压缩感知恢复算法及应用研究
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摘要
基于信号的稀疏性结构，集采样和压缩为一体，压缩感知突破了香农采样定理，能够运用远少于香农采样定理所界定的采样数目来精确恢复原始稀疏信号。压缩感知具有广泛的应用背景，包括误差校正、图像处理、通信工程、盲信号分离、模式识别等。压缩感知的研究，促进了信号处理理论和工程应用的发展，已经成为该领域的研究热点之一。
信号恢复算法是压缩感知理论的重要组成部分。针对不同的稀疏信号，选择合适的恢复算法，运用尽可能少的测量数目，压缩感知致力于精确恢复原稀疏信号。本文以此为目标，针对恢复算法展开研究，主要贡献如下：
1.基于硬阈值追踪算法（hard thresholding pursuit，HTP），提出了一种新的贪婪算法，旨在解决信号稀疏度未知情况下的恢复性问题。该算法采用了渐近估计稀疏度的技巧，解决真实稀疏度未知造成的困难。将限制等距性质（restricted isometryproperty，RIP）作为理论分析工具，给出了算法收敛的充分条件，并且给出了恢复的信号与原始信号之间的误差上界。在信号稀疏度未知的前提下，合成信号和自然图像的恢复实验表明该算法具有良好的恢复性能。
2.目前，针对块正交匹配追踪算法（block orthogonal matching pursuit，BOMP）精确恢复原始块稀疏信号的条件大多以块-互相关度（block mutual coherence）为判别准则。利用块-RIP，本文给出了保证BOMP算法精确恢复原信号的充分条件，并且说明了给出基于块-RIP的精确恢复条件是必要的；针对人脸识别等工程应用中可能出现冗余块的情况，提出了一种解决冗余块问题的算法，并且给出了保证算法精确恢复的条件；在多重测量向量（multiple measurement vectors，MMV）模型的基础上，本文所提的算法能够实现同时处理多个样本。最后，通过人脸识别的实验，表明了所提算法的有效性。
3.针对所求稀疏信号部分支撑信息已知的情况，提出了加权L2,1最小化方法。该方法可以利用信号序列中帧与帧之间的相关性，将上一帧的支撑信息作为恢复下一帧信号的先验信息，使得进一步降低采样数目成为可能。利用RIP，给出了恢复的信号与原始信号之间的误差上界。另外，由于该方法将二维的信号直接看做矩阵来处理，而不是将其向量化，这样大大的减少了运行时间。通过恢复Larynx图像序列的实验，验证了算法的有效性。
4.针对贪婪块坐标下降算法（greedy block coordinate descent，GBCD），在加性噪声和乘性噪声干扰下，运用RIP理论工具，分析了该算法的性能。给出了保证GBCD算法精确恢复原始信号的支撑集合的充分条件，并且给出了满足该充分条件的例子；讨论了该充分条件的上界，通过例子，指出了在不满足该充分条件时，存在着GBCD算法不能精确恢复的情况。最后，通过仿真实验，验证了GBCD算法的性能。
Based on the signal sparsity structure, integrating sampling and compression, theShannon sampling theorem is broken through by the theory of compressed sensing, whichcan recover the sparse signals exactly from far less samples than those required by theclassical Shannon theorem. Compressed sensing has a wide range of applications thatinclude error correction, image processing, communication engineering, blind source sep-aration, pattern recognition and so on. It promotes the development of the theory andengineering application and has been one of the hottest topics in the feld of signal pro-cessing.
The recovery algorithm is an important part of compressed sensing. In compressedsensing model, selecting the appropriate algorithm for diferent signals, recovering theoriginal signal exactly by using as few measurements as possible is the goal. Taking thisas the target and carrying out research on the recovery algorithms, the main contributionsof this paper are as follows.
1. Based on the hard thresholding pursuit (HTP), this paper proposes a new greedyalgorithm, aiming at resolving the recovery of sparse signals without the sparsity levelinformation. The algorithm estimates the sparsity level stage by stage, thus it can over-come the difculty caused by the unknown sparsity level information. By using restrictedisometry property (RIP), the sufcient condition for the convergence of the algorithmis presented, and the error between the recovered signal and the original signal is alsogiven. By recovering the synthetic signals and natural images, this paper shows that theproposed algorithm has good performances even without the sparsity level information.
2. Based on block mutual coherence, the existing work analyzes the recovery ofsparse signals using block orthogonal matching pursuit (BOMP). This paper presents asufcient condition, based on block RIP, which can ensure to recover the original signalusing BOMP. Also, this paper explains that it is necessary to give the sufcient condition.Redundant blocks may be found in face recognition. This paper proposes an algorithmto solve this problem, and gives a sufcient condition for exactly recovering the supportof unknown sparse vector. Based on multiple measurement vectors (MMV) model, thispaper proposes an algorithm that can simultaneously process many samples. Finally, theefectiveness of the proposed algorithm is shown in the experiment of face recognition.
3. The weighted L2,1minimization method is proposed when a part of a sparsesignal’s support is known. By utilizing the correlation of the consecutive frames of signalsequence, this method sees the support of the last frame as prior information, thus, it is possible to recover the original signal using less samples. Based on RIP, the error betweenthe recovered signal and the original signal is presented. In addition, this method directlyregards2D signal as matrix, rather than1D vector, it greatly reduces the running time.The efectiveness of the proposed method is indicated through the recovery of the Larynxsequence.
