IMAGE COMPRESSION-II. Week IX. 03/6/2003 Image Compression-II 1
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1 IMAGE COMPRESSION-II Week IX 3/6/23 Image Compression-II 1
2 IMAGE COMPRESSION Data redundancy Self-information and Entropy Error-free and lossy compression Huffman coding Predictive coding Transform coding 3/6/23 Image Compression-II 2
3 Data Redundancy CODING: Fewer bits to represent frequent symbols. INTERPIXEL / INTERFRAME: Neighboring pixels have similar values. PSYCHOVISUAL: Human visual system can not simultaneously distinguish all colors. 3/6/23 Image Compression-II 3
4 Coding Redundancy (contd.) Consider equation (A): It makes sense to assign fewer bits to those r k for which p r (r k ) are large in order to reduce the sum. this achieves data compression and results in a variable length code. More probable gray levels will have fewer # of bits. L 1 Lavg = l( rk ) pr( rk) ( A) k= 3/6/23 Image Compression-II 4
5 General Model General compression model f(x,y) Source encoder Channel encoder Channel Channel decoder Source decoder g(x,y) Source encoder f(x,y) Mapper Quantizer Symbol encoder To channel 3/6/23 Image Compression-II 5
6 Predictive coding To reduce / eliminate interpixel redundancies Lossless predictive coding: ENCODER Image f n e + Σ n Compressed - f ^ n Symbol encoder Image P Q Predictor Prediction error e = f fˆ n n n 3/6/23 Image Compression-II 6
7 Decoder Compressed image symbol decoder e n + Σ + f n Predictor Prediction error: e n = f n -f ^ n e n is coded using a variable length code (symbol encoder) ^ f n = e n + f n ^ f n original image 3/6/23 Image Compression-II 7
8 Example m ˆ Example 1: f n=int αi fn i i= 1 Linear predictor ; m = order of predictor m Example 2: f ˆ n(x,y)=int αi f( x, y i) i= 1 In 2- D Past information x f(x,y) y W xy X Current pixel f (x,y) = Σ α (x, y ) f (x, y) (x, y ) ε W xy 3/6/23 Image Compression-II 8
9 An example of first order predictor f ˆ( x, y) = Int[ f( x, y 1) Prediction error image Histograms of original and error images. 3/6/23 Image Compression-II 9
10 f n Lossy Compression (Section 8.5) Lossy compression: uses a quantizer to compress further the number of bits required to encode the error. First consider this: P e n e Σ Q Enc ^ f n. e e! f" ~ f n n n Dec. e n + ~ f n Σ P. f n Notice that, unlike in the case of loss-less prediction, in lossy prediction the predictors P see different inputs at the encoder and decoder 3/6/23 Image Compression-II 1
11 Quantization error This results in a gradual buildup of error which is due to the quantization error at the encoder site. In order to minimize this buildup of error due to quantization we should ensure that Ps have the same input in both the cases. f n f ^ n = f n-1 e n = e n = f n = /6/23 Image Compression-II 11
12 Predictive Coding With Feedback f(x,y) + ^ f n - Compressed image e n P. f n Q Symbol decoder ^ f n = e n + f n e n symbol encoder 3/6/23 Image Compression-II 12. e n + ^ f n +.. ^ f n = e n + f n Compressed image This feedback loop prevents error building.. uncompressed f n P image
13 Example Example: f" = α f! n and e! n n 1 en = + ξ > ξ en < < α < 1 prediction coefficient f! = e! + f" n n n = e! f! n + α n 1 3/6/23 Image Compression-II 13
14 Example with feedback f n = e n = _ e n = _ 2 2. f n = ^ f n = _ 2 2 f(x,y) ^ f n e n + - Q P. f n + + e n Note: The quantizer used here is-- floor (e n /2)*2. This is different from the one used in the earlier example. Note that this would result in a worse response if used without Feedback (output will be flat at ). 3/6/23 Image Compression-II 14
