This page is a part of CVprimer.com, a wiki devoted to computer vision. It focuses on low level computer vision, digital image analysis, and applications. It is designed as an online textbook but the exposition is informal. It geared towards software developers, especially beginners, and CS students. The wiki contains mathematics, algorithms, code examples, source code, compiled software, and some discussion. If you have any questions or suggestions, please contact me directly.

# The Mathematics of Computer Vision

### From Computer Vision Primer

**MATH 482**

Spring 2009

**Instructor:** Peter Saveliev

An introduction into the mathematics of computer vision and digital image analysis: topology and geometry.

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

**Prerequisites:** Calculus I, computer language preferred but not required.

**Outline:**

Introduction: analysis of visual information

- 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, Pixcavator)

- Topology and geometry of binary images
- Cell decomposition of images
- Detection of objects (connected components)
- Measuring digitally 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
- Main characteristics of objects (average 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

**Evaluation:**

- Two tests (open book)
- Homework and quizzes
- Choice of the final exam or a programming project (based on the course material)

**Types of 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).

**Textbook**:

- Lecture notes posted here
- This wiki

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