Opportunities for the Mathematical Sciences

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Table of Contents
Preface
Summary Article
Individual Contributions
  Statistics as the information science
  Statistical issues for databases, the internet, and experimental data
  Mathematics in image processing, computer graphics, and computer vision
  Future challenges in analysis
  Getting inspiration from electrical engineering and computer graphics to develop interesting new mathematics
  Research opportunities in nonlinear partial differential equations
  Risk assessment for the solutions of partial differential equations
  Discrete mathematics for information technology
  Random matrix theory, quantum physics, and analytic number theory
  Mathematics in materials science
  Mathematical biology: analysis at multiple scales
  Number Theory and its Connections to Geometry and Analysis
  Revealing hidden values: inverse problems in science and industry
  Complex stochastic models for perception and inference
  Model theory and tame mathematics
  Beyond flatland: the future of space and time
  Mathematics in molecular biology and medicine
  The year 2000 in geometry and topology
  Computations and numerical simulations
  Numbers, insights and pictures: using mathematics and computing to understand mathematical models
List of Contributors with Affiliations


Mathematics in Image Processing, Computer Graphics, and Computer Vision

T. Chan

The research areas of Image Processing (IP), Computer Graphics (CG) and Computer Vision (CV) are emerging inter-related computer science subdisciplines that offer tremendous intellectual opportunities for the mathematical sciences. In fact, one can argue that together they offer a unique opportunity. There are several reasons for this.

First, IP/CG/CV has many applications, ranging from medical imaging (PET, MRI, fMRI), to astronomical imaging, to virtual reality and special effects in digital entertainment, to robotics. It is a fundamental component of the ongoing information technology revolution. Second, as can be seen from the few examples mentioned above, the applications cut across many different areas of science and technology. Third, just as importantly, this application domain also cuts across many different areas of mathematics, including analysis, geometry, computational mathematics, probability and statistics, and discrete mathematics.

Research in this area has traditionally been conducted by computer scientists and electrical engineers, many of whom are also adept at mathematical skills. In spite of this, mathematicians can bring new ideas, techniques and perspectives to this important area. One of the powers of mathematics is to act as a medium to translate ideas from one scientific area to another. A striking example of this is the recent emergence of the use of PDE and CFD techniques in IP/CG/CV, which have had a tremendous impact in the field. Mathematical scientists can also step back from the immediacy of the applications and study the more fundamental, structural foundation of the key concepts and techniques, which will pave the way for a deeper understanding and future breakthroughs. Conversely, the IP/CG/CV area offers many challenges and new problems and concepts for mathematics. Thus, strengthening the interaction between mathematics and IP/CG/CV is mutually beneficial.

It should be noted that the Board on Mathematical Sciences of the National Academy of Sciences recently (April 2000) conducted a two-day workshop on "The Interface of Three Areas of Computer Science with the Mathematical Sciences" in which IP/CG/CV was one of three topics selected for panel discussions (the author was the moderator). (See http://www.cs.umd.edu/~oleary/nasworkshop.html for a summary.)

The three areas of IP, CG and CV are fundamentally related. CG attempts to generate two-dimensional images from a three-dimensional world and CV consists of the inverse problem of reconstructing models of the three-dimensional world from single or multiple images of different views. IP is used to either enhance the images generated in the forward CG problem, or as pre-processing to help solve the inverse CV problem.

 

Last Modified:
Jan 24, 2013
 

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