CN108846430B - Image signal sparse representation method based on multi-atom dictionary - Google Patents

Image signal sparse representation method based on multi-atom dictionary Download PDF

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CN108846430B
CN108846430B CN201810549855.8A CN201810549855A CN108846430B CN 108846430 B CN108846430 B CN 108846430B CN 201810549855 A CN201810549855 A CN 201810549855A CN 108846430 B CN108846430 B CN 108846430B
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郭俊锋
李育亮
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Abstract

The invention belongs to the field of dictionary updating, and particularly relates to a multi-atom dictionary updating method. The method aims to solve the problems that the SGK dictionary is updated through a single-column dictionary, the calculation complexity is high, and the time consumption is high. The invention discloses a method for updating a multi-atom dictionary, which comprises the following steps: carrying out sparse coding on a signal to be processed to obtain a sparse coefficient matrix; updating residual errors of a preset number of atom columns in a preset dictionary based on the sparse coefficient matrix; shrinking the sparse coefficient matrix and the updated residual error items in the dictionary; and iteratively updating a preset number of atom columns in the dictionary according to the shrunk sparse coefficient matrix and the shrunk residual error items, and obtaining an updated dictionary under the condition that an iteration stop condition is reached. The updating method based on the multi-atom dictionary further improves the learning efficiency of the multi-atom dictionary, reduces the calculated amount of single iteration and accelerates the speed of dictionary learning.

Description

Image signal sparse representation method based on multi-atom dictionary
Technical Field
The invention belongs to the field of updating of a multi-atom dictionary, and particularly relates to a sparse representation method of an image signal based on the multi-atom dictionary.
Background
The intelligent robot communicates with the outside world through various sensors, of which about 70% of external information is shown from visual images. The traditional visual image sampling is based on the Shannon sampling theorem that the sampling frequency cannot be lower than 2 times of the highest frequency in an analog signal frequency spectrum, a huge amount of data can be obtained by the sampling mode, the post-processing workload is large, the speed is low, and the robot runs slowly. In recent years, the brand new signal compression and sampling method proposed by Donoho and Candes, namely compressed sensing, can greatly reduce the dimensionality of data and reduce the data volume, and the obtained measured value contains rich information capable of reconstructing the original image signal. For robot vision, the main purpose of image processing is to acquire feature information of an image, and a complete original image is not required, so that only low-dimensional data needs to be processed in a compressed domain and the feature information of the low-dimensional data is acquired, and further the response and the running speed of the robot are improved. The precondition of compressed sensing is that an image signal has sparsity on a transformation base or a dictionary, which directly influences the number of compressed measurements, and the more sparse an image is on a selected sparse dictionary, the less the number of compressed measurements is needed, and the faster the post-processing is, whereas if the selected sparse dictionary cannot meet the requirements well, the more the data amount is and the worse the quality of the reconstructed image is. Only if the best sparse matrix is obtained can the original signal be represented most efficiently and concisely, and the accuracy of the recovered signal can be guaranteed.
Sparse representation of signals is generally classified into two main categories: the first type is an analysis dictionary, such as a fourier basis, a DCT basis, a wavelet basis, a ridge wave dictionary, a curvelet dictionary, and the like; the second category is an overcomplete dictionary that is trained on the data or the signal itself. The former has simple structure, but the dictionary has single atomic form and cannot effectively and sparsely represent complex natural images; the redundant dictionary obtained through learning has better self-adaptive image reconstruction capability, atoms in the dictionary have correlation with an image to be reconstructed, and the local structure characteristics of the image can be more effectively expressed, so that signals are more sparsely represented on the dictionary. Moreover, since part of noise is non-sparse on the dictionary, and the image signal has sparsity on the dictionary, the sparse representation of the image based on the learning dictionary has a certain denoising effect. In conclusion, the method of learning the dictionary to perform sparse representation on the image has great advantages.
