[Paper Review] Persistence Lenses: Segmentation, Simplification, Vectorization, Scale Space and Fractal Analysis of Images

Martin Brooks|arXiv (Cornell University)|Apr 25, 2016

Topological and Geometric Data Analysis33 references3 citations

TL;DR

This paper introduces persistence lenses—a contrast-invariant topological framework for image analysis that hierarchically segments images using level sets of the Reeb graph. By leveraging varilet transforms and persistence-based simplification, it enables vectorized segmentation, scale space analysis, and fractal region detection through a stable, continuous-domain representation of luminance images.

ABSTRACT

A persistence lens is a hierarchy of disjoint upper and lower level sets of a continuous luminance image's Reeb graph. 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 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.

Motivation & Objective

• To develop a contrast-invariant, topological representation of image contrast variation using the Reeb graph's upper and lower level sets.
• To enable hierarchical image segmentation via disjoint facets bounded by Jordan curves, suitable for vector graphics rendering.
• To support image simplification through subspace projection in a varilet basis, preserving structural integrity across scales.
• To construct a scale space and detect fractal regions by applying scale measures (e.g., total variation, contrast) to basis functions.
• To ensure stability and trackability of critical points through persistence-based conflict resolution and extremal tracking.

Proposed method

• Constructs a persistence lens by extracting persistence birth-death pairs from the Reeb graph of a continuous luminance image.
• Applies bilinear interpolation with triangular patch refinement to eliminate saddle points, ensuring topological monotonicity per patch.
• Defines varilet basis functions as characteristic functions of lens facets, enabling image simplification via orthogonal projection.
• Uses the SVG even-odd fill rule to render vectorized segmentations with holes, where each facet is a connected open set bounded by Jordan curves.
• Applies scale measures (e.g., topological total variation, contrast) to basis functions to generate image scale space and identify fractal regions.
• Employs extremal tracking and persistence stability proofs to ensure robustness against noise and perturbations in the image domain.

Experimental results

Research questions

• RQ1 How can a hierarchical, contrast-invariant segmentation of images be achieved using topological features of the Reeb graph?
• RQ2 Can varilet transforms and their basis functions enable stable, multiresolution image simplification and vectorization?
• RQ3 How can scale space and fractal structure detection be derived from a topological image representation?
• RQ4 What conditions ensure the stability and trackability of critical points across different scales in the persistence lens framework?
• RQ5 To what extent can persistence lenses detect power-law distributed fractal regions in natural images using contrast or geometric measures?

Key findings

• The persistence lens generates 43,572 facets across 29,298 lens regions with a maximum nesting depth of 19, enabling fine-grained multiresolution segmentation.
• The first level of the hierarchy contains 6,308 pairwise disjoint facets, each bounded by one or more Jordan curves, suitable for vector graphics rendering.
• Image simplification is achieved via subspace projection in the varilet basis, with persistence stability proven through extremal tracking and non-increasing persistence over refinement.
• Scale space is constructed by applying scale measures (e.g., topological total variation) to basis functions, revealing power-law distributions in large image regions.
• Fractal structure is identified by detecting power-law distributions in facet contrast, with additional potential via area and total variation measures.
• Theoretical proofs establish that persistence is non-increasing under refinement and that extremal tracking preserves stability, ensuring robustness in segmentation and simplification.