Image Compression. Fundamentals: Coding redundancy. The gray level histogram of an image can reveal a great deal of information about the image
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1 Fundamentals: Coding redundancy The gray level histogram of an image can reveal a great deal of information about the image That probability (frequency) of occurrence of gray level r k is p(r k ), p n n k ( r ) = k =,,..., L k
2 Fundamentals: Coding redundancy If the number of bits used to represent each value of r k is l(r k ), the average number of bits required to represent each pixel is L avg L = k = l ( r ) p( r ) k To code an MxN image requires MNL avg bits k
3 Fundamentals: Coding redundancy some pixel values more common than others 3
4 Fundamentals: Coding redundancy L avg L = k = l ( r ) p( r ) To code an MxN image requires MNL avg bits k k If m-bit natural binary code is used to represent the the gray levels, then L L ( r ) p( r ) = mp( r ) m = Lavg l k k k = k = k = 4
5 Fundamentals: Coding redundancy L avg L = k = l ( r ) p( r ) To code an MxN image requires MNL avg bits k k If m-bit natural binary code is used to represent the the gray levels, then L L ( r ) p( r ) = mp( r ) m = Lavg l k k k = k = k = 5
6 Fundamentals: Coding redundancy Variable length Coding: assign fewer bits to the more probable gray levels than to less probable ones can achieve data compression 6
7 Fundamentals: Interpixel (spatial) redundancy neighboring pixels have similar values Binary images Gray scale image (later) 7
8 Fundamentals: Interpixel (spatial) redundancy: Binary images Run-length coding Mapping the pixels along each scan line into a sequence of pairs (g, r ), (g, r ),, Where g i is the ith gray level, r i is the run length of ith run 8
9 Example: Run-length coding Row : (, 6) Row : (, 6) Row 3: (, 7) (, ) (, 7) Row 4: (, 4), (, 8) (, 4) Row 5: (, 3) (, ) (, 6) (, 3) (, ) Row 6: (,) (, ) (,8) (, ) (, ) Row 7: (, ) (,) (, ) (,) (, ) Row 8: (, 3) (, ) (,3) Row 9: (, 3) (, ) (, 3) Row : (,) (, ) (,) (, ) (, ) Row : (, ) (, ) (, 8) (, ) (, ) Row : (, 3) (, ) (, 6) (, 3) (, ) Row 3: (, 4) (,8) (, 4) Row 4: (, 7) (, ) (, 7) Row 5: (, 6) Row 6: (, 6) encode decode
10 Fundamentals: Psychovisual redundancy some color differences are imperceptible
11 Fidelity criteria Image Compression Root mean square error (e rms ) and signal to noise ratio (SNR): Let f(x,y) be the input image, f (x,y) be reconstructed input image from compressed bit stream, then e ( f '( x, y) ) ( f '( x, y) f ( x, y) ) x= M N / M N x= y= = ( ( ) ( )) rms f ' x, y f x, y SNR = M N MN x= y= y=
12 Fidelity criteria e rms and SNR are convenient objective measures Most decompressed images are view by human beings Subjective evaluation of compressed image quality by human observers are often more appropriate
13 Fidelity criteria Image Compression 3
14 Image compression models 4
15 Exploiting Coding Redundancy These methods, from information theory, are not limited to images, but apply to any digital information. So we speak of symbols instead of pixel values and sources instead of images. The idea: instead of natural binary code, where each symbol is encoded with a fixed-length code word, exploit nonuniform probabilities of symbols (nonuniform histogram) and use a variable-length code. 5
16 Exploiting Coding Redundancy Entropy H L = k = p( r k )log p( r k ) is a measure of the information content of a source. If source is an independent random variable then you can t compress to fewer than H bits per symbol. Assign the more frequent symbols short bit strings and the less frequent symbols longer bit strings. Best compression when redundancy is high (entropy is low, histogram is highly skewed). 6
17 Exploiting Coding Redundancy Two common methods Huffman coding and, LZW coding 7
18 Exploiting Coding Redundancy Huffman Coding Codebook is precomputed and static. Compute probabilities of each symbol by histogramming source. Process probabilities to precompute codebook: code(i). Encode source symbol-by-symbol: symbol(i) -> code(i). The need to preprocess the source before encoding begins is a disadvantage of Huffman coding 8
19 Exploiting Coding Redundancy Huffman Coding 9
20 Exploiting Coding Redundancy Huffman Coding
21 Exploiting Coding Redundancy Huffman Coding Average length of the code is..bits/symbol The entropy of the source is.4 bits/symbol
22 Exploiting Coding Redundancy Huffman Coding
23 Exploiting Spatial/Interpixel Redundancy Predictive Coding Image pixels are highly correlated (dependent) Predict the image pixels to be coded from those already coded 3
24 Exploiting Spatial/Interpixel Redundancy Predictive Coding Differential Pulse-Code Modulation (DPCM) Simplest form: code the difference between pixels Original pixels: 8, 83, 86, 88, 56, 55, 56, 6, 58, 55, 5, DPCM: 8,, 3,, -3, -,, 4, -, -3, -5, 4
25 Exploiting Spatial/Interpixel Redundancy Predictive Coding Key features: Invertible, and lower entropy (why?) H I (k) H D (k) K- -K K- image histogram high entropy (high image entropy) DPCM reduced histogram entropy (low image entropy) 5
