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Deconvolution with a Box

Pedro Felzenszwalb

TL;DR

A direct proof that improves on the reconstruction bound that follows from previous results is given and it is shown the bound is tight and matches an information theoretic limit.

Abstract

Deconvolution with a box (square wave) is a key operation for super-resolution with pixel-shift cameras. In general convolution with a box is not invertible. However, we can obtain perfect reconstructions of sparse signals using convex optimization. We give a direct proof that improves on the reconstruction bound that follows from previous results. We also show our bound is tight and matches an information theoretic limit.

Deconvolution with a Box

TL;DR

A direct proof that improves on the reconstruction bound that follows from previous results is given and it is shown the bound is tight and matches an information theoretic limit.

Abstract

Deconvolution with a box (square wave) is a key operation for super-resolution with pixel-shift cameras. In general convolution with a box is not invertible. However, we can obtain perfect reconstructions of sparse signals using convex optimization. We give a direct proof that improves on the reconstruction bound that follows from previous results. We also show our bound is tight and matches an information theoretic limit.
Paper Structure (3 theorems, 8 equations, 1 figure)

This paper contains 3 theorems, 8 equations, 1 figure.

Key Result

Proposition 1

$z \in \ker(A)$ if and only if $z_j = z_{j \bmod k}$ and $\sum_{j=1}^k z_j = 0$.

Figures (1)

  • Figure 1: (a) 256x256 high-resolution target; (b) "valid" part of the convolution of the target with a box of width 20, obtained by "scanning" the target using a low-resolution sensor (each sensor pixel is 20 times the size of the target pixel size); (c) reconstruction using convex optimization.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof