The Discrete Cosine Transform (UMAP)

TARGET AUDIENCE:
Students in introductory linear algebra, particularly students in engineering and communication theory. The Module can be also used by motivated high school students studying trigonometry and matrices.

ABSTRACT:
To handle huge amounts of information efficiently, the information must be digitized and compressed. A few basic concepts of linear algebra, together with highschool trigonometry, let us efficiently compress and decompress information. We give details of the Discrete Cosine Transform, a major ingredient in the compression used in JPEG images and YouTube videos.

Table of Contents

1. Introduction
2. Why You Should Be Interested
3. Transmitting Optical Images
3.1 Color
3.2 Pixels
4. Discrete Cosine Transform
4.1 Introduction
4.2 DCT
4.3 One-Dimensional DCT
4.3.1 n = 2
4.3.2 n = 4
4.3.3 n = 8
4.4 Trigonometric Interpolation
4.5 Inverse DCT (IDCT)
4.6 Quantization and Rounding
4.7 Two-Dimensional DCT
5. Project
6. Solutions to Selected Exercises
7. Appendix
7.1 One-Dimensional DCT
7.1.1 DCT Maple
7.1.2 DCT Mathematica
7.1.3 DCT MATLAB
7.2 Two-Dimensional DCT
References
Acknowledgments
About the Authors