Abstract :
The restoration of images which have been degraded is a key problem in image processing. This is normally required following the acquisition of sensor data and prior to detailed image analysis and understanding. Many physical processes responsible for image distortion or degradation may be modeled by systems which are linear (shift-invariant or shift-variant) and various restoration procedures exist in these situations. Very recently, an efficient recursive scheme has been described[1] to recover images corrupted by blurs that may be modeled by linear shift-variant systems. For any 2-D discrete first quadrant quarter-plane causal linear shift-variant (LSV) system, whose impulse response is a K-th order degenerate sequence, a K-th order state-space model was obtained. This model is recursive and is based on a three-term recurrence formula relating any point in the state-space to its three closest neighboring points and the current input. The state-space model was extended in order to model 2-D discrete LSV systems with support on a causality cone. Subsequently, the 2-D quarter-plane causal and weakly causal discrete models were generalized to the n-D(n>2) case. The resulting state-space models are recursive and are based on a (2n-1)-points recurrence formula, which for the causal case uses the (2n-1)-closest neighboring points in addition to the input in order to compute any current output state. For the weakly causal case, the (2n-1) computed outputs required are not, in general, the closest neighbors to the output presently being computed. Conditions for the existence of a 2-D state-space model for the inverse system were obtained and with these conditions satisfied, a state-space model of the inverse system is readily derivable from the original one. Models for the 2-D LSV system and its inverse can be used to perform analysis and deconvolution problems very efficiently. This was substantiated from derived expressions for space-time computational complexities.
