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Chapter 8 Lossy Compression 8.1 Introduction 81 Algorithms 8.1 8 1 Introduction Lossless compression algorithms do not 8.2 Distortion Measures deliver compression ratios that are high 8.3 8 3 The Rate-Distortion Theory Rate Distortion enough. Hence, most multimedia 8.4 Quantization compression algorithms are lossy. p g y 8.5 8 5 Transform Coding 8.6 Wavelet-Based Coding What is lossy compression ? 8.7 W l t P k t 8 7 Wavelet Packets ◦ The compressed data is not the same as the 8.8 Embedded Zerotree of Wavelet Coefficients original data, but a close approximation of it. 8.9 S Partitioning i Hierarchical T 8 9 Set P i i i in Hi hi l Trees (SPIHT) ◦ Yi ld a much hi h compression ratio than Yields h higher i i h 8.10 Further Exploration that of loss-less compression. 1 2 8.2 Distortion Measures 82 8.3 The Rate-Distortion Theory 83 Rate- The three most commonly used distortion measures in image compression are: Provides a framework for the study of ◦ mean square error (MSE) σ 2, tradeoffs between Rate and Distortion. where xn yn and N are the input data sequence xn, yn,and sequence, reconstructed data sequence, and length of the data sequence respectively. ◦ signal to noise ratio (S ) in decibel units (dB), (SNR), ( ) ◦ peak signal to noise ratio (PSNR), 3 4

8.4 Quantization 84 Uniform Scalar Quantization Reduce the number of distinct output A uniform scalar quantizer partitions the domain of input values into equally spaced intervals, values to a much smaller set, via except p p possibly at the two outer intervals. y quantization. ◦ The output or reconstruction value corresponding to each interval is taken to be the midpoint of the Main source of the "loss" in lossyy interval. i l compression. ◦ The length of each interval is referred to as the step Three diff Th different f t forms of quantization. f ti ti size, size denoted by the symbol ∆ ∆. Two types of uniform scalar quantizers: ◦ Uniform: midrise and midtread quantizers. ◦ Midrise quantizers have even number of output levels levels. ◦ Nonuniform: companded quantizer. ◦ Midtread quantizers have odd number of output ◦ Vector Quantization. Q levels, including zero as one of them (see Fig. 8.2). 5 6 For the special case where ∆ = 1, we can 1 simply compute the output values for these q quantizers as: Performance of an M level quantizer. Let B = {b0,b1,...,bM }be the set of decision boundaries and Y = { 1,y2,. ..,yM } the set of {y }be reconstruction or output values. Suppose the input is uniformly di ib d i S h i i if l distributed in the interval [−Xmax, Xmax]. The rate of the quantizer is: 7 8

Quantization Error of Uniformly Distributed Source Granular distortion: quantization error caused by the quantizer for q y q bounded input. To get an overall figure for granular distortion, notice that decision boundaries bi for a midrise quantizer are [(i − 1)∆, i∆], i =1..M/2, covering positive data X ( d another h lf for negative X values). i ii d (and h half f i l ) Output values yi are the midpoints i∆ − ∆/2, i =1..M/2, again just considering the positive data. The total distortion is twice the sum over the positive data or data, where we divide by the range of X to normalize to a value of at most 1. Since the reconstruction values yi are the midpoints of each interval, the quantization error must lie within the values. For a uniformly distributed source, the graph of quantization error is shown in Fig. 8 3 Fig 8.3. 9 10 Therefore, Therefore the average squared error is Signal variance is , so the same as the variance of the if the quantizer is n bits, M=2n, then from quantization error calculated f l l d from just Eq. (8.2) we have the interval [0, Δ] with error values in . The error value at x is e(x) = x – Δ/2 so Δ/2, the variance of errors is given by 11 12

