Image and Multidimensional Signal Processing

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1 Image and Multidimensional Signal Processing Professor William Hoff Dept of Electrical Engineering &Computer Science

2 Image Compression 2

3 Image Compression Goal: Reduce amount of data, for transmission or storage Try to preserve information, so that image can be reconstructed: Exactly (loss-less compression) Approximately (lossy compression) Compression ratio C b b ' where b = # bits uncompressed b = #bits compressed 3

Redundancy We take advantage of redundancy in the input image: Coding redundancy Spatial and temporal redundancy Irrelevant information Coding redundancy: Only a few gray values are present; we can

4 Redundancy We take advantage of redundancy in the input image: Coding redundancy Spatial and temporal redundancy Irrelevant information Coding redundancy: Only a few gray values are present; we can represent using short code words (with few bits) Spatial redundancy: All gray values are present, but pixels along each row are the same; if you know the value of the first pixel you know the whole row Irrelevant information: Only a few gray values are present, clustered around value 128; could represent using a constant 128 and would look the same 4

5 Coding Redundancy Use short code words instead of long ones Can use variable length codes, so that most common values have shortest codes Average bit length L 1 L l( r ) p ( r ) avg k r k k 0 5

6 Measuring Image Information We can compute the theoretical minimum number of bits needed to code an image First, consider an event E If the event is unlikely to happen, then its probability P(E) is small, and 1/P(E) is large When the event does occur, it is significant and provides us with a lot of information We define the self information of event E I 1 E) log 2 log ( P( E)) P( E) ( 2 This is how much information is attached to E Example: If P(E) = 1, I = 0 bit If P(E) = ½, I = 1 bit If P(E) = ¼, I = 2 bits 6

7 Measuring Image Information Say we have a set of random events, drawn from a possible set of values {a 1, a 2,, a J } We ll call these the source symbols They could be randomly generated from a transmitter The probabilities of generating these symbols are {P(a 1 ), P(a 2 ),, P(a J )} The information carried by a single symbol is -log 2 (P(a j )) Thus, the average information per symbol is J H P( a j)log P( a j) j 1 Assumes that events are independent 7

8 Entropy of an image For images, the events {r 0, r 1,, r L-1 } are possible gray level values We can use the histogram to estimate the probabilities of the symbols The average information per pixel is (also called the entropy of the image) Examples: L 1 H pr( rk )log 2 pr ( rk ) k 0 8 gray levels, same probability for all values: p0=p1= = p7=1/8. H=? This is the best you can do (for uncorrelated values) 8 gray levels, but only one has nonzero probability, say level 1: p0=p2= = p7 = 0, p1=1. H=? 8

9 Entropy - example L 1 H pr( rk )log 2 pr ( rk ) k 0 rk P(ai) log P(ai) -P log P H (bits)

10 Fidelity Criteria RMS (root mean square) error an objective measure of error between original and the compressed image e rms 1 MN M 1N 1 x 0 y 0 fˆ( x, y) f ( x, y) 2 1/ 2 Can also use SNR SNR And subjective measures M 1N 1 x 0 y 0 ms M 1N 1 2 x 0 y 0 fˆ( x, y ) fˆ( x, y ) f ( x, y ) E.g., rate quality on a scale of 1 to

11 Image Compression & Decompression Model Mapper: Transforms data to a form that can be more easily compressed (eg, Fourier or wavelet transform) Quantizer: Reduces amount of data (eg., throws away smallest transform coefficients) Symbol coder: Codes the resulting data using the shortest code words (eg., variable length coding) 11

12 Loss-less Compression Methods Loss-less : the compressed image can be reconstructed exactly We ll look at these methods: Huffman coding Arithmetic coding LZW coding Run-length coding 12

13 Huffman Coding Takes advantage of coding redundancy Generates a variable length code, as close as possible to the theoretical minimum length Doesn t take advantage of inter-pixel redundancy Widely used as a component (the symbol coder) in many compression methods Algorithm: (1) Find the gray level probabilities (2) Order the probabilities, from smallest to largest (3) Combine the smallest two by addition (4) Repeat steps 2-3 until only two probabilities are left (5) By working backward along the tree, generate code by alternating assignment of 0 and 1 13

