Project 5
Compression of an Image
By Austin Alderete
Introduction
This project explores using the Discrete Cosine Transform (DCT) alongside a method of quantization to reduce the number of bits required to store a distinguishable image. In other words, the aim of this project will be to develop a method of image compression with information loss acceptable to the human eye. In particular, we will use the Discrete Cosine Transform and quantization to perform the compression. When we refer to the YQ matrix, we are refering to the quantitized transformed matrix. For more information on the DCT and its application, click here.
Main Problem
- For computer problem 11.2.3, we are asked to obtain a grayscale image of our choice and use the imread command to import the image into MATLAB. The image in question was cropped using ImageJ to be 400 by 440 pixels in dimension. The quantities chosen were not special, but did ensure that the image dimensions were multiples of 8.
Code for this problem.
As requested, we begin by observing a single 8x8 block before and after the compression. See here.
The compression was then carried out for the entire grayscale image. See here.
The same sort of compression was also carried out for the individual R,G, and B channels and then recombined to produce a complete image. See here.
- We also are requested to provide the YQ matrix for p=1, p=2, and p=4. This is done here for the first 8x8 block only. Thus:
\(p=1\):
\[
\left(
\begin{matrix}
34 & -11 & 0 & 0 & 0 & 0 & 0 & 1\\
4 & -4 & -1 & 1 & 0 & 0 & 0 & 0\\
-2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & -1 & 0 & 0 & 1 & 0 & 0 & 0\\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{matrix}
\right)
\]
\(p=2\):
\[
\left(
\begin{matrix}
17 & -6 & 0 & 0 & 0 & 0 & 0 & 0\\
2 & -2 & -1 & 0 & 0 & 0 & 0 & 0\\
-1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{matrix}
\right)
\]
\(p=4\):
\[
\left(
\begin{matrix}
8 & -3 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{matrix}
\right)
\]
- We are then asked to repeat the previous problem, but only for the grayscale image using the JPEG-suggested matrix. See here for the matrix and results respectively.
- Finally, the color image has its RGB values transformed to luminance and color difference coordinates. Following the guidelines presented on page 514, the image is decomposed, compressed, decompressed, and recombine as a color image.
Doing so, we obtain the following:
Uncompressed
Baseline JPEG Compression
The code used can be found here.
Conclusion
All told, the 2D-DCT was effectively used to carry out the compressions. The heavier the compression, the easier it is to store the data, but the more information is lost. However, the JPEG baseline applies a heavy compression with less human perceptive loss.