A Transportation $$L^p$$ Distance for Signal Analysis

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 non-negative and normalised. 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 multichannelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multichannelled non-positive one-dimensional signals and two-dimensional images, and colour transfer.