Improved Recovery Guarantees and Sampling Strategies for TV Minimization in Compressive Imaging

In this paper, we consider the use of total variation (TV) minimization for compressive imaging, that is, image reconstruction from subsampled measurements. Focusing on two important imaging modalities---namely, Fourier and structured binary via Walsh--Hadamard transform---we derive uniform recovery guarantees asserting stable robust arbitrary random sampling strategies. Using this, then a class strategies which are theoretically near-optimal approximately gradient-sparse images. For sampling, show such an $m \gtrsim_d s \cdot \log^2(s) \log^4(N)$ measurements, in $d \geq 1$ dimensions. When = 2$, improves current state-of-the-art result by factor $\log(s) \log(N)$. It also extends it to dimensions 2$. Walsh prove \log^2(N/s) \log^3(N) $ measurements suffice 2$ To best our knowledge, is first guarantee with TV minimization.