Perceptual convergence of discrete clamped fractal operator

Abstract

While clamping the fractal operator to the grayscale range of values of the images preserves its contractivity, performing the rounding process to discrete levels spoils this property and the given iterative sequence is generally not convergent. In practice this lack of convergence is not observed by HVS (Human Visual System) on decoder’s output. We explain this phenomenon by presenting a strict notion of the perceptual convergence at the given threshold. It is used to prove that any iterative sequence for a fractal operator which is contractive in l _∞ norm with contractivity c ^*<1, after clamping and rounding to integer levels, is perceptually convergent at the threshold τ≥1/(1−c ^*).