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