Projects
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.
Fusion Frames
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).
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
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 has proved to be one of the most intractable problems in mathematics. Recently it was shown that this problem is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and engineering. The link connecting all these areas of research is Hilbert space frame theory. Our interest here is in the fact that this problem represents a fundamental open problem in frame theory whose ultimate solution will have applications to a broad spectrum of applications of frame theory. Also, the solution to this problem will require a fundamental new method for constructing frames and this will also have broad application to many areas of research.