20060811-Discrete cosine transforms, JPEG, and software compatibility

来源：互联网发布：尼康官方调焦软件编辑：程序博客网时间：2024/05/22 12:15

原文：http://blogs.mathworks.com/steve/2006/08/11/discrete-cosine-transforms-jpeg-and-software-compatibility/

The function dct2, which computes the two-dimensional discrete cosine transform (DCT), appeared in the original 1993 version of the toolbox. The authors of the toolbox relied on several authoritative references, includingTwo-Dimensional Signal and Image Processing by Lim. This book contains the following one-dimensional DCT equation:

$C_x(k)=\sum_{n=0}^{N-1} 2x(n) \cos \frac{\pi}{2N} k(2n+1), 0 \leq k \leq N-1$

The original implementation of dct2 used this formula.

At roughly the same time as the first toolbox release, the newly-standardized JPEG image compression method and format was becoming popular. The DCT is the mathematical heart of the JPEG algorithm. Here's the 8-point one-dimensional DCT equation that appears in JPEG: Still Image Data Compression Standard by Pennebaker and Mitchell.

$S(u)=\frac{C(u)}{2}\sum_{x=0}^7 s(x) \cos [(2x+1)u \pi/16]$

where

$C(u)=\left\{\begin{matrix} 1/ \sqrt{2} & \textrm{for} \ u=0 \\ 1 & \textrm{for} \ u>0 \end{matrix}\right.$

The variable name differences are trivial, but the scale factor differences are very important! The JPEG DCT is an orthonormal transform, whereas the DCT defined in Lim's book is not.

0 0