Vision, whether *bi**ological *or*
computational* , is the science and technology of perception
generation from observed optical signals (or popularly referred to as *images*).
It allows a human being or an intelligent robot to sense,
interpret, communicate with, and react to the environment (i.e., time+space).

From quantum mechanics, Einstein's relativity theories, to nanotechnology
and photonics, the role of light or photons has always been critical.
Light (or electromagnetic waves), traveling in space and time, allows
us to __ see__ the images of distant galaxies and stars millions of light-years
away (or rather, ago), to

Beneath all these sciences and technologies, lie the
fundamental, counterintuitively technical, and often even more profoundly philosophical, questions: What do
we mean by __ seeing__? What do we mean by seeing some

This digital and information age further pushes those questions
to the frontier. What is the chance in this universe for a natural
area on the Mars' surface to bear the pattern of a human face? How
much trust shall we invest in a doctor's words when s/he observes
abnormalities from CT/MRI/PET images? Could a new bright spot in an
astronomical image be a star missed by all the previous observations
(e.g., the tenth newly discovered planet-**Sedna**- of our solar system)?

Naturally, vision modeling has to be interdisciplinary, cutting
across vision psychology, cognitive science, computational neuron
science, learning theory, pattern theory, image processing, computer
vision, artificial intelligence, and so on. As applied mathematicians,
our goals are to develop models based on all the experimental
results and data, analyze the models (existence, uniqueness,
well-posedness, stability etc.), efficiently compute the models, and
validate and improve them.

We closely follow and are deeply inspired by the pioneering works of Brown's Pattern Theory Group.

__On the Foundations of Vision Modeling V. Noncommutative Monoids of Occlusive Preimages__

__From the abstract:__: A significant cue for visual perception is the occlusion pattern in 2-D images projected onto biological or digital retinas, which allows humans or robots to successfully sense and navigate the 3-D environments. There have been many works on modeling and studying the role of occlusion in image analysis and visual perception, mostly from analytical or statistical points of view. The current paper presents a new theory of occlusion based on simple topological definitions of*preimages*and a*binary operation*on them called ``occlu." We study many topological as well as algebraic structures of the resultant preimage monoids (a monoid is a semigroup with identity). The current paper is intended to foster the connection between mathematical ways of abstract thinking and realistic modeling of human and computer vision. (*UCLA CAM Tech Report 04-22,*April, 2004, by Jianhong Shen.)

[Keywords: Depth, occlusion, preimages, segmentation, monoids (semi-groups), topology, invariants, knot theory]

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

__From the 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*. 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 the law in faithfully modelling both human and computer vision, and attempted to integrate it into 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 distribution 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 \Gamma-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 confirm and highlight the light adaptivity feature of the new model. (*IMA Tech. Preprint No. 1949*, December, 2003, by Jianhong Shen and Yoon-Mo Jung.)

__On the Foundations of Vision Modeling III. Pattern-Theoretic Analysis of Hopf and Turing's Reaction-Diffusion Patterns__

__From the abstract__: After Turing's ingenious work on the chemical basis of morphogenesis fifty years ago, reaction-diffusion patterns have been extensively studied in terms of modelling and analysis of pattern formations (both in chemistry and biology), pattern growing in complex laboratory environments, and novel applications in computer graphics. But one of the most fundamental elements has still been missing in the literature. That is, what do we mean exactly by (reaction-diffusion)*patterns*? When presented to human vision, the patterns usually look deceptively simple and are often tagged by household names like*spots*or*stripes*. But are such split-second pattern identification and classification equally simple for a computer vision system? A confirmative answer does not seem so obvious, just as in the case of face recognition, one of the greatest challenges in contemporary A.I. and computer vision research.Inspired and fuelled by the recent advancement in mathematical image and vision analysis (Miva), as well as modern pattern theory, the current paper develops both statistical and geometrical tools and frameworks for identifying, classifying, and characterizing common reaction-diffusion patterns and pattern formations. In essence, it presents a data mining theory for the scientific simulations of reaction-diffusion patterns. (

*CAM Tech. Report 03-19*, by Jianhong Shen and Yoon-Mo Jung. May, 2003.)__On the Foundations of Vision Modeling II. Mining of Mirror Symmetry of 2-D Shapes__

__From the abstract:__Vision can be considered as a feature mining problem. Visually meaningful features are often geometrical, e.g., boundaries (or edges), corners, T-junctions, and symmetries. Mirror symmetry or near mirror symmetry is one of the most common and useful symmetry types in image and vision analysis. The current paper proposes several different approaches for studying 2-dimensional (2-D) mirror symmetric shapes. Proper mirror symmetry metrics are introduced based on Lebesgue measures, Hausdorff distance, and lower-dimensional feature sets. Theory and computation of these approaches and measures are developed. (*CAM Tech. Report 02-62*, by Jianhong Shen, 2003.)__On the Foundations of Vision Modeling I. Weber's Law and Weberized TV Restoration__

__From the abstract__: Most conventional image processors consider little the influence of human vision psychology.*Weber's Law*in psychology and psychophysics claims that human perception and response to the intensity fluctuation*du*of visual signals should be weighted by the background stimulus*u*, instead of being plainly uniform. This paper attempts to integrate this well known perceptual law into the classical total variation (TV) image restoration model of Rudin, Osher, and Fatemi [*Physica D*, 60:259-268, 1992]. We study the issues of existence and uniqueness for the proposed Weberized nonlinear TV restoration model, making use of the direct method in the space of functions with bounded variations. We also propose an iterative algorithm based on the linearization technique for the associated nonlinear Euler-Lagrange equation. (*CAM Tech. Report 02-20*, by Jianhong Shen.*Physica D*,**175**(3/4), pp.241-251, 2003.)

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