CN110912564A - An Image Measurement Matrix Optimization Method Based on Unit Norm Compact Framework - Google Patents

Abstract

The invention proposes an image measurement matrix optimization method based on a unit norm compact frame. Starting from the boundary conditions of the perception matrix itself, it is extremely decomposed to obtain an initialized α-compact frame, and a projection algorithm is used to separate the initialized α-compact frame. Projecting onto the structural constraint set and the spectral constraint set newly defined in this paper, then normalize the obtained frame, and finally obtain a unit-norm compact frame approximating the boundary conditions. The newly obtained frame is not restricted by the matrix dimension, Not only the row vectors are orthogonal, but also many characteristics of the perception matrix are preserved. From the spectral constraints, the constructed framework is a positive semi-definite matrix set, which makes it easier to solve the measurement matrix, greatly reducing the mutual interference coefficient and the impact on sparsity. requirements, improve the reconstruction accuracy of the image, and also increase the robustness of the image compressed sensing system to a certain extent.

Description

An Image Measurement Matrix Optimization Method Based on Unit Norm Compact Framework

technical field

The invention belongs to the field of signal processing, in particular to an image measurement matrix optimization method based on a unit norm compact frame.

Background technique

As a brand-new theory in the field of signal processing in recent years, compressed sensing has attracted more and more scholars at home and abroad. The theory shows that as long as the signal itself is sparse or sparse in a certain transform domain, it can be A measurement matrix irrelevant to the sparse dictionary projects the sampled high-dimensional signal onto a low-dimensional space, and performs sampling and compression at the same time. The projected signal contains enough information of the original signal, and then through some reconstruction algorithms, Using these small amounts of projection information, the original signal can be reconstructed with high probability. Compressed sensing technology breaks through the Nyquist sampling theory in a certain sense, effectively utilizes the sparsity of the original signal, and has nothing to do with the sampling frequency and bandwidth of the original signal.

Compressed sensing is a linear measurement process. Let X∈R ^N be the original signal and the length is N. By multiplying with the measurement matrix Φ∈R ^M×N , the observation value Y of length M is obtained,

Y=ΦX (1)

If X is not a sparse signal, it can be sparsely transformed to obtain X=Ψθ, where Ψ is the sparse basis, θ is the sparse coefficient, D=ΦΨ is the perception matrix, and the observation process of the CS image signal can be understood as the signal X is reduced from N dimension. To the process of M dimension, the specific process is shown in Figure 1. When θ satisfies ||θ|| ₀ ≤S, X is said to be an S-sparse signal. ||θ|| ₀ represents the number of non-zero elements. The sparse signal X can be recovered by the following formula:

表示噪声估计水平。在稀疏基已知的情况下，测量矩阵的性能不仅影响对初始信号进行的压缩，也影响着测量值重构原始信号。在压缩感知的研究中，通常使用的是随机测量矩阵，因为这类矩阵能够以很高的概率满足约束等距性质(RIP),这个性质要求测量矩阵和稀疏基之间满足一定的不相干性。in,

Indicates the noise estimate level. When the sparse basis is known, the performance of the measurement matrix affects not only the compression of the original signal, but also the reconstruction of the original signal from the measured values. In the research of compressed sensing, random measurement matrices are usually used, because such matrices can satisfy the constrained equidistant property (RIP) with a high probability, which requires a certain incoherence between the measurement matrix and the sparse basis. .

To design measurement matrices with better performance, most scholars refer to frame theory. The frame is a redundant set composed of vectors. The linear independence between elements is not required in the frame. It is more natural and broader when representing elements in the space. The frame is a generalization of the orthonormal basis. Using the frame to represent the signal is not only robust but also numerically stable during reconstruction. The specific definitions are as follows:

为M维Hilbert空间中的一组向量(L＞M)，对于Hilbert空间中的任意向量f_l∈R^M，如果满足下式：Assume

is a set of vectors (L>M) in the M-dimensional Hilbert space, for any vector f _l ∈ R ^M in the Hilbert space, if the following formula is satisfied:

称为一个框架，α和β分别称为下界和上界，框架冗余度定义为L/M。如果框架向量在

范数下是单位化的，对

均成立，则

称为单位范数框架(Unit-Norm Frame,UNF)。对于一个UNF，其不同列向量间内积的绝对值定义为原子相干性，相干性反映一个框架的结构特性。若存在常数0≤η＜1使得where 0<α≤β<∞, then

Called a frame, α and β are called the lower and upper bounds, respectively, and the frame redundancy is defined as L/M. If the frame vector is in

is unitized under the norm, yes

are established, then

It is called the Unit-Norm Frame (UNF). For a UNF, the absolute value of the inner product between its different column vectors is defined as atomic coherence, which reflects the structural properties of a frame. If there is a constant 0≤η<1 such that

为等角框架。established, then the

Isometric frame.