26 Exploiting Spatial/Interpixel Redundancy Higher Order (Pattern) Prediction Use both D and D patterns for prediction D Causal: D Causal: D Non-causal: D Non-Causal: 6
27 Exploiting Spatial/Interpixel Redundancy D Causal: 7
28 Quantization Quantization: Widely Used in Lossy Compression Represent certain image components with fewer bits (compression) With unavoidable distortions (lossy) Quantizer Design Find the best tradeoff between maximal compression minimal distortion 8
29 Quantization Scalar quantization Uniform scalar quantization: Non-uniform scalar quantization:
30 Quantization Scalar quantization 3
31 Quantization Vector quantization and palletized images (gif format) 3
32 Palletized color image (gif) A true colour image 4bits/pixel, R 8 bits, G 8 bits, B 8 bits 6776 possible colours A gif image - 8bits/pixel 56 possible colours 3
33 Palletized color image (gif) Exploits psychovisual redundancy A true colour image 4bits/pixel, R 8 bits, G 8 bits, B 8 bits 6776 possible colours A gif image - 8bits/pixel 56 possible colours 33
34 Palletized color image (gif) (r, g, b) r g b r g b Record the index of that colour (for storage or transmission) For each pixel in the original image Find the closest colour in the Colour Table r 55 g 55 b 55 Colour Table To reconstruct the image, place the indexed colour from the Colour Table at the corresponding spatial location 34
35 Palletized color image (gif) (r, g, b) r g b r g b How to choose the colours in the table/pallet? Record the index of that colour (for storage or transmission) For each pixel in the original image Find the closest colour in the Colour Table r 55 g 55 b 55 Colour Table To reconstruct the image, place the indexed colour from the Colour Table at the corresponding spatial location 35
36 Construct the pallet (vector quantization, k-means algorithm) (r, g, b) B A pixel corresponding to A point in the 3 dimensional R, G, B space R G 36
37 Construct the pallet (vector quantization, k-means algorithm) B Map all pixels into the R, G, B space, clouds of pixels are formed R G 37
38 B Image Compression Group pixels that are close to each other, and replace them by one single colour Construct the pallet (vector quantization, k-means algorithm) G R 38
39 B Representative colours placed in the colour table r g b r g b G R r 55 g 55 b 55 Colour Table 39
40 4 Image Compression Discrete Cosine Transform (DCT) D-DCT Inverse D-DCT = = + + = ) ( cos ) ( )cos, ( ) ( ) ( 4 ), ( N m N n N v n N u m n m x N v C u C v u X π π = = + + = ) ( cos ) ( )cos, ( ) ( ) ( ), ( N u N v N v n N u m v u X v C u C n m x π π = = =,, ) ( N u u u C L where
41 4 Image Compression D-DCT image block DCT block low frequency high frequency low frequency high frequency DC component
42 4 JPEG Compression DCT scalar quantization zig-zag scan x8x blocks
43 JPEG Compression Bit-stream Dequantizer Quantization Table De-zigzag De-DC coding Huffman Decoding Huffman Decoding Chrominance Upsampling (4:: or 4::) 8 X 8 IDCT YVU color coordinate R G B Decoded Image 43
44 JPEG Compression The Baseline System Quantization X '( u, v) = Round X ( u, v) Q( u, v) X(u,v): original DCT coefficient X (u,v): DCT coefficient after quantization Q(u,v): quantization value 44
45 Why quantization? JPEG Compression to achieve further compression by representing DCT coefficients with no greater precision than is necessary to achieve the desired image quality Generally, the high frequency coefficients has larger quantization values Quantization makes most coefficients to be zero, it makes the compression system efficient, but it s the main source that make the system lossy 45
46 JPEG Compression Quantization is the step where we actually throw away data. Luminance and Chrominance Quantization Table Smaller numbers in the upper left direction larger numbers in the lower right direction The performance is close to the optimal condition Quantization Dequantization Fuv (, ) F( u, v) Quantization = round Quv (, ) Fuv (, ) deq = Fuv (, ) Quantization Quv (, ) 46
47 JPEG Compression Quantization Dequantization Fuv (, ) F( u, v) Quantization = round Quv (, ) Fuv (, ) deq = Fuv (, ) Quantization Quv (, ) QY = QC =
48 JPEG Compression Exploiting Psychovisual Redundancy Exploit variable sensitivity of humans to colors: We re more sensitive to differences between dark intensities than bright ones. Encode log(intensity) instead of intensity. We re more sensitive to high spatial frequencies of green than red or blue. Sample green at highest spatial frequency, blue at lowest. We re more sensitive to differences of intensity in green than red or blue. Use variable quantization: devote most bits to green, fewest to blue. 48
49 JPEG Compression Exploiting Psychovisual Redundancy NTSC Video Y bandlimited to 4. MHz I to.6 MHz Q to.6 MHz 49
50 JPEG Compression Exploiting Psychovisual Redundancy In JPEG and MPEG Cb and Cr are sub-sampled 5
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