Image Compression. Image Data

Transcription

1 Image Compression Image Data Inherently large M*N*bytes_per_pixel Consider this 30 frames per second video 640x480 (R-G-B-A) frame 4 bytes per pixel equals ~ MB per frame 30MB for second Over GB for minute

2 Image Compression We need a way to compress image data Conserve bandwidth For backing store (disk drive, CD-ROM) Network transmission Data Compression Two categories Information Preserving Error free compression Original data can be recovered completely Lossy Original data is approximated Less than perfect Generally allows much higher compression 2

3 Basics Data Compression Process of reducing the amount of data required to represent a given quantity of information Data vs. Information Data and Information are not the same thing Basics Data vs. Information Data the means by which information is conveyed various amounts of data can convey the same information Information A signal that contains no uncertainty 3

4 Example of data vs. information Dataset Dataset 2 NxN image circle(0,0,r) NxN*byte (pixel data) 24* byte (24 ascii characters to store the above text) Redundancy Redundancy data that provides no relevant information data that restates what is already known For example say N and N2 denote the number of data units in two sets that represent the same information Relative redundancy of the first set of data N is R D = - /C r where C r is the Compression Ratio C r = n / n2 4

5 Redundancy Relative Redundancy of N to N2 R D = - /C r Compression Ratio C r = n / n2 if (n2 = n) C r =, R D = 0 Compression Ratio is, implying there is no redundancy if (n2 << n) Highly redundant data N2 is much more compact than N i.e., high compression ratio N = 0 & N2 = Example Compression Ratio C R = N/N2 = 0 (or 0:) R D = - (/0) or 0.9 Implying 90% of the data in N is redundant 5

6 Compression Given a set of N data units Try to remove redundant data that is not needed to express the underlying information Hopefully your new set, N N << N Image Compression Three basic types of redundancies Coding Redundancy Interpixel Redundancy Psycho-visual Redundancy Image compression Reduce one or more of these redundancies 6

7 Coding Redundancy Recall the Histogram p r (r k ) = n k /n k=0,,2,.., L- L is the number of gray levels n k is the number of pixels with gray level r k n is the total number of pixels p r (r k ) is the probability that r k will occur in this image Let L(r k ) = the number of bits needed to represent the gray level r k Coding Redundancy Let L(r k ) = the number of bits needed to represent the gray level r k The average number of bits needed to represented each pixel is Lavg = Σ L(r k ) p r (r k ) Thus, the total number of bits needed to code an image N*M*Lavg 7

8 Coding Redundancy Example Consider an image with 8 gray levels L(r k ) is a natural binary coding set m-bit counting sequence ie (2 m ) thus, to encode 8 distinct codes needs m = 3 Each pixel uses 3 bits Lavg = 3bits Coding Redundancy However, consider using the codes in this table Lavg = Sum l 2 (r k )p r (r k ) = 2(9) + 2(0.25) + 2(0.2) + 3(6) + 4(0.08) + 5(0.06) + 6(0.03) + 6(0.02) Lavg = 2.7 bits 8

9 Coding Redundancy Lavg = 3 bits L2avg = 2.7 bits Cr = Lavg/L2avg = 3/2.7 or. Rd = - /. = Thus approximately 0 percent of the data in L is redundant Variable Length Coding Assigning fewer bits to more probable symbols achieves compression This process is commonly referred to as variable-length-coding What is the pitfall? Equally probable symbols Generally no gain.. Then don t encode... Need to know how to choose the symbol 9

10 Variable Length Coding and Compression If gray-levels of an image are coded using more symbols than absolutely necessary It is said to contain coding redundancy Variable length coding is very powerful and is frequently applied Not just for images Characters strings too Gzip, Zip, etc use a form of variable length coding Interpixel Redundancy Think of an average binary image The value of a pixel can be reasonably predicted from its neighbors Sometimes called Spatial Redundancy Geometric Redundancy Interframe Redundancy Well call it interpixel redundancy 0

