Image restoration in the presence of compatible convex constraints can be carried out by the method of convex projections [1]-[3]. In a recent interesting paper [4], Goldburg and Marks have used a modified version of the above technique to solve an optimization problem involving the synthesis of a signal subject to two inconsistent constraints. We complete this result and also show that their restriction to a real Hilbert space setting is unnecessary. A unique generalization of the above optimization problem to the case of more than two constraints does not seem possible. Nevertheless, considerations of symmetry have led us to a formulation which identifies "minimizers" as "nodes" on closed "greedy" paths and an important and potentially useful property of such paths is proven in Theorem 4.
[1]
I. M. Glazman,et al.
Theory of linear operators in Hilbert space
,
1961
.
[2]
A. Lent,et al.
An iterative method for the extrapolation of band-limited functions
,
1981
.
[3]
Henry Stark,et al.
Image Restoration by the Method of Projection onto Convex Sets. Part I
,
1982
.
[4]
D. Youla,et al.
Image Restoration by the Method of Convex Projections: Part 1ߞTheory
,
1982,
IEEE Transactions on Medical Imaging.
[5]
R. Marks,et al.
Signal synthesis in the presence of an inconsistent set of constraints
,
1985
.