Room 3102, Engineering I
Department of Electrical and Computer Engineering
University of California
Santa Barbara, CA 93106-9560
phone: (805) 893-4682
fax: (805) 893-3262
Ph.D. University of Maryland
Most of my effort in this area involves the solution of algebraic Riccati equations and stable Lyapunov equations. Although there are many ways to approach these problems, for large systems with hundreds or thousands of states the matrix sign function method seems to offer the best economy. This method is directed toward finding a basis for the negative invariant subspace of an associated Hamiltonian matrix and then using this basis to construct the desired solution. The matrix sign function provides a means of finding this negative invariant subspace and has been used in control theory since the work of Roberts in the early 1970's. Since then there have been many contributions to the use of the matrix sign function in control. My work in this area has been done in collaboration with Alan Laub and has focused on rational iterations which start with the Hamiltonian matrix and then converge to its associated sign matrix. Several years ago we developed a rather complete theory for such rational iterations based on Pade approximations of a related hypergeometric function. This was followed by a marvelous discovery with Pradeep Pandey of an exact partial fraction decomposition of the so-called principal Pade approximants. I call this discovery marvelous because it opens the door to parallel evaluation of matrix sign function. This method has been used by Nick Higham and co-workers at the University of Manchester for the related problem of polar decomposition. Just recently, we have been working with a surprising hyperbolic tangent identity that Alan discovered for the Pade approximants; this identity gives easy proofs for most of the properties of the Pade recursions, such as global convergence, and leads to new viewpoints on the relationship between scaling the matrix sequence and eigenvalue assignment methods.
Alan Laub and I have also been working in the area of estimating the sensitivity of matrix functions such as the matrix exponential. Several years ago we published a paper on condition estimation for matrix functions that can be expressed as power series. This work showed that the usual inverse power method for estimating the condition number of a matrix with respect to inversion could be adapted to provide condition estimates for analytic matrix functions. Although this approach applies to exponential and logarithmic matrix functions it does not extend to more general matrix maps such as the one between the system matrices (A,B,C) and the Riccati solution X. Fortunately, we were able to find a means of condition estimation that avoids the transpose step of the usual power method and can be applied to any differentiable matrix function. This new procedure is statistical in the sense that the probability of a poor condition estimate can be determined exactly as a function of the number of function evaluations. This approach, which we refer to as the small-sample statistical method, relies on the theory of beta distributions and gives remarkably good condition estimates for only a few (2 or 3) extra function evaluations. Most importantly, the condition estimates can be obtained in a "black-box"' mode without extensive analysis of the underlying function.
In conjunction with Dr. Gary Hewer at the Naval Air Warfare Center in China Lake, California and B. S. Manjunath at the Vision Research Laboratory, I have been looking into nonlinear partial differential equations arising in image processing. Our research has focused on optical flow estimation, variational image segmentation and noise removal via a new procedure that we call peer group averaging (PGA).
In optical flow, we have had great success using aggregate velocity methods which assume that distant targets can be modeled as having constant velocity vectors over their visible surface area. This eliminates the problems associated with aperture effects and results in very accurate velocity estimates that can be computed in real time.
In image segmentation, by using continuous boundary functions we produced a general framework for variational image segmentation. The result is general enough to include a wide variety of approaches including Mumford-Shah formulations and Geman type functionals. For such problems the optimal boundary function can be found explicitly and a PDE descent procedure yields the desired approximation function for a given image. more...
For noise removal, we have devised a procedure in which we identify a peer group for each pixel. This peer group consists of nearby pixels with similar intensity values. Averaging over the peer group eliminates noise, especially speckle noise, but does not blur edges. This type of processing is related to both median filtering and anisotropic diffusion with the drawbacks of either method. It is easy to implement with very fast execution time.