11 Interpixel Redundancy In order to reduce redundancy in 2D pixel arrays We can convert them into more efficient, non-visual format Transforms of this type are called mappings they are reversible if the original image elements can be reconstructed from the transformed data sets Run-length Encoding Each scan line, y, of an binary image f(:,y) is encoded as a sequence of 0 and s f(0,y) f(,y) f(2,y)... f(n-, y) An alternative encode as a pair of sequences (g,r) (g2,r2).. (gi,ri)... (gn,rn) where g encodes the gray-level (0 or ) and ri encodes the run length in pixels

12 Example Threshold Example The entire image in previous example 024x343 pixels ( bit) required only 2,66 runs each run requires bits to encode N = 024x343 (the binary image) N2 = 2,66 * (run length encoded image) Cr = (024)(343)/(2,66*) = 2.63 Rd = - /2.63 = 0.62 N (the binary image) is roughly 62% redundant 2

13 Psychovisual Redundancy Brightness as perceived by the eye is dependent on many things The fact is, the eye does not respond with equal sensitivity to all visual information Some information has less relative importance than other information This information is said to be psychovisually redundant It can be eliminated without significantly impairing visual quality Psychovisual Redundancy Unlike Coding and Interpixel Redundancy Psychovisual is associated with real or quantifiable visual information It is possible only because the elimination of this data does not effect the visual process But we lose quantifiable information Thus, this type of reduction is called quantification And it is lossy 3

14 Lots of examples Improved Gray-Scale Quantization (IGS) Maps 8 bits to 4 bits Adds some random noise to pixels before quantization This noise helps removes our sensitivity to edges created by quantization Original Straight Quantization IGS Quantization Lots of examples Color Quantization is very common MPEG and (digital video) DV Standards Terms like 4:2:2 For every 4 intensity pixels only 2 color pixels are stored (in X and Y direction) 4:: For every 4 intensity pixels only color pixels are stored (in X and Y direction) 4

15 Temporal Interlacing TV signal is interlaced frame is composed of 2 fields one with odd scanlines one with even scanlines Frame rate is 29.9 per second (NTSC) 24 per second (PAL) But ~60 fields per second are displayed Exploits temporal and spatial coherency progressive scan Full frame Interlacing Odd Even (De-interlacing) 5

16 Fidelity Criteria Psychovisual data reduction results in a loss of quantitative visual information Because information of interest may be lost a repeatable or reproducible means of quantifying the nature and extent of information loss is desirable Fidelity Criteria Two class of criteria are used Objective fidelity criteria Subjective fidelity criteria Objective loss can be expressed as a function of the input image and the resulting compressed (then decompressed) image 6

17 Objective Fidelity Criteria Let f(x,y) be the input image Let f (x,y) be the approximated f(x,y) from compression/decompression For any value of x and y, the error e(x,y) between f(x,y) and f (x,y) is e(x,y) = f (x,y) - f(x,y) Objective Fidelity Criteria The total error between the two images e = M x= 0 N y= 0 [ f '( x, y) f ( x, y)] The Root-Mean-Square Error e rms = M 2 [ [ '(, ) (, )] ] N f x y f x y 2 MN x= 0 y= 0 7

18 Objective Fidelity Criteria Commonly used fidelity criteria is mean-square signal-to-noise ratio defined as: SNR x= 0 y= 0 ms = M N x= 0 y= 0 M N f '( x, y) [ f '( x, y) f ( x, y)] 2 2 signal noise (Mean squared error) Subjective Fidelity Criteria Human-based criteria Usually side-by-side comparison 8

19 Image Compression Models f(x,y) Source encoder Channel encoder Channel Channel decoder Source decoder f (x,y) f(x,y) Mapper Quantizer Symbol Encoder Symbol decoder Inverse Mapper f (x,y) Loss Error-Free Image Compression (Lossless Compression) 9

