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

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

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

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

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)

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

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

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

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

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 11011101111100111110 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

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.

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

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

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

(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. 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1

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.

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

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