A Transportation $L^p$ Distance for Signal Analysis
This work addresses a bottleneck in signal and image analysis for researchers and practitioners by extending transport-based methods to more diverse signal types, though it is incremental as it builds on existing frameworks.
The paper tackles the limitations of transport-based distances like Wasserstein, which require nonnegative and normalized signals, by introducing the TL^p distance that handles general, non-positive, and multi-channelled signals, enabling applications in classification and color transfer.
Transport based distances, such as the Wasserstein distance and earth mover's distance, have been shown to be an effective tool in signal and image analysis. The success of transport based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However these distances require the signal to be nonnegative and normalized. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here we study a transport-based distance, called the $TL^p$ distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multi-channelled signals. The framework allows the application of existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multi-channelled, non-positive one and two-dimensional signals, and color transfer.