CN110113607A - A kind of compressed sensing video method for reconstructing based on part and non-local constraint - Google Patents

A kind of compressed sensing video method for reconstructing based on part and non-local constraint Download PDF

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CN110113607A
CN110113607A CN201910336663.3A CN201910336663A CN110113607A CN 110113607 A CN110113607 A CN 110113607A CN 201910336663 A CN201910336663 A CN 201910336663A CN 110113607 A CN110113607 A CN 110113607A
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宋云
章登勇
邓泽林
余飞
梁惠
兰林涛
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Changsha University of Science and Technology
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    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/102—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the element, parameter or selection affected or controlled by the adaptive coding
    • H04N19/132—Sampling, masking or truncation of coding units, e.g. adaptive resampling, frame skipping, frame interpolation or high-frequency transform coefficient masking
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/169—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding
    • H04N19/177—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding the unit being a group of pictures [GOP]
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    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/189—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the adaptation method, adaptation tool or adaptation type used for the adaptive coding
    • H04N19/19—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the adaptation method, adaptation tool or adaptation type used for the adaptive coding using optimisation based on Lagrange multipliers
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    • H04—ELECTRIC COMMUNICATION TECHNIQUE
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Abstract

The present invention relates to a kind of compressed sensing video method for reconstructing based on part and non-local constraint, it is rebuild in compressed sensing and introduces part and non-local constraint in Video Model, using based on picture frame minimum total variation (Total Variation, TV) constrain with the low-rank constraint based on the similar block matrix of sequence space-time, guide to optimize the inherent feature that solution to model tends to three-dimensional video sequence.Rebuild frame compression of images perceptual image every in video using conventional method first during optimization problem solving, one initial estimation of obtained picture frame, then it introduces auxiliary variable and is that a TV optimizes subproblem and a low-rank optimizes subproblem by PROBLEM DECOMPOSITION, alternately solve, and inside circulation in further by TV subproblem and low-rank optimization subproblem be decomposed into can micro- and solution part solve respectively.Due to the method for invention fully considered picture frame space-time part and non local feature, thus improve the reconstruction quality of image.

Description

A kind of compressed sensing video method for reconstructing based on part and non-local constraint
Technical field
The present invention relates to a kind of compressed sensing video method for reconstructing based on part and non-local constraint.
Background technique
Compressed sensing (Compressed Sensing, CS) is the new signal sampling mode of one kind proposed in recent years, it It points out when signal is sparse or compressible, it can be by way of non-adaptive linear projection, with far below Nai Kuisi The all information for the frequency acquisition signal that special sampling thheorem requires.Signal sampling method based on compressive sensing theory is not only broken through The frequency limit of signal sampling, while also changing the mode that recompression is first sampled in classical signal acquisition methods.Its letter Number sampling process is integrated with data compression, thus sensor and storage quantity needed for reducing signal acquisition and storage, thus New technical thought is provided for high efficiency low power consumption signal sampling method, the extensive concern by domestic and foreign scholars.
Specifically, a N-dimensional signal is givenThe target of CS theory is only from the random of the M of signal (M < < N) Linear measurement y=Φ x reconstruction signal x, in formulaIt is measurement,It is calculation matrix.Due to M < < N, rebuild It is one and owes fixed problem, it means that x there are multiple solutions.CS theory thinks, if signal is sparse enough and calculation matrix Φ is full Foot is limited equidistant property (Restricted Isometry Property, RIP), then solves a l by formula (1)0Optimization Problem obtains least square solution x, can restore signal well.
Wherein, | | | |0Indicate the l of a vector0Pseudonorm, it is the number of nonzero term in vector, and Ψ is a transformation Matrix, s are the expression coefficients of signal on the transform domain.However, l0Least norm is NP-hard problem, it is just difficult in this way With by direct solution and it is very sensitive to noise.Fortunately, it has been proved under certain condition, the l in formula (1)0Model Number minimization problem is equivalent to the l in formula (2)0Norm problem.
