  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.
