基于主成分变换的图像稀疏度估计方法

摘要

In compressive sensing, signal sparsity is an important parameter which influences the number of data sampling in reconstruction process and the quantity of the reconstructed result. But in practice, undersampled and oversampled phenomenon will occur because of the unknown sparsity, which may lose the advantages of compressive sensing. So how to determine the image sparsity quickly and accuratly is significant in the compressive sensing process. In this paper, we calculate the image sparsity based on the data acquired during compressive sensing recontruction projection which sparses the origin image in wavelets domain, but we find that its procession is complex, and the final results are seriously influenced by wavelet basis function and the transform scales. We then introduce the principle component analysis (PCA) theory combined with compressive sensing, and establish a linear relationship between image sparsity and coefficient founction variance based on the assumption that PCA is of approximately normal distribution. Multiple sets of experiment data verify the correctness of the linear relationship mentioned above. Through previous analysis and simulation, the sparsity estimation based on PCA has an important practical value for compressive sensing study.%压缩感知理论基于信号的稀疏性和可压缩性，突破传统Nyquist采样频率的限制，以较低的数据量对信号进行采样和高概率重构。在压缩感知理论中，信号的稀疏度确定了稀疏采样的最低数据量，是验证采样方法及重构方法优劣的重要参数。在实际研究过程中，图像稀疏度通常未知，这就可能导致过采样或欠采样的情况，从而无法验证采样方法及重构方法的优劣。因此，快速而客观地估计图像的稀疏度对于压缩感知理论研究来说意义重大。本文分析了基于小波变换的图像稀疏化表示方法，通过遍历采样和重构得到基于小波变换方法的图像稀疏度，但过程复杂，而且结果的准确性依赖于小波基和变换尺度的选择。本文通过压缩感知理论对主成分变换进行阐述，在基于主成分变换系数近似为正态函数的假设下，建立了图像稀疏度与系数函数方差间的线性关系，并通过多组图像数据进行仿真验证，结果表明线性关系的正确性。通过分析和仿真可以看出，基于主成分变换的稀疏度估计方法比小波变换简单、快速、客观，对压缩感知理论研究有重要的应用价值。