Compressive video recovery with upper and lower bound constraints

The recovery of sparse images from noisy, blurry, and potentially low-dimensional observations can be accomplished by solving an optimization problem that minimizes the least-squares error in data fidelity with a sparsity-promoting regularization term (the so-called ℓ₂ - ℓ₁ minimization problem). This paper focuses on the reconstruction of a video sequence of images where known pixel-intensity bounds exist at each video frame. It has been established that the ℓ₂ - ℓ₁ minimization problem can be solved efficiently using gradient projection, which was recently extended to solve general bound-constrained ℓ₂ - ℓ₁ minimization problems. Furthermore, the video reconstruction can be made more efficient by exploiting similarities between consecutive frames. In this paper, we propose a method for reconstructing a video sequence that takes advantage of the inter-frame correlations while constraining the solution to satisfy known a priori bounds, offering a higher potential for increasingly accurate reconstructions. To demonstrate the effectiveness of this approach, we have included the results of our numerical experiments.