15 Another example {14, 15, 14, 15, 13, 15, 15, 14, 2, 26, 27, 28, 27, 27, 29, 37, 37, 62, 75, 77, 78, 79, 8, 81, 81, 82, 82} 3/6/23 Image Compression-II 15
16 A comparison (Fig 8.23) 3/6/23 Image Compression-II 16
17 Four linear predictors 3/6/23 Image Compression-II 17
18 Transform Coding Feb 28, 22 3/6/23 Image Compression-II 18
19 Transform coding N X N images Construct n X n subimages Forward transform Quantizer Symbol encoder Compressed image Compressed image Symbol decoder Inverse transform Merge n X n subimages Uncompressed image Blocking artifact: boundaries between subimages become visible 3/6/23 Image Compression-II 19
20 Transform Selection DFT Discrete Cosine Transform (DCT) Wavelet transform Karhunen-Loeve Transform (KLT). 3/6/23 Image Compression-II 2
21 Fig 8.31 DFT, Hadamard and DCT 3/6/23 Image Compression-II 21
22 Discrete Cosine Transform Ahmed, Natarajan, and Rao, IEEE T-Computers, pp. 9-93, /6/23 Image Compression-II 22
23 1-D Case: Extended 2N Point Sequence Consider 1- D first; Let x(n) be a N point sequence n N N point DFT 2 N point N point x(n) y(n) Y(u) C(u) yn ( ) = xn ( ) + x( 2N 1 n) = x(n) xn ( ), n N 1 x( 2N 1 n), N n 2N 1 y(n) /6/23 Image Compression-II 23
24 DCT & DFT 2N 1 2π Y( u) = y( n)exp j un n= 2N N 1 2N 1 2π 2π = x( n)exp j un + x(2n 1 n)exp j un n= 2N n= N 2N N 1 N 1 2π 2π = x( n)exp j un + x( m)exp j u(2n 1 m) n= 2N m= 2N N 1 π π 2π = exp j u x( n) exp j u j un 2N n= 2N 2N N 1 π π 2π + exp j u x( n) exp j u + j un 2N n= 2N 2N N 1 π π = exp j u 2 x( n) cos u(2n + 1). 2N n= 2N 3/6/23 Image Compression-II 24
25 Cu ( ) = DCT The N - point DCTof x(n), C(u), is given by R S T π exp j u N -1 N uyu K ( ), 2 otherwise. The unitary DCT transformations are: N -1 π Fu ( ) α( u) Â f( n)cos ( 2 N n 1) u K, u N -1, 2 n= where 1 2 α( ) =, α( u) = for 1 k N -1. The inverse transformation is N N F I N-1 π f( n) = Âα( u) F( u)cos H ( 2 N n + 1) u K, u N n= F - H F = H + I I 3/6/23 Image Compression-II 25
26 Discrete Cosine Transform in 2-D N 1 2x+ 1) uπ Cuv (,) = α() uα() v f(, xy)cos cos x= N 1 y= for uv, = 1,,2,, N 1, where ( ( 2y+ 1) vπ 2 N 2 N α( u) = R S T 1 N 2 N for u = for u = 1,2,.., N -1. and N 1 2x+ 1) uπ f( x, y) = α( u) α( v) C( u, v)cos cos u= N 1 v= for x, y = 1,,2,, N 1 ( ( 2y+ 1) vπ 2 N 2 N 3/6/23 Image Compression-II 26
27 DCT Basis functions 3/6/23 Image Compression-II 27
28 Implicit Periodicity-DFT vs DCT (Fig 8.32) 3/6/23 Image Compression-II 28
29 Why DCT? Blocking artifacts less pronounced in DCT than in DFT. Good approximation to the Karhunen-Loeve Transform (KLT) but with basis vectors fixed. DCT is used in JPEG image compression standard. 3/6/23 Image Compression-II 29
30 Karhunen-Loeve Transform READ pp. 476 and Section 11.4 Text (if you are working on the face recognition project, you must!) Also called the Hotelling Transform. Transform is data dependent. Let X denote the random data, and let C be the covariance matrix of X, i.e. C = E{(X-m)(X-m) T } where m is the mean vector of the data. The matrix C is real and symmetric, and hence can be diagonalized using its eigenvectors. 3/6/23 Image Compression-II 3
