
x R ＝x C -x S (19)
wherein x is R Is an intermediate restored image x C And corresponding source image x S Residual errors between; c is the number of cascaded layers, and when c=1,
Figure BDA0003666655280000152
x S =y; when c=c, ">
Figure BDA0003666655280000153
x S =y. Then, reconstruct x by solving the objective function in equation (19) R The method comprises the following steps:
Figure BDA0003666655280000154
wherein U is R Is an introduced lagrangian multiplier; λ and γ are coefficients of the data fidelity term and regularization term, respectively, in equation (19). By x R The singular value decomposition is carried out and,
Figure BDA0003666655280000155
the problem in equation (19) can be further solved by the PINNM algorithm of the present invention:
Figure BDA0003666655280000156
finally, the reconstructed residual error
Figure BDA0003666655280000157
And adding back the intermediate recovery image to obtain a result image. Wherein (1)>
Figure BDA0003666655280000158
Is->
Figure BDA0003666655280000159
Is the i-th singular value estimate of (c).
Figure BDA0003666655280000161
By employing the above steps, i.e., equations (19) - (23), the intermediate restored image can be enhanced layer by layer until the desired result is obtained. Wherein U is updated by ADMM R ，
Figure BDA0003666655280000162
The invention also evaluates the image restoration method based on probability induction nuclear norm minimization and records objective numerical indexes to verify the performance of PINNM.
Specifically, the present invention adopts peak-signal-to-noise-ratio (PSNR) and structural similarity (structural similarity, SSIM) as objective evaluation indexes to measure the performance of different image restoration algorithms, and in the present invention, the optimal index values are highlighted in bold. Basic parameters of the PINNM algorithm in the invention are set as follows: the image block size d is set to 4×4; number of similar blocks p=80; setting a step length to 3 pixels to extract an image block; the singular value threshold s is set as 1.01,1.02 and 1.03 respectively, and the corresponding amplification coefficients are 2,3 and 4;1.02,1.03 and 1.05 correspond to noise level sigma n =20, 30 and 40. The regularization term parameters η and γ are both set to 1.0; lagrangian multiplier U j And U R The parameters μ and λ of (2) are initialized to 2, increasing by 5% per iteration. All experimental methods tested were evaluated on 10 images shown in fig. 3.
TABLE 1 PSNR and SSIM comparison of different image super-resolution algorithms
Figure BDA0003666655280000163
TABLE 2 PSNR and SSIM comparison for different image denoising algorithms
Figure BDA0003666655280000171
In the experiments of the invention, the PINNM algorithm is compared with 7 more advanced super-resolution methods at present, including Bicubic, RFI, FALGSS, A+, CRC, SRCNN and LRGSS.
As shown in Table 1, the algorithm of the invention is superior to all other testing methods for different amplification factors, and the highest PSNR and SSIM average values are obtained on 10 testing images, so that the method has certain robustness to various image super-resolution tasks, and the effectiveness of the method is further verified. Fig. 4 is an experimental result of 7 super-resolution methods on the test image plants. As can be seen from fig. 4, the method of the present invention has visually better recovery performance than other methods. Since interpolation-based methods are limited by their own limitations, significant blocking artifacts appear in the resulting images of Bicubic and RFI. Although this phenomenon does not occur in the a+, CRC and srcan reconstructed images, the above method relies heavily on external image data and the presented image appears to be excessively smooth in the texture region. Although LRGSS and FALGSS can reconstruct more satisfactory images, some image details are lost.
The present invention also compares the proposed method with 7 excellent image denoising algorithms, including BM3D algorithm, EPLL algorithm, NCSR algorithm, WNNM algorithm, PCLR algorithm, LSM-NLR algorithm and LRJS algorithm. As shown in Table 2, the best performance of the process of the present invention was still achieved. For different noise levels sigma n =20, 30 and 40, compared to the suboptimal denoising algorithm WNNM, the inventive method increases the PSNR average over 10 images by 0.88,0.87 and 1.04dB, respectively.
To further illustrate the effectiveness of the method of the present invention, 7 denoising methods were demonstrated on the experimental results of the test image camera. As can be seen from fig. 5, the EPLL algorithm, the NCSR algorithm, the LSM-NLR algorithm, and the LRJS algorithm all exhibit significant image blurring. Although the BM3D algorithm, the WNNM algorithm and the PCLR algorithm can suppress image blurring to some extent, both of the above methods have image textures that are excessively smoothed, resulting in poor visual effects. As shown in FIG. 5, the method of the invention can restore clearer textures and improve the visual quality of restored images.
In summary, the image recovery method based on probability induction nuclear norm minimization provided by the invention is applied to an image recovery task, solves the problem of low-rank matrix approximation through the statistical characteristic of maximum posterior modeling singular values, adopts a probability induction singular value estimator to better estimate the singular values of a potential low-rank matrix, and suppresses the interference of factors such as noise and the like through low-rank truncation. The invention provides a residual cascade scheme for further compensating the lost image details in low-rank cut-off, and gradually improves the quality of the result image by utilizing an intermediate recovery image.