In 1996, Olshausen et al proposed a famous Sparsenet dictionary learning algorithm on Nature, and solved the corresponding optimization problem by using a maximum likelihood estimation and gradient descent algorithm, thereby laying the theoretical basis of dictionary learning. Inspired by a generalized clustering algorithm, Engan and the like provide an MOD (method of optimal directions) dictionary learning algorithm on the basis of a sparse dictionary learning algorithm, the optimization problem of an objective function is solved by directly adopting a least square method, but the process needs multiplication and inversion calculation of a large matrix, and the requirement on storage capacity is high. In order to reduce the computational complexity, especially the spatial complexity, of the MOD algorithm, Aharon et al propose a K-SVD (K-singular value decomposition) algorithm to update a dictionary column by column and improve the dictionary learning speed, but the computation complexity is still large by applying SVD decomposition. Wangqiang [7] and the like propose sparse decomposition by using an improved K-SVD algorithm, and assign values to multiple columns of dictionary atoms simultaneously after singular value decomposition every time, so that the iteration time is reduced, but the signal reconstruction quality is reduced to some extent. The optimized dictionary learning algorithm is provided for constructing a sparse dictionary in compressed data collection, so that the adaptability of the compressed data collection to diversified sensing data is improved, the influence of environmental noise on data collection precision is restrained, and the dictionary learning efficiency is not improved. In the dictionary atom training process, certain specific sequencing rules are added, so that each image dictionary has image attributes and atoms have similar arrangement sequence, the interference of difference among images is reduced, and the traditional K-SVD algorithm is still used for updating the dictionary, so that the calculation complexity is higher.
The method does not greatly improve the dictionary updating speed, has high calculation complexity, and cannot be applied to sparse representation of robot images. Recently, a dictionary learning algorithm called SGK (Sequential generation of K-means) has been used as an effective substitute for the classic K-SVD algorithm, which has a faster execution speed and a dictionary training effect comparable to that of K-SVD. However, the SGK algorithm is only updated through a single-column dictionary, the calculation complexity is high, and the time consumption is still high.
Aiming at the problems of high computational complexity and high time consumption caused by the fact that only a single-column dictionary is updated in an algorithm for learning a multi-atom dictionary such as SGK in compressed sensing, an effective solution is not provided at present.
Disclosure of Invention
In order to solve the problems of high computational complexity and much time consumption caused by updating of a single-column dictionary only by an algorithm (for example, an SGK algorithm) for learning a multi-atom dictionary in the prior art, a sparse representation method of an image signal based on the multi-atom dictionary is provided, and the sparse representation method comprises the following steps:
carrying out sparse coding on a signal to be processed to obtain a sparse coefficient matrix;
updating residual errors of a preset number of atom columns in a preset dictionary based on the sparse coefficient matrix;
shrinking the sparse coefficient matrix and the updated residual error items in the dictionary;
and iteratively updating a preset number of atom columns in the dictionary according to the shrunk sparse coefficient matrix and the shrunk residual error terms, and obtaining the updated dictionary under the condition that an iteration stop condition is reached.
Further, in the above technical solution, the preset number of atom columns is predetermined by a first parameter, where the first parameter includes at least one of: the size of an image corresponding to a signal to be processed, the size of a dictionary and the dimension number of a sparse coefficient matrix;
alternatively, the predetermined number of atom columns is a predetermined constant.
Further, updating the residual error for a preset number of atom columns in a preset dictionary comprises:
in the preset dictionary, fixing K-r terms
Figure BDA0001680105900000041
The remaining term of r is
Figure BDA0001680105900000042
Wherein r is the number of atom columns to be residual updated,
Figure BDA0001680105900000043
the jth column of the initial dictionary is obtained for the t-1 th iteration,
Figure BDA0001680105900000044
and j, a j row of the sparse matrix of the t iteration, and performing residual error updating by the following formula:
Figure BDA0001680105900000045
wherein, X is a signal to be processed,
Figure BDA0001680105900000046
are residual terms except the i +1 th to i + r th atoms to be solved.
Further, in the above technical solution, iteratively updating a preset number of atom columns in the dictionary according to the shrunk sparse coefficient matrix and the shrunk residual term includes:
formula and objective function combined with updated residual error
Figure BDA0001680105900000047
And the least squares method obtains a cost function as follows:
Figure BDA0001680105900000048
wherein the content of the first and second substances,
Figure BDA0001680105900000049
for the residual terms after the shrinkage, the residual terms,
Figure BDA00016801059000000410
is a matrix of the sparse coefficients after the shrinkage,
Figure BDA00016801059000000411
is a square matrix of r multiplied by r,
Figure BDA00016801059000000412
is the full rank of the row, in each iteration, using the value of the cost function
Figure BDA00016801059000000413
Replacing the dictionary atom to be updated obtained from the last iteration
Figure BDA00016801059000000414
And finishing the updating of the r column dictionary atoms until the r column atoms are completely updated.
