Joint Estimation of Dictionary and Image from Compressive Samples

Abstract

Effective compressed sensing of images relies on the prior knowledge of a well-suited dictionary for sparse representation of the target image. In the absence of such knowledge, which is a typical scenario in real-world applications, the following question arises: Would it be advantageous to take an off-the-shelf overcomplete dictionary and fine-tune the dictionary with respect to (the observed part of) the image? A primary obstacle in this approach is overfitting, i.e., the loss in model generalization to the whole image. In this paper, we establish that local dictionary optimization using the compressive samples reduces the image recovery error—relative to the off-the-shelf recovery—with an overwhelming probability that depends on the sampling matrix. We present joint estimation of dictionary and image (JEDI), an iterative algorithm for dictionary fine-tuning from compressive samples and analyze its performance for image recovery. Our algorithmic analysis is supplemented with numerical simulations under different random sampling patterns and off-the-shelf dictionary initializations.