14 Example Start with gray level probabilities Sort the probabilities, from largest to smallest Combine the smallest two by addition Symbol Probability a1 0.1 a2 0.4 a a4 0.1 a a

15 Example 15

16 Example Work backward along the tree, generate code by alternating assignment of 0 and 1 Final result: Sym Code 16

17 Example generate code sequence for symbols a2 a1 a3 a1 Example Example decode the sequence Average length of code? 17

18 Example Compare to theoretical minimum H (z) J j 1 P( a j )log2 P( a j ) P(ai) log P(ai) -P log P a2 0.4 a6 0.3 a1 0.1 a4 0.1 a a H 18

19 Example Compare to theoretical minimum H (z) J j 1 P( a j )log2 P( a j ) P(ai) log P(ai) -P log P a a a a a a H

20 Example Find Huffman code for Algorithm: (1) Find the gray level probabilities (2) Sort the probabilities (3) Combine the smallest two by addition (4) Repeat steps 2-3 until only two are left (5) Work backward, generate code Gray Level Probability Lavg =? H =? 20

21 Example Find Huffman code for Algorithm: (1) Find the gray level probabilities (2) Sort the probabilities (3) Combine the smallest two by addition (4) Repeat steps 2-3 until only two are left (5) Work backward, generate code Gray Level Probability

22 Arithmetic Coding A sequence of values is assigned a single arithmetic code word The code word is a fractional number between 0 and 1 (e.g., ) Each symbol is assigned an interval based on its probability of occurrence Code words are fixed length 22

23 Example Message: a 1 a 2 a 3 a 3 a 4 With each new interval, you find the subintervals by multiplying the total interval by the probability of each symbol, and then add it to the lower bound of the interval to get the upper bound. 23

24 Example Message: a 1 a 2 a 3 a 3 a ( )(0.8) ( )(0.8) Can use (0.4)

25 Example Final code word: Three decimal digits for five symbols, or 3/5 = 0.6 digits per symbol Equivalently, 1.99 bits = 2 x -> x = 1.99 bits Theoretical minimum: 1.92 bits or 0.58 digits P(ai) log P(ai) -P log P a a a a H

26 Example The sequence a1 a1 a1 can be encoded as what number? 26

27 LZW Coding Stands for Lempel-Ziv-Welch Works by coding short strings of data Used in GIF, TIFF, and PDF file formats Creates a dictionary of code words For an 8-bit image, the first 256 words are assigned to the gray values 0,1,2,, 255 As sequences are discovered, new code words (i.e., 256 through 511) are assigned to represent them Eg: The sequence may be assigned to code word

28 Example Original image Initial Dictionary Dictionary Index Entry : : ? : : 511? 28

29 29

30 Final Dictionary Original image Coded sequence Dictionary Index Entry : :

31 Run Length Coding Inter-pixel redundancy Usually used for binary images Output the number of consecutive 0 s along a row, then the # of 1 s, etc : Output code Best case for compression: a row is all zeros (or all ones) Worst case for compression? 31

32 RLC Applied to Non-binary Images For an m-bit image, the gray values are a m-1 2 m-1 + a m-2 2 m a a Apply RLC separately to each bit plane This can be a problem in areas where values fluctuate about certain transition points Example: = But = So each bit keeps flipping from 0 to 1 32

33 Example 33

34 Better: Use Gray Code First code the image using the Gray Code rather than the normal binary code Gray code: Each adjacent code word differs only by one bit So if a pixel differs from its neighbor by one gray level, only one bit in the gray code is different 34

35 Gray Codes 2 bit binary 2 bit Gray 3 bit Gray

36 36

ECE472/572 - Lecture 11. Roadmap. Roadmap. Image Compression Fundamentals and Lossless Compression Techniques 11/03/11.

ECE472/572 - Lecture 11. Roadmap. Roadmap. Image Compression Fundamentals and Lossless Compression Techniques 11/03/11. ECE47/57 - Lecture Image Compression Fundamentals and Lossless Compression Techniques /03/ Roadmap Preprocessing low level Image Enhancement Image Restoration Image Segmentation Image Acquisition Image