Abstract
<p style='text-indent:20px;'>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 [<xref ref-type="bibr" rid="b29">29</xref>] to arbitrary (signed) signals on <inline-formula><tex-math id="M1">\begin{document}$ \overline {\mathbb{R}} $\end{document}</tex-math></inline-formula>. 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.</p>
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