匹配追踪的最佳原子选择策略和压缩感知盲稀疏度重建算法改进
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摘要
压缩感知(CS)理论能对信号实施远低于奈奎斯特速率的采样,是信号处理领域的又—重要成果。其中基于l0范数最小的匹配追踪重构算法,原子搜索主要采用贪婪优化,其关键点是：(1)最佳原子的选择策略和匹配准则；(2)信号稀疏度的准确估计。原子选择方式分为串行选择和并行选择两种；匹配准则分为当前残差逼近、当前最佳观测和全局重建误差最小准则。选择方式关系到匹配的效率和容错性,准则决定原子选择的方向和全局最优解的逼近程度。本文主要改善最佳原子的选择方法,并改进稀疏度未知信号的鲁棒性重建方案。
本文的主要贡献包括：
1)为提高原子选择准确度,提出了前向预测与回溯结合的正交匹配追踪(LABOMP)算法。在迭代初期用前向预测策略串行选择候选原子,在迭代后期通过触发阈值启动一个间隔回溯机制,剔除错选原子。实验证明,用LABOMP算法重建稀疏信号和可压缩语音信号,平均精确重建概率(MERP)比只使用前向预测策略的LAOMP算法平均提高了12.5%～31.3%,已选原子的全局最优解逼近能力明显提高。
2)为提高LABOMP算法的收敛速度,提出正则化LABOMP算法(RLABOMP)。鉴于CS重建具有不适定的多个可用解,而且单次迭代的残差匹配有异于全局重建最优化匹配,保留多个性能相当的备选解是合理的。因此我们以最佳原子集内候选原子的公平选入为正则约束,保证同时选入性能相近的最优原子,以便在提高容错性的同时减少迭代次数。实验证明,对稀疏信号和可压缩语音信号,RLABOMP选取K个原子所需迭代次数为LABOMP的42.90%～54.28%,为LAOMP的62.85%～71.57%,但MERP相对LABOMP降低9.27%～25.12%,相对LAOMP提高1.91～10.80%,明显优于LAOMP算法,与LABOMP算法比则取得迭代效率与重建性能的折中。
3)为同时提高原子选择准确度和计算速度,提出广义最佳正交匹配追踪(GOOMP)算法。其改进思路是：(1)采用当前观测信号最佳的匹配准则,保证所求解能使当前观测信号的匹配误差最小。(2)成批选入L个最优原子,保证选入原子集具备较高的算法容错能力。对稀疏信号,使用当前最佳观测准则的gOMP算法,其MERP提高了3.16%～10.8%,使用成批选入L个最优原子的OOMP算法完成精确重建所需迭代次数为OOMP算法的1/3,MERP提高31.26%～37.54%。
此外,本文证明了GOOMP算法在K次迭代内完全重建原始信号的充分条件是：在保证正确原子不相关的前提下,后续待选正确原子与错误原子在己选原子张成的补空间中各自投影并归一化,投影并归一化后错误原子在所有投影并归一化后正确原子构成空间上的投影系数向量l1范数小于1。
4)基于残差匹配和原子系数阈值判断,提出改进的盲稀疏度信号重构方案。首先分析了回溯型自适应正交匹配追踪(BAOMP)算法的原子选入阈值参数与稀疏度的关系,然后针对BAOMP算法迭代后期误选原子,引起的残差模值陡增,估计信号误差周期骤升的缺点,提出了改进的算法IBAOMP。当估计的观测信号能量与目标能量接近时,IBAOMP算法按80-20准则提高原子选入阈值,控制错误原子的选入和候选原子集的容量,非常有效地消除了误差周期性突增的现象。
总的来说,本文原子选择方法和盲稀疏度压缩重建的研究工作给出了如下结果：
具有间隔回溯机制的LABOMP算法有利于提高击中正确原子的概率,RLABOMP算法由于使用正则化原子选择方法可以明显提高LABOMP算法的收敛速度,而GOOMP算法既可以提高MERP又能减少计算时间。另一方面,对于盲稀疏度信号重建IBAOMP算法通过阈值调节机制,避免了稀疏度信息缺失时的重建盲目性,保证了估计信号逐步逼近目标信号。
Compressed sensing can sample signals at a rate far below Nyquist rate, and it's a new tool in signal processing. In the l0norm based algorithms, the atom matching pursuit mainly applied greedy search, their key techniques are (1) optimal atom selection and matching criteria;(2) accurate estimate of signal sparsity. Atom selection strategies can be divided into single atom hit and a batch selection of atoms, while criteria include residual matching, current best observation and global error minimization. In addition to the atom selection related to efficiency and fault tolerance, matching criteria determine directions of atom search which approaches an optimal solution. This thesis mainly focuses on efficient atom selection and robust reconstruction of sparsity unknown signals.
