The Signed Cumulative Distribution Transform for 1-D signal analysis and classification

Akram Aldroubi; Rocio Diaz Martin; Ivan Medri; Gustavo K. Rohde; Sumati Thareja

doi:10.3934/fods.2022001

Article Contents

2022, Volume 4, Issue 1: 137-163. Doi: 10.3934/fods.2022001

This issue Previous Article Smart Gradient - An adaptive technique for improving gradient estimation Next Article

The Signed Cumulative Distribution Transform for 1-D signal analysis and classification

1.
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
2.
Department of Biomedical Engineering, Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904, USA

^*Corresponding author: Ivan Medri
^*Corresponding author: Ivan Medri

Received: June 2021

Revised: October 2021

Early access: January 2022

Published: March 2022

This work is partially supported by NIH award GM130825. The second author was partially supported by CONICET.

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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 [29] 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.

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Mathematics Subject Classification: Primary: 94A12, 94A16; Secondary: 68T01, 68T10.

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Figure 1. The (modified) Cumulative Distribution Transform (CDT) enables data representation that facilitates learning. Hand signal images are preprocessed for edge map extraction and their respective X and Y projections are computed. The X, Y projections are then transformed using the CDT. Linear classification methods are then applied to the data in CDT space, as well as in original (projection) space for comparison. The middle row displays the 2D linear discriminant embedding [41] of test data in original signal space and transform space. Test (held out from training) data in transform space is clearly more convex and linearly separable than data in original signal space. This is confirmed by test accuracy results of 3 different linear classifiers (bottom row)

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Figure 2. Translation property of the SCDT. Left panel: A signal and its translation. Right panel: A transforms of the signals and its translation (the reference measure $ \mu_0 $ was taken with uniform distribution on $ [0,1] $)

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Figure 3. Scaling property of the SCDT. Left panel: A signal and its scaling. Right panel: A transforms of the signals and its scaling (the reference measure $ \mu_0 $ was taken with uniform distribution on $ [0,1] $)

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Figure 4. The set of signals generated from the algebraic generative model stated in Proposition 2 becomes convex in the SCDT space

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Figure 5. Three signal classes: A Gabor wave, an apodized sawtooth wave, and an apodized square wave are randomly translated and scaled to form three signal classes. Seven example training signals are shown per class

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Figure 6. Classification of test signals (from the three classes depicted in Figure 5): Projection to LDA subspace learned from training data. Left panel: The linear classification method is unsuccessful classifying signal data in its original space. Right panel: Test data is much better separated in SCDT space

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