AI 中文总结
研究比较剪切波与小波在傅里叶压缩感知中的性能，探讨稀疏性差异对重建所需傅里叶测量数量的影响，通过理论分析和数值实验发现，虽剪切波逼近率高，但实际样本复杂度与小波相当。
AI 中文摘要
本文探讨压缩感知中信号恢复的测量要求，比较剪切波框架与传统小波系统的性能。剪切波等方向表示系统以能稀疏表示具有各向异性特征的图像而闻名，有助于高效非线性逼近。核心问题是这种稀疏性差异能否使成功重建所需的傅里叶测量数量成比例减少。理论上，研究将基于框架的标准恢复结果应用于剪切波系统时遇到的障碍，包括傅里叶测量与锥适应剪切波之间的局部相干性衰减慢，管理稀疏系统冗余依赖评估定位因子且下限估计可能指数级小。数值实验表明，虽剪切波能以更少系数准确表示图像，但稀疏性优势未带来所需傅里叶样本的成比例减少。结论是，尽管剪切波非线性逼近率更高，但在傅里叶测量下的实际样本复杂度与传统小波相当。
英文摘要
This paper explores the measurement requirements for signal recovery in compressed sensing, comparing the performance of shearlet frames with traditional wavelet systems. Directional representation systems such as shearlets are known for their ability to sparsely represent images with anisotropic features, which allows for efficient nonlinear approximations. The central question we address is whether this difference in sparsity allows for a proportional reduction in the number of Fourier measurements needed for successful reconstruction. On the theoretical front, we study the obstacles encountered when trying to apply standard frame-based recovery results to shearlet systems. First, we show that the (optimal) local coherence between Fourier measurements and cone-adapted shearlets decays more slowly than the corresponding local coherence for wavelets. Second, managing the sparsifying system's redundancy relies on evaluating a localization factor, which requires lower frame bound estimates that can become exponentially small. Thus, even under optimal theoretical conditions, the number of samples required for shearlets scales quadratically with sparsity, which offers no substantial theoretical reduction over the standard wavelet benchmark. These theoretical limitations are assessed through a series of numerical experiments on a dataset of piecewise smooth images. While empirical observations confirm that shearlets can accurately represent these images using fewer coefficients than wavelets, phase diagrams indicate that this advantage in sparsity does not yield a proportional reduction in required Fourier samples. Ultimately, we conclude that despite the superior nonlinear approximation rates of shearlets, their practical sample complexity in compressed sensing scenarios with subsampled Fourier measurements remains comparable to that of traditional wavelets.
Comments38 pages, 3 figures