31 What are eigenvectors? T Let C =Ε[( x mx)( x mx) ] where C is the N N covariance matrix, x is an N dimensional vector, and mx is the mean vector of the samples. The eigenvectors ei of C are given by Cei = λiei where λi are the corresponding eigenvalues. Now consider a matrix A whose columns correspond to the eigenvectors of C, arranged such that the first column corresponds to the eigenvector with the largest eigenvalue, and second with the second largest and so on. Then, consider a transformation on the vectors x such that T y= A ( x mx). Note that y is zero mean, and its covariance matrix is T T T T Cy =Ε ( yy ) =Ε[ A ( x mx)( x mx) A] = A CA=Λ = diagnoal matrix of eigenvalues, in decreasing order. Note that the elements of the transformed vector are uncorrelated (non-diagnoal elements are zero). 3/6/23 Image Compression-II 31
32 What is KLT The transformation from X to Y is called the KLT. In computing the transformation matrix A, we assume that the columns are made up of orthonormal eigenvectors (i.e., the inverse of A is also its transpose.) Thus the basis vectors of the KLT are the orthonormal eigenvectors of the covriance matrix C. The KLT yields uncorrelated coefficients. For compression, we only keep the top K coefficients corresponding to the K largest eigenvectors. Then the mean squared error in reconstruction is given by MSE N = j= K+ 1 λ j 3/6/23 Image Compression-II 32
33 Sub-image size selection 3/6/23 Image Compression-II 33
34 Different sub-image sizes 3/6/23 Image Compression-II 34
35 Bit Allocation/Threshold Coding # of coefficients to keep How to quantize them Threshold coding Threshold coding Zonal coding For each subimage i Arrange the transform coefficients in decreasing order of magnitude Keep only the top X% of the coefficients and discard rest. Code the retained coefficient using variable length code. 3/6/23 Image Compression-II 35
36 Zonal Coding Zonal coding:!!!!!!!!!! i th coefficient in each sub image (1) Compute the variance of each of the transform coeff; use the subimages to compute this. (2) Keep X% of their coeff. which have maximum variance. (3) Variable length coding (proportional to variance) Bit allocation: In general, let the number of bits allocated be made proportional to the variance of the coefficients. Suppose the total number of bits per block is B. Let the number of retained coefficients be M. Let v(i) be variance of the i-th coefficient. Then M M 2 i= 1 B bi () = + log vi () log vi () M 3/6/23 Image Compression-II 36
37 3/6/23 Image Compression-II 37 Zonal Mask & bit allocation: an example
38 Typical Masks (Fig 8.36) 3/6/23 Image Compression-II 38
39 Image Approximations 3/6/23 Image Compression-II 39
40 The JPEG standard The following is mostly from Tekalp s book. Digital Video Processing by M. Tekalp (Prentice Hall). MPEG Video compression standard, edited by Mitchell, Pennebaker, Fogg and LeGall (Chapman and Hall). For the new JPEG-2 check out the web site 3/6/23 Image Compression-II 4
41 JPEG (contd.) JPEG is a lossy compression standard using DCT. Activities started in 1986 and the ISO in Four modes of operation: Sequential (basline), hierarchical, progressive, and lossless. Arbitrary image sizes; DCT mode 8-12 bits/sample. Luminance and chrominance channels are separately encoded. We will only discuss the baseline method. 3/6/23 Image Compression-II 41
