Sparse Linear Operator identification without sparse regularization? Applications to mixed pixel problem in Time-of-Flight/Range imaging

In this paper, we consider the problem of Sparse Linear Operator identification which is also linked with the topic of Sparse Deconvolution. In its abstract form, the problem can be stated as follows: Given a well behaved probing function, is it possible to identify a Sparse Linear Operator from its response to the function? We present a constructive solution to this problem. Furthermore, our approach is devoid of any sparsity inducing penalty term and explores the idea of parametric modeling. Consequently, our algorithm is non-iterative by design and circumvents tuning of any regularization parameter. Our approach is computationally efficient when compared the ℓ₀/ℓ₁-norm regularized counterparts. Our work addresses a problem of industrial significance: decomposition of mixed-pixels in Time-of-Flight/Range imaging. In this case, each pixel records range measurements from multiple contributing depths and the goal is to isolate each depth. Practical experiments corroborate our theoretical set-up and establish the efficiency of our approach, that is, speed-up in processing with lesser mean squared error. We also derive Cramér-Rao Bounds for performance characterization.