Persistence Lenses: Segmentation, Simplification, Vectorization, Scale Space and Fractal Analysis of Images

Abstract

A persistence lens is a hierarchy of disjoint upper and lower level sets of a continuous luminance image's Reeb graph, providing a contrast-invariant topological representation of image contrast variation. Pulled back to the image, the boundary components of a persistence lens's interior components are Jordan curves that serve as a hierarchical segmentation of the image, and may be rendered as vector graphics. A persistence lens determines a varilet basis [10] for the luminance image, in which image simplification is a realized by subspace projection. Image scale space, and image fractal analysis, result from applying a scale measure to each basis function.

Varilets: Additive Decomposition, Topological Total Variation, and Filtering of Scalar Fields

“…Looking forward to those results, the varilet transform's twin additive decomposition of function f and topological total variation TTV (f ) is a universal property of the varilet transform, i.e. it holds for all C. Whereas, in the contexts of image segmentation, simplification and fractal analysis [4], we will require lens C to have special properties.…”

A Topological Approach to Scaling in Financial Data

“…Lemire et al [20], Brooks [21] and Seversky et al [22] have used TDA for analyzing various data sets. In particular, Brooks has developed a method consisting of decomposing the data into components of its total variation and has applied it to the analysis of medical images [23].…”