Nonuniform Scalar Quantization Minimize the total distortion by setting the derivative of Eq. (8.12) to zero. Two common approaches for nonuniform ◦ Differentiating with respect to yi yields the set of quantization: Lloyd-Max quantizer and reconstruction values companded quantizer. Lloyd-Max quantizer y q ◦ Th optimal reconstruction value i the weighted The i l i l is h i h d ◦ The probability distribution fX(x), the decision centroid of the x interval. boundaries bi and the reconstruction values yi ◦ Differentiating with respect to bi and setting the Total distortion measure result to zero yields ◦ The decision boundary at the midpoint of two adjacent reconstruction values 13 14 Companded Quantizer Algorithm 8 1 8.1 LLOYD-MAX QUANTIZATION BEGIN Choose initial level set y0 I=0 Repeat Compute bi using Eq. 8.14 I=I+1 Companded quantization is nonlinear nonlinear. Compute yi using Eq. 8.13 As shown above, a compander consists of a Until | yi - yi-1 | < ε compressor function G a uniform quantizer G, quantizer, and an expander function G−1. END The two commonly used companders are 15 the µ-law and A-law companders. 16

Vector Quantization (VQ) According to Shannon’s original work on Shannon s information theory, any compression system p performs better if it operates on vectors or p groups of samples rather than individual symbols or samples. Form vectors of input samples b simply F f l by l concatenating a number of consecutive samples into a single vector vector. Instead of single reconstruction values as in scalar quantization, in VQ code vectors with quantization n components are used. A collection of these code vectors form the codebook. 17 18 8.5 Transform Coding 85 Spatial Frequency and DCT The rationale behind transform coding: Spatial frequency indicates how many times If Y is the result of a linear transform T of the input pixel values change across an image block. vector X in such a way that the components of Y are much less correlated then Y can be coded more correlated, The DCT formalizes this notion with a efficiently than X. measure of how much the image contents If most information is accurately described by the y y change in correspondence to the number of first few components of a transformed vector, then the remaining components can be coarsely quantized, cycles of a cosine wave per block. or even set to zero, with little signal distortion. The role of the DCT is to decompose the Discrete Cosine Transform (DCT) will be studied first. original signal into its DC and AC In addition, we will examine the Karhunen-Loeve Transform (KLT) which optimally d T f hi h ti ll decorrelates th l t the components; the role of the IDCT is to components of the input X. reconstruct (re-compose) the signal. 19 20

Definition of DCT: DCT: Given an input function f(i j) over two f(i, integer variables i and j (a piece of an image), the 2D DCT transforms it into a new function F(u, v), with integer u and v running over the same range as i and j. The general definition of the transform is: 21 22 23 24

25 26 27 28

The DCT is a linear transform transform. In general, a transform T (or function) is linear, iff T (αp + βq)= αT (p)+ βT (q) βq) (8.21) (8 21) where α and β are constants, p and q are any ffunctions, variables or constants. bl From the definition in Eq. 8.17 or 8.19, q , this property can readily be proven for the DCT because it uses only simple 29 arithmetic operations. 30 The Cosine Basis Functions 31 32

2D Separable Basis Comparison of DCT and DFT The 2D DCT can be separated into a sequence of The discrete cosine transform is a close counterpart two, 1D DCT steps: to the Discrete Fourier Transform (DFT). DCT is a transform that only involves the real part of the DFT. For a continuous signal we define the continuous signal, Fourier Because the use of digital computers requires us to It is straightforward to see that this simple change discretize the input signal we define a DFT that signal, saves many arithmetic steps. The number of operates on 8 samples of the input signal {f0,f1,...,f7} as: iterations required is reduced from 8×8 to 8+8. 8 8 8 8. 33 34 Writing the sine and cosine terms explicitly, we have The formulation of the DCT that allows it to use only the cosine basis functions of the DFT is that we can cancel out the imaginary part of the DFT by making a symmetric copy of the original input signal. DCT of 8 input samples corresponds to DFT of the 16 samples made up of original 8 input samples and a symmetric copy of these, as shown in Fig. 8.10. 35 36