Wherein, | | | |1Indicate the l of a vector1Pseudonorm, it is all in vector the sum of absolute values, and Ψ is one A transformation matrix, s are the expression coefficients of signal on the transform domain.Here l1Least norm problem is that a convex optimization is asked Topic, can be efficiently solved with linear programming method, iterative shrinkage algorithm and other a variety of methods.With l0Least norm Greedy algorithm is compared, convex optimization method computation complexity usually with higher, but they can converge to globally optimal solution, and And less measurement is only needed, to realize more accurate reconstruct.
l0Least norm and l1Least norm is considered as the standard or classical model of CS recovery.Wright and The CS that Chartrand is proposed, which restores problem, can also be modeled as a non-convex lpNorm (0 < p < 1) minimization, such as formula (3), More accurate restore is realized by cost of higher computation complexity.
Wherein, vector s=s [s1,s2,…,sn] lpNorm is defined as:
In fact, this is the unified model that CS restores under sparse prior constraint.Optimized model (1) and (2) are its p=0 With the special case of p=1.
Other than above-mentioned master pattern, another well-known CS Image restoration is that full variation is minimum Change, as shown in formula (5).
Wherein, TV (x) is the full variation of signal, for a 2D signalIt is defined as follows,
Or
Wherein, 1≤i≤n, 1≤j≤m indicate the coordinate of sample,Indicate calculus of differences.WithIt respectively indicates Image is located at the horizontal and vertical difference of the position pixel i, j.Formula (6) and formula (7) respectively indicate the isotropism of 2D signal With anisotropy total variation.Unlike traditional CS model of the focus in signal sparsity, TV-CS model is to utilize figure Sectionally smooth structure as in.It can be regarded as the sparsity in difference domain using image to l1Norm optimization's model It promotes.The success of the model means that the higher order dependencies of signal and sparsity are very helpful to CS recovery.Based on this It has been observed that a variety of in recent years rely on high-order is suggested with the method for structural sparse.
Summary of the invention
The embodiment of the present invention is to provide a kind of compressed sensing video method for reconstructing based on part and non-local constraint, can be with More accurately compressed sensing video is rebuild, improves and rebuilds video quality.
Technical proposal that the invention solves the above-mentioned problems is as follows: a kind of to be regarded based on the compressed sensing of part and non-local constraint Frequency method for reconstructing, it is characterised in that: rebuild in compressed sensing and introduce part and non-local constraint in Video Model, using based on figure As frame minimum total variation (Total Variation, TV) constraint and the low-rank constraint based on the similar block matrix of sequence space-time, guidance The solution of Optimized model tends to the inherent feature of three-dimensional video sequence, i.e., carries out compressed sensing video weight using following Optimized models It builds:
Wherein, function TV (xl) be l frame in image group GOP (Group of Picture, GOP) full variation, function LowRank(Si) indicate the low-rank objective function of the i-th similar block matrix, and pass through formula Come approximate.Herein, SiIndicate the initial estimation of the i-th similar block matrix, PiIt is its low-rank matrix, ΩXIt is one The set of all similar block matrix of this group, Rank (P in GOPi) indicate PiOrder, η > 0 and λ > 0 are regularization parameters, and | |·||FIndicate Frobenius norm.All x are minimized using formula (8) frame by framel, S is also minimized block by blocki, can also Handle the image and block in all GOP as a whole.Assuming that the sum of the similar block matrix of all GOP is K, that is, exist Model conversation in formula (8) is formula (9) solution by the sum that sample is extracted in GOP:
Wherein,Indicate the total variation of entire GOP.
Auxiliary variable is introduced, problem is converted are as follows:
Wherein:
Wherein, β > 0, θ > 0 are regularization parameters, and α and γ are Lagrange multipliers.So formula (9) optimization problem can wait Valence is in formula (10).
Using alternating direction, X, U are solved respectively, solve X using following formula (12)(k+1)::
For fixed X(k+1),U(k+1)Using following formula (13) approximation:
Wherein,It is initial using the image group of the fixed base compressed sensing method for reconstructing acquisition of tradition Estimation.
Solution for formula (12) X, definitionFor vector image xlJth element discrete difference Value, thenIt is isotropic TV value for image,Be for The anisotropic TV value of image.UsingBe converted to formula (14) solution:
It introduces and decomposes auxiliary variableIt obtains,
After an estimation for solving the image group for obtaining image in TV subproblem, using based on sequence space-time similar block The weight kernel normal form minimum value-based algorithm and reconstruction solution formula is iterated to all frames with singular value thresholding algorithm that low-rank constrains (13).Further formula (13) is divided into about GOP image group U and similar block PiLow-rank matrix two sub-problems, then Alternately solve.The optimization for first considering similar block matrix, for each similar block group Si, they optimize low-rank matrixPass through formula (16) approximation obtains.
