Transport Transforms for Solving Signal Estimation and Classification Problems

The ability to solve signal estimation and classification problems has wide-ranging practical applications. With the advent of advanced sensor technology, signal data generation has increased in various fields, creating a need for effective and efficient solutions to the ever-growing signal processing problems. Prior literature offers several approaches to solving estimation and classification problems. However, they often fail to address the underlying nonlinearity of the system of interest, resulting in poor performance when solving nonlinear problems. The proposed research explores the use of transport-based transforms to estimate signal parameters and classify signal data. A transport-based generative model is adopted to define an estimation or classification problem. The mathematical properties of transport transforms are then used to simplify the problem in the transform domain, leading to an effective and efficient solution. The specific objectives of the thesis include: (1) proposing a closed-form solution to certain nonlinear estimation problems using the cumulative distribution transform (CDT), a transport-based signal transformation technique, (2) proposing an end-to-end signal classifier using the signed cumulative distribution transform (SCDT), an extension of the CDT, and (3) exploring a transport-based modeling approach to identify the governing partial differential equation (PDE) of a dynamical system.