In the above technical solution, the dictionary is an SGK dictionary.
According to an aspect of the present invention, there is further provided a sparse representation method for image signals based on a multi-atom dictionary, configured to perform sparse representation on image signals by using a dictionary updated according to the update method for a multi-atom dictionary in the foregoing scheme, where the sparse representation method includes:
acquiring a signal to be processed;
and performing sparse representation on the signal to be processed by using the updated dictionary.
According to an aspect of the present invention, there is also provided a storage medium storing a program for executing the above-described method for updating a multi-atom dictionary.
According to an aspect of the present invention, there is also provided a storage medium, where the processor is configured to execute a program, where the program executes the method for updating a multi-atom dictionary.
According to an aspect of the present invention, there is also provided a storage medium including a stored program, wherein the program executes the above sparse representation method.
According to another aspect of the present invention, there is also provided a processor for executing a program, wherein the program executes the sparse representation method.
The technical scheme adopted by the invention has the advantages that:
(1) residual error updating is carried out on a preset number of columns in the dictionary, iterative updating is carried out on the dictionary, the calculated amount of single iteration and the time of sparse representation can be reduced, the complexity of dictionary learning is effectively reduced, the time consumed by learning is reduced, and the efficiency of dictionary learning is greatly improved;
(2) determining the preset number r of atom columns according to the size of the image corresponding to the signal to be processed, the size of the dictionary and the dimensionality number of the sparse coefficient matrix, and updating the preset number r by using the dictionary
Figure BDA0001680105900000051
Replacing dictionary atoms to be updated
Figure BDA0001680105900000052
The method has the advantages that the r-column dictionary atoms are updated, the dictionary updating is completed once until the r-column atoms are updated, the computation time complexity of repeated iteration during the updating of the dictionary single atoms can be further reduced, and the problems that the SGK algorithm is only updated through a single-column dictionary, the computation complexity is high, and the time consumption is large are solved.
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FIG. 1 is a diagram illustrating steps of a method for updating a multi-atom dictionary according to the present invention;
fig. 2 is a step of a sparse representation method of an image signal based on a multi-atom dictionary according to the present invention.
Detailed Description
The following embodiments are merely examples for illustrating the technical solutions of the present invention more clearly, and therefore, the technical solutions of the present invention are not limited to the following embodiments.
Aiming at the problems of higher computational complexity and more time consumption of the SGK algorithm only through updating of a single-column dictionary in the prior art, the SGK algorithm provides an updating method based on a multi-atom dictionary,
with reference to fig. 1, includes:
step 102: to-be-processed signal X, X ═ X1,x2,···,xN]∈Rn×NSparse coding is carried out to obtain a sparse coefficient matrix A(t),A(t)＝[α1,α2...αN]∈RK×N，αiReferred to herein as matrix A(t)The column (c). (ii) a
Step 104: based on sparse coefficient matrix A(t)Updating residual errors for a preset number of atom columns r in a preset dictionary;
step 106: for sparse coefficient matrix A(t)And updated residual terms in the dictionary
Figure BDA0001680105900000061
Shrinking;
step 108: from the shrunk sparse coefficient matrix
Figure BDA0001680105900000062
And residual terms after shrinkage
Figure BDA0001680105900000063
And carrying out iterative updating on the atom columns r with the preset number in the dictionary, and obtaining the updated dictionary under the condition of reaching the iteration stop condition.
Further, in a specific embodiment, the preset number r of atom columns is predetermined by a first parameter, wherein the first parameter includes at least one of: the size of an image corresponding to a signal to be processed, the size of a dictionary and the dimension number of a sparse coefficient matrix;
alternatively, the preset number of atom columns is a predetermined constant (fixed empirical value).
In order to update all columns of the dictionary without repetition and completely, the number of columns of atoms in the updated dictionary must be divisible by the number of columns of the dictionary, and in one embodiment, the number of atoms in the dictionary is 256, so the number of atoms in one update can be 2, 4, 8, 16, etc. When the number of atomic columns is updated at the same time is small, the number of updates is large, but the computational complexity is small.
