#### Technical Report USC-IPI-1060

“Nonstationary Recursive Restoration of Images With Signal- Dependent Noise With Application to Speckle Reduction”

by Darwin Ta-Wen Kuan

August 1982

A two-dimension recursive image restoration filter is developed for images degraded by blur and a class of uncorrelated, signal- dependent noise. Unlike conventional image restoration techniques, the filter does not require any a priori information of the original image and adopts a nonstationary image model. All the parameters needed for the filter are estimated from the noisy image. The filter has a simple recursive structure, and is able to adapt itself to the nonstationary content of the image and to different types of signal-dependent noise.

The second major subject area of this dissertation is on speckle reduction techniques. Speckle noise inherently exists in all types of coherent imagery such as synthetic aperture radar imagery, acoustic imagery, and laser illuminated imagery. Past work on speckle reduction assumed that speckle noise is multiplicative and uncorrelated. We model the speckle according to the exact physical process of coherent image formation. The problem of how to generate discrete speckle images accurately without aliasing is discussed in detail. A local linear minimum mean square error speckle reduction filter is developed for intensity speckle images, where only the speckle intensity is observable. Unlike other existing approaches, this filter considers the second order statistics of speckle and uses a nonstationary image model. The two-dimensional recursive implementation of this filter is also developed as a fast computation algorithm. In some applications, both the amplitude and phase of the speckle image are observable. In the past, the additional phase information is ignored in designing the speckle reduction filter. Here, we develop a nonlinear maximum a posteriori (MAP) filter for complex amplitude speckle images. The MAP equations can be expressed in terms of the filtered estimate and filtered covariance matrix of a nonstationary two-dimensional recursive filter and a cubic equation. Thus the MAP estimate can be solved iteratively by using the recursive filter as a fast computation algorithm and using the cubic equation as a constraint to optimize the estimate can be solved iteratively by using the recursive filter as a fast computation algorithm and using the cubic equation as a constraint to optimize the estimate at each iteration.

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