Shapes From Pixels
Mitra Fatemi, Arash Amini, Loic Baboulaz, and Martin Vetterli

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…
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