1. Digital Image Processing Lecture 22: Image Compression
Prof. Charlene Tsai
*Section 8.4 in Gonzalez

2. Starting with Information Theory
Data compression: the process of reducing the amount of data required to represent a given quantity of information.
Data information
Data convey the information; various amount of data can be used to represent the same amount of information.
E.g. story telling (Gonzalez pg411)
Data redundancy
Our focus will be coding redundancy

3. Coding Redundancy
Again, we’re back to gray-level histogram for data (code) reduction
Let rk be a graylevel with occurrence probability pr(rk).
If l(rk) is the # of bits used to represent rk, the average # of bits for each pixel is

4. Example on Variable-Length Coding
Average for code 1 is 3, and for code 2 is 2.7
Compression ratio is 1.11 (3/2.7), and level of reduction is

5. Information Theory
Information theory provides the mathematical framework for data compression
Generation of information modeled as a probabilistic process
A random event E that occurs with probability p(E) contain
units of information (self-information)

6. Some Intuition I(E) is inversely related to p(E)
If p(E) is 1 => I(E)=0
No uncertainty is associated with the event, so no information is transferred by E.
Take alphabet “a” and “q” as an example. p(“a”) is high, so, low I(“a”); p(“q”) is low, so high I(“q”).
The base of the logarithm is the unit used to measure the information.
Base 2 is for information in bit

7. Entropy Measure of the amount of information
Formal definition: entropy H of an image is the theoretical minimum # of bits/pixel required to encode the image without loss of information
where i is the grayscale of an image, and pi is the probability of graylevel i occurring in the image.
No matter what coding scheme is used, it will never use fewer than H bits per pixel

8. Variable-Length Coding
Lossless compression
Instead of fixed length code, we use variable-length code:
Smaller-length code for more probable gray values
Two methods:
Huffman coding
Arithmetic coding
We’ll go through the first method

9. Huffman Coding
The most popular technique for removing coding redundancy
Steps:
Determine the probability of each gray value in the image
Form a binary tree by adding probabilities two at a time, always taking the 2 lowest available values
Now assign 0 and 1 arbitrarily to each branch of the tree from the apex
Read the codes from the top down

10. Example The average bit per pixel is 2.7 How to decode the string
Much better than 3,
originally
Theoretical minimum
(entropy) is 2.7
How to decode the string
Huffman codes are uniquely decodable.
Gray value
Huffman code
0 (0.19) 00
1 (0.25) 10
2 (0.21) 01
3 (0.16)
110
4 (0.08)
1110
5 (0.06)
11110
6 (0.03)
111110
7 (0.02)
111111

11. LZW (Lempel-Ziv-Welch) Coding
Lossless Compression
Compression scheme for Gif, TIFF and PDF
For 8-bit grayscale images, the first 256 words are assigned to grayscales 0, 1, …255
As the encoder scans the image, the grayscale sequences not in the dictionary are placed in the next available location.
The encoded output consists of dictionary entries.

12. Example Consider the 4x4, 8-bit image of a vertical edge 39 39 126 126
A 512-word dictionary
starts with the content
Dictionary location
Entry 1
255
256
----
511
--- … … … …

13. To decode, read the 3rd column from top to bottom

14. Run-Length Encoding (1D)
Lossless compression
To encode strings of 0s and 1s by the number or repetitions in each string.
A standard in fax transmission
There are many versions of RLE

15. (con’d) Consider the binary image on the right Method 1:
(123)(231)(0321)(141)(33)(0132)
Method 2:
(22)(33)(1361)(24)(43)(1152)
For grayscale image, break up the image first into the bit planes.

16. Problem with grayscale RLE
Long runs of very similar gray values would result in very good compression rate for the code.
Not the case for 4 bit image consisting of randomly distributed 7s and 8s.
One solution is to use gray codes.

17. Example in pg 400 For 4 bit image, Bit planes are:
Binary encoding: 8 is 1000, and 7 is 0111
Gray code encoding: 8 is 1100 and 7 is 0100
Bit planes are:
Uncorrelated
Highly correlated 1 1 1
0th, 1st, and 2nd binary bit plane
3rd binary bit plane
0th and 1st gray code bit plane
(replace 0 by 1 for 2nd plane)
3rd gray code bit plane

18. Summary Information theory Lossless compression schemes
Measure of entropy, which is the theoretical minimum # of bits per pixel
Lossless compression schemes
Huffman coding
LZW
Run-Length encoding