压缩感知中优化投影矩阵的研究
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摘要
压缩感知(Compressed Sensing, CS)可以把稀疏信号以小于Shannon-Nyquist率的采样率恢复。近年来,压缩感知引起信号处理学界广泛的关注,在通信,图像处理,盲信号分离和模式识别等领域都有着广泛的应用。实际上,由于其重要的理论价值和广泛的应用前景,压缩感知一直是信号处理领域最热门的研究方向之一。
在压缩感知模型中,投影矩阵和字典会影响到稀疏信号的恢复精度。二者的乘积被称为等价字典。传统意义上,一直采用随机矩阵作为投影矩阵,因为它几乎与所有的正交字典都是不相关的,这被证明在概率意义上是最优的。近几年来,研究人员通过设计优化的投影矩阵使其得到的等价字典具有较低的相关度,从而增强压缩感知的重构效果。而在一些新的信号稀疏模型中,当字典具有特殊结构时,对优化投影矩阵的设计也提出新的要求,比如高维稀疏误差校正模型和分布式压缩感知(Distributed Compressed Sensing, DCS)模型要求优化投影矩阵设计需要适应字典的特殊结构。本论文针对压缩感知中优化投影矩阵设计问题展开研究,主要贡献如下。
1.提出一种基于粒子群优化(Particle Swarm Optimization, PSO)的优化投影矩阵设计算法,以增强基于压缩感知的高维稀疏误差校正的精度。基于压缩感知的CAB (Cross-And-Bouquet)模型由Wright J.等人提出用于降低稀疏误差校正的复杂度,这里使用的是随机高斯矩阵。为提高基于压缩感知的校正效果,本文以等价字典的平均互相关度(mutual coherence)最小化为目标函数,基于粒子群算法构造优化投影矩阵。该平均相关度一个是由Elad M.提出,但不适用于高维情形。另一个是本文专为高维情况下提出的。本文提出的优化投影矩阵设计算法无需高维奇异值分解。适用于CAB模型解决高维稀疏误差校正问题。最后,通过高维情形下的解码问题证实该算法的有效性。
2.针对一般压缩感知模型,本文提出一种基于低秩自相关矩阵模型的优化投影矩阵设计算法。已有的研究提出把投影矩阵和字典的乘积矩阵接近于等角度紧框架(Equiangular Tight Frame, ETF)这一想法应用在优化投影矩阵设计问题中。通过引进低秩Gram矩阵模型实现这一想法,并且基于低秩矩阵接近问题,本文提出一种优化投影矩阵设计算法。通过基于稀疏表示的图像融合实验和图像去噪实验表明该算法在性能上优于已有的一些算法。
3.DCS理论建立在多个信号具有联合稀疏表示(Joint Sparse Representation, JSR)这一基础之上。这些信号组成一个信号族。三种联合稀疏模型被提出：JSM-1,JSM-2和JSM-3。在JSM-1模型中,一个信号族中的所有信号具有共同的稀疏分量,每个信号拥有各自唯一的特有稀疏分量。联合稀疏表示比一般稀疏表示在处理多个信号上具有较低的计算复杂度。本文给出一种适用于JSM-1的字典学习算法MODJSR(Method of Optimal Directions for Joint Sparse Representation)。根据JSM-1的结构,其字典更新步骤只需一次特征值分解运算。该算法本质上是把基于稀疏表示的最优方向法(Method of Optimal Directions, MOD)推广到联合稀疏表示。MODJSR较常用的K-SVD(K-Singular Value Decomposition)具有更低的计算复杂度。为更有效地提取图像细节,本文把JSR推广到广义联合稀疏表示(Generalized Joint Sparse Representation, GJSR)。JSR下的信号族具有共同分量和唯一分量,这两个分量在同一个字典下表示系数是稀疏的,而GJSR下这两个分量在各自的字典下具有稀疏表示。MODJSR也被推广到MODGJSR (Method of Optimal Directions for Generalized Joint Sparse Representation)。在基于JSR的图像融合中,本文提出一种新的融合规则。MODJSR/MODGJSR可以同时完成有噪声源图像的字典学习、去噪和融合等过程。图像融合实验结果表明,本文提出的GJSR模型,MODJSR/MODGJSR字典学习算法和融合规则的优越性。
4.基于GJSR,本文把分布式压缩感知推广到广义分布式压缩感知(Generalized Distributed Compressed Sensing, GDCS)。根据GDCS中等价字典的结构,本文把最小化互相关度问题归结于非凸函数的最小化问题,首次提出一种解决GDCS中的优化投影矩阵设计算法。该算法属于梯度下降法,其步长选择采用BB步长(Barzilai-Borwein stepsize)。该算法被推广到分块稀疏模型中。其有效性在合成信号实验和真实图像的融合实验中得以证实。
综上所述,本文主要研究压缩感知中优化投影矩阵的设计算法。首先,针对高维稀疏误差校正模型提出一种优化投影矩阵设计算法。然后,针对一般压缩感知模型,提出一种基于低秩矩阵模型的优化投影矩阵设计算法。此外,本文提出一种联合稀疏表示模型下的字典学习算法,并把联合稀疏表示推广到广义联合稀疏表示。最后,本文推导出广义分布式压缩感知下的优化投影矩阵设计算法。同时,通过大量的仿真实验,验证了本文所提算法的有效性。
Compressed sensing (CS) has shown that sparse signals can be recovered from far less samples than those required by the classical Shannon-Nyquist Theorem. In recent years. CS has attracted widely attention of signal processing society. It has widely application in communication, image processing, blind source separation and pattern recognition. Practically, CS is one of the most popular research areas in signal processing society, due to its important theoretical value and widely application.
