On Cartesian line sampling with anisotropic total variation regularization

TL;DR

This paper demonstrates that anisotropic total variation regularization enables exact recovery of 2D signals from Fourier samples along Cartesian lines, with sampling complexity depending on sparsity and separation rather than ambient dimension.

Contribution

It provides theoretical guarantees for exact recovery using anisotropic TV from Fourier samples along Cartesian lines, with improved dependence on separation rather than ambient dimension.

Findings

Exact recovery is possible with sampling along s1 horizontal and s2 vertical lines.

Recovery guarantees depend on sparsity and separation, not on the full dimension N^2.

Log factors in the sampling bounds depend on separation distances.

Abstract

This paper considers the use of the anisotropic total variation seminorm to recover a two dimensional vector $x\in \mathbb{C}^{N\times N}$ from its partial Fourier coefficients, sampled along Cartesian lines. We prove that if $(x_{k,j} - x_{k-1,j})_{k,j}$ has at most $s_1$ nonzero coefficients in each column and $(x_{k,j} - x_{k,j-1})_{k,j}$ has at most $s_2$ nonzero coefficients in each row, then, up to multiplication by $\log$ factors, one can exactly recover $x$ by sampling along $s_1$ horizontal lines of its Fourier coefficients and along $s_2$ vertical lines of its Fourier coefficients. Finally, unlike standard compressed sensing estimates, the $\log$ factors involved are dependent on the separation distance between the nonzero entries in each row/column of the gradient of $x$ and not on $N^2$, the ambient dimension of $x$.