Function Reconstruction, the MomEnt+ Project
Headed by Mark A. Limber, the
MomEnt+ project
is the computational
engine behind the convex entropy optimization research at the CECM.
The name "MomEnt+" is derived from "Moment problems solved via Entropy
maximization with Positivity". However, we are not exclusively working
on moment problems, except in a general sense, nor are we restricting
ourselves to maximum entropy methods.
The underlying problem we study is Ax = b where A : X -> R^n is a
continuous linear operator and X is some function space. Since this is
generally an underdetermined problem, we can pick a solution via an
optimization criterion, for example, the solution to
inf { f(x) : Ax = b, x >= 0 }
where f:X -> R is a suitable convex functional. We have implemented
the theory of convex duality as developed in [2] in both a Maple and C
environment [3] to solve such problems. We are also concentrating on
other computation methods, including projection [1] and multigrid [4]
methods.
We have made contacts with the medical imaging groups at Vancouver
General Hospital and TRIUMF, the government research centre located
in Vancouver. These and other contacts will keep this project
application oriented and force us to address real world issues that
arise in applications, in particular, extremely noisy data.
- [1] H. H. Bauschke and J.M. Borwein, "On projection algorithms for solving convex feasibility problem," SIAM Review, submitted.
- [2] J.M. Borwein and A. S. Lewis, "Duality relationships for entropy-like minimization problems," SIAM J. Control and Optim., 29(1991), 325-338.
- [3] J.M. Borwein, R. K. Goodrich, and M. A. Limber, "A comparison of entropies in the underdetermined moment problem," Numerical Functional Analysis and Optimization, submitted.
- [4] M. A. Limber, T. A. Manteuffel, S. F. McCormick and D. S. Sholl, "Optimal Resolution in Maximum Entropy Image Reconstruction from Projections with Multigrid Acceleration," Proceedings of the Sixth Annual Copper Mountain Conference on Multigrid Methods, Eds. S. F. McCormick and T. A. Manteuffel, 1993