Mathematics of computer vision: course

1 Description

This is an introduction into the mathematics of computer vision and digital image analysis: topology and geometry. (For a more advanced approach see Introductory algebraic topology: course, more math here.)

The exposition is geared towards CS students and potential and current math majors. The material is self-contained beyond high school math (no calculus). Special attention is paid to the algorithmic implementation of the mathematics. A variety of applications is considered in detail.

2 Prerequisites

Calculus I, computer language preferred but not required.

3 Lectures

A large part of the lectures is now a part of a draft of a book called Topology Illustrated.

Introduction: analysis of visual information and Examples of image analysis

• Digital images
  • Raster representation of images: binary, gray scale, and color
  • Basics of image processing: noise, blur, contrast, sharpness etc
  • Available software (MS Paint, Photoshop, ImageJ, MATLAB, CHomP, jPlex, Pixcavator)
• Topology and geometry of binary images
  • Cell decomposition of images
  • Detection of objects (connected components)
  • Digital measuring and the main geometric characteristics of objects: area, perimeter, roundness, centroid, etc
  • Applications
• Topology and geometry of gray scale images
  • Thresholding and detection of objects
  • Graph representation of the topology - the topology graph
  • More digital measuring and characteristics of objects: intensity, center of mass, standard deviation, etc
  • Applications
• 3D image analysis
  • Euler number
  • Betti numbers
  • Homology of cubical complexes
  • Homology software
• Miscellaneous topics
  • Motion tracking
  • Other image segmentation methods
  • Stereo vision
  • Fourier transform
• Analysis of color images and other multi-parameter images
• Robustness of image analysis
  • Robustness under image transformations: noise, blur, translation, rotation
  • Robustness of topology under dilation and erosion
  • Robustness of geometry under dilation and erosion

4 Review

• What is the difference between binary and gray scale images?
• How do you count holes in a binary image?

5 Projects

C++, Java, or MATLAB

• Modification of the algorithms (construction of the topology graph, analysis of the topology graph);
• Implementation of the algorithms (motion, color, stereo, 3D, etc);
• Practical applications of the algorithms (based on Pixcavator SDK or CHomP).

6 Notes

Software: Pixcavator Student Edition (free)

This page can also be reached by typing CVmath.com or mathCV.com.