The Cumulative Distribution Transform and Linear Pattern Classification

Discriminating data classes emanating from sensors is an important problem with many applications in science and technology. We describe a new transform for pattern identification that interprets patterns as probability density functions, and has special properties with regards to classification. The transform, which we denote as the Cumulative Distribution Transform (CDT) is invertible, with well defined forward and inverse operations. We show that it can be useful in `parsing out' variations (confounds) that are `Lagrangian' (displacement and intensity variations) by converting these to `Eulerian' (intensity variations) in transform space. This conversion is the basis for our main result that describes when the CDT can allow for linear classification to be possible in transform space. We also describe several properties of the transform and show, with computational experiments that used both real and simulated data, that the CDT can help render a variety of real world problems simpler to solve.