# Lecture 5: Compression I. This Week s Schedule

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1 Lecture 5: Compression I Reading: book chapter 6, section 3 &5 chapter 7, section 1, 2, 3, 4, 8 Today: This Week s Schedule The concept behind compression Rate distortion theory Image compression via DCT Wed.: Speech compression via Prediction Video compression via IPB and motion estimation/ compensation #&

2 Motivation A simple example: a 10x10 colored image, how many bits are required to represent this image: Without compression With compression The concept behind compression is to extract and remove redundancy that naturally exists in media data; In many cases, we can remove information without affecting the visual/ audio effect, because human eyes/ears have limited sensitivity Redundancy in Media Data Medias (speech, audio, image, video) are not random collection of signals, but exhibit a similar structure in local neighborhood Temporal redundancy: current and next signals are very similar (smooth media: speech, audio, video) Spatial redundancy: the pixels intensities and colors in local regions are very similar (image, video) Spectral redundancy: When the data is mapped into the frequency domain, a few frequencies dominate over the others \$&

3 Lossless Compression Lossless compression Compress the signal but can reproduce the exact original signal Used for archival purposes and often medical imaging, technical drawings Example 1: Run Length Encoding (BMP, PCX) BBBBEEEEEEEECCCCDAAAAA! 4B8E4C1D5A Example 2: Lempel-Ziv-Welch (LZW): adaptive dictionary, dynamically create a dictionary of strings to efficiently represent messages, used in GIF & TIFF Example 3: Huffman coding: the length of the codeword to present a symbol (or a value) scales inversely with the probability of the symbol s appearance, used in PNG, MNG, TIFF Huffman Coding Fixed-length coding Symbol Probability Binary Code Code length A B C D E F G H Symbol Probability Binary Code Code length A B C D E F G H Average symbol length=3 Average symbol length=2.63 The length of the codeword to present a symbol (or a value) scales inversely with the probability of the symbol s appearance. '&

4 Lossy Compression The compressed signal after de-compressed, does not match the original signal Compression leads to some signal distortion Suitable for natural images such as photos in applications where minor (sometimes imperceptible) loss of fidelity is acceptable to achieve a substantial reduction in bit rate. Types Color space reduction: reduce 24!8bits via color lookup table Chrominance subsampling: from 4:4:4 to 4:2:2, 4:1:1, 4:2:0 eye perceives spatial changes of brightness more sharply than those of color, by averaging or dropping some of the chrominance information Transform coding (or perceptual coding): Fourier transform (DCT, wavelet) followed by quantization and entropy coding Today s focus Rate Distortion Theory As the degree of compression increases, the number of bits used to represent the image reduces, and this increases the distortion Rate Distortion!&

5 Distortion Measures The three most commonly used distortion measures in image are: mean square error (MSE)! 2, " 2 = 1 #(x N n \$ y n ) 2 where x n, y n, and N are the input data sequence, reconstructed data sequence, and length of the data sequence respectively. signal to noise ratio (SNR), in decibel units (db), peak signal to noise ratio (PSNR), in decibel units (db), N n =1 N 1 2 # x N n n =1 SNR =10" log 10 ( ) \$ 2 PSNR =10" log 10 ( 2552 # 2 ) = 20" log 10 ( 255 # ) Lossy Compression The compressed signal after de-compressed, does not match the original signal Compression leads to some signal distortion Suitable for natural images such as photos in applications where minor (sometimes imperceptible) loss of fidelity is acceptable to achieve a substantial reduction in bit rate. Types Color space reduction: reduce 24!8bits via color lookup table Chrominance subsampling: from 4:4:4 to 4:2:2, 4:1:1, 4:2:0, eye perceives spatial changes of brightness more sharply than those of color, by averaging or dropping some of the chrominance information Transform coding (or perceptual coding): Fourier transform (DCT, wavelet) followed by quantization and entropy coding Today s focus (&

6 A Typical Compression System Transform original data into a new representation that is easier to compress Use a limited number of levels to represent the signal values Find an efficient way to represent these levels using binary bits Transformation Quantization Binary Encoding Temporal Prediction DCT for images Model fitting Scalar quantization Vector quantization Uniform quantization Nonuniform quantization Fixed length Variable length (Run-length coding Huffman coding ) Why Transformation? Goal of transformation: To yield a more efficient representation of the original samples. The transformed parameters should require fewer bits to code. Types of transformation used: For speech coding: prediction Code predictor and prediction error sample For audio coding: subband decomposition Code subband samples For image coding: DCT and wavelet transforms Code DCT/wavelet coefficients )&

