The University of North Carolina at Chapel Hill
                     Department of Computer Science

                      FALL 1994 COURSE ANNOUNCEMENT

COMP 290-066:  Theory and Implementation of Ridges
Instructor: Dr. David Eberly
Times: 11:00-12:15 T Th SN 325 (115 available as needed)
Credits:  2 or 3 
(it's scheduled to permit 3, but actual credit may be negotiated with instructor)
Call number: 10043

Prerequisites:  Background in Calculus and Linear Algebra.  
  Numerical Analysis (COMP 151) will be helpful, but not essential.

Description:
  The definition of "ridges" for a real-valued function of N independent
  variables is an extension of the definition for strict local maxima.
  Intuitively, a strict local maximum is a point for which the function has
  a local maximum in N linearly independent directions.  These points are
  isolated and can be thought of as 0-dimensional sets.  A D-dimensional
  ridge point is a point for which the function has a local maximum in N-D
  linearly independent directions.  The choices of ridge dimension and the
  directions are typically made based on the geometric nature of the
  application.  The selection of the word "ridge" is motivated by the case
  N=2 and D=1.  If you consider the graph of the function as mountainous
  terrain, the strict local maxima correspond to mountain peaks.  The
  1-dimensional sets obtained by finding local maxima in special directions
  resemble the ridges of the terrain.

  The concept of ridges appears to be useful in many applications.  In
  particular, the Medical Image Display Research Group at UNC uses ridges for
  determining medial and boundary representations for objects in medical
  images.  Other applications which use ridge theory include building
  representations of arterial structures from angiograms, determining
  molecular structure from potential energy data, and finding locations of
  high pressure in air flow over wings.

Outline:
  (1) Present the basic theory of ridges (as a continuous model).
  (2) Discuss how to formulate the definitions in a way that allow
      computation of ridges from discrete data.
  (3) Numerical implementation of ridge construction.  Topics which will
      be covered in detail are:
      (a) B-spline interpolation (continuous representation of discrete data)
      (b) Householder method for solving eigensystems
      (c) Gradient descent algorithms
      (d) Methods for solving systems of ordinary differential equations
  (4) Application to medical image analysis
  (5) Application to molecular modeling
  (6) Application to fluid dynamics (if time permits)