For an M×L UNF, the maximum value of the coherence of different atoms can be defined as the maximum frame coherence. When the maximum frame coherence is minimized, the frame at this time is called a Grassmannian Frame (GF). The main purpose is to minimize the maximum frame coherence of all UNFs, which can be obtained from the above formula. The value of η depends on M and L, and the lower bound is defined as:

则称其为最佳Grassmannian框架，如果α＝β，则称此时的框架为α-紧框架，α称为紧致常数，定义为：When a UNF is made

Then it is called the optimal Grassmannian frame. If α=β, then the frame at this time is called the α-compact frame, and α is called the compact constant, which is defined as:

The above formula shows that the Rayleigh quotient of any FFT is equal to α, and the eigenvalue of the matrix FFT is equal to α ^. The decision matrix ^{F∈R M×L} to be α-TF must meet the following three conditions:

①M non-zero singular values of matrix F are equal to

② The M non-zero eigenvalues of the matrix F ^T F are all equal to α;

③ The row vector of matrix α ^-1/2 F is standard orthogonal.

From the above three conditions, the SVD form corresponding to F of α-TF is as follows:

Among them, U∈R ^M×M and V∈R ^L×L are arbitrary standard orthogonal matrices. The compactness reflects the spectral properties of the matrix. Unit-Norm Tight Frame (UNTF) has the following properties:

When a UNTF satisfies Equation (8), the frame is called an Equiangular Tight Frame (ETF). The SVD form of an ETF is as follows:

ETFs have very desirable structural and spectral properties. However, not all ETFs of corresponding sizes can be constructed for all values of N and K. The necessary conditions for constructing an equiangularly compact frame are:

It can be seen from the previous discussion that GF focuses on the coherence of the frame, while TF focuses on the compactness of the frame. Both are important properties of the frame and play a crucial role in its application in the field of signal processing. However, the structure of ETF is easily affected by the matrix dimension, and it is very difficult to prove theoretically. In view of the above problems, the present invention starts from the boundary conditions of the Gram matrix itself, uses the matrix projection and the structural and spectral characteristics of the frame to construct We propose a novel unit-norm compact framework that can be used to optimize measurement matrices of arbitrary dimensions.

SUMMARY OF THE INVENTION

The purpose of the present invention is to propose a new method for constructing a unit norm compact frame in view of the deficiencies of the prior art. , constructs a new unit norm compact framework, the internal atomic coherence of the framework is smaller, and the compactness of the framework is well maintained. The optimized measurement matrix of the framework is closer to the boundary conditions, and the mutual relationship between the sparse basis and the Dryness is significantly reduced.

The technical solution of the present invention is the optimization of image measurement matrix based on the unit norm compact frame. The optimization algorithm of the traditional measurement matrix mainly approximates the Gram matrix to the ETF or the identity matrix I. However, the ETF is not only limited by the matrix dimension, but also the ETF is a full-rank framework, and the Gram matrix used to analyze the cross-correlation coefficient of the perception matrix. It is not full rank, and ETF cannot guarantee the orthogonality between the frame rows, that is, the compactness of the frame itself is not considered. This scheme starts from the boundary conditions of the Gram matrix itself, decomposes it polarly, and obtains the initialization The α-compact frame of , uses the projection algorithm to project the initialized α-compact frame onto the structural constraint set, and obtains an F-compact frame whose diagonal elements are 1 and whose off-diagonal elements are approximate to μ, and then project the F-compact frame to On the spectral constraint set newly defined in this paper, normalization is performed, and finally a unit-norm compact frame F~ that approximates the boundary condition μ is obtained. The newly obtained frame F~ is not only orthogonal to row vectors, but also a positive semi-definite matrix. , it is easier to solve the measurement matrix Φ.

Description of drawings

Fig. 1 is a signal observation process diagram of compressed sensing in an embodiment of the present invention

FIG. 2 is a convergence diagram of mutual interference coefficients of a unit-norm compact frame according to an embodiment of the present invention.

Fig. 3 is a graph showing the variation of the mutual interference coefficient of the unit norm compact frame with the value of M in the embodiment of the present invention

Fig. 4 is the abstract drawing in the embodiment of the present invention

Detailed ways

The specific construction method of the unit norm compact frame and the optimized measurement matrix are given below to further illustrate the specific implementation of the present invention.