7 Image: Transformation Represent an image as the linear combination of some basis images and specify the linear coefficients Instead of storing the original image, now store the linear coefficients {t k } Image: Transformation II Optimality Criteria: Energy compaction: a few basis images are sufficient to represent a typical image. Decorrelation: coefficients for separate basis images are uncorrelated. Karhunen Loeve Transform (KLT) is the Optimal transform for a given covariance matrix of the underlying signal. Discrete Cosine Transform (DCT) is close to KLT for images that can be modeled by a first order Markov process (i.e., a pixel only depends on its previous pixel). *&

8 1D Transformation # s 0 & % ( s 1 s = % ( % : ( % ( \$ ' s N "1 The inverse transform says that s can be represented as the sum of N basis vectors s = t 0 u 0 + t 1 u t N "1 u N "1 # % u k = % % % \$ u k,0 u k,1 : u k,n "1 & ( ( ( ( ' where u k corresponds to the k-th transform kernel : t k = [ s 0 s 1.. s N "1 ] # % % % % \$ u k,0 u k,1 : u k,n "1 * & ( N "1 ( * = ) u ( k,n * s n n =1 ( ' The forward transform says that the expansion coefficient t k can be determined by the inner product of s with u k 1D Discrete Cosine Transform N "1 * t k = # u k,n \$ s n n =1 \$ u k,n = "(k)cos #k ' & (2n +1) ) % 2N ( 1 "(0) = N, "(k) = 2, k =1,2,...,N *1 N, \$ cos #k ' /. & ) 1. % 2N ( \$ u k = "(k) cos & #k 1. % 2N + 3 ' ) 1. ( 1. : 1. \$ cos & #k ' 1. (2N +1) ) - % 2N ( 1 0 +&

9 Example 4-Point DCT Group Exercise! 2D Tranformation,&

10 2D DCT MATLAB! dct2 4x4 DCT #%&

11 Matlab Demo x=imread( avatar1.jpg ); y=x(:,:,1)* x(:,:,2) * x(:,:,3)*0.114; %convert RGB " Y subplot(2,2,1); imshow(y); [m,n]=size(y); m=floor(m/8); n=floor(n/8); for i=1:m, for j=1:n, z=y((i-1)*8+1:i*8, (j-1)*8+1:j*8); d(i,j,:,:)=dct2(z); zz((i-1)*8+1:i*8, (j-1)*8+1:j*8)=d(i,j,:,:); end; end; y(9*8+1:80,9*8+1:80) squeeze(d(10,10,:,:)) surf(squeeze(d(10,10,:,:))) subplot(2,2,2); imshow(zz); ##&

12 Recover via IDCT % No compression for i=1:m, for j=1:n, u=squeeze(d(i,j,:,:)); w((i-1)*8+1:i*8, (j-1)*8+1:j*8)=idct2(u); end; end; subplot(2,2,3); imshow(w,[0,255]); % Some level of compression for i=1:m, for j=1:n, u=squeeze(d(i,j,:,:)); f= zeros(8,8); k=1; f(1:k,1:k)=u(1:k,1:k); % only consider the first kxk coefficient v((i-1)*8+1:i*8, (j-1)*8+1:j*8)=idct2(f); end; end; subplot(2,2,4); imshow(v,[0,255]); Matlab Results an original image block ans = its DCT coefficients ans = recovered image block ans = recovered with first 4x4 coefficient ans = #\$&

13 Compression via DCT Zig-Zag Ordering of DCT Coefficients DC AC #'&

14 After Zig-Zag Ordering Original block DCT results Zig-Zag ordering , , , , , , , , , , , , , , , , , , , , , , , , , , , Recap: A Typical Compression System Transform original data into a new representation that is easier to compress Use a limited number of levels to represent the signal values Find an efficient way to represent these levels using binary bits Transformation Quantization Binary Encoding DCT for images Scalar quantization Vector quantization Uniform quantization Nonuniform quantization Fixed length Variable length (Run-length coding Huffman coding ) #!&

15 Quantization Reduce the number of distinct output values to a much smaller set. Main source of the loss" in lossy compression. Three different forms of quantization. Uniform: midrise and midtread quantizers. Nonuniform: companded quantizer. Vector Quantization. Uniform Quantization A uniform scalar quantizer partitions the domain of input values into equally spaced intervals, except possibly at the two outer intervals. The output or reconstruction value corresponding to each interval is taken to be the midpoint of the interval. The length of each interval is referred to as the step size Q #(&

16 Quantizing DCT Coefficients Use uniform quantizer on each coefficient Different coefficient is quantized with different step-size (Q): Human eye is more sensitive to low frequency components Low frequency coefficients with a smaller Q High frequency coefficients with a larger Q Specified in a normalization matrix Normalization matrix can then be scaled by a scale factor (QP), i.e. Actual quantization = QP * Q, QP=1, 2,3, 4. JPEG Quantization For Luminance component Q = Original block DCT results Quantized & dequantized DCT results Recovered block #)&