A Simple Comparison of DCT and DFT Table 8 1 and Fig. 8.11 show the comparison 8.1 Fig 8 11 of DCT and DFT on a ramp function, if only the first three t th fi t th terms are used. d 37 38 Karhunen- Karhunen-Loeve Transform (KLT) (KLT) Our goal is to find a transform T such that the g components of the output Y are uncorrelated, i.e ,if t ≠s. Thus, the autocorrelation The Karhunen-Loeve transform is a matrix of Y takes on the form of a positive diagonal matrix. reversible linear transform that exploits the Since any autocorrelation matrix is symmetric and statistical properties of the vector non-negative d fi i there are k orthogonal i definite, h h l representation. eigenvectors u1,u2,...,uk and k corresponding real and nonnegative eigenvalues λ1 ≥ λ2 ≥ ···≥λk ≥0. g g It optimally decorrelates the input signal signal. If we define the Karhunen-Loeve transform as To understand the optimality of the KLT, consider the autocorrelation matrix RX of the input vector X defined as Then, the autocorrelation matrix of Y becomes 39 40

KLT Example To illustrate the mechanics of the KLT KLT, The eigenvalues of RX are λ1=6 1963 =6.1963, consider the four 3D input vectors x1 =(4, λ2=0.2147, and λ3=0.0264. The 4, 5), x2 =(3, 2, 5), x3 =(5, 7, 6), and x4 =(6, corresponding eigenvectors are ) 7, 7). 41 42 8.6 Wavelet-Based Coding 8 6 Wavelet- Subtracting the mean vector from each input vector and g p The objective of the wavelet transform is to apply the KLT decompose the input signal into components that are easier to deal with, have special interpretations, or have some components that can be thresholded away away, for compression purposes. We want to be able to at least approximately reconstruct the original signal given these h i i l i l i h Since the rows of T are orthonormal vectors, the inverse components. transform is just the transpose: T−1= TT ,and The basis functions of the wavelet transform are localized in both time and frequency. There are two types of wavelet transforms: the In I general, after the KLT most of the "energy" of the l f h f h " " f h continuous wavelet t ti l t transform (CWT) and th f d the transform coefficients are concentrated within the first few discrete wavelet transform (DWT). components. This is the "energy compaction" property of the KLT. KLT 43 44

The Continuous Wavelet Transform The continuous wavelet transform (CWT) of f L2(R)at time u and scale s is defined as: The inverse of the continuous wavelet transform is: t f i 45 46 Multiresolution Analysis in the The Discrete Wavelet Transform Wavelet Domain Discrete wavelets are again formed from a Multiresolution analysis provides the tool to adapt mother wavelet, but with scale and shift in signal resolution to only relevant details for a particular task. discrete steps. p The approximation component is then recursively The DWT makes the connection between decomposed into approximation and detail at wavelets in the continuous time domain and successively coarser scales. "filter banks" "f l b k " in the discrete time domain in h d d Wavelet functions ψ(t) are used to characterize detail a multiresolution analysis framework. information. The averaging (approximation) information is formally determined by a kind of dual It is possible to show that the dil t d and i ibl t h th t th dilated d to the mother wavelet, called the "scaling function" translated family of wavelets ψ φ(t). Wavelets W l t are set up such that the approximation at t h th t th i ti t resolution 2−j contains all the necessary information to compute an approximation at coarser resolution form f rm an orthonormal basis of L2(R) rth n rmal f (R). 2−(j+1) (j+1) 47 48