42 JPEG-baseline. DCT: The image is divided into 8x8 blocks. Each pixel is level shifted by 2 n-1 where 2 n is the maximum number of gray levels in the image. Thus for 8 bit images, you subtract 128. Then the 2-D DCT of each block is computed. For the baseline system, the input and output data precision is restricted to 8 bits and the DCT values are restricted to 11 bits. Quantization: the DCT coefficients are threshold coded using a quantization matrix, and then reordered using zig-zag scanning to form a 1-D sequence. The non-zero AC coefficients are Huffman coded. The DC coefficients of each block are DPCM coded relative to the DC coefficient of the previous block. 3/6/23 Image Compression-II 42
43 JPEG -color image RGB to Y-Cr-Cb space Y =.3R +.6G +.1B Cr =.5 (B-Y) +.5 Cb = (1/1.6) (R Y) +.5 Chrominance samples are sub-sampled by 2 in both directions. Y1 Y2 Y3 Y4 Cr1 Cr2 Cb1 Cb2 Y5 Y6 Y7 Y8 Cr3 Cr4 Cb3 Cb4 Y9 Y13 Y1 Y14 Y11 Y15 Y12 Y16 Non-Interleaved Scan 1:Y1,Y2,,Y16 Scan 2: Cr1, Cr2, Cr3, Cr4 Scan 3: Cb1, Cb2, Cb3, Cb4 Interleaved: Y1, Y2, Y3, Y4, Cr1, Cb1,Y5,Y6,Y7,Y8,Cr2,Cb2, 3/6/23 Image Compression-II 43
44 JPEG quantization matrices Check out the matlab workspace (dctex.mat). Quantization table for the luminance channel. Quantization table for the chrominance channel. JPEG baseline method Consider the 8x8 image (matlab: array s.) Level shifted (s-128=sd). 2d-DCT: dct2(sd)= dcts After dividing by quantiization matrix qmat: dctshat. Zigzag scan as in threshold coding. [2, 5, -3, -1, -2,-3, 1, 1, -1, -1,,, 1, 2, 3, -2, 1, 1,,,,,,, 1, 1,, 1, EOB]. 3/6/23 Image Compression-II 44
45 An 8x8 sub-image (s) s = (8x8block) sd =(level shifted) /6/23 Image Compression-II 45
46 2D DCT (dcts) and the quantization matrix (qmat) dcts= qmat= /6/23 Image Compression-II 46
47 Division by qmat (dctshat)=dcts/qmat dcthat= dcts= /6/23 Image Compression-II 47
48 Zig-zag scan of dcthat dcthat= Zigzag scan as in threshold coding. [2, 5, -3, -1, -2,-3, 1, 1, -1, - 1,,, 1, 2, 3, -2, 1, 1,,,,,,,1, 1,, 1, EOB]. 3/6/23 Image Compression-II 48
49 Threshold coding -revisited Zig-zag scanning of the coefficients. The coefficients along the zig-zag scan lines are mapped into [run,level] where the level is the value of non-zero coefficient, and run is the number of zero coeff. preceding it. The DC coefficients are usually coded separately from the rest. 3/6/23 Image Compression-II 49
50 JPEG baseline method example Zigzag scan as in threshold coding. [2, 5, -3, -1, -2,-3, 1, 1, -1, -1,,, 1, 2, 3, -2, 1, 1,,,,,,,1, 1,, 1, EOB]. The DC coefficient is DPCM coded (difference between the DC coefficient of the previous block and the current block.) The AC coef. are mapped to run-level pairs. (,5), (,-3), (, -1), (,-2),(,-3),(,1),(,1),(,-1),(,-1), (2,1), (,2), (,3), (,- 2),(,1),(,1),(6,1),(,1),(1,1),EOB. These are then Huffman coded (codes are specified in the JPEG scheme.) The decoder follows an inverse sequence of operations. The received coefficients are first multiplied by the same quantization matrix. (recddctshat=dctshat.*qmat.) Compute the inverse 2-D dct. (recdsd=idct2(recddctshat); add 128 back. (recds=recdsd+128.) 3/6/23 Image Compression-II 5
51 Decoder Recddcthat=dcthat*qmat Recdsd= /6/23 Image Compression-II 51
52 Received signal Reconstructed S= s = (8x8block) /6/23 Image Compression-II 52
53 Example 3/6/23 Image Compression-II 53
54 Image Compression: Summary Data redundancy Self-information and Entropy Error-free compression Huffman coding, Arithmetic coding, LZW coding, Run-length encoding Predictive coding Lossy coding techniques Predictive coing (Lossy) Transform coding DCT, DFT, KLT, JPEG image compression standard 3/6/23 Image Compression-II 54
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