Then, it solves
I.e.
Wherein,It is a diagonal matrix, wherein the overlapping of each component identification covering respective pixel location The number of block, so the optimal reconstruction image solution of formula (18) is to be divided by efficiently to be calculated one by one by matrix element , without matrix inversion operation.
The beneficial effects of the present invention are: a kind of compressed sensing video based on part and non-local constraint of the present invention Method for reconstructing, this method are first divided into an image group (GOP), several frames in video sequence, pass through classical compression Perceptual image method for reconstructing estimates one group of reconstruction imageAs the initial pictures of this method, introduce part with it is non local about Beam compressed sensing video method for reconstructing, using alternating direction Multiplier Algorithm (ADMM), Problems of Reconstruction be divided into TV subproblem and Low-rank subproblem, while by the way of iteration.ADMM algorithm is applied in TV subproblem and subproblem is separated into definite result Non-differentiability part and differentiable part carry out inner iteration calculating.Low-rank subproblem can be further divided into image group U and similar block Li The two subproblems of low-rank matrix carry out inner iteration calculating.Using the mode of multilayer nest alternating iteration, therefore this method is It can be preferably in conjunction with the time non local block similarity of the space local smoothing method and video of image.In addition, using iteration weight It builds, so that the reconstruction image of each frame has carried out multiple refining, further improves the quality for rebuilding video.
Based on the above technical solution, the present invention can also be improved as follows.
Further, the Optimized model is carried out based on the compressed sensing video method for reconstructing of part and non-local constraint , it is obtaining in GOP after the initial estimation image of all frames, using the friendship of minimum total variation (TV) constraint based on picture frame Reconstruction is iterated to all frames for direction Multiplier Algorithm (ADMM), the specific steps are as follows:
1) it carries out outer loop first and the value of its number of iterations parameter k is set.Secondly it calculates and solves TV subproblem, it should Subproblem applies ADMM algorithm, and subproblem is further separated into non-differentiability part and differentiable part alternately solves djAnd xl, into The value of the internal the number of iterations parameter t for recycling and it being set of row, initiation parameter
2) for the calculating of non-differentiability partial parameters,djIndicate vector image xlJth The discrete difference of pixel.Then vector image x is calculatedlThe discrete difference of (t+1) secondary iteration of middle jth pixel, calculating side Formula is as follows:
Wherein,Indicate vector image xlThe Lagrange multiplier of (t+1) secondary iteration of middle jth pixel, μj> 0 indicates just Then change parameter.Therefore, by (t+1) secondary interior circulation time iteration, vector image x is obtainedlThe discrete difference of all pixels, i.e.,
3) fixed for differentiable partWithxlOptimal value can pass through the square of (t+1) secondary iterative approximation image Battle array asks local derviation to be equal to zero, acquires a closing solution:
In order to reduce computation complexity, acceleration convergence is carried out using Barzilai-Borwein step.Setting xlIt is secondary at (t+1) The result of iteration is obtained by approximation, is calculated as follows:
4) value for updating t, enables t=t+1, repeats step 2) -4), until meeting termination condition, terminates iterative cycles and obtain more New image group matrixThus the Image estimation value of outer circulation is obtained
Further, the Optimized model is carried out based on the compressed sensing video method for reconstructing of part and non-local constraint , it is solved by TV subproblem, after the estimation image for obtaining vector image, using the low-rank based on sequence space-time similar block The weight kernel normal form minimum value-based algorithm of constraint and with singular value thresholding algorithm to all frames be iterated rebuild solution formula (13), tool Steps are as follows for body:
1) initialEvery time with image group U(k)In image carry out piecemeal and rope as interior loop iteration image Draw.Provide the size of sample block and by image group U(k)In each reconstruction image be divided into the sample block s of overlapping1,si…,sn, Each sample s in the group of imagesiSimilar block retrieval is carried out, later to each sample siIn all similar block be grouped to obtain Si (k)。
