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Restriction and interpolation operators for digital images and their boundaries

Janosch Rieger, Kyria Wawryk

TL;DR

The paper develops a coherent, operator-based framework to transfer boundary representations of digital images across resolutions without full-image data, enabling effective multiresolution processing and boundary reduction in computational dynamics and control contexts. It introduces inner/outer boundary layer geometry, bijective boundary-pair mappings via the trace operator, and restriction/ interpolation operators $R$ and $I$ with strong approximation and connectivity properties, plus their lifted counterparts $oldsymbol{partial R}$ and $oldsymbol{partial I}$ acting on boundary spaces. It further analyzes parity-driven exactness versus approximation behavior of the operators and provides practical algorithms to compute lifted boundary mappings directly from boundary data, validated by a computational example. Overall, the framework enables boundary-centric multiresolution computations with guarantees on boundary complexity, facilitating scalable invariant-set and reachability analyses.

Abstract

The aim of this paper is to provide a coherent framework for transforming boundary pairs of digital images from one resolution to another without knowledge of the full images. It is intended to facilitate the simultaneous usage of multiresolution processing and boundary reduction, primarily for algorithms in computational dynamics and computational control theory.

Restriction and interpolation operators for digital images and their boundaries

TL;DR

The paper develops a coherent, operator-based framework to transfer boundary representations of digital images across resolutions without full-image data, enabling effective multiresolution processing and boundary reduction in computational dynamics and control contexts. It introduces inner/outer boundary layer geometry, bijective boundary-pair mappings via the trace operator, and restriction/ interpolation operators and with strong approximation and connectivity properties, plus their lifted counterparts and acting on boundary spaces. It further analyzes parity-driven exactness versus approximation behavior of the operators and provides practical algorithms to compute lifted boundary mappings directly from boundary data, validated by a computational example. Overall, the framework enables boundary-centric multiresolution computations with guarantees on boundary complexity, facilitating scalable invariant-set and reachability analyses.

Abstract

The aim of this paper is to provide a coherent framework for transforming boundary pairs of digital images from one resolution to another without knowledge of the full images. It is intended to facilitate the simultaneous usage of multiresolution processing and boundary reduction, primarily for algorithms in computational dynamics and computational control theory.
Paper Structure (9 sections, 23 theorems, 171 equations, 5 figures, 2 algorithms)

This paper contains 9 sections, 23 theorems, 171 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and outline
  3. Definitions
  4. Outline
  5. Straight paths
  6. Geometry of digital images
  7. Boundary pairs of digital images
  8. Restriction and interpolation operators
  9. Algorithms evaluating lifted operators

Key Result

Lemma 5

The specific rounding function has the following properties:

Figures (5)

  • Figure 1: Anatomy of a digital image from Definition \ref{['eq:bdrylayers']}.
  • Figure 2: Examples of pairs $(D_0,D_1)\in S_\rho^+\times S_\rho^+$ satisfying axioms \ref{['bdry:axiom:0']} through \ref{['bdry:axiom:3']}, but not axiom \ref{['bdry:axiom:4']} from Definition \ref{['def:boundary:pairs']}.
  • Figure 3: Illustration of covering property encoded in \ref{['def:V']} from Definition \ref{['def:collections']} with $\rho=\check\rho$ and $\rho'=\hat{\rho}=2\check\rho$: Set from (d) covers set from (b). White lines in (b) and (d) are visual aids only.
  • Figure 4: Overview of spaces and mappings.
  • Figure 5: Small computational example with layout as in commutative diagram in Figure \ref{['fig:overview']}. Pair (b) computed from (a) by Algorithm \ref{['Alg:refine']}, pair (c) computed from (b) by Algorithm \ref{['Alg:coarsen']}. Also illustrates stability properties \ref{['new:boundary:in:old:boundary']} and \ref{['I:new:boundary:in:old:boundary']}, as well as the effect from equation \ref{['I:R:almost:id']}.

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 40 more