Experiments of the invention show that the method can realize excellent image recovery performance in quantitative and qualitative comparison.
In addition to the above image restoration method based on probability induction kernel norm minimization, the present invention further provides a terminal for implementing the image restoration method based on probability induction kernel norm minimization, which comprises:
the memory is used for storing a terminal program and an image recovery method based on probability induction kernel norm minimization;
and the processor is used for executing the terminal program and the image restoration method based on the probability induction nuclear norm minimization so as to realize the image restoration method based on the probability induction nuclear norm minimization.
The terminal may be implemented in various forms. For example, the terminals described in the embodiments of the present invention may include mobile terminals such as mobile phones, smart phones, notebook computers, personal digital assistants (Personal Digital Assistant, PDAs), tablet computers (PADs), and the like, and fixed terminals such as digital TVs, desktop computers, and the like.
The terminal implementing the image restoration method based on probability-induced kernel-norm minimization is the units and algorithm steps of each example described in connection with the embodiments disclosed in the present invention, and can be implemented in electronic hardware, terminal software, or a combination of both, and to clearly illustrate the interchangeability of hardware and software, each example's composition and steps have been generally described in terms of functions in the above description. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the solution. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention.
Those skilled in the art will appreciate that aspects of a terminal implementing the probability-induced kernel-norm minimization-based image restoration method may be implemented as a system, method, or program product. Accordingly, various aspects of the disclosure may be embodied in the following forms, namely: an entirely hardware embodiment, an entirely software embodiment (including firmware, micro-code, etc.) or an embodiment combining hardware and software aspects may be referred to herein as a "circuit," module "or" system.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims (5)

1. An image recovery method based on probability induction kernel norm minimization is characterized by comprising the following steps:
restoring a required image x from a degraded image y based on a low-rank model of the block, and utilizing an image non-local self-similarity constraint reconstruction process through a low-rank regularization term;
if the similar blocks extracted from the image are remodeled into vector form, then a low-rank matrix based on the similar blocks is formed by collecting block vectors;
constructing an augmented lagrangian equation by adopting an alternate direction multiplier method;
prior modeling using each singular value in equation (11), and then adding the estimates of all singular values as
Figure QLYQS_1
Is determined by the final estimate of (a);
Figure QLYQS_2
(11)
σ i (L j ) Is L j Is the first of (2)jSingular in natureA value;
h i is a neighborhood Ω i Standard deviation of all singular values in the interior;
by combining the first of the formulae (10) and (11)mIndividual components
Figure QLYQS_3
Substituted into the formula (9),
Figure QLYQS_4
(9)
Figure QLYQS_5
(10)
obtaining:
Figure QLYQS_6
(12)
let (12) relate to
Figure QLYQS_7
The partial derivative of (c) is 0,
Figure QLYQS_8
(13)
by using each component in the formula (11) as a priori model, we obtain
Figure QLYQS_9
And then adding and averaging the estimated values to obtain:
Figure QLYQS_10
(14)
wherein,,
Figure QLYQS_11
representing neighborhood->
Figure QLYQS_12
The number of internal singular values; omega shape i Represent the firstiA group singular value;f j is a known matrix I j And the matrix to be solved->
Figure QLYQS_13
Torsion degree of the contained element;
pair matrix
Figure QLYQS_14
The singular value decomposition is carried out and,
Figure QLYQS_15
(15)
then, before choosingsSingular value estimation value formation
Figure QLYQS_16
Is>
Figure QLYQS_17
；
Wherein,,
Figure QLYQS_18
，/>
Figure QLYQS_19
；
by truncating singular values
Figure QLYQS_20
Compression matrix->
Figure QLYQS_21
；
Solving to obtain
Figure QLYQS_22
Is estimated by (a):
Figure QLYQS_23
(16)
wherein,,
Figure QLYQS_24
，/>
Figure QLYQS_25
the method comprises the steps of carrying out a first treatment on the surface of the Define weight +.>
Figure QLYQS_26
；/>
Figure QLYQS_27
Is the firstkEstimated first in the second iterationsSingular values>
Figure QLYQS_28
Is a positive number with smaller value;
performing enhancement processing on the intermediate reconstructed image by using a residual cascade mechanism;
wherein the magnitude of the singular value division is based on e, e=10 4 ,...,10 -2 ；
u j Meaning of (2): matrix I j Left singular vectors of (a);
v j meaning of (2): matrix I j Right singular vectors of (a);
L j meaning of (2): representing a low-rank image block matrix of an original image;
Figure QLYQS_29
is a neighborhood Ω i Inner I firstmSingular values;
Figure QLYQS_30
meaning of (2): image block matrix L j Is the first of (2)iThe first singular value ofmEstimating components;
n=min(d,p) In (a) and (b)dMeaning of (2): the size of the extracted image block;
pmeaning the number of selected image blocks;
Figure QLYQS_31
in (a) and (b)U s Meaning of matrix L j Left singular vector group of (a);
Figure QLYQS_32
in (a) and (b)V s Meaning of matrix L j Right singular vector group of (a).