2) calculating of low-rank subproblem is solved, first progress Initialize installation, iteration is for the first time with image group U(0)It is first Beginning image, obtains Pi (0)=Si (k), wherein Pi (0)For the optimised low-rank matrix of iteration for the first time.Secondly it carries out internal circulation and sets The subproblem, further can be divided into image group U and similar block P by the value for setting its number of iterations parameter tiLow-rank matrix this Two sub-problems,
3) similar block matrix optimizing subproblem is considered first, for each similar block group Si, optimised low-rank matrix Pi (t)Iterative calculation method it is as follows:
Using matrix PiWeight nuclear norm function | | Pi||w,*=∑c||ωcσc(Pi)||1As Rank (Pi) function Alternative functions convert formula (24) for formula (23) and solve:
Wherein, σc(Pi) indicate the c times singular value, wc> 0 is σc(Pi) and τ=η/λ weight.Then singular value threshold is used Value-based algorithm (SVT) carrys out effective solution low-rank subproblem, and calculation method is as follows:
Pi (t+1)=Γ Sw,τ(∑)ΛT (25)
Wherein, Γ, ∑, Λ indicate SiThe result of singular value decomposition, that is to say, that Si=Γ ∑ ΛT, wherein ∑ is unusual Value matrix, diagonal element are the singular value S of matrixi, every other element is all zero.That is, ∑c,c=σc(Pi)。Sw,τ(∑) is The threshold function table of ∑, is defined as Sw,τ(∑c,c)=max (∑c,c-τωc,0).To obtain (t+1) secondary iteration optimization Low-rank matrix.Update weight vector wcValue.
4) the subproblem Solve problems about image group U, are obtainingLater, all images in GOP are considered as by we One entirety, and obtain the estimation image of the secondary interior loop iteration of image group U (t+1), calculation method is as follows:
Wherein U ', γ(k)′,X(k+1)′Respectively indicate vector U, γ(k),X(k+1).The similar block of each sample is in GOP It finds.It is fixedγ(k)And X(k+1), obtain the solution of a closing formItsMinimum value is secondary by (t+1) Iterative approximation image is asked local derviation to be equal to zero and is acquired, and is effectively calculated by the division of element one by one, calculation method is as follows:
Wherein,It is a diagonal matrix, wherein the overlapping of each element mark covering respective pixel location The quantity of block.
5) value for updating t enables t=t+1, repetition step 3), 4), until meeting termination condition, terminate iterative cycles.
Detailed description of the invention
Fig. 1 is a kind of compressed sensing video method for reconstructing main-process stream based on part and non-local constraint of the present invention Figure；
Fig. 2 obtains image group (Group of Picture, GOP) just starting weight using TVAL3 algorithm to be of the present invention The step flow chart built；
Fig. 3 is the step flow chart that acquisition GOP of the present invention is finally rebuild；
Specific embodiment
It is described below in conjunction with attached drawing and to principles and features of the present invention, the given examples are served only to explain the present invention, It is not intended to limit the scope of the present invention.
Fig. 1 is a kind of compressed sensing video method for reconstructing main-process stream based on part and non-local constraint of the present invention Figure；Fig. 2 is the step of the present invention that image group (Group of Picture, GOP) original reconstruction is obtained using TVAL3 algorithm Rapid flow chart；Fig. 3 is the step flow chart that acquisition GOP of the present invention is finally rebuild；As shown in Figure 1, 2, 3, one kind is based on Specific step is as follows for the compressed sensing video method for reconstructing of part and non-local constraint:
Video to be tested is provided by user.
Step 1: it is first divided into an image group (Group of Picture, GOP), several in video sequence Frame makes the original image in this image group estimate one group of reconstruction image by classical compressed sensing image rebuilding methodAnd be stored in this image group according to the sequence in video sequence, i.e.,Wherein L is indicated Image group frame number,It indicates the vector image of image group, and using this group of image as the initial pictures before iteration, is denoted asWherein, X(0)Initial pictures group before referring to iteration, secondly, carrying out outer loop and its number of iterations being arranged The value of parameter k.