20 Error-Free Compression To reduce only coding redundancy Most popular technique is Huffman Encoding This is a variable-length coding technique Huffman encoding yields the smallest possible number of code symbols per source symbol Optimal for a fixed number of source symbols subject that source symbols are coded one at a time Huffman Encoding Huffman encoding Perform a series of source reductions Ordering the probabilities of the symbols occurrence combine the lowest probable symbols into a single symbol that replaces them in the next reduction assigns codes based on these reduction This is variable-length coding So, we will have to build variable-length codes 20

21 Huffman Encoding (Reduction) a2 a6 a a4 a3 a5 Original Source Symbol Probability Source Reduction 3 4 Huffman Encoding (Reduction) a2 a6 a a4 a3 a5 Original Source Symbol Probability Source Reduction 3 4 2

22 Huffman Encoding (Reduction) Symbol a2 a6 a a4 a3 a5 Original Source Source Reduction Probability Huffman Encoding (Reduction) a2 a6 a a4 a3 a5 Original Source Symbol Source Reduction Probability

23 Huffman Encoding (Reduction) a2 a6 a a4 a3 a5 Original Source Symbol Source Reduction Probability Code assignment Original Source Source Reduction Sym Prob Code a a6 a 0.2 a4 a a

24 24 Code assignment 0.04 a a3 a a 00 a a Code Prob Sym Source Reduction Original Source Code assignment 0.04 a a3 0 a a a a Code Prob Sym Source Reduction Original Source

25 25 Code assignment 0.04 a a a a a a Code Prob Sym Source Reduction Original Source Code assignment a a a a a a Code Prob Sym Source Reduction Original Source

26 Huffman Encoding Uniquely Decodable a2 Symbol Table consider a6 a a a3 a a2 a2 a6 a3 a Huffman Encoding Very powerful encoding scheme Produces the optimal codes Given an input symbol set And its known distribution Need to store symbol table along with compressed data! 26

27 Error Free Interpixel Encoding Run Length Encoding d d2 d3 We call this a run d,d2,d3,d4,d5,d6,d7,.... Distances alternate between white and black 27

28 Run-length coding variations Divide image into 2D regions If all white, assign a If all black, assign a 00 Otherwise, code the block using some technique (D RLE, etc) Previous Row RAC Relative address coding c Current Row e c Look at white to black transitions In current row, position c is the first transition This run is length ec Examine previous row at position c Find the first transition before c, c this distance is c c 28

29 Previous Row RAC Relative address coding c Current Row e c Encode the smallest distance, d = min( ec, c c ) You ll need to a flag to denote which distance is used (, or 0) Need to handle boundaries The idea is to reduce the number of bits needed to store the run Predictive Encoding Consider a gray-scale image Try to predict what a pixel s value is going to be Only encode the difference between the predicted value and the real value 29

30 Predictive Encoding f(x,y) + e(x,y) Compressed Image Predictor Nearest integer f (x,y) e(x,y) = f(x,y) f (x,y) Predictive Decoding Compressed Image e(x,y) + f(x,y) Predictor f (x,y) 30

31 Predictor Various local, global, and adaptive methods can be used Most common is a linear combination of m previous pixels f '( m x, y) = round α i f i= ( x i, y) fixed weights Predictor Differential Coding If previous pixel is used we call this differential coding m= and α= I Encode the difference Can reduce symbol set size Helps variable length coding x 3

32 Example Original f(x,y) e(x,y) Differential Coding Interpixel Coding Encoding Interpixel coding Is usually followed by variable-length coding RLE or Predictive Huffman Source Encoder 32

33 Image Compression Models f(x,y) Source encoder Channel encoder Channel Channel decoder Source decoder f (x,y) f(x,y) Mapper Quantizer Symbol Encoder Symbol decoder Inverse Mapper f (x,y) Loss Image Compression Schemes Error Free Compression (often called lossless compression) Data can be recovered completely Some applications require this type of image compression Medical imagery Digital Preservation Digital archives of black and white documents However, most compression schemes allow for loss in order to achieve high compression rates 33

34 Next Lecture Lossy Compression Quantization Transform-based compression With Quantization JPEG standard 34