In CS, projection matrix and dictionary influence the sparse recovery accuracy. Ran-dom projections were used since they present small coherence with almost any orthogonal dictionary. It was proved to be optimal in probabilistic sense. Recent researches show that optimizing the projection matrix toward decreasing the coherence is possible and can improve the reconstruction performance of CS. For a new sparse model, when the dictio-nary has special structure, the optimization of the projection matrix should be developed. For example, high-dimensional error correcting model and distributed compressed sens-ing (DCS) require the the optimization of the projection matrix should adapt the special structure of the dictionary. This thesis researches on the optimization of the-projection matrix in CS, the main contributions are as follows.
1. To enhance the performance of CS-based high-dimensional sparse error correcting, PSO (Particle Swarm Optimization)-based algorithm to optimize the projection matrix is used to overcome the computational difficulty in high-dimensional cases. CS-based cross-and-bouquet (CAB) model was proposed by J. Wright et al. to reduce the com-plexity of sparse error correcting, where the random Gaussian projections are used. For the sake of leading to better performance of CS-based decoding for the CAB model, an algorithm is proposed in this thesis for constructing a well-designed projection matrix to minimize the average measures of mutual coherence. One was proposed by M. Elad, but it is not suitable for the high-dimensional cases. Another is proposed by this thesis for high dimensional cases. Using the equivalent dictionary, the dimensionality is reduced. Also, high-dimensional Singular Value Decomposition (SVD) is avoided in the procedure of constructing a well-designed projection matrix. The high-dimensional CAB model of sparse error correcting can be solved by the proposed algorithm without computational difficulty. At last, the validity of the proposed algorithm is illustrated by decoding ex-periments in high-dimensional cases.
2. For the common CS model, this thesis gives a low-rank Gram matrix-based algorithm to optimize the projection matrix. Bring the multiplication of the projection matrix and the dictionary to be near an equiangular tight frame (ETF) was proposed as an idea in some previous works. Here, a low-rank Gram matrix model is introduced to realize it. Also, an algorithm is presented motivated from the computation method of the matrix nearness for low-rank cases. Simulations show that the proposed algorithm is better than some other algorithms to optimize the projection matrix in terms of image fusion and image denoising via sparse representation.