Step 2: first time outer loop, i.e. k=1, initiation parameter are carried out WhereinIt indicates to carry out the initial pictures before TV problem iteration.It carries out inside circulation and is arranged its number of iterations parameter t's Value solves the calculating of TV subproblem, which also applies ADMM algorithm, subproblem is separated into definite result not Differentiable part and differentiable part, the specific steps are as follows:
1) firstly for the calculating of non-differentiability partial parameters, definitionWherein, djIt indicates Vector image xlThe dispersive difference of jth pixel.Then vector image x is calculatedlThe discrete difference of (t+1) secondary iteration of middle jth pixel Different, calculation is as follows:
Wherein,Indicate vector image xlThe Lagrange multiplier of (t+1) secondary iteration of middle jth pixel, μj> 0 table Show regularization parameter.Therefore, by (t+1) secondary interior circulation time iteration, vector image x is obtainedlThe dispersive difference of all pixels, i.e.,
2) the secondly calculating for differentiable part,It is isotropic TV for image Value,It is the anisotropic TV value for image.UsingAs system One TV is indicated, is then introduced and is decomposed auxiliary variableThe just estimation figure after available (t+1) secondary interior loop iteration Picture, calculating process are as follows:
Wherein, β > 0, θ > 0 indicate regularization parameter, α(k)、γ(k)Indicate the glug of the kth updated time outer loop iteration Bright day multiplier.The estimation image of l frame image is derivative to (t+1) secondary iteration, fixedWithObtain a closing form Solution xl, xlMinimum value is asked local derviation to be equal to zero and is acquired by (t+1) secondary iterative approximation image, and calculation method is as follows:
In order to reduce computation complexity, acceleration convergence is carried out using Barzilai-Borwein step.Setting xlIt is secondary at (t+1) The result of iteration can approximation obtain, calculate it is as follows:
Final updating vector image xlThe Lagrange multiplier of (t+1) secondary iteration of middle jth pixel updates calculating side Method is as follows:
The value for updating t enables t=t+1, and repeating step 2 until meeting termination condition terminates internal circulation.Update figure As group X, i.e.,
Step 3: an image group U is divided in outer circulation iteration for the first time, initiallyEvery time with image group U(k)In Image carry out piecemeal and index as interior loop iteration image.Provide the size of sample block and by image group U(k)In each Reconstruction image is divided into the sample block s of overlapping1,si…,sn, each sample s in the group of imagesiSimilar block retrieval is carried out, later To each sample siIn all similar block be grouped to obtain Si (k)=RiU(k)′, wherein RiIt is 1 to determine in GOP for diagonal element The diagonal matrix of similar block coordinate, U(k)′For loop iteration in last time in the secondary outer circulation iteration of low-rank subproblem (k-1) Image group afterwards.
Step 4: solving the calculating of low-rank subproblem, first progress Initialize installation, and iteration is for the first time with image group U(0)For initial pictures, P is obtainedi (0)=Si (k), wherein Pi (0)For the optimised low-rank matrix of iteration for the first time, that is, Pi (0)=Si (k)。 Wherein, Pi (0)Indicate the low-rank matrix of initial pictures, Si (k)Indicate i-th of similar block matrix of kth time outer loop iteration.Its The subproblem, further can be divided into image group U and phase by circulation and the value for the number of iterations parameter t that it is arranged inside secondary progress Like block PiLow-rank matrix the two subproblems, and alternately solve it.Specific step is as follows:
1) similar block matrix optimizing subproblem is considered first, for each similar block group Si, optimised low-rank matrix Pi (t)Iterative calculation method it is as follows:
This is a classical order minimization problem, minimizes algorithm (WNNM) using weight nuclear norm.Make | | Pi||w,* =∑c||ωcσc(Pi)||1, matrix PiWeight nuclear norm function, as Rank (Pi) function replace-conceive function, by formula (33) It is converted into formula (34) solution:
Wherein, σc(Pi) indicate the c times singular value, wc> 0 is σc(Pi) and τ=η/λ weight.Then singular value threshold is used Value-based algorithm (SVT) carrys out effective solution low-rank subproblem, and calculation method is as follows:
Pi (t+1)=Г Sw,τ(∑)ΛT (36)
Wherein, Г, ∑, Λ indicate SiThe result of singular value decomposition, that is to say, that Pi (t)=Si=Γ ∑ ΛT, wherein ∑ It is singular value matrix, diagonal element is the singular value S of matrixi, every other element is all zero.That is, ∑c,c=σc(Pi)。Sw,τ (∑) is the threshold function table of ∑, is defined as Sw,τ(∑c,c)=max (∑c,c-τωc,0).To obtain (t+1) secondary iteration The low-rank matrix of optimization.Update weight vector wcValue.