Further, step 104: the step of updating the residual errors of the atom columns with the preset number in the preset dictionary comprises the following steps:
in the preset dictionary, fixing K-r terms
Figure BDA0001680105900000071
The remaining term of r is
Figure BDA0001680105900000072
Wherein r is the number of atom columns to be residual updated,
Figure BDA0001680105900000073
the jth column of the initial dictionary is obtained for the t-1 th iteration,
Figure BDA0001680105900000074
for the jth row of the sparse matrix for the t iteration,and residual updating is performed through the following formula:
Figure BDA0001680105900000075
wherein, X is a signal to be processed,
Figure BDA0001680105900000076
are residual terms except the i +1 th to i + r th atoms to be solved.
Further, in the above embodiment, step 108: from the shrunk sparse coefficient matrix
Figure BDA0001680105900000077
And residual terms after shrinkage
Figure BDA0001680105900000078
The iterative updating of the atom columns r with the preset number in the dictionary comprises the following steps:
formula and objective function combined with updated residual error
Figure BDA0001680105900000079
And the least squares method obtains a cost function as follows:
Figure BDA00016801059000000710
wherein the content of the first and second substances,
Figure BDA00016801059000000711
for the residual terms after the shrinkage, the residual terms,
Figure BDA00016801059000000712
is a matrix of the sparse coefficients after the shrinkage,
Figure BDA00016801059000000713
is a square matrix of r multiplied by r,
Figure BDA00016801059000000714
is the full rank of the row, in each iteration, using the value of the cost function
Figure BDA00016801059000000715
Replacing the dictionary atom to be updated obtained from the last iteration
Figure BDA00016801059000000716
And finishing the updating of the r column dictionary atoms until the r column atoms are completely updated.
Optionally, in the foregoing technical solution, the dictionary is an SGK dictionary.
The solving process of the cost function comprises the following steps: first, the formula of the residual error will be updated
Figure BDA0001680105900000082
Second, the objective function of the dictionary update
Figure BDA0001680105900000083
Thereby to obtain
Figure BDA0001680105900000084
Converted into residual error solving terms
Figure BDA0001680105900000085
Is equivalent to solving the residual error pair variable
Figure BDA0001680105900000086
In the case where the derivative is 0, the value,
Figure BDA0001680105900000087
because it does not contain
Figure BDA0001680105900000088
The derivative is taken to be zero,
Figure BDA0001680105900000089
to pair
Figure BDA00016801059000000810
All derivatives are
Figure BDA00016801059000000811
To pair
Figure BDA00016801059000000812
Is derived as
Figure BDA00016801059000000813
Therefore, the formula is updated, the residual formula is deformed, and the derivative is obtained:
Figure BDA00016801059000000814
the equivalent deformation of the deformation derivative of the updated residual error formula is as follows:
Figure BDA00016801059000000815
Figure BDA00016801059000000816
Figure BDA00016801059000000817
according to an aspect of this embodiment, a sparse representation method for an image signal based on a multi-atom dictionary is further provided, which is used for performing sparse representation on the image signal by using a dictionary updated according to the updating method for the multi-atom dictionary in the foregoing scheme, and with reference to fig. 2, the sparse representation method includes the following steps:
step 002: acquiring a signal X to be processed;
step 004: and performing sparse representation on the signal X to be processed by using the updated dictionary.
The learning efficiency of the SGK algorithm dictionary is further improved, the calculated amount of single iteration is reduced, and the dictionary learning speed is accelerated.
In the case of the same image size and the same dictionary size, the dictionary can be used multiple times after updating, without having to perform the dictionary updating steps of steps 102 to 108 again each time for a different image signal.
Verified by experiments, use
Figure BDA0001680105900000091
In place of D(t-1)From i +1 to i + r columns of (1), repeating all D's in sequence(t-1)And completing dictionary updating once by using the atoms in the group. According to the invention, through alternately repeating iteration of sparse coding and dictionary updating until a preset iteration number or an error requirement is met, the structure of a sparse coefficient is not damaged, the dictionary atoms are updated sequentially, the dictionary learning speed is greatly increased, multiple atoms are updated simultaneously, and the complexity of the calculation time of repeating multiple iterations when a single dictionary atom is updated is reduced.
In the description of the present invention, numerous specific details are set forth. It is understood, however, that embodiments of the invention may be practiced without these specific details. In some instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
In a typical configuration, a computing device includes one or more processors (CPUs), input/output interfaces, network interfaces, and memory.