The scaling function must satisfy the so-called dilation equation: The wavelet at the coarser level is also expressible as a sum of translated scaling functions: The vectors h0[n] and h1[n] are called the low pass low-pass and high-pass analysis filters. To reconstruct the original input, an inverse operation is needed. The inverse filt i filters are called synthesis filt ll d th i filters. 49 50 Wavelet Transform Example Suppose we are given the following input sequence. pp g g p q Form a new sequence having length equal to that of the original sequence by concatenating the two sequences {xn−1,i } and {dn−1,i }. The resulting sequence Consider h C id the transform that replaces the original sequence f h l h i i l is with its pairwise average xn−1,i and difference dn−1,i defined as follows: This sequence has exactly the same number of elements as the input sequence — the transform did not increase the amount of data data. Since the first half of the above sequence contain averages from the original sequence, we can view it as g g q The Th averages and d ff d differences are applied only on consecutive l d l a coarser approximation to the original signal. The pairs of input sequences whose first element has an even second half of this sequence can be viewed as the index. Therefore, the number of elements in each set {xn−1,i } details or approximation errors of the first half. pp and {dn−1,i } i exactly half of th number of elements i th d is tl h lf f the b f l t in the original sequence. 51 52

It is easily verified that the original sequence y g q can be reconstructed from the transformed sequence using the relations This transform is the discrete Haar wavelet transform. 53 54 55 56

57 58 Biorthogonal Wavelets For orthonormal wavelets, the forward transform and its inverse are transposes of each other and the analysis filters are identical to the synthesis filters. filters Without orthogonality, the wavelets for analysis and synthesis are called “biorthogonal”. The biorthogonal . synthesis filters are not identical to the analysis filters. We denote them as and . To specify a biorthogonal wavelet transform, we require both 59 60

2D Wavelet Transform For an N by N input image, the two-dimensional two dimensional DWT proceeds as follows: ◦ Convolve each row of the image with h0[n] and h1[n], discard the odd numbered columns of the resulting arrays arrays, and concatenate them to form a transformed row. ◦ After all rows have been transformed, convolve each column of the result with h0[n]and h1[n] Again discard the [n]. odd numbered rows and concatenate the result. After the above two steps, one stage of the DWT is complete. Th transformed i l The f d image now contains four i f subbands LL, HL, LH, and HH, standing for low-low, high-low, etc. g The LL subband can be further decomposed to yield yet another level of decomposition. This process can be continued until the desired number of decomposition levels is reached. 61 62 2D Wavelet Transform Example The input image is a sub-sampled version of the image Lena. The size of the input is 16×16. 16×16 The filter used in the example is the Antonini 9/7 filter set 63 64

The input image is shown in numerical form below. Convolve the first row with both h0[n] and h1[n] and discarding the values with odd-numbered index. The results of these two operations are: p Form the transformed output row by concatenating the resulting coefficients. The first g g row of the transformed image is then: First, First we need to compute the analysis and synthesis high-pass filters. Continue the same process for the remaining rows. 65 66 Apply the filters to the columns of the resulting image. image Apply both h0[n] and h1[n] to each column and discard the odd indexed results: The result after all rows have been processed Concatenate the above results into a single column and apply the same procedure to each of the remaining columns. 67 68

This completes one stage of the discrete wavelet transform. We can perform another transform stage of the DWT by applying the same transform procedure illustrated above to the f d ill d b h upper left 8 × 8 DC image of I12(x, y). The resulting two-stage transformed image is 69 70 8.7 Wavelet Packets 87 Discrete Wavelet Transform In the usual dyadic wavelet decomposition, only the low-pass filtered subband is recursively decomposed and thus can be represented by a logarithmic l arithmic tree str ct re structure. A wavelet packet decomposition allows the decomposition to be represented by any pruned subtree of the full tree topology. The wavelet packet decomposition is very ﬂexible since a best wavelet basis in the sense of some cost metric can be found within a large library of permissible bases bases. The computational requirement for wavelet packet decomposition is relatively low as each decomposition can be computed in the order of NlogN using fast filter banks. 71 72

