Regularization of Van Cittert's algorithm

Van Cittert's iteration is:

$$O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha{R}^{(n)}(x,y)$$ (14.115)

with R⁽ⁿ⁾(x,y) = I(x,y) - P(x,y) * O⁽ⁿ⁾ (x,y). The regularization by the significant structures leads to:

$$O^{(n+1)} (x,y) = O^{(n)} (x,y) + \alpha {\bar{R}}^{(n)}(x,y)$$ (14.116)

The basic idea of our method consists of detecting, at each scale, structures of a given size in the residual R⁽ⁿ⁾(x,y) and putting them in the restored image O⁽ⁿ⁾(x,y). The process finishes when no more structures are detected. Then, we have separated the image I(x,y) into two images $\tilde O(x,y)$ and R(x,y). $\tilde O$ is the restored image, which does not contain any noise, and R(x,y) is the final residual which does not contain any structure. R is our estimation of the noise N(x,y).