On Cartesian line sampling with anisotropic total variation regularization

This paper considers the use of the anisotropic total variation seminorm to recover a two dimensional vector x ∈ CN×N from its partial Fourier coefficients, sampled along Cartesian lines. We prove that if (xk,j − xk−1,j)k,j has at most s1 nonzero coefficients in each column and (xk,j − xk,j−1)k,j has at most s2 nonzero coefficients in each row, then, up to multiplication by log factors, one can exactly recover x by sampling along s1 horizontal lines of its Fourier coefficients and along s2 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, the ambient dimension of x.