17 A Typical Compression System Transform original data into a new representation that is easier to compress Use a limited number of levels to represent the signal values Find an efficient way to represent these levels using binary bits Transformation Quantization Binary Encoding DCT for image Scalar quantization Fixed length Variable length (Run-length coding Huffman coding ) Coding Quantized DCT coefficient DC coefficient: Predictive coding The DC value of the current block is predicted from that of the previous block, and the error is coded using Huffman coding AC Coefficients: Run length coding Many high frequency AC coefficients are zero after first few lowfrequency coefficients Runlength Representation: Ordering coefficients in the zig-zag order Specify how many zeros before a non-zero value Each symbol=(length-of-zero, non-zero-value) Code all possible symbols using Huffman coding More frequently appearing symbols are given shorter codewords One can use default Huffman tables or specify its own tables #*&

18 An Illustrative Example DC Zig-zag + run length encoding DC Coding DC component Current quantized DC index: 2 Previous block DC index: 4 Prediction error: -2 The prediction error is coded in two parts: Which category it belongs to (in the Table of JPEG Coefficient Coding Categories), and code using a Huffman code (JPEG Default DC Code) DC= -2 is in category 2, with a codeword 100 Which position it is in that category, using a fixed length code, length=category number -2 in category 2, has a fixed length code of 10. The overall codeword is #+&

19 JPEG Tables for Coding DC JPEG Tables for Coding DC 100 #,&

20 JPEG Tables for Coding DC Representation inside the category 11 for -3, 10 for -2, 01 for 2, 00 for Coding AC components First symbol (0,5) is represented in two parts: Which category it belongs to (Table of JPEG Coefficient Coding Categories), and code the (runlength, category) using a Huffman code (JPEG Default AC Code) AC=5 is in category 3, Symbol (0,3) has codeword 100 Which position it is in that category, using a fixed length code, length=category number 5 is the number 5 (starting from 0) in category 3, with a fixed length code of 101 The overall codeword for (0,5) is Second symbol (0,9) 9 in category 4, (0,4) has codeword is number 9 in category 4 with codeword 1001 overall codeword for (0,9) is \$%&

21 (runlength, category) Coding Table for AC symbol (0,5) JPEG Coefficient Coding Categories 5 JPEG Default AC Codes Recap: A Typical Image Compression System Transform original data into a new representation that is easier to compress Use a limited number of levels to represent the signal values Find an efficient way to represent these levels using binary bits Transformation Quantization Binary Encoding DCT for images +Zigzag ordering Scalar quantization (Run-length coding Huffman coding ) DC: prediction + Huffman AC: run-length + Huffman \$#&

22 JPEG The Joint Photographic Expert Group (JPEG), under both the International Standards Organization (ISO) and the International Telecommunications Union Telecommunication Sector (ITU-T) Has published several standards JPEG: lossy coding of continuous tone still images Based on DCT JPEG-LS: lossless and near lossless coding of continuous tone still images Based on predictive coding and entropy coding JPEG2000: scalable coding of continuous tone still images (from lossy to lossless) Based on wavelet transform 1992 JPEG Support several modes Baseline system (what is commonly known as JPEG!): lossy Can handle grayscale or colorimages, with 8bits per color component Extended system: can handle higher precision (12 bit) images, providing progressive streams, etc. Lossless version Baseline version Each color component is divided into 8x8 blocks For each 8x8 block, three steps are involved: Block DCT Perceptual-based quantization Variable length coding: Run length and Huffman coding \$\$&

23 Coding Colored Images Color images are typically stored in (R,G,B) format JPEG standard can be applied to each component separately Does not make use of the correlation between color components Does not make use of the lower sensitivity of the human eye to chrominance samples Alternate approach Convert (R,G,B) representation to a YCbCr representation Y: luminance, Cb, Cr: chrominance Down-sample the two chrominance components Because the peak response of the eye to the luminance component occurs at a higher frequency than to the chrominance components Chrominance Subsampling \$'&

24 Quantization Tables for Y, Cr Cb Summary The concept behind compression and transformation How to perform 2D DCT: forward and inverse transform Manual calculation for small sizes, using inner product notation Using Matlab: dct2, idct2 Why DCT is good for image coding Real transform, easier than DFT Most high frequency coefficients are nearly zero and can be ignored Different coefficients can be quantized with different accuracy based on human sensitivity How to quantize & code DCT coefficients Varying step sizes for different DCT coefficients based on visual sensitivity to different frequencies; A quantization matrix specifies the default quantization stepsize for each coefficient; The matrix can be scaled using a user chosen parameter (QP) to obtain different trade-offs between quality and size DC: prediction + huffman; AC: run-length + huffman \$!&

25 Next Lecture Speech/Audio/Video Compression! \$(&

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