This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Mathematics of computer vision: course

### From Mathematics Is A Science

## 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
- 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*.