We will introduce a mathematical signal transform based on Optimal Transport Theory. It builds upon the existing Cumulative Distribution Transform by Park, Kolouri, Kundu, and Rohde (2018). We will present both forward (analysis) and inverse (synthesis) formulas for this nonlinear transform, and describe several of its properties including translation, scaling, convexity, and linear separability. Indeed, since this tool is a new signal representation based on Optimal Transport theory, it has suitable properties for decoding information related to certain signal displacements. We will describe a Wasserstein-type metric in transform space, and show applications in classifying (detecting) signals under random displacements, and parameter estimation problems for certain types of generative models.