At the Frame Research Center, we generally work on 10 or more projects at any given time. Here we will discuss a few of the larger long term projects we are working on at this time. Research related to these projects can be found in the publication section of this web site.
A number of new applications of frame theory have emerged where the set-up can hardly be modeled naturally by one single frame system. They generally share a common property that requires distributed processing such as sensor networks. Furthermore, we are often overwhelmed by a deluge of data assigned to one single frame system, which becomes simply too large to be handled numerically. In these cases it would be highly beneficial to split a large frame system into a set of (overlapping) much smaller systems, and being able to process effectively locally within each sub-system. Interestingly, preliminary research indicates that our brain may store and processes information by using distributed processing techniques.
Fusion frames were introduced to address problems on the modeling of sensor networks. In (wireless) sensor networks, resource-constrained sensor nodes are spread over a (potentially large) area to measure environmental characteristics such as temperature, sound, vibration, pressure, motion, and/or pollutants. Such networks are frequently quite redundant in the sense that there is a high degree of overlap among the environmental conditions being sensed by neighboring sensors. Also, due to the unpredictable nature of low-cost devices and hostile geographical factors, certain local sensors can be less reliable than others. Fusion frames are able to handle both the overlap problems and the problems with loss of information. Recent work by Calderbank, Kutyniok and Pezeshki has established a close connection with Grassmannian packings. Our goal here is to construct fusion frames for this area as well as to develop new applications of the theory. Parseval fusion frames are particularly difficult to construct as this problem is equivalent to a little understood question in operator theory. There is a website reporting advances in fusion frame theory and its applications (www.fusionframe.org).
Defense Threat Reduction Agency
Recent advances in sensor hardware provide an unprecedented capability to detect the production, deployment and use of chemical and biological weapons. At the same time, the cost of misdetecting such threats is extremely high. Moreover, human intelligence analysts are increasingly overwhelmed by ever-larger data sets. The purpose of this project is to develop new mathematics which will help automate as much of this threat detection process as is possible, freeing human analysts to focus on the most high-level tasks. In particular, a good understanding of sparse representations allows one to break very complicated phenomena into a relatively small number of much-easier-to-understand pieces. Meanwhile, a good understanding of multimodal data allows one to combine data from many different sensors, in order to make a decision which is wiser than could be made from the data of any single sensor alone.
This project addresses several emerging problems in the mathematical field of Hilbert space frames. This research has two thrusts. The first thrust involves using frame theory to provide sparse representations of data. Here, the project is concerned with the paramount problems of the field: to provide verifiable constructions of restricted isometry property (RIP) matrices of implementable size; to find algorithmic constructions of real and complex Grassmannian frames; to find frames which, in the sparsest possible manner, represent a given set of data. The second thrust involves using frame theory to fuse information gathered from disparate sensor modalities. Here, the work focuses on several fundamental open problems of multimodal data processing, such as how to mathematically determine what given collections of modalities yield optimal detection accuracies, and how data from different modes should be registered.
Signal Reconstruction Without Phase
This project originated when Siemens Corporate Research contacted us with a problem which was severely hindering one of their important projects. In this project they were faced with signal reconstruction in which there was significant noise in the phase. Although there were some existing methods in the engineering community for doing signal reconstruction without phase, these methods were not able to handle the type and quantity of noise encountered in Siemens' particular applications. Our approach to the problem was to first show mathematically that there is an unlimited supply of frames for doing signal reconstruction without phase. The next step was to develop algorithms for carrying this out in practice. The second step turned out to be more difficult than the first. Over time we have managed to develop some very efficient algorithms for doing signal reconstruction without phase but there is work here left to be done.
Equiangular frames (i.e. frames for which the modulus of the angle between frame vectors is a constant) first appeared in discrete geometry but today have applications to signal processing, communications, coding theory and quantum physics. The main problem hindering their application is the fact that we know very few of them. This lacking is due to the fact that the starting point for constructing equiangular tight frames is the existence of maximal equiangular line sets. The construction of equiangular line sets has proved exceptionally difficult and in sixty years of research in the real case, the maximal number of equiangular lines is known only for 35 dimensions. Our goal is to develop completely new tools for constructing equiangular line sets and applying these methods to the construction of equiangular tight frames. The ideal situation would be to find a constructive method for constructing such families one vector at a time as has been done previously for constructing tight frames.
The Kadison-Singer Problem
The Frame Research Center is dedicated to all applications of frame theory, including applications to pure mathematics. The 1959 Kadison-Singer Problem in C*-algebras proved to be one of the most intractable problems in mathematics. This problem is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics, and engineering, and the link connecting all these areas of research is Hilbert space frame theory. The Kadison-Singer Problem was recently solved by Marcus, Spielman, and Srivastava, but there is much work to be done to find the best constants for the solution.
Quantum Information Theory
Frame theory is the right platform for dealing with problems in quantum information theory. Due to the massive development of frame theory over the last few years (including frame potentials, spectral tetris constructions, phase retrieval, fusion frames, probabilistic frames, etc) with hundreds of papers and numerous new construction methods, many problems in quantum information theory are now accessible. Our goal is to bring all these tools to the area to address a number of fundamental open problems.