A Theory of Coprime Blurred Pairs

Abstract

We present a new Coprime Blurred Pair (CBP) theory that may benefit a number of computer vision applications. A CBP is constructed by blurring the same latent image with two unknown kernels, where the two kernels are co-prime when mapped to bivariate polynomials under the z-transform. We first show that the blurred contents in a CBP are difficult to restore using conventional blind deconvolution methods based on sparsity priors. We therefore introduce a new coprime prior for recovering the latent image in a CBP. Our solution maps the CBP to bivariate polynomials and sample them on the unit circle in both dimension. We show that coprimality can be derived in terms of the rank of the Bézout Matrix [2] formed by the sampled polynomials and we present an efficient algorithm to factor the Bézout Matrix for recovering the latent image. Finally, we discuss applications of the CBP theory in privacy-preserving surveillance and motion deblurring, as well as physical implementations of CBPs using flutter shutter cameras.