The Signed Cumulative Distribution Transform for 1-D Signal Analysis and Classification

TL;DR

This paper introduces a novel signed cumulative distribution transform for 1-D signal analysis, enabling effective decoding of non-rigid displacements and improving classification performance.

Contribution

It extends the Cumulative Distribution Transform to signed signals with a measure theoretic framework, including analysis, synthesis, and properties for signal classification.

Findings

Effective classification of signals under random displacements.

The transform exhibits translation, scaling, and convexity properties.

Demonstrated improved signal separation in transform space.

Abstract

This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [ACHA 45 (2018), no. 3, 616-641] to arbitrary (signed) signals on $\overline{\mathbb{R}}$. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.