3. The DCS theory rests on a new concept called multiple signals have joint sparse representation (JSR). These signals form a signal ensemble. Three joint sparsity models JSM-1, JSM-2and JSM-3were presented. In JSM-1model, all signals in one ensemble have a common sparse component, and each individual signal owns an innovation sparse component. The JSR offers lower computational complexity compared with the SR in dealing with multiple signals. This thesis proposed a novel dictionary learning method (MODJSR) whose dictionary updating procedure is derived employing the JSR structure with only once eigen value decomposition operation. Indeed, the MODJSR is the SR-based method of optimal directions (MOD) extended for the JSR. The MODJSR has lower complexity than the K-SVD algorithm which often used. To capture the image details more efficiently, this thesis extended the JSR to the GJSR (generalized joint sparse representation). The JSR models the common component and the innovation component by one dictionary while the GJSR depends on two dictionaries. The MODJSR is extended to MODGJSR in this case. For the JSR-based image fusion, this thesis gives a new fusion rule. MODJSR/MODGJSR can carry out dictionary learning, denoising and fusion of noisy source images, simultaneously. Some experiments on image fusion are given to demonstrate the validity of the proposed GJSR model, the MODJSR/MODGJSR and the new fusion rule.
4. Based on the GJSR, the DCS is extended to generalized distributed compressed sensing (GDCS) in this letter. In this thesis, the minimization of the mutual coherence is summarized in the minimization of a non-convex function by the structure of the equiv-alent dictionary. Originally, this thesis proposed an algorithm to optimize the projection matrix for the GDCS using the structure of its equivalent dictionary. The algorithm belongs to the gradient method, its stepsize is chosen as Barzilai-Borwein stepsize. The algorithm is extended to block sparse model. The validity of the proposed algorithm is illustrated by some experiments for synthesized signals and real-world image fusion.
Above all, the optimized projection matrix designing algorithms in CS are studied in this thesis. Firstly, an optimized projection matrix designing algorithm is given for CS-based high-dimensional sparse error correcting model. Secondly, a low-rank model based optimized projection matrix designing algorithm is proposed for the common CS model. Moreover, this thesis gives a dictionary learning method for the JSR. The JSR is extended to the GJSR. At last, an algorithm to optimize the projection matrix for the GDCS is proposed. Simultaneously, numerical experiments are given to illustrate the efficiency of the proposed algorithms in this thesis.
在压缩感知模型中,投影矩阵和字典会影响到稀疏信号的恢复精度。二者的乘积被称为等价字典。传统意义上,一直采用随机矩阵作为投影矩阵,因为它几乎与所有的正交字典都是不相关的,这被证明在概率意义上是最优的。近几年来,研究人员通过设计优化的投影矩阵使其得到的等价字典具有较低的相关度,从而增强压缩感知的重构效果。而在一些新的信号稀疏模型中,当字典具有特殊结构时,对优化投影矩阵的设计也提出新的要求,比如高维稀疏误差校正模型和分布式压缩感知(Distributed Compressed Sensing, DCS)模型要求优化投影矩阵设计需要适应字典的特殊结构。本论文针对压缩感知中优化投影矩阵设计问题展开研究,主要贡献如下。
