Abstract
Most image formation systems involve a built-in nonlinear transformation.
Furthermore, many image processing applications involve longtailed
noise processes, which introduce outliers in the gray-level distribution
of the image. The performance of conventional restoration algorithms
is highly degraded by such noise processes. A robust-design approach
aims to eliminate the effects of outliers in the performance of the
algorithm. The concept of M-estimation leads to robust estimators, which,
however, require iterative evaluation. Such estimators have been employed
in recursive image restoration, mainly for the robust parameter
identification. In this paper, the exploitation of M-estimators in iterative
nonlinear restoration algorithms is considered. These algorithms are designed
to optimize the robust least-squares-error criterion, which limits
the effects of large residuals through an influence function. The robust
version of the steepest-descent (SD) algorithm is introduced. This algorithm
overcomes the linear search required by the basic SD technique
through a particular procedure for the selection of the iteration parameter.
Based on the convergence analysis of this algorithm a new influence
function, the redescending nonzero function, is introduced. Structural
modifications of the robust SD technique result in two hybrid SD algorithms,
which achieve fast convergence without degrading the quality of
the final estimate. The algorithms introduced can be easily implemented
through Fourier transform techniques. The convergence properties of the
robust algorithms are rigorously studied through their singular-value
analysis and the global convergence theorem. The application of these
algorithms on image restoration problems is demonstrated through examples.