Wavelet Packet Decomposition 8.8 Embedded Zerotree of Wavelet (WPD) Coefficients Effective and computationally efficient for image coding. The EZW algorithm addresses two problems: ◦ obtaining the best image quality for a given bit-rate, and ◦ accomplishing this task in an embedded fashion fashion. Using an embedded code allows the encoder to terminate the encoding at any point. Hence, the g yp encoder is able to meet any target bit-rate exactly. Similarly, a decoder can cease to decode at any point and can produce reconstructions corresponding to all lower-rate encodings. embedded code contains all lower-rate codes “embedded” at the beginning of the bitstream 73 74 The Zerotree Data Structure The EZW algorithm efficiently codes the "significance map" which indicates the locations of nonzero quantized wavelet coefficients. This Th is achieved using a new data structure called h d d ll d the zerotree. Using the hierarchical wavelet decomposition presented earlier, we can relate every coefficient at a given scale to a set of coefficients at the next finer scale of similar orientation. The coefficient at the coarse scale is called the "parent" while all corresponding coefficients are parent the next finer scale of the same spatial location and similar orientation are called "children". 75 76

Given a threshold T, a coefficient x is an element of the zerotree if it is insignificant and all of its descendants are insignificant as well. The significance map is coded using the zerotree with a four-symbol alphabet: ◦ The zerotree root: The root of the zerotree is encoded with a special symbol indicating that the insignificance of the coefficients at finer scales is completely predictable. ◦ Isolated zero: The coefficient is insignificant but has some g significant descendants. ◦ Positive significance: The coefficient is significant with a p positive value. ◦ Negative significance: The coefficient is significant with a negative value. 77 78 Successive Approximation Dominant Pass Quantization Motivation: Coefficients having their coordinates on the ◦ Takes advantage of the efficient encoding of the dominant list implies that they are not yet significance map using the zerotree data structure by g significant. allowing it to encode more significance maps maps. ◦ Produce an embedded code that provides a coarse- Coefficients are compared to the threshold Ti to to-fine, multiprecision logarithmic representation of p g p determine their significance. If a coefficient is the scale space corresponding to the wavelet- found to be significant, its magnitude is appended transformed image. to the subordinate list and the coefficient in the The SAQ method sequentially applies a sequence wavelet t l t transform array i set to 0 to enable the f is t t t bl th of thresholds T0,...,TN−1 to determine the possibility of the occurrence of a zerotree on significance of each coefficient. future dominant passes at smaller thresholds thresholds. A dominant list and a subordinate list are maintained during the encoding and decoding The resulting significance map is zerotree coded. process. process 79 80

Subordinate Pass All coefficients on the subordinate list are scanned and their magnitude (as it is made available to the EZW Example decoder) is refined to an additional bit of precision. The width of the uncertainty interval for the true magnitude of the coefficients is cut in half. For each magnitude on the subordinate list, the g , refinement can be encoded using a binary alphabet with a "1" indicating that the true value falls in the upper half of the uncertainty interval and a "0" 0 indicating that it falls in the lower half. After the completion of the subordinate pass, the magnitudes on the subordinate li t are sorted i it d th b di t list t d in decreasing order to the extent that the decoder can perform the same sort. 81 82 Encoding The Th coefficient −29 i i i ifi ffi i 29 is insignificant, b contains a significant but descendant 33 in LH1. Therefore, it is coded as z. i i ifi Since the largest coefficient is 57 the initial 57, The coefficient 30 is also insignificant, and all its descendant are insignificant, so it is coed as t. threshold T0 is 32. Continuing in this manner, the dominant pass outputs the At the beginning, the dominant list contains following symbols: g y the coordinates of all the coefficients. {57(p), −37(n), −29(z), 30(t), 39(p), −20(t), 17(t), 33(p), 14(t), 6(z), 10(t), 19(t), 3(t), 7(t), 8(t), 2(t), 2(t), 3(t), 12(t), −9(t), The following is the list of coefficients visited g 33(p), 20(t), 2(t), 4(t)} (p), ( ), ( ), ( )} in the order of the scan: {57, −37, −29, 30, 39, −20, 17, 33, 14, 6, 10, 19, There are five coefficients found to be significant: 57, -37, 39, 33, 33 and another 33. Since we know that no coefficients are 33 3, 7, 8, 2, 2, 3, 12, −9, 33, 20, 2, 4} } greater than 2T0 = 64 and the threshold used in the first With respect to the threshold T0 = 32, it is dominant pass is 32, the uncertainty interval is thus [32, 64). easy to see that the coefficients 57 and -37 h h ffi i d 37 The subordinate pass following the dominant pass refines the magnitude of these coefficients by indicating whether they lie are significant. Thus, we output a p and a n in the first half or the second half of the uncertainty interval. to represent them them. 83 84