1.提出一种基于粒子群优化(Particle Swarm Optimization, PSO)的优化投影矩阵设计算法,以增强基于压缩感知的高维稀疏误差校正的精度。基于压缩感知的CAB (Cross-And-Bouquet)模型由Wright J.等人提出用于降低稀疏误差校正的复杂度,这里使用的是随机高斯矩阵。为提高基于压缩感知的校正效果,本文以等价字典的平均互相关度(mutual coherence)最小化为目标函数,基于粒子群算法构造优化投影矩阵。该平均相关度一个是由Elad M.提出,但不适用于高维情形。另一个是本文专为高维情况下提出的。本文提出的优化投影矩阵设计算法无需高维奇异值分解。适用于CAB模型解决高维稀疏误差校正问题。最后,通过高维情形下的解码问题证实该算法的有效性。
2.针对一般压缩感知模型,本文提出一种基于低秩自相关矩阵模型的优化投影矩阵设计算法。已有的研究提出把投影矩阵和字典的乘积矩阵接近于等角度紧框架(Equiangular Tight Frame, ETF)这一想法应用在优化投影矩阵设计问题中。通过引进低秩Gram矩阵模型实现这一想法,并且基于低秩矩阵接近问题,本文提出一种优化投影矩阵设计算法。通过基于稀疏表示的图像融合实验和图像去噪实验表明该算法在性能上优于已有的一些算法。
3.DCS理论建立在多个信号具有联合稀疏表示(Joint Sparse Representation, JSR)这一基础之上。这些信号组成一个信号族。三种联合稀疏模型被提出：JSM-1,JSM-2和JSM-3。在JSM-1模型中,一个信号族中的所有信号具有共同的稀疏分量,每个信号拥有各自唯一的特有稀疏分量。联合稀疏表示比一般稀疏表示在处理多个信号上具有较低的计算复杂度。本文给出一种适用于JSM-1的字典学习算法MODJSR(Method of Optimal Directions for Joint Sparse Representation)。根据JSM-1的结构,其字典更新步骤只需一次特征值分解运算。该算法本质上是把基于稀疏表示的最优方向法(Method of Optimal Directions, MOD)推广到联合稀疏表示。MODJSR较常用的K-SVD(K-Singular Value Decomposition)具有更低的计算复杂度。为更有效地提取图像细节,本文把JSR推广到广义联合稀疏表示(Generalized Joint Sparse Representation, GJSR)。JSR下的信号族具有共同分量和唯一分量,这两个分量在同一个字典下表示系数是稀疏的,而GJSR下这两个分量在各自的字典下具有稀疏表示。MODJSR也被推广到MODGJSR (Method of Optimal Directions for Generalized Joint Sparse Representation)。在基于JSR的图像融合中,本文提出一种新的融合规则。MODJSR/MODGJSR可以同时完成有噪声源图像的字典学习、去噪和融合等过程。图像融合实验结果表明,本文提出的GJSR模型,MODJSR/MODGJSR字典学习算法和融合规则的优越性。
4.基于GJSR,本文把分布式压缩感知推广到广义分布式压缩感知(Generalized Distributed Compressed Sensing, GDCS)。根据GDCS中等价字典的结构,本文把最小化互相关度问题归结于非凸函数的最小化问题,首次提出一种解决GDCS中的优化投影矩阵设计算法。该算法属于梯度下降法,其步长选择采用BB步长(Barzilai-Borwein stepsize)。该算法被推广到分块稀疏模型中。其有效性在合成信号实验和真实图像的融合实验中得以证实。
综上所述,本文主要研究压缩感知中优化投影矩阵的设计算法。首先,针对高维稀疏误差校正模型提出一种优化投影矩阵设计算法。然后,针对一般压缩感知模型,提出一种基于低秩矩阵模型的优化投影矩阵设计算法。此外,本文提出一种联合稀疏表示模型下的字典学习算法,并把联合稀疏表示推广到广义联合稀疏表示。最后,本文推导出广义分布式压缩感知下的优化投影矩阵设计算法。同时,通过大量的仿真实验,验证了本文所提算法的有效性。
Compressed sensing (CS) has shown that sparse signals can be recovered from far less samples than those required by the classical Shannon-Nyquist Theorem. In recent years. CS has attracted widely attention of signal processing society. It has widely application in communication, image processing, blind source separation and pattern recognition. Practically, CS is one of the most popular research areas in signal processing society, due to its important theoretical value and widely application.
In CS, projection matrix and dictionary influence the sparse recovery accuracy. Ran-dom projections were used since they present small coherence with almost any orthogonal dictionary. It was proved to be optimal in probabilistic sense. Recent researches show that optimizing the projection matrix toward decreasing the coherence is possible and can improve the reconstruction performance of CS. For a new sparse model, when the dictio-nary has special structure, the optimization of the projection matrix should be developed. For example, high-dimensional error correcting model and distributed compressed sens-ing (DCS) require the the optimization of the projection matrix should adapt the special structure of the dictionary. This thesis researches on the optimization of the-projection matrix in CS, the main contributions are as follows.
