The Geometric Refinement Transform: A Novel Uncountably Infinite Transform Space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917975514152960 |
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| author | Mullaghy, Zachary |
| author_facet | Mullaghy, Zachary |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20096 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Geometric Refinement Transform: A Novel Uncountably Infinite Transform Space Mullaghy, Zachary Numerical Analysis Mathematical Physics Functional Analysis 65D15, 65T60, 52C99, 42C15, 94A12, 94A08, 94A17, 44A12 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. |
| title | The Geometric Refinement Transform: A Novel Uncountably Infinite Transform Space |
| topic | Numerical Analysis Mathematical Physics Functional Analysis 65D15, 65T60, 52C99, 42C15, 94A12, 94A08, 94A17, 44A12 |
| url | https://arxiv.org/abs/2503.20096 |