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

by   Ben Adcock, et al.

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 α-Hölder functions. In this setting, we show the suboptimality of random Gaussian sampling, exhibit a provably optimal sampling strategy and prove that Fourier sampling outperforms random Gaussian sampling when the Hölder exponent α is large enough. This resolves the claimed paradox, and provides a clear theoretical justification for the practical success of compressed sensing techniques in imaging problems.


Non-uniform recovery guarantees for binary measurements and infinite-dimensional compressed sensing

Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclea...

On oracle-type local recovery guarantees in compressed sensing

We present improved sampling complexity bounds for stable and robust spa...

Compressed sensing photoacoustic tomography reduces to compressed sensing for undersampled Fourier measurements

Photoacoustic tomography (PAT) is an emerging imaging modality that aims...

Predictive refinement methodology for compressed sensing imaging

The weak-ℓ^p norm can be used to define a measure s of sparsity. When we...

Learning Sub-Sampling and Signal Recovery with Applications in Ultrasound Imaging

Limitations on bandwidth and power consumption impose strict bounds on d...

Theoretical links between universal and Bayesian compressed sensing algorithms

Quantized maximum a posteriori (Q-MAP) is a recently-proposed Bayesian c...