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

TL;DR

This paper introduces varilets, a new basis for scalar fields that decomposes functions into piecewise monotone components with invariant topological total variation, enabling effective filtering and analysis.

Contribution

It proposes the concept of varilets and a varilet transform, providing a novel additive decomposition method based on topological total variation for scalar fields.

Findings

Varilet basis allows decomposition with invariant topological total variation.

The varilet transform enables filtering by coefficient manipulation.

The approach generalizes 1D total variation to topological settings.

Abstract

Continuous interpolation of real-valued data is characterized by piecewise monotone functions on a compact metric space. Topological total variation of piecewise monotone function f:X->R is a homeomorphism-invariant generalization of 1D total variation. A varilet basis is a collection of piecewise monotone functions { $g_i$ |i = 1...n}, called varilets, such that every linear combination $\sum a_ig_i$ has topological total variation $\sum |a_i|$. A varilet transform for $f$ is a varilet basis for which $f =\sum \alpha_ig_i$. Filtered versions of $f$ result from altering the coefficients $\alpha_i$.