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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