  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.