1. To enhance the performance of CS-based high-dimensional sparse error correcting, PSO (Particle Swarm Optimization)-based algorithm to optimize the projection matrix is used to overcome the computational difficulty in high-dimensional cases. CS-based cross-and-bouquet (CAB) model was proposed by J. Wright et al. to reduce the com-plexity of sparse error correcting, where the random Gaussian projections are used. For the sake of leading to better performance of CS-based decoding for the CAB model, an algorithm is proposed in this thesis for constructing a well-designed projection matrix to minimize the average measures of mutual coherence. One was proposed by M. Elad, but it is not suitable for the high-dimensional cases. Another is proposed by this thesis for high dimensional cases. Using the equivalent dictionary, the dimensionality is reduced. Also, high-dimensional Singular Value Decomposition (SVD) is avoided in the procedure of constructing a well-designed projection matrix. The high-dimensional CAB model of sparse error correcting can be solved by the proposed algorithm without computational difficulty. At last, the validity of the proposed algorithm is illustrated by decoding ex-periments in high-dimensional cases.
2. For the common CS model, this thesis gives a low-rank Gram matrix-based algorithm to optimize the projection matrix. Bring the multiplication of the projection matrix and the dictionary to be near an equiangular tight frame (ETF) was proposed as an idea in some previous works. Here, a low-rank Gram matrix model is introduced to realize it. Also, an algorithm is presented motivated from the computation method of the matrix nearness for low-rank cases. Simulations show that the proposed algorithm is better than some other algorithms to optimize the projection matrix in terms of image fusion and image denoising via sparse representation.
3. The DCS theory rests on a new concept called multiple signals have joint sparse representation (JSR). These signals form a signal ensemble. Three joint sparsity models JSM-1, JSM-2and JSM-3were presented. In JSM-1model, all signals in one ensemble have a common sparse component, and each individual signal owns an innovation sparse component. The JSR offers lower computational complexity compared with the SR in dealing with multiple signals. This thesis proposed a novel dictionary learning method (MODJSR) whose dictionary updating procedure is derived employing the JSR structure with only once eigen value decomposition operation. Indeed, the MODJSR is the SR-based method of optimal directions (MOD) extended for the JSR. The MODJSR has lower complexity than the K-SVD algorithm which often used. To capture the image details more efficiently, this thesis extended the JSR to the GJSR (generalized joint sparse representation). The JSR models the common component and the innovation component by one dictionary while the GJSR depends on two dictionaries. The MODJSR is extended to MODGJSR in this case. For the JSR-based image fusion, this thesis gives a new fusion rule. MODJSR/MODGJSR can carry out dictionary learning, denoising and fusion of noisy source images, simultaneously. Some experiments on image fusion are given to demonstrate the validity of the proposed GJSR model, the MODJSR/MODGJSR and the new fusion rule.
4. Based on the GJSR, the DCS is extended to generalized distributed compressed sensing (GDCS) in this letter. In this thesis, the minimization of the mutual coherence is summarized in the minimization of a non-convex function by the structure of the equiv-alent dictionary. Originally, this thesis proposed an algorithm to optimize the projection matrix for the GDCS using the structure of its equivalent dictionary. The algorithm belongs to the gradient method, its stepsize is chosen as Barzilai-Borwein stepsize. The algorithm is extended to block sparse model. The validity of the proposed algorithm is illustrated by some experiments for synthesized signals and real-world image fusion.
Above all, the optimized projection matrix designing algorithms in CS are studied in this thesis. Firstly, an optimized projection matrix designing algorithm is given for CS-based high-dimensional sparse error correcting model. Secondly, a low-rank model based optimized projection matrix designing algorithm is proposed for the common CS model. Moreover, this thesis gives a dictionary learning method for the JSR. The JSR is extended to the GJSR. At last, an algorithm to optimize the projection matrix for the GDCS is proposed. Simultaneously, numerical experiments are given to illustrate the efficiency of the proposed algorithms in this thesis.
引文
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