The Flat Norm for Shapes and Images

Washington State University, Pullman WA

Investigators

Abstract

In this project, the Principal Investigator and his collaborators pursue an integrated program of research and education aimed at elucidating and exploiting their recent discovery connecting the flat norm from geometric measure theory and the Chan-Esedoglu (CE) variational functional from image analysis. The several threads in the research program cover application of the new multiscale flatnorm to the study of shapes, integrated with supporting theoretical and computational work. The central discovery making this program possible is the realization by Vixie and his collaborator Simon Morgan that (1) minimizers of the CE functional translate directly to minimizers for the flatnorm decomposition, (2) this identification generalizes both the flat norm and the CE functional and (3) the minimal values are the same. This opened the way for (1) denoising of non-boundary and/or higher codimension sets (a generalization of the CE functional), (2) a multiscale flat norm (a generalization of the flatnorm), and 3) a practical way to calculate the flat norm through methods for the CE functional. Taken together, they present us with new methods to analyze shape and image data. This research is a part of a larger program promoted by the PI and collaborators in which insights and innovations in geometric measure theory and geometric analysis are exploited in the pursuit of solutions to various data challenges. In less technical terms, the investigator and his collaborators are developing new ways to represent and characterize image and image-like data. The technical aspects of the research program are both very interesting and amenable for the training new generations of mathematicians who are capable of both serious technical contributions and practical work on pressing problems related to the information and data sciences. The mathematics, arising from an area originally invented to study minimal surface problems related to question "What shape will a bubble be if we dip this wire into a soap solution?", turns out to have practical applications to the understanding of shapes and images. The investigators focus on both the theoretical developments that support practical applications and the development of practical algorithms for computing the multiscale flatnorm, the centerpiece of this project. Applications of this new multiscale flatnorm extends into many areas like image and shape denoising, shape recognition and the detection of anomalies in image streams. It is generally accepted that intelligent, efficient extraction of information from massive data streams is a very important, largely unsolved problem. Examples of such data streams include hyperspectral imagery from satellites and image streams from extended time high resolution microscopic observations of dynamic, living systems, to name just two. The tools being developed in this project promise new ways to understand and analyze such large scale data sets.