Kernel compressive sensing

Compressive sensing allows us to recover signals that are linearly sparse in some basis from a smaller number of measurements than traditionally required. However, it has been shown that many classes of images or video can be more efficiently modeled as lying on a nonlinear manifold, and hence described as a non-linear function of a few underlying parameters. Recently, there has been growing interest in using these manifold models to reduce the required number of compressive sensing measurements. However, the complexity of manifold models has been an obstacle to their use in efficient data acquisition. In this paper, we introduce a new algorithm for applying manifold models in compressive sensing using kernel methods. Our proposed algorithm, kernel compressive sensing (KCS), is the kernel version of classical compressive sensing. It uses dictionary learning in the feature space to build an efficient model for one or more signal manifolds. It then is able to formulate the problem of recovering the signal's coordinates in the manifold representation as an underdetermined linear inverse problem as in traditional compressive sensing. Standard compressive sensing recovery methods can thus be used to recover these coordinates, avoiding additional computational complexity. We present experimental results demonstrating the efficiency and efficacy of this algorithm in manifold-based compressive sensing.