2. The method for image restoration based on probability-induced kernel norm minimization according to claim 1,
the step of recovering the required image x from the degraded image y based on the low-rank model of the block and utilizing the non-local self-similarity constraint reconstruction process of the image through the low-rank regularization term comprises the following steps:
is provided with
Figure QLYQS_33
Representing image block +.>
Figure QLYQS_34
Is included in the group of similar blocks to sample block->
Figure QLYQS_35
Most similar frontpA number of similar blocks;
wherein,,
Figure QLYQS_36
is selected by pixel positionjIs center and size ∈>
Figure QLYQS_37
Is a similar block of (2);
to construct image similarity blocks
Figure QLYQS_38
Mapping between the original image x, defining +.>
Figure QLYQS_39
；
Wherein,,
Figure QLYQS_40
is a binary index matrix if the image block +.>
Figure QLYQS_41
Is a pixel of (2)aPixels corresponding to the original image xbThen
Figure QLYQS_42
The method comprises the steps of carrying out a first treatment on the surface of the Otherwise->
Figure QLYQS_43
；
Using low rank matrices
Figure QLYQS_44
To approximate a high quality image x, will +.>
Figure QLYQS_45
As image priors;
Figure QLYQS_46
(3)
Figure QLYQS_47
is a degradation matrix, and is predefined as an identity matrix for denoising an analog image or a super-resolution fuzzy downsampling composite matrix of the image;
Figure QLYQS_48
is a regularized item parameter;
Figure QLYQS_49
is an image block matrix I j An estimated amount of (2);
introducing an auxiliary variable
Figure QLYQS_50
Characterizing the low rank matrix of image x and re-estimating the block matrix by probability induction>
Figure QLYQS_51
Is a singular value of (2);
wherein, define
Figure QLYQS_52
Is->
Figure QLYQS_53
Is estimated by (a):
Figure QLYQS_54
(4) 。
3. the method for image restoration based on probability-induced kernel norm minimization according to claim 2,
in the method, if there is
Figure QLYQS_55
There is a real matrix->
Figure QLYQS_56
，
Figure QLYQS_57
Figure QLYQS_58
Figure QLYQS_59
Is->
Figure QLYQS_60
Is used to determine the singular value of (c),sis defined as:
Figure QLYQS_61
(17)
finally, the augmented Lagrangian multiplier is multiplied by (18)
Figure QLYQS_62
The updating is performed such that,
Figure QLYQS_63
(18)。
4. the method for image restoration based on probability-induced kernel norm minimization according to claim 2,
the step of utilizing a residual cascade mechanism to enhance the intermediate reconstructed image, wherein the enhancement of the image quality of the result further comprises:
first, an image is restored from the middle
Figure QLYQS_64
The residual characteristics are extracted from the obtained product,
Figure QLYQS_65
(19)
wherein,,
Figure QLYQS_67
is an intermediate restored image +.>
Figure QLYQS_70
And corresponding source image->
Figure QLYQS_74
Residual errors between; />
Figure QLYQS_66
Is the number of cascade layers, when->
Figure QLYQS_71
When (I)>
Figure QLYQS_72
，/>
Figure QLYQS_75
The method comprises the steps of carrying out a first treatment on the surface of the When->
Figure QLYQS_68
，/>
Figure QLYQS_69
，/>
Figure QLYQS_73
；
Then, reconstructing by solving the objective function in the equation (19)
Figure QLYQS_76
The method comprises the following steps:
Figure QLYQS_77
(20)
wherein,,
Figure QLYQS_78
is an introduced lagrangian multiplier; />
Figure QLYQS_79
And->
Figure QLYQS_80
The sum of the data fidelity terms in the formula (19)Coefficients of the regular term;
by aligning
Figure QLYQS_81
The singular value decomposition is carried out and,
Figure QLYQS_82
(21)
solving the formula (19) based on the PINNM algorithm:
Figure QLYQS_83
(22)
finally, the reconstructed residual error
Figure QLYQS_84
Adding back the intermediate recovery image to obtain a result image;
wherein,,
Figure QLYQS_85
is->
Figure QLYQS_86
Is the first of (2)iEstimating singular values;
Figure QLYQS_87
(23)
wherein, update through ADMM
Figure QLYQS_88
，
Figure QLYQS_89
(24)。
5. The terminal machine for realizing the image recovery method based on probability induction kernel norm minimization is characterized by comprising the following steps:
the memory is used for storing a terminal program and an image recovery method based on probability induction kernel norm minimization;
a processor for executing the terminal program and the image restoration method based on probability-induced kernel norm minimization to implement the steps of the image restoration method based on probability-induced kernel norm minimization according to any one of claims 1 to 4.
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