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A signature result in compressed sensing is that Gaussian random samplingachieves stable and robust recovery of sparse vectors under optimal conditionson the number of measurements. However, in the context of image reconstruction,it has been extensively documented that sampling strategies based on Fouriermeasurements outperform this purportedly optimal approach. Motivated by thisseeming paradox, we investigate the problem of optimal sampling for compressedsensing. Rigorously combining the theories of wavelet approximation andinfinite-dimensional compressed sensing, our analysis leads to new error boundsin terms of the total number of measurements $m$ for the approximation ofpiecewise $\alpha$-H\"{o}lder functions. Our theoretical findings suggest thatFourier sampling outperforms random Gaussian sampling when the H\"olderexponent $\alpha$ is large enough. Moreover, we establish a provably optimalsampling strategy. This work is an important first step towards the resolutionof the claimed paradox, and provides a clear theoretical justification for thepractical success of compressed sensing techniques in imaging problems.

Do log factors matter? On optimal wavelet approximation and the foundations of compressed sensing