{
  "id": "2305.13255",
  "version": "v1",
  "published": "2023-05-22T17:26:51.000Z",
  "updated": "2023-05-22T17:26:51.000Z",
  "title": "The Geometric Approach to the Classification of Signals via a Maximal Set of Signals",
  "authors": [
    "Leon A. Luxemburg",
    "Steven B. Damelin"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we study the scale-space classification of signals via the maximal set of kernels. We use a geometric approach which arises naturally when we consider parameter variations in scale-space. We derive the Fourier transform formulas for quick and efficient computation of zero-crossings and the corresponding classifying trees. General theory of convergence for convolutions is developed, and practically useful properties of scale-space classification are derived as a consequence also give a complete topological description of level curves for convolutions of signals with the maximal set of kernels. We use these results to develop a bifurcation theory for the curves under the parameter changes. This approach leads to a novel set of integer invariants for arbitrary signals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-22T17:26:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "94A63",
      "42A16",
      "42A20",
      "94A12"
    ],
    "keywords": [
      "maximal set",
      "geometric approach",
      "scale-space classification",
      "fourier transform formulas",
      "integer invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}