Main contributions include:
1) To improve the accuracy of atom selection, a look-ahead and backtracking orthogonal matching pursuit (LABOMP) algorithm is proposed. In early iterations a set of candidate atoms is chosen by look ahead verification, on the contrary, an interval backtracking mechanism is triggered by a threshold in later iterations to prune error atoms. Experiments show that, for sparse signals and compressive voice, the mean exact reconstruction probability (MERP) of LABOMP is12.5%～31.3%higher than that of LAOMP, where the latter uses only the look-ahead strategy, which indicates the probability of finding a global solution is improved.
2) For rapid convergence of LABOMP, a regularized LABOMP (RLABOMP) algorithm is proposed. Note that the CS reconstruction is an ill-posed solution and the difference between criterion of residual matching and criterion of global matching, multiple'good' atoms could be regarded as equivalent candidate. Thus an equal opportunity constraint is introduced to regularize the recruitment of candidate atoms to include the atoms with similar performances, to promote fault tolerance and accelerate the convergence. Experiments show that the number of iterations of RLABOMP for choosing K atoms is42.90%～54.28%of that of LABOMP and62.85%～71.57%of that of LAOMP. However, MERP of RLABOMP is9.27%～25.12%lower than that of LABOMP and1.91%～10.80%higher than LAOMP. Obviously RLABOMP is better than LAOMP and balances the efficiency and the reconstruction performance comparing to LABOMP.
3) To increase the accuracy and speed of atom selection, a generalized optimal orthogonal matching pursuit (GOOMP) algorithm is proposed. Improvements are (1) using optimal matching criterion of current observation signal to guarantee the solution can minimize matching error.(2) selecting L optimal atoms in batch to guarantee higher fault-tolerant ability. For sparse signals gOMP algorithm using current best observation criteria improves its MERP by3.16%～10.80%, OOMP algorithm selecting L optimal atoms in batch exactly reconstruct signals with number of iterations which is one third of that of OOMP, and improves MERP by31.26%～37.54%.
Furthermore, this thesis proves the sufficient condition of GOOMP algorithm exactly reconstructing original signals in K iterations is:On the premise of the independence of correct atoms, unselected atoms are projected into complement space spanned by selected atoms and then normalized, then each projected and normalized error atom is projected into space spanned by all projected and normalized error atoms, each l1norm of projecting coefficient vector is smaller than1.
4) Based on residual matching and atom coefficients thresholding, an improved reconstruction algorithm for blind sparsity signal is proposed. Firstly, the relationship between atom selecting threshold and sparsity in backtracking-based adaptive orthogonal matching pursuit (BAOMP) algorithm is analyzed. Then aimed at shortcomings of steep rise of residual norms and estimated signal errors aroused by selected error atoms in later iterations of BAOMP, an improved algorithm IBAOMP is proposed. When the energy of estimated signal is closed to that of target signal, by increasing atom selecting threshold according to80-20principle to control error atom selections and capacity of candidate atom set, IBAOMP effectively eliminate the phenomenon of periodic steep rise of errors.