4. Under the additive noise and multiplicative noise, using RIP, this paper aimsat analyzing the theoretical performance of greedy block coordinate descent (GBCD)algorithm. A sufcient condition is presented and an example that satisfes this conditionis also given. The upper bound of this sufcient condition is also analyzed. By giving anexample, this paper shows that GBCD can not exactly recover the support of the originalsignal. Finally, the performance of GBCD is illustrated by experimental results.
信号恢复算法是压缩感知理论的重要组成部分。针对不同的稀疏信号，选择合适的恢复算法，运用尽可能少的测量数目，压缩感知致力于精确恢复原稀疏信号。本文以此为目标，针对恢复算法展开研究，主要贡献如下：
1.基于硬阈值追踪算法（hard thresholding pursuit，HTP），提出了一种新的贪婪算法，旨在解决信号稀疏度未知情况下的恢复性问题。该算法采用了渐近估计稀疏度的技巧，解决真实稀疏度未知造成的困难。将限制等距性质（restricted isometryproperty，RIP）作为理论分析工具，给出了算法收敛的充分条件，并且给出了恢复的信号与原始信号之间的误差上界。在信号稀疏度未知的前提下，合成信号和自然图像的恢复实验表明该算法具有良好的恢复性能。
2.目前，针对块正交匹配追踪算法（block orthogonal matching pursuit，BOMP）精确恢复原始块稀疏信号的条件大多以块-互相关度（block mutual coherence）为判别准则。利用块-RIP，本文给出了保证BOMP算法精确恢复原信号的充分条件，并且说明了给出基于块-RIP的精确恢复条件是必要的；针对人脸识别等工程应用中可能出现冗余块的情况，提出了一种解决冗余块问题的算法，并且给出了保证算法精确恢复的条件；在多重测量向量（multiple measurement vectors，MMV）模型的基础上，本文所提的算法能够实现同时处理多个样本。最后，通过人脸识别的实验，表明了所提算法的有效性。
3.针对所求稀疏信号部分支撑信息已知的情况，提出了加权L2,1最小化方法。该方法可以利用信号序列中帧与帧之间的相关性，将上一帧的支撑信息作为恢复下一帧信号的先验信息，使得进一步降低采样数目成为可能。利用RIP，给出了恢复的信号与原始信号之间的误差上界。另外，由于该方法将二维的信号直接看做矩阵来处理，而不是将其向量化，这样大大的减少了运行时间。通过恢复Larynx图像序列的实验，验证了算法的有效性。
4.针对贪婪块坐标下降算法（greedy block coordinate descent，GBCD），在加性噪声和乘性噪声干扰下，运用RIP理论工具，分析了该算法的性能。给出了保证GBCD算法精确恢复原始信号的支撑集合的充分条件，并且给出了满足该充分条件的例子；讨论了该充分条件的上界，通过例子，指出了在不满足该充分条件时，存在着GBCD算法不能精确恢复的情况。最后，通过仿真实验，验证了GBCD算法的性能。
Based on the signal sparsity structure, integrating sampling and compression, theShannon sampling theorem is broken through by the theory of compressed sensing, whichcan recover the sparse signals exactly from far less samples than those required by theclassical Shannon theorem. Compressed sensing has a wide range of applications thatinclude error correction, image processing, communication engineering, blind source sep-aration, pattern recognition and so on. It promotes the development of the theory andengineering application and has been one of the hottest topics in the feld of signal pro-cessing.
The recovery algorithm is an important part of compressed sensing. In compressedsensing model, selecting the appropriate algorithm for diferent signals, recovering theoriginal signal exactly by using as few measurements as possible is the goal. Taking thisas the target and carrying out research on the recovery algorithms, the main contributionsof this paper are as follows.
1. Based on the hard thresholding pursuit (HTP), this paper proposes a new greedyalgorithm, aiming at resolving the recovery of sparse signals without the sparsity levelinformation. The algorithm estimates the sparsity level stage by stage, thus it can over-come the difculty caused by the unknown sparsity level information. By using restrictedisometry property (RIP), the sufcient condition for the convergence of the algorithmis presented, and the error between the recovered signal and the original signal is alsogiven. By recovering the synthetic signals and natural images, this paper shows that theproposed algorithm has good performances even without the sparsity level information.
2. Based on block mutual coherence, the existing work analyzes the recovery ofsparse signals using block orthogonal matching pursuit (BOMP). This paper presents asufcient condition, based on block RIP, which can ensure to recover the original signalusing BOMP. Also, this paper explains that it is necessary to give the sufcient condition.Redundant blocks may be found in face recognition. This paper proposes an algorithmto solve this problem, and gives a sufcient condition for exactly recovering the supportof unknown sparse vector. Based on multiple measurement vectors (MMV) model, thispaper proposes an algorithm that can simultaneously process many samples. Finally, theefectiveness of the proposed algorithm is shown in the experiment of face recognition.
3. The weighted L2,1minimization method is proposed when a part of a sparsesignal’s support is known. By utilizing the correlation of the consecutive frames of signalsequence, this method sees the support of the last frame as prior information, thus, it is possible to recover the original signal using less samples. Based on RIP, the error betweenthe recovered signal and the original signal is presented. In addition, this method directlyregards2D signal as matrix, rather than1D vector, it greatly reduces the running time.The efectiveness of the proposed method is indicated through the recovery of the Larynxsequence.
4. Under the additive noise and multiplicative noise, using RIP, this paper aimsat analyzing the theoretical performance of greedy block coordinate descent (GBCD)algorithm. A sufcient condition is presented and an example that satisfes this conditionis also given. The upper bound of this sufcient condition is also analyzed. By giving anexample, this paper shows that GBCD can not exactly recover the support of the originalsignal. Finally, the performance of GBCD is illustrated by experimental results.
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