The University of Montana
Department of Mathematical Sciences

Technical report #4/2006

The Stabilizing Properties of Nonnegativity Constraints
in Image Deblurring Problems


J. Bardsley
Department of Mathematical Sciences
The University of Montana (USA)

J.K. Merikoski
Department of Mathematics, Statistics and Philosophy
University of Tampere (Finland)

and

R. Vio
Chip Computers Consulting s.r.l. (Venice, Italy)

Abstract

Aims. It is well known from practice that incorporating nonnegativity constraints in image deblurring algorithms often yields solutions that are much more stable with respect to errors in the data. In the current literature, no formal explanation of the stabilizing effects of nonnegativity constraints has been given. In this paper, we present both theoretical and computational results in support of this empirical finding.
Methods. Our arguments are developed in the context of the least-squares approach. For our analysis, we express the solution of a nonnegatively constrained least squares-problem as a pseudo-solution of a linear system. The conditioning of the corresponding coefficient matrix is then compared with the conditioning of the coefficient matrix of the linear system without constraints.
Results. In general, the matrix corresponding to the nonnegatively constrained problem is better conditioned than that of the unconstrained problem, and as a result, the corresponding solutions are typically more stable with respect to errors in the data.
Conclusions. In astronomical imaging, some form of regularization is either implicitly or explicitly used in all algorithms. The most standard regularization techniques, e.g., Tikhonov and iterative regularization, bias solutions in a way that is not compatible with the true solution. The incorporation of nonnegativity constraints, on the other hand, provides stability in a way that is fully compatible with the true solution. When incorporated into the least-squares framework, the resulting algorithms often yield reconstructions that are either on par, or are of a higher quality, than those obtained with more sophisticated approaches. This suggests that incorporating prior information about the object has a greater impact on results than does the sophistication of the algorithm that is used. We emphasize that this is not an academic result. In fact, simple algorithms such as those based on linear least-squares approaches are easy to implement, more flexible regarding the incorporation of constraints and are available in the most popular packages. Consequently, we believe that in many situations they should be the first choice in deblurring problems.

Keywords: Methods: data analysis Methods: statistical Techniques: Image processing

AMS Subject Classification: 65F20, 65F30

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