2) the subproblem Solve problems about image group U, are obtainingLater, all images in GOP are considered as by we One entirety, and obtain the estimation image of the secondary interior loop iteration of image group U (t+1), calculation method is as follows:
Wherein U ', γ(k)′,X(k+1)′Respectively indicate vector U, γ(k),X(k+1).The similar block of each sample is in GOP It finds.It is fixedγ(k)And X(k+1), obtain the solution of a closing formItsMinimum value is secondary by (t+1) Iterative approximation image is asked local derviation to be equal to zero and is acquired, and is effectively calculated by the division of element one by one, calculation method is as follows:
Wherein,It is a diagonal matrix, wherein the overlapping of each element mark covering respective pixel location The quantity of color lump.The value for updating t enables t=t+1, and repeating step 5 until meeting termination condition terminates internal circulation.It obtainsObtain the interior circulation primary iteration image group U of next outer circulation(k+1), i.e.,Under we enter An iteration updates Lagrange's multiplier and calculates α(k+1)andγ(k+1), calculating side is as follows:
α(k+1)=α(k)-β(Y-ΦX(k+1)) (39)
γ(k+1)=γ(k)-θ(X(k+1)-U(k+1)) (40)
Step 5: updating the value of k, enables k=k+1, and repeat step 2 three or four terminates outside and follow until meeting termination condition Ring.The image group recycled in (t+1) of obtained (k+1) outer circulationIt is to rebuild graphical set U.
It is sub that the foregoing is merely presently preferred embodiments of the present invention, is not intended to limit the invention, all in spirit of the invention Within principle, any modification, equivalent replacement, improvement and so on be should all be included in the protection scope of the present invention.

Claims (3)

1. a kind of compressed sensing video method for reconstructing based on part and non-local constraint, it is characterised in that: in compressed sensing weight It builds and introduces part and non-local constraint in Video Model, using based on picture frame minimum total variation (Total Variation, TV) Constrain with the low-rank constraint based on the similar block matrix of sequence space-time, guide to optimize solution to model and tend to the intrinsic of three-dimensional video sequence Feature carries out the reconstruction of compressed sensing video using following Optimized models:
Wherein, function TV (xl) be l frame in image group GOP (Group of Picture, GOP) full variation, function LowRank(Si) indicate the low-rank objective function of the i-th similar block matrix, and pass through formula Come approximate.Here SiIndicate the initial estimation of the i-th similar block matrix, PiIt is its low-rank matrix, ΩXIt is one The set of all similar block matrix of this group, Rank (P in GOPi) indicate PiOrder, η > 0 and λ > 0 are regularization parameters, and | | ||FIndicate Frobenius norm.All x are minimized using formula (1) frame by framek, S is minimized block by blocki, whole as one Body handles the image and block in all GOP.Assuming that the sum of the similar block matrix of all GOP is K, i.e., sample is extracted in GOP Model conversation in formula (1) is formula (2) solution by this sum:
Wherein,Indicate the total variation of entire GOP.Auxiliary variable is introduced, problem is converted are as follows:
Wherein:
Wherein, β > 0, θ > 0 are regularization parameters, and α and γ are Lagrange multipliers.
X, U are solved using alternating direction respectively, first solve X using following formula (5)(k+1):
For fixed X(k+1),U(k+1)It is then approximate by following formula (6):
Wherein,For the image group initial estimation obtained using the fixed base compressed sensing method for reconstructing of tradition.