In conclusion, the researches in this thesis on atom selecting methods and blind sparsity signal reconstruction in CS give the following results:
LABOMP with the interval backtracking mechanism improves the probability of hitting correct atoms, by using regular atom selection method, RLABOMP algorithm significantly improves convergence speed of LABOMP, and GOOMP algorithm can not only improve MERP but also reduce the computation time. Besides, for blind sparsity signals, IBAOMP algorithm avoids reconstruction blindness when sparsity information is missing by threshold adjustment mechanism, and ensures the estimated signal gradually approaching target signal.
本文的主要贡献包括：
1)为提高原子选择准确度,提出了前向预测与回溯结合的正交匹配追踪(LABOMP)算法。在迭代初期用前向预测策略串行选择候选原子,在迭代后期通过触发阈值启动一个间隔回溯机制,剔除错选原子。实验证明,用LABOMP算法重建稀疏信号和可压缩语音信号,平均精确重建概率(MERP)比只使用前向预测策略的LAOMP算法平均提高了12.5%～31.3%,已选原子的全局最优解逼近能力明显提高。
2)为提高LABOMP算法的收敛速度,提出正则化LABOMP算法(RLABOMP)。鉴于CS重建具有不适定的多个可用解,而且单次迭代的残差匹配有异于全局重建最优化匹配,保留多个性能相当的备选解是合理的。因此我们以最佳原子集内候选原子的公平选入为正则约束,保证同时选入性能相近的最优原子,以便在提高容错性的同时减少迭代次数。实验证明,对稀疏信号和可压缩语音信号,RLABOMP选取K个原子所需迭代次数为LABOMP的42.90%～54.28%,为LAOMP的62.85%～71.57%,但MERP相对LABOMP降低9.27%～25.12%,相对LAOMP提高1.91～10.80%,明显优于LAOMP算法,与LABOMP算法比则取得迭代效率与重建性能的折中。
3)为同时提高原子选择准确度和计算速度,提出广义最佳正交匹配追踪(GOOMP)算法。其改进思路是：(1)采用当前观测信号最佳的匹配准则,保证所求解能使当前观测信号的匹配误差最小。(2)成批选入L个最优原子,保证选入原子集具备较高的算法容错能力。对稀疏信号,使用当前最佳观测准则的gOMP算法,其MERP提高了3.16%～10.8%,使用成批选入L个最优原子的OOMP算法完成精确重建所需迭代次数为OOMP算法的1/3,MERP提高31.26%～37.54%。
此外,本文证明了GOOMP算法在K次迭代内完全重建原始信号的充分条件是：在保证正确原子不相关的前提下,后续待选正确原子与错误原子在己选原子张成的补空间中各自投影并归一化,投影并归一化后错误原子在所有投影并归一化后正确原子构成空间上的投影系数向量l1范数小于1。
4)基于残差匹配和原子系数阈值判断,提出改进的盲稀疏度信号重构方案。首先分析了回溯型自适应正交匹配追踪(BAOMP)算法的原子选入阈值参数与稀疏度的关系,然后针对BAOMP算法迭代后期误选原子,引起的残差模值陡增,估计信号误差周期骤升的缺点,提出了改进的算法IBAOMP。当估计的观测信号能量与目标能量接近时,IBAOMP算法按80-20准则提高原子选入阈值,控制错误原子的选入和候选原子集的容量,非常有效地消除了误差周期性突增的现象。
总的来说,本文原子选择方法和盲稀疏度压缩重建的研究工作给出了如下结果：
具有间隔回溯机制的LABOMP算法有利于提高击中正确原子的概率,RLABOMP算法由于使用正则化原子选择方法可以明显提高LABOMP算法的收敛速度,而GOOMP算法既可以提高MERP又能减少计算时间。另一方面,对于盲稀疏度信号重建IBAOMP算法通过阈值调节机制,避免了稀疏度信息缺失时的重建盲目性,保证了估计信号逐步逼近目标信号。
Compressed sensing can sample signals at a rate far below Nyquist rate, and it's a new tool in signal processing. In the l0norm based algorithms, the atom matching pursuit mainly applied greedy search, their key techniques are (1) optimal atom selection and matching criteria;(2) accurate estimate of signal sparsity. Atom selection strategies can be divided into single atom hit and a batch selection of atoms, while criteria include residual matching, current best observation and global error minimization. In addition to the atom selection related to efficiency and fault tolerance, matching criteria determine directions of atom search which approaches an optimal solution. This thesis mainly focuses on efficient atom selection and robust reconstruction of sparsity unknown signals.