Now the dominant list contains the coordinates of all Before we move on to the second round of the coefficients except those found to be significant dominant and subordinate passes, we need and the subordinate list contains the values: to set the values of the significant g dominant list: { 29 30 39, −20, 17, 14, 6, 10, 19, 3, 7, 8, {−29, 30, 39 20 17 14 6 10 19 3 7 8 coefficients to 0 in the wavelet transform 2, 2, 3, 12, −9, 20, 2, 4} subordinate list: {57, 37, 39, 33, 33}. { , , , , } array so that they do not prevent the Now, we attempt to rearrange the values in the emergence of a new zerotree zerotree. subordinate list such that larger coefficients appear The new threshold for second dominant before smaller ones with the constraint that the ones, pass is T1 = 16 Using the same procedure as 16. decoder is able do exactly the same. above, the dominant pass outputs the The decoder is able to distinguish values from [32, 48) g [ ) following symbols g y and [48, 64). Since 39 and 37 are not distinguishable in the decoder, their order will not be changed. The subordinate list is now: 85 86 The subordinate pass that follows will halve each of the three current uncertainty Decoding intervals [48, 64), [32, 48), and [16, 32). The Suppose we only received information from the first pp y subordinate pass outputs the following bits: b d h f ll b dominant and subordinate pass. From the symbols in D0 we can obtain the position of the significant coefficients. Then, using the bits decoded from S0, we can reconstruct the value of these coefficients using the center of the uncertainty The output of the subsequent dominant and p q interval. subordinate passes are shown below: 87 88

8.9 Set Partitioning in Hierarchical If the decoder received only D0, S0, D1, S1, Trees (SPIHT) The SPIHT algorithm is an extension of the EZW D2, and only the first 10 bits of S2, then algorithm. the reconstruction is The SPIHT algorithm significantly improved the performance of its predecessor by changing the way subsets of coefficients are partitioned and how refinement information is conveyed. A unique property of the SPIHT bitstream is its compactness. The resulting bitstream from the SPIHT algorithm is so compact that passing it through an entropy coder would only produce very marginal gain in compression. No d i i f N ordering information i explicitly transmitted t ti is li itl t itt d to the decoder. Instead, the decoder reproduces the execution path of the encoder and recovers the ordering information. f 89 90 8.10 Further Explorations 8 10 Text books: ◦ Introduction to Data Compression by Khalid Sayood ◦ Vector Quantization and Signal Compression by Allen Gersho and Robert M. Gray ◦ Digital Image Processing by Rafael C. Gonzales and Richard E. Woods ◦ Probability and Random Processes with Applications to Signal Processing by Henry Stark and John W. Woods ◦ A Wavelet Tour of Signal Processing by Stephane G. Mallat Web sites: → Link to Further Exploration for Chapter 8 8.. including: ◦ An online graphics-based demonstration of the wavelet transform. transform ◦ Links to documents and source code related to quantization, Theory of Data Compression webpage, FAQ for comp.compression, etc. ◦ A link to an excellent article Image Compression – from DCT to Wavelets : A Review. 91

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