Step 1: Initialize the compact frame and construct an equivalent matrix. Generate a random Gaussian measurement matrix Φ and a sparse basis Ψ, and normalize the measurement matrix Φ, where Φ∈R ^M×N , Ψ∈R ^N×L , where M<N<L. The perception matrix is:

D=ΦΨ (11)

According to the polar decomposition of the matrix, the perception matrix D can be decomposed into the product of two matrices. The theorem is as follows:

Theorem 1 Suppose the perception matrix D∈R ^M×L and M<L, there is a P and Q, such that D=PQ. Among them, P∈R ^M×M is a semi-positive definite matrix, rank(P)=rank(Q) and Q∈R ^M×L , and is a row orthogonal vector, QQ ^T =I _M .

prove:

It can be obtained from the singular value decomposition of the perception matrix D, D=USV ^T , S is a diagonal matrix, U and V are the corresponding orthogonal matrices, then D=USU ^T UV ^T , let

P=USU ^T , Q=UV ^T (12)

P ² =USU ^T USU ^T =US ² U ^T =DD ^T (13)

QQ^T＝I_M矩阵Q为矩阵D的正交矩阵。由于矩阵Q具有良好的特性，可利用该正交矩阵构造初始化的α紧框架。From the above formula, P is a positive semi-definite matrix, and

QQ ^T = _IM Matrix Q is an orthogonal matrix of matrix D. Since the matrix Q has good properties, an initialized α-compact frame can be constructed using this orthogonal matrix.

利用Frobenius范数，最接近感知矩阵D的α紧框架可以表示为：

Theorem 2 Suppose the perception matrix D∈R ^M×L , and M<L, the matrix is decomposed into D=PQ. The perceptual matrix redundancy is

Using the Frobenius norm, the α-compact frame closest to the perceptual matrix D can be expressed as:

The proof is as follows:

which is

From the above conditions, it can be seen that the F-compact frame is very close to the boundary conditions of the perception matrix D, and the obtained equivalent matrix is:

H=F ^T F (16)

Step 2: Structural constraint set projection. Project H=F ^T F onto the structural constraint set, and mainly perform shrinking transformation on the matrix H.

Where h=H(i,j), sign represents the sign function.

Step 3: Spectral constraint set projection, normalized compact frame. Project He obtained in step 2 onto the spectral constraint set S _tf newly defined in this _paper to ensure that the final unit norm compact frame is positive semi-definite and a set of matrices whose rank is less than or equal to M.

The set of spectral constraints is as follows:

S _tf ={S∈R ^L×L :S=S ^T ,X ^T SX≥0,rank(S)≤M} (18)

_{①Eigenvalue} decomposition of matrix He

He = _VΛV (19)

②Threshold value processing. Arrange the eigenvalues from large to small, and correspond the eigenvectors one by one, and perform threshold processing on the arranged eigenvalue matrix.

并标准化，且rank(H_e)≤M。Step 4: Update

and normalized, and rank(H _e )≤M.

为H_e平方根，即

Step 5: Solve

is the square _root of He, that is

Step 6: Solve the Measurement Matrix

The advantages of the present invention: starting from the boundary conditions of the perception matrix itself, decompose it extremely to obtain the initialized α-compact frame, and use the projection algorithm to project the initialized α-compact frame onto the structural constraint set and the spectral constraint set newly defined in this paper. In the above, normalization is performed, and finally a unit-norm compact frame F~ that is close to the boundary condition μ is obtained. The newly obtained frame F~ is not limited by the matrix dimension, not only the row vectors are orthogonal, but also retains many perceptual matrices. characteristic. According to the spectral constraints, the constructed framework is a positive semi-definite matrix set, which makes it easier to solve the measurement matrix Φ, greatly reduces the mutual interference coefficient, and reduces the dependence on sparse basis.

Claims (4)

1. An image measurement matrix optimization method based on a unit norm tight frame is characterized in that the redundancy characteristic of the frame is utilized, the edge condition of a sensing matrix is considered, the sensing matrix is subjected to polar decomposition to obtain an initialized α tight frame, the initialized α tight frame is projected onto a structure constraint set and a newly defined spectrum constraint set by using a projection algorithm to be subjected to normalization processing, finally, a unit norm tight frame F which is close to the edge condition mu is obtained, and then the frame F is optimized to obtain a measurement matrix phi.

2. The unit-norm tight-frame-based image measurement matrix optimization of claim 1, wherein: the newly obtained frame F is not limited by the matrix dimension, the row vectors are orthogonal, the characteristics of a plurality of sensing matrixes are kept, and the constructed frame is a semi-positive matrix set and is easier to solve the measurement matrix phi according to the spectrum constraint condition.

3. The unit norm tight frame of claim 1 is formed by the intersection of a structural constraint set and a spectral constraint set, wherein the structural constraint set mainly reduces α elements in the tight frame to make them closer to boundary conditions, and the spectral constraint set mainly reduces the rank of the matrix and retains larger eigenvalues, so that the final unit norm tight frame contains more matrix information and also plays a role in noise immunity.

4. The method for optimizing the image measurement matrix based on the unit norm tight box according to claim 1, wherein: the sensing matrix is close to a frame F, so that the non-correlation between the measurement matrix and a sparse base is greatly reduced, the requirement on sparsity is reduced, the performance of the measurement matrix is further optimized, the PSNR of a recovered image is improved, and the robustness of an image compression sensing system is increased to a certain extent.

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