2. a kind of compressed sensing video method for reconstructing based on part and non-local constraint according to claim 1, feature It is, is obtaining in GOP after the initial estimation image of all frames, using minimum total variation (TV) constraint based on picture frame Alternating direction Multiplier Algorithm (ADMM) is iterated reconstruction to all frames, the specific steps are as follows:
1) it definesFor vector image xlJth element discrete difference, It is isotropic TV value of image, applicationIt is indicated as unified TV, (5) is converted into formula (7):
It introduces and decomposes auxiliary variableIt obtains,
2) subproblem is further separated into non-differentiability part and differentiable part alternately solves djAnd xl, carry out internal circulation simultaneously The value of its number of iterations parameter t, initiation parameter are setIt is right In the calculating of non-differentiability partial parameters,djIndicate vector image xlThe discrete difference of jth pixel Value, calculation are as follows:
Wherein,Indicate vector image xlThe Lagrange multiplier of (t+1) secondary iteration of middle jth pixel, μj> 0 indicates just Then change parameter.Therefore, by (t+1) secondary interior circulation time iteration, vector image x is obtainedlThe discrete difference of all pixels, i.e.,
3) fixed for differentiable partAnd vj (t),xlOptimal value can enable the matrix by (t+1) secondary iterative approximation image It asks local derviation to be equal to zero and acquires a closing solution:
In order to reduce computation complexity, acceleration convergence is carried out using Barzilai-Borwein step.Setting xlIt is obtained in the result of (t+1) secondary iteration by approximate It arrives, calculates as follows:
Final updating vector image xlThe Lagrange multiplier of (t+1) secondary iteration of middle jth pixel, more new calculation method is such as Under:
4) value for updating t, enables t=t+1, repeats this step, until meeting termination condition, terminates the figure that iterative cycles are updated As groupIt enables
3. a kind of compressed sensing video method for reconstructing based on part and non-local constraint according to claim 1, feature It is, after the estimation that the image group for obtaining image is solved in TV subproblem, using based on the low of sequence space-time similar block The weight kernel normal form minimum value-based algorithm of order constraint and with singular value thresholding algorithm to all frames be iterated rebuild solution formula (6), Further formula (6) is divided into about GOP image group U and similar block PiLow-rank matrix two sub-problems, then alternately solve Certainly.The optimization for first considering similar block matrix, for each similar block group Si, they optimize low-rank matrixIt is close by formula (14) Seemingly:
Then, it solves
Specific step is as follows:
1) initialEvery time with image group U(k)In image carry out piecemeal and index as interior loop iteration image.Rule Determine the size of sample block and by image group U(k)In each reconstruction image be divided into the sample block s of overlapping1,si…,sn, in image Each sample s in groupiSimilar block retrieval is carried out, later to each sample siIn all similar block be grouped to obtain Si (k)。
2) calculating of low-rank subproblem is solved, first progress Initialize installation, iteration is for the first time with image group U(0)For initial graph Picture obtains Pi (0)=Si (k), wherein Pi (0)For the optimised low-rank matrix of iteration for the first time.Secondly it carries out internal circulation and it is set The number of iterations parameter t value, the subproblem is further divided into image group U and similar block PiTwo sons of low-rank matrix ask Topic,
3) similar block matrix optimizing subproblem is considered first, for each similar block group Si, optimised low-rank matrix Pi (t)'s Iterative calculation method is as follows:
Using matrix PiWeight nuclear norm function | | Pi||w,*=∑c||ωcσc(Pi)||1As Rank (Pi) function substitution Function converts formula (17) for formula (16) and solves:
Wherein, σc(Pi) indicate c-th of singular value, wc> 0 is σc(Li) and τ=η/λ weight.Then it is calculated using singular value threshold value Method (SVT) solves low-rank subproblem, and calculation method is as follows:
Pi (t+1)=Γ Sw,τ(∑)ΛT (18)
Wherein, Γ, ∑, Λ indicate SiThe result of singular value decomposition, that is to say, that Si=Γ ∑ ΛT, wherein ∑ is singular value square Battle array, diagonal element is the singular value S of matrixi, every other element is all zero.That is, ∑c,c=σc(Pi)。Sw,τ(∑) is ∑ Threshold function table is defined as Sw,τ(∑c,c)=max (∑c,c-τωc,0).To obtain the low-rank of (t+1) secondary iteration optimization Matrix.Update weight vector wcValue.
4) it is obtainingLater, all images in GOP are considered as an entirety, and show that image group U (t+1) is followed in secondary The estimation image of ring iterative, calculation method are as follows:
Wherein U ', γ(k)′,X(k+1)′Respectively indicate vector U, γ(k),X(k+1).The similar block of each sample is sought in GOP It looks for.It is fixedγ(k)And X(k+1), obtain the solution of a closing formItsMinimum value is secondary repeatedly by (t+1) It asks local derviation to be equal to zero for reconstruction image to acquire, effectively calculate by the division of element one by one, calculation method is as follows:
Wherein,It is a diagonal matrix, wherein the overlapping block of each component identification covering respective pixel location Number, that is to say, that it is to be divided by efficiently to be calculated one by one by matrix element that the optimal reconstruction image of formula (19), which solves, is not had There is matrix inversion operation.
5) value for updating t enables t=t+1, repetition step 3), 4), until meeting termination condition, terminate iterative cycles.
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