#### Technical Report USC-IPI-1090

“A State Space Approach to Image Modeling Restoration and Identification”

by Fabrice Jean Clara

October 1983

We address three problems of major importance in the field of image processing, namely image modeling, restoration and system identification. This study was motivated by recent developments on a class of 2-D causal, separable in denominator state space (CRSD) models and on identification of 1-D space-varying systems.

We first extend a previous study on space-invariant CRSD models. Identification and model reduction techniques are developed, to obtain the model parameters from 1) an ARMA input/output representation of a 2-D system, and 2) from the autocorrelation function of a 2-D stochastic process. The CRSD model identification can thus be performed from most major statistical definitions of an image.

In a second part of this work, we derive a 2-D state-space model for space-varying systems, which closely parallels the CRSD model. A study of its dynamic properties is presented. Model identification and reduction procedures, using the concept of balancing, are derived and applied on practical examples. Suboptimal approaches to identification are also suggested, which provide efficient tools for the design and approximation of 2-D space-varying filters.

Finally, the image restoration problem is studied in the context of space-varying systems. A ``piecewise stationary'' model is assumed for the image, and a recursive restoration algorithm is derived for images degraded with additive noise. The error criterion used in the restoration includes properties of the human visual system, to produce estimates of good visual quality. The identification of the image model and the estimation procedure are implemented recursively, on a line-by-line basis. This algorithm is then generalized to include both space-invariant and space-varying blur degradations.

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