The memory may include forms of volatile memory in a computer readable medium, Random Access Memory (RAM) and/or non-volatile memory, such as Read Only Memory (ROM) or flash memory (flash RAM). The memory is an example of a computer-readable medium.
Computer-readable media, including both non-transitory and non-transitory, removable and non-removable media, may implement information storage by any method or technology. The information may be computer readable instructions, data structures, modules of a program, or other data. Examples of computer storage media include, but are not limited to, phase change memory (PRAM), Static Random Access Memory (SRAM), Dynamic Random Access Memory (DRAM), other types of Random Access Memory (RAM), Read Only Memory (ROM), Electrically Erasable Programmable Read Only Memory (EEPROM), flash memory or other memory technology, compact disc read only memory (CD-ROM), Digital Versatile Discs (DVD) or other optical storage, magnetic cassettes, magnetic tape magnetic disk storage or other magnetic storage devices, or any other non-transmission medium that can be used to store information that can be accessed by a computing device. As defined herein, a computer readable medium does not include a transitory computer readable medium such as a modulated data signal and a carrier wave.
It should also be noted that the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other identical elements in the process, method, article, or apparatus that comprises the element.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The above are merely examples of the present application and are not intended to limit the present application. Various modifications and changes may occur to those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the scope of the claims of the present application.

Claims (5)

1. A sparse representation method of an image signal based on a multi-atom dictionary is characterized in that the image signal is sparsely represented by a dictionary updated by an updating method of the multi-atom dictionary, and the sparse representation method comprises the following steps:
acquiring a signal to be processed;
carrying out sparse representation on the signal to be processed by using the updated dictionary;
the updating method of the multi-atom dictionary comprises the following steps:
carrying out sparse coding on a signal to be processed to obtain a sparse coefficient matrix;
updating residual errors for a preset number of atom columns in a preset dictionary based on the sparse coefficient matrix;
shrinking the sparse coefficient matrix and the updated residual error items in the dictionary;
iteratively updating a preset number of atom columns in the dictionary according to the shrunk sparse coefficient matrix and the shrunk residual error terms, and obtaining an updated dictionary under the condition that an iteration stop condition is reached;
the preset number of atom columns is predetermined by a first parameter, wherein the first parameter comprises at least one of: the size of an image corresponding to the signal to be processed, the size of a dictionary and the dimensionality number of the sparse coefficient matrix;
or the preset number of the atom columns is a preset constant;
iteratively updating a preset number of atom columns in the dictionary according to the shrunk sparse coefficient matrix and the shrunk residual error terms comprises:
formula and objective function combined with updated residual error
Figure FDA0003457167050000011
And the least squares method obtains a cost function as follows:
Figure FDA0003457167050000021
wherein the content of the first and second substances,
Figure FDA0003457167050000022
for the residual terms after the shrinkage, the residual terms,
Figure FDA0003457167050000023
is a matrix of the sparse coefficients after the shrinkage,
Figure FDA0003457167050000024
is a square matrix of r multiplied by r,
Figure FDA0003457167050000025
is the full rank of the row, in each iteration, using the value in the cost function
Figure FDA0003457167050000026
Replacing the dictionary atom to be updated obtained from the last iteration
Figure FDA0003457167050000027
And finishing the updating of the r column dictionary atoms until the r column atoms are completely updated.
2. The method of claim 1, wherein updating residuals for a preset number of atom columns in a preset dictionary comprises:
fixing K-r terms in the preset dictionary
Figure FDA0003457167050000028
The remaining term of r is
Figure FDA0003457167050000029
Wherein r is the number of atom columns to be residual updated,
Figure FDA00034571670500000210
for the t-1 th iterationThe jth column of the original dictionary is obtained,
Figure FDA00034571670500000211
and j, a j row of the sparse matrix of the t iteration, and performing residual error updating by the following formula:
Figure FDA00034571670500000212
wherein, X is a signal to be processed,
Figure FDA00034571670500000213
are residual terms except the i +1 th to i + r th atoms to be solved.
3. The method for sparse representation of image signals based on a polyatomic dictionary according to claim 1, wherein the dictionary is an SGK dictionary.
4. A storage medium characterized in that it comprises a stored program, wherein said program executes the method of sparse representation of multi-atom dictionary-based image signals of claim 1.
5. A processor, characterized in that the processor is configured to run a program, wherein the program is configured to perform the method of sparse representation of multi-atom dictionary-based image signals of claim 1 when running.
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