  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 $x$.
