A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions on the number of measurements. However, in the context of image reconstruction, it has been extensively documented that sampling strategies based on Fourier measurements outperform this purportedly optimal approach. Motivated by this seeming paradox, we investigate the problem of optimal sampling for compressed sensing. Rigorously combining the theories of wavelet approximation and infinite-dimensional compressed sensing, our analysis leads to new error bounds in terms of the total number of measurements m for the approximation of piecewise \(\alpha \)-Hölder functions. Our theoretical findings suggest that Fourier sampling outperforms random Gaussian sampling when the Hölder exponent \(\alpha \) is large enough. Moreover, we establish a provably optimal sampling strategy. This work is an important first step towards the resolution of the claimed paradox and provides a clear theoretical justification for the practical success of compressed sensing techniques in imaging problems.

压缩感测的一个标志性结果是，高斯随机采样可在最佳条件下根据测量次数实现稀疏矢量的稳定且稳健的恢复。但是，在图像重建的背景下，已有大量文献证明，基于傅立叶测量的采样策略要优于这种据称的最佳方法。由于这种看似矛盾的原因，我们研究了压缩感测的最佳采样问题。严格地将小波逼近和无限维压缩感测的理论结合起来，我们的分析导致了新的误差界，即针对分段\（\ alpha \）逼近的测量总数m-Hölder功能。我们的理论发现表明，当Hölder指数\（\ alpha \）足够大时，傅立叶采样优于随机高斯采样。此外，我们建立了可证明的最佳抽样策略。这项工作是朝着解决所主张的悖论迈出的重要第一步，并为压缩传感技术在成像问题中的实际成功提供了明确的理论依据。