Main contributions include:
1) To improve the accuracy of atom selection, a look-ahead and backtracking orthogonal matching pursuit (LABOMP) algorithm is proposed. In early iterations a set of candidate atoms is chosen by look ahead verification, on the contrary, an interval backtracking mechanism is triggered by a threshold in later iterations to prune error atoms. Experiments show that, for sparse signals and compressive voice, the mean exact reconstruction probability (MERP) of LABOMP is12.5%～31.3%higher than that of LAOMP, where the latter uses only the look-ahead strategy, which indicates the probability of finding a global solution is improved.
2) For rapid convergence of LABOMP, a regularized LABOMP (RLABOMP) algorithm is proposed. Note that the CS reconstruction is an ill-posed solution and the difference between criterion of residual matching and criterion of global matching, multiple'good' atoms could be regarded as equivalent candidate. Thus an equal opportunity constraint is introduced to regularize the recruitment of candidate atoms to include the atoms with similar performances, to promote fault tolerance and accelerate the convergence. Experiments show that the number of iterations of RLABOMP for choosing K atoms is42.90%～54.28%of that of LABOMP and62.85%～71.57%of that of LAOMP. However, MERP of RLABOMP is9.27%～25.12%lower than that of LABOMP and1.91%～10.80%higher than LAOMP. Obviously RLABOMP is better than LAOMP and balances the efficiency and the reconstruction performance comparing to LABOMP.
3) To increase the accuracy and speed of atom selection, a generalized optimal orthogonal matching pursuit (GOOMP) algorithm is proposed. Improvements are (1) using optimal matching criterion of current observation signal to guarantee the solution can minimize matching error.(2) selecting L optimal atoms in batch to guarantee higher fault-tolerant ability. For sparse signals gOMP algorithm using current best observation criteria improves its MERP by3.16%～10.80%, OOMP algorithm selecting L optimal atoms in batch exactly reconstruct signals with number of iterations which is one third of that of OOMP, and improves MERP by31.26%～37.54%.
Furthermore, this thesis proves the sufficient condition of GOOMP algorithm exactly reconstructing original signals in K iterations is:On the premise of the independence of correct atoms, unselected atoms are projected into complement space spanned by selected atoms and then normalized, then each projected and normalized error atom is projected into space spanned by all projected and normalized error atoms, each l1norm of projecting coefficient vector is smaller than1.
4) Based on residual matching and atom coefficients thresholding, an improved reconstruction algorithm for blind sparsity signal is proposed. Firstly, the relationship between atom selecting threshold and sparsity in backtracking-based adaptive orthogonal matching pursuit (BAOMP) algorithm is analyzed. Then aimed at shortcomings of steep rise of residual norms and estimated signal errors aroused by selected error atoms in later iterations of BAOMP, an improved algorithm IBAOMP is proposed. When the energy of estimated signal is closed to that of target signal, by increasing atom selecting threshold according to80-20principle to control error atom selections and capacity of candidate atom set, IBAOMP effectively eliminate the phenomenon of periodic steep rise of errors.
In conclusion, the researches in this thesis on atom selecting methods and blind sparsity signal reconstruction in CS give the following results:
LABOMP with the interval backtracking mechanism improves the probability of hitting correct atoms, by using regular atom selection method, RLABOMP algorithm significantly improves convergence speed of LABOMP, and GOOMP algorithm can not only improve MERP but also reduce the computation time. Besides, for blind sparsity signals, IBAOMP algorithm avoids reconstruction blindness when sparsity information is missing by threshold adjustment mechanism, and ensures the estimated signal gradually approaching target signal.
引文
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