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.
This project explored the use of mathematical geometric ideas in understanding the content of signal and image data. The research was carried out with the additional goals of weaving research threads into both undergraduate and graduate education, and establishing collaborations and professional networks for graduate students and other researchers. This work brought together the tools of geometric measure theory, graph theory, optimization and scientific computing. Integrating Research with Education This grant supported two mathematics PhD students. Several graduate and undergraduate summer students were also partially supported in related research. As part of the grant concept, the PI, co-PI and graduate students offered and taught extra courses at Washington State University. These courses ranged from advanced mathematics courses not typically taught at WSU to informal offerings such as a weekly data interpretation challenge. Graduate students often taught or co-taught these courses as a means of obtaining breadth and experience. Building Research Networks This grant partially supported and benefitted from several educational and collaborative workshops, two summer schools and several visits to WSU by external faculty and graduate students. The results of these efforts include a new extended research network among graduate students and faculty researchers at all levels of experience. Three several-day workshops were organized around finding practical solutions to specific immediate data problems for researchers at Washington State University, University of Oregon and wildlife biologists at Red Rock Lakes National Wildlife Refuge in Montana. The workshops included a data interpretation task in the noninvasive measurement of hormone levels in northwest salmon, automatic movement and shape tracking in video of c. elegans worms, and automatic vegetation type identification from aerial photography. The PI was the principle lecturer, and the co-PI the computational workshop lead, for the 2010 Summer Park City Mathematics Institute Undergraduate Faculty Program. The theme was image processing, and the concept was to foster richer undergraduate program instruction in smaller colleges and universities. The PI also sponsored a week of visiting faculty and graduate students for the purpose of tutorials and explorations in Geometric Analysis - the mathematical heart of the research grant. Real-Time Analysis of Signals and Images The research accomplishments include theoretical advancement in the geometric quantification of data and new computational methods for application to real data challenges. Theoretical research was aimed at exploring how geometric measures (length, area, volume) can be used to differentiate, classify and order collections of data. Central ideas are the comparison of shapes and quantification of the presence of various sized objects found in data such as images. An intuitive tool for these goals is the flat norm which is a combined comparison of proximity, area and perimeter. The flat norm can also be used to find the least convoluted shape closest to any other given shape. New properties of flat norm derived shapes were discovered. New similar measures for data were developed which mimicked the flat norm properties but were simpler to compute. Computational research was aimed at implementing the flat norm and related ideas on images and data. A central goal was the real-time implementation on video which demands versatile and efficient computer algorithms. Methods for computing on regularly arrayed data (such as images stored as pixels) and on irregularly arrayed data were explored. As a result, the flat norm itself can be efficiently computed for data on a triangulated array and data on a regular gridded array. Our similar (or surrogate) measures have been applied to video processing for simple object identification. The flat norm computations have been applied to image classification tasks.
