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The Geometric Refinement Transform: A Novel Uncountably Infinite Transform Space

Zachary Mullaghy

TL;DR

The paper tackles the limitation of classical transforms in handling nonconvex and geometrically complex domains by introducing a geometric refinement transform based on hierarchical Voronoi refinements parameterized by refinement multiplicity $N$, dispersion $\delta$, and rotation. It develops a rigorous framework establishing completeness, uniqueness, invertibility, closure, and stability via measure-weighted frame bounds and derives a natural inner product from the refinement geometry, with the transform densely representing $L^2(\Omega,\mu)$ via a BV-to-L^2 density argument. The authors connect the new transform to existing constructs, showing how Radon-like behavior and wavelet-like decompositions arise as special cases, and analyze artifacts through the Radon Zone concept, proposing averaging over angular configurations to ensure faithful recovery in higher dimensions. They also provide an entropy-based interpretation of the coefficient spectrum, suggesting a variational principle for energy-minimizing dynamics and highlighting broad potential applications in PDEs, imaging, physics, and beyond, while noting avenues for numerical and non-Euclidean development.

Abstract

This work introduces a novel and general class of continuous transforms based on hierarchical Voronoi based refinement schemes. The resulting transform space generalizes classical approaches such as wavelets and Radon transforms by incorporating parameters of refinement multiplicity, dispersion, and rotation. We rigorously establish key properties of the transform including completeness, uniqueness, invertibility, closure, and stability using frame bounds over functions of bounded variation and define a natural inner product structure emerging in L2. We identify regions of parameter space that recover known transforms, including multiscale wavelet decompositions and the generalized Radon transform. Applications are discussed across a range of disciplines, with particular emphasis on entropy formulations. Notably, the transform remains well behaved on geometrically complex and even non convex domains, where traditional methods may struggle. Despite the complexity of the underlying geometry, the coefficient spectrum reveals structure, offering insight even in highly irregular settings.

The Geometric Refinement Transform: A Novel Uncountably Infinite Transform Space

TL;DR

The paper tackles the limitation of classical transforms in handling nonconvex and geometrically complex domains by introducing a geometric refinement transform based on hierarchical Voronoi refinements parameterized by refinement multiplicity , dispersion , and rotation. It develops a rigorous framework establishing completeness, uniqueness, invertibility, closure, and stability via measure-weighted frame bounds and derives a natural inner product from the refinement geometry, with the transform densely representing via a BV-to-L^2 density argument. The authors connect the new transform to existing constructs, showing how Radon-like behavior and wavelet-like decompositions arise as special cases, and analyze artifacts through the Radon Zone concept, proposing averaging over angular configurations to ensure faithful recovery in higher dimensions. They also provide an entropy-based interpretation of the coefficient spectrum, suggesting a variational principle for energy-minimizing dynamics and highlighting broad potential applications in PDEs, imaging, physics, and beyond, while noting avenues for numerical and non-Euclidean development.

Abstract

This work introduces a novel and general class of continuous transforms based on hierarchical Voronoi based refinement schemes. The resulting transform space generalizes classical approaches such as wavelets and Radon transforms by incorporating parameters of refinement multiplicity, dispersion, and rotation. We rigorously establish key properties of the transform including completeness, uniqueness, invertibility, closure, and stability using frame bounds over functions of bounded variation and define a natural inner product structure emerging in L2. We identify regions of parameter space that recover known transforms, including multiscale wavelet decompositions and the generalized Radon transform. Applications are discussed across a range of disciplines, with particular emphasis on entropy formulations. Notably, the transform remains well behaved on geometrically complex and even non convex domains, where traditional methods may struggle. Despite the complexity of the underlying geometry, the coefficient spectrum reveals structure, offering insight even in highly irregular settings.
Paper Structure (26 sections, 7 theorems, 37 equations)

This paper contains 26 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Key contributions.
  3. Mathematical Preliminaries
  4. Voronoi Diagrams
  5. Bounded Variation and Function Averages
  6. Function Spaces and Domain of the Transform
  7. The Geometric Refinement Transform
  8. Building the Transform
  9. Refinement Multiplicity. N
  10. Dispersion. $\delta$
  11. Rotation.
  12. Coefficient Definition
  13. The Role of Simplex Geometry in Refinement
  14. Connection to Existing Transforms
  15. Radon Transform
  16. ...and 11 more sections

Key Result

Theorem 1

There exist constants $A, B > 0$ such that for all $f \in L^2(\Omega)$,

Theorems & Definitions (7)

  • Theorem 1: Frame Bounds
  • Theorem 2: Generalized Geometric Refinement Radon Transform
  • Proposition 1: Symmetric Refinements
  • Theorem 3: Completeness
  • Theorem 4: Uniqueness
  • Theorem 5: Invertibility
  • Proposition 2: Closure