Shapes From Pixels

TL;DR

This paper addresses the challenge of reconstructing shape images with smooth boundaries from discrete samples by formulating and relaxing a shape perimeter minimization problem, introducing a reducibility condition linked to sampling density.

Contribution

It proposes a novel shape reconstruction method based on total variation minimization with a reducibility condition ensuring equivalence to the original shape perimeter problem.

Findings

The relaxed total variation approach effectively reconstructs shapes from samples.

Reducibility condition relates to sampling density, ensuring accurate reconstruction.

Numerical experiments demonstrate the method's practical effectiveness.

Abstract

Continuous-domain visual signals are usually captured as discrete (digital) images. This operation is not invertible in general, in the sense that the continuous-domain signal cannot be exactly reconstructed based on the discrete image, unless it satisfies certain constraints (\emph{e.g.}, bandlimitedness). In this paper, we study the problem of recovering shape images with smooth boundaries from a set of samples. Thus, the reconstructed image is constrained to regenerate the same samples (consistency), as well as forming a shape (bilevel) image. We initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We also introduce a requirement (called reducibility) that guarantees equivalence between the two problems. We illustrate that the reducibility property effectively sets a requirement on the minimum sampling density. One can draw analogy between the reducibility property and the so-called restricted isometry property (RIP) in compressed sensing which establishes the equivalence of the $\ell_0$ minimization with the relaxed $\ell_1$ minimization. We also evaluate the performance of the relaxed alternative in various numerical experiments.