On the Foundations of Vision Modelling, IV. Weberized Mumford-Shah Model with Bose-Einstein Photon Noise: Light Adapted Segmentation Inspired by Vision Psychology, Retinal Physiology, and Quantum Statistics

Human vision works equally well in a large dynamic range of light intensities, from only a few photons to typical midday sunlight. Contributing to such remarkable flexibility is a famous law in perceptual (both visual and aural) psychology and psychophysics known as Weber’s Law. There has been a great deal of efforts in mathematical biology as well to simulate and interpret the law in the cellular and molecular level, and by using linear and nonlinear system modelling tools. In terms of image and vision analysis, it is the first author who has emphasized the significance of Weber’s Law in faithfully modelling both human and computer vision, and attempted to integrate it into low level visual processors such as image denoising (Physica D, 175, pp. 241-251, 2003). The current paper develops a new segmentation model based on the integration of both Weber’s Law and the celebrated Mumford-Shah segmentation model (Comm. Pure Applied Math., 42, pp. 577-685, 1989). Explained in details are issues concerning why the classical Mumford-Shah model lacks light adaptivity, and why its “weberized” version can more faithfully reflect human vision’s superior segmentation capability in a variety of illuminance conditions from dawn to dusk. It is also argued that the popular Gaussian noise model is physically inappropriate for the weberization procedure. As a result, the intrinsic thermal noise of photon ensembles is introduced based on Bose and Einstein’s distributions in quantum statistics, which turns out to be compatible with weberization both analytically and computationally. The current paper then focuses on both the theory and computation of the weberized Mumford-Shah model with Bose-Einstein noise. In particular, Ambrosio-Tortorelli’s -convergence approximation theory is adapted (Boll. Un. Mat. Ital., 6-B, pp. 105-123,1992), and stable numerical algorithms are developed for the associated pair of nonlinear Euler-Lagrange PDEs. Numerical results further confirm and highlight the light adaptivity feature of the new model.