压缩感知信号重建算法研究和应用

发布时间：2018-09-03 06:31

【摘要】：传统的信号采样是以Nyquist定理为理论指导,并要求采样速率达到带宽的两倍以上。随着科技的发展,在信号采样的实际应用中,带宽变得越来越大,传统的理论采样已经不能满足人们的要求。一个新的采样理论应运而生,即压缩感知理论。它实现了信号采样过程中的采样和压缩的并行操作,将二者合二为一,不需要先得到海量采样数据,节省了大量时间和存储空间。目前,压缩感知理论已成为国际上研究的热点,它在很多领域的实用价值非常高,应用前景十分广阔。本文对压缩感知理论进行了系统的介绍,就其中的关键环节:信号的稀疏表示、测量矩阵设计及重建算法作了详细描述,着重对重建算法中典型的几种算法作了介绍,并从实验仿真中比较了它们的性能。而基于光滑0l范数最小化问题的重建算法在相同或更好精度情况下比其他算法快2至3倍,本文针对光滑0l范数最小化算法作了以下研究工作:SL0算法选用高斯函数作为近似估计0l范数的函数,本文提出了用复合三角函数近似估计0l范数,函数图像显示了其比已有的高斯函数更陡峭,因此逼近性能更加优良。针对最速下降法的搜索路径为锯齿状和牛顿法在远离最优解时计算较慢的缺点,本文采用最速下降法和牛顿法相结合的方法,对优化问题迭代过程中前数次迭代用最速下降法,之后用阻尼牛顿法。并通过数值实验表明了改进算法的有效性,且与其他算法相比,本算法在压缩图像重构精度上有明显提高。基于光滑0l范数最小化的NSL0重建算法,针对该算法中用到的阻尼牛顿法在远离最优解时收敛速度慢的缺点,本文采用前数次迭代用最速下降法,之后用阻尼牛顿法。且在用阻尼牛顿法迭代求解中设计了有效迭代步长,第一次迭代步长由一维精确搜索得到。通过设计迭代步长的更新方案,使得迭代中每一步的计算更为有效,在保证重构精度的同时能够提升算法的收敛速度。并且在改进算法中加入了支撑集,部分支撑集是用前次迭代得到的稀疏向量来估计的,而后建立了基于支撑集的近似0l范数最小化问题。通过人工数据实验和机器图像压缩重构实验,表明了改进算法的有效性。
[Abstract]:The traditional signal sampling is guided by the Nyquist theorem, and the sampling rate is required to be more than twice the bandwidth. With the development of science and technology, in the practical application of signal sampling, the bandwidth becomes larger and larger, and the traditional theoretical sampling can no longer meet the needs of people. A new sampling theory emerged as the times require, that is, the theory of compressed perception. It realizes the parallel operation of sampling and compressing in the process of signal sampling. It combines the two operations and saves a lot of time and storage space. At present, the theory of compressed perception has become the focus of international research, it has a very high practical value in many fields, and has a very broad application prospects. In this paper, the theory of compression perception is systematically introduced, and the key links are described in detail, such as the sparse representation of signal, the design of measurement matrix and the reconstruction algorithm, with emphasis on several typical algorithms in the reconstruction algorithm. Their performances are compared by simulation. The reconstruction algorithm based on smooth 0l norm minimization problem is 2 to 3 times faster than other algorithms under the same or better precision. In this paper, the following research work is done for the smooth 0l norm minimization algorithm: SL0 algorithm chooses Gao Si function as the function of approximate estimation of 0l norm. In this paper, a compound trigonometric function is proposed to approximate estimate 0l norm. The function image shows that it is steeper than the existing Gao Si function, so the approximation performance is better. Because the search path of the steepest descent method is sawtooth and Newton's method is slow to calculate far from the optimal solution, this paper adopts the method of the combination of the steepest descent method and Newton's method, and uses the steepest descent method for the first several iterations in the iterative process of the optimization problem. Then the damped Newton method is used. Numerical experiments show the effectiveness of the improved algorithm, and compared with other algorithms, the proposed algorithm can significantly improve the accuracy of image reconstruction. Based on the smooth 0l norm minimization NSL0 reconstruction algorithm, the damped Newton method used in this algorithm has the disadvantage of slow convergence rate when it is far from the optimal solution. In this paper, the first iteration uses the steepest descent method, and then the damped Newton method is used. The effective iteration step size is designed in the iterative solution with damped Newton method. The first iteration step size is obtained by one dimensional exact search. By designing the update scheme of iteration step size, the calculation of each step in the iteration is more efficient, and the convergence speed of the algorithm can be improved while the reconstruction accuracy is guaranteed. The support set is added to the improved algorithm, and the partial support set is estimated by the sparse vector obtained from the previous iteration, and then the approximate 0l norm minimization problem based on the support set is established. The effectiveness of the improved algorithm is demonstrated by artificial data experiments and machine image compression and reconstruction experiments.
【学位授予单位】：西安电子科技大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TN911.7

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