Appl Math Optim 53:331–358 (2006)
2006 Springer Science+Business Media, Inc.
Weberized Mumford–Shah Model with
Bose–Einstein Photon Noise
Jianhong Shen and Yoon-Mo Jung
School of Mathematics, University of Minnesota,
Minneapolis, MN 55455, USA
Abstract. 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. The current paper devel-
ops a new segmentation model based on the integration of Weber’s Law and the
celebrated Mumford–Shah segmentation model (Comm. Pure Appl. Math., vol. 42,
pp. 577–685, 1989). Explained in detail 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 com-
patible with weberization both analytically and computationally. The current paper
focuses on both the theory and computation of the weberized Mumford–Shah model
with Bose–Einstein noise. In particular, Ambrosio–Tortorelli’s -convergence ap-
proximation theory is adapted (Boll. Un. Mat. Ital. B, vol. 6, pp. 105–123, 1992),
and stable numerical algorithms are developed for the associated pair of nonlinear
Key Words. Weber’s Law, Light adaptivity, Retina, Mumford–Shah segmenta-
tion, Bose–Einstein distribution, Noise, Bayesian, Variational, Free boundary, -
convergence, Computational PDE.
AMS Classiﬁcation. 94A08, 92C20, 49N45.
This research was supported by (USA) NSF Program in Applied Mathematics under Grant Number