# Multimedia Communications. Transform Coding

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1 Multimedia Communications Transform Coding

2 Transform coding Transform coding: source output is transformed into components that are coded according to their characteristics If a sequence of inputs is transformed into another sequence in which most of the information is contained in only a few elements, we can encode and transmit those elements resulting in data compression Issues: transform quantization and encoding of the transformed coefficients Transform Q E CHANNEL D Q -1 Inverse Transform

3 Transform coding Transform coding consists of three steps: 1. Data sequence x[n] is divided into blocks of size N and each block is mapped into a transform sequence y[n] 2. Quantization of transformed sequence which depends of three factors: 1. desired average bit rate 2. statistics of various elements of transformed sequence 3. effects of distortion in the transform coefficients on the reconstructed sequence 3. Quantized values need to be encoded using some binary encoding technique (e.g., run length coding, Huffman coding)

4 Linear Transforms Linear transform: Desired properties of a transform: invertible energy preserving decorrelating energy compacting: information contained in only a few elements

5 2-D Transforms Most useful 2-D transforms are separable: the 2-D transform of a 2-D signal can be obtained by applying the corresponding 1-D transform to the rows and columns. where x is the matrix holding the samples of the 2-D signal x.

6 Orthogonal transforms Orthogonal transforms: inverse of the transformation matrix is its transpose because the rows of the transform matrix form an orthogonal basis set Orthogonal transforms are energy preserving:

7 Performance measures To choose among several transforms, we need some performance measures. Let s assume the autocorrelation matrix of the input is W and the transform is V. Decorrelation efficiency: Energy packing efficiency: Coding gain:

8 Karhunen-Loève Transform Karhunen-Loève transform (Hotelling transform or method of principle component analysis) decorelates the input sequence Sample-to-sample correlation of the transformed sequence is zero

9 Advantages: Karhunen-Loève Transform the transform samples are completely uncorrelated KLT maximizes transform coding gain Disadvantages: data dependent: most be computed for each data set and sent to receiver (significant overhead) computationally intensive

10 The Discrete Cosine Transform It can be computed from the DFT. Data independent Fast transform exists, O(Nlog 2 N) Approximates well the KLT in terms of energy compaction especially Markov sources with high correlation coefficient Used in JPEG, MPEG, MJPEG, etc.

11 The DCT Basis Vectors (N=8) The basis functions of the DCT are real sinusoids. Similar Fourier-type frequencydomain interpretation holds.

12 Discrete Walsh-Hadamard Transform Hadamard matrix of order N: HH T =NI Hadamard matrices with dimensions of power of two can be constructed in the following manner: The Walsh-Hadamard transform: order the rows of Hadamard matrix in increasing order of sequency (number of zero crossings). Advantage: very efficient multiplier-less implementation. Disadvantage: less energy compacting.

13 Transform Coding x Transform y 1 y^ Q E Q D 1 y 2 y^ Q Q -1 2 y N Q E 2 E N CHANNEL D 2 D N Q -1 y^ N Inverse Transform ^ x

14 Bit allocation Bit allocation: which transform coefficients are kept and what is the precision that is used to represent them. Retaining the coefficients: 1. Maximum variance (zonal coding) 2. Maximum magnitude (threshold coding)

15 Zonal Coding Uses the information theory concept of viewing information as uncertainty. Transform coefficients of maximum variance carry the most picture information and should be retained. How to calculate the variance: 1. From the (N/n)(N/n) transformed subimage arrays 2. An assumed image model Coefficients of maximum variance are located around the origin of the transform The retained coefficients must be quantized and coded.

16 Threshold coding Threshold coding is adaptive: the location of the transform coefficients retained for each subimage vary from one subimage to another one. For each subimage, the transform coefficients of largest magnitude make the most significant contribution to reconstructed subimage quality. What is the threshold and how it is obtained? 1. A single global threshold for all subimages 2. A single threshold for each subimage 3. The threshold can be varied as a function of the location of each coefficient within the subimage.

17 JPEG JPEG: Joint Photographic Experts Group Scope: development of international standard for the compression and decompression and encoding of digital continuous tone still pictures. Benefit: To make use of continuous tone digital images more economical during both storage and transmission Applicable fields: Databases, electronic mail and photoediting Medical imaging and scientific imaging

18 Requirements: Generic still image compression JPEG Modest to low software/hardware complexity Sequential, progressive and layered coding Solution: Differential and Huffman coding (lossless) DCT, qunatization run-length and Huffman/ arithmetic coding (lossy) Features: Psychovisual-based quantization Sequential, progressive and hierarchical modes Interleaving between color components

19 JPEG

20 JPEG Input: 1-4 color components (8 bits/pixel) DCT: the most computationally demanding Quantization: uniform, midtread, quantization Quantizers step sizes are arranged in a table called Q-table The Q-table can be changed by scaling the prototype with a quality factor.

21 JPEG DC coefficients: first coded differentially (difference between neighboring labels are coded) The number of values that the difference can get is large It is difficult to manage a Huffman code The possible values of the difference are partitioned into categories Size of these categories grow as power of two Category number is Huffman coded Elements within each category are specified by tacking on extra bits to the end of the Huffman code for that category

22 JPEG

23 JPEG

24 JPEG

25 JPEG As the categories are of different size, we need differing number of bits to identify the value in each category For AC coefficients the category number (C) and the number of zero-valued labels Z since the last non-zero label form a pointer to the Huffman code Extra bits are added to the Huffman code to determine the value If a particular coefficient is the last nonzero value along the zigzag scan, the code for it is followed by EOB.

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