This paper considers the use of the anisotropic total variation seminorm to recover a two dimensional vector @math from its partial Fourier coefficients, sampled along Cartesian lines. We prove that if @math has at most @math nonzero coefficients in each column and @math has at most @math nonzero coefficients in each row, then, up to multiplication by @math factors, one can exactly recover @math by sampling along @math horizontal lines of its Fourier coefficients and along @math vertical lines of its Fourier coefficients. Finally, unlike standard compressed sensing estimates, the @math factors involved are dependent on the separation distance between the nonzero entries in each row column of the gradient of @math and not on @math , the ambient dimension of @math .
Compressed Sensing (CS) is an appealing framework for applications such as Magnetic Resonance Imaging (MRI). However, up-to-date, the sensing schemes suggested by CS theories are made of random isolated measurements, which are usually incompatible with the physics of acquisition. To reflect the physical constraints of the imaging device, we introduce the notion of blocks of measurements: the sensing scheme is not a set of isolated measurements anymore, but a set of groups of measurements which may represent any arbitrary shape (parallel or radial lines for instance). Structured acquisition with blocks of measurements are easy to implement, and provide good reconstruction results in practice. However, very few results exist on the theoretical guarantees of CS reconstructions in this setting. In this paper, we derive new CS results for structured acquisitions and signals satisfying a prior structured sparsity. The obtained results provide a recovery probability of sparse vectors that explicitly depends on their support. Our results are thus support-dependent and offer the possibility for flexible assumptions on the sparsity structure. Moreover, the results are drawing-dependent, since we highlight an explicit dependency between the probability of reconstructing a sparse vector and the way of choosing the blocks of measurements. Numerical simulations show that the proposed theory is faithful to experimental observations.