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Multimedia Compression Algorithms

Wen-Shyang HwangKUAS EE.

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Outline

Introduce to Compression Lossless Compression Algorithm Lossy Compression Algorithm Image Compression Standards

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Compression

Compression: the process of coding that will effectively reduce the total number of bits needed to represent certain information.

If compression and decompression processes induce no information loss, then the compression scheme is lossless; otherwise, it is lossy.

Basics of Information Theory Entropy of an information source with alphabet S ={s1,s2,..,sn} is

pi : probability that symbol si will occur in S. indicates amount of information contained in si, which corresponds to the

number of bits needed to encode si. The entropy specifies the lower bound for the average number of bits to

code each symbol in S

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Lossless Compression

Variable-Length Coding (VLC): the more frequently-appearing symbols are coded with fewer bits per symbol, and vice versa.

Shannon-Fano Algorithm Sort symbols according to the frequency of occurrences. Recursively divide symbols into two parts, each with approximately same

counts, until all parts contain only one symbol. Example: Frequency count of the symbols in "HELLO"

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Huffman Coding

Initialization: Put all symbols on a list sorted according to frequency. Repeat until the list has only one symbol left:

1. From the list pick two symbols with the lowest frequency counts. Form a Human subtree that has these two symbols as child nodes and create a parent node.

2. Assign the sum of the children's frequency counts to the parent and insert it into the list such that the order is maintained.

3. Delete the children from the list. Assign a codeword for each leaf based on the path from the root.

The contents in the list:

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Adaptive Huffman Coding

statistics are gathered and updated dynamically as data stream arrives. increments the frequency counts for the symbols Example:Initial code assignment for AADCCDD

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Dictionary-based Coding

Lempel-Ziv-Welch (LZW) algorithm employs an adaptive, dictionary-based compression technique.

LZW uses fixed-length codewords to represent variable-length strings of symbols/characters that commonly occur together.

Example: LZW compression for string “ABABBABCABABBA"

Output codes are: 1 2 4 5 2 3 4 6 1. Instead of sending 14 characters,only 9 codes need to be sent (compression ratio = 14/9 = 1.56).

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Lossless Image Compression

Approaches of Differential Coding of Images: Given an original image I(x, y), using a simple difference operator we can

define a difference image d(x, y) as follows:

Due to spatial redundancy existed in normal images I, the difference image d will have a narrower histogram and hence a smaller entropy

Distributions for Original versus Derivative Images. (a,b): Original gray-level image and its partial derivative image; (c,d): Histograms for original and derivative images.

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Lossless JPEG

The Predictive method:

1. Forming a differential prediction: A predictor combines the

values of up to three neighboring pixels as the predicted value for

the current pixel, indicated by `X' in Figure. The predictor can use

any one of the seven schemes listed in the below Table.

2. Encoding: The encoder compares the prediction with the actual

pixel value at the position `X' and encodes the difference using one

of the lossless compression techniques, e.g., the Human coding

scheme.

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Lossy Compression Algorithms lossy compression

Compressed data is not the same as the original data, but a close approximation of it.

Yields a much higher compression ratio than that of lossless compression.

Distortion Measures mean square error (MSE) 2, where xn, yn, and N are the input data sequence, reconstructed data sequence, and l

ength of the data sequence respectively. signal to noise ratio (SNR),

in decibel units (dB), where is the average square value of the original data sequence and is the MSE.

peak signal to noise ratio (PSNR),

Which measures the size of the error relative to the peak value of the signal Xpeak

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Rate-Distortion Theory

Rate: average number of bits requiredto represent each source symbol.

Provides a framework for the study of tradeoffs between Rate and Distortion.

Typical Rate Distortion Function.

D is a tolerable amount of distortion,R(D) specifies the lowest rate at which the source data can be encoded while keeping the distortion bounded above by D.

D=0, have a lossless compression of sourceR(D)=0 (Dmax), max. amount of distortion

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Quantization

Three different forms of quantization. Uniform: partitions the domain of input values into equally spaced interv

als. Two types - Midrise: even number of output levels (a) Midtread: odd number of output levels (b); zero: one of output

Nonuniform: companded (Compressor/Expander) quantizer. Vector Quantization.

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Companded and Vector quantization

A compander consists of a compressor function G, a uniform quantizer, and an expander function G−1.

Vector Quantization (VQ)

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Transform Coding

If Y is the result of a linear transform T of the input vector X in such a

way that the components of Y are much less correlated, then Y can

be coded more efficiently than X. Discrete Cosine Transform (DCT)

to decompose the original signal into its DC and AC components Spatial frequency: how many times pixel values change across an

image block. IDCT is to reconstruct (re-compose) the signal.

2D DCT and 2D IDCT

(Definition of DCT)

(2D DCT)

(2D IDCT)

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1D DCT basis functions

Fourier analysis !

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DFT (Discrete Fourier Transform)

DCT is a transform that only involves the real part of the DFT.

Continuous Fourier transform:

Euler’s formula

Discrete Fourier Transform:

Graphical illustration of 8 X 8 2D-DCT basis.White (1), Black (0)

To obtain DCT coefficients, just form the inner product of each of these 64 basis image with an 8 X 8 block from an origial image.

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Wavelet-Based Coding

Objective: to decompose input signal (for compression purposes) into components that are easier to deal with, have special interpretations, or have some components that can be thresholded away.

Its basis functions are localized in both time and frequency. Two types of wavelet transforms: continuous wavelet transform (CWT) and th

e discrete wavelet transform (DWT) Discrete wavelets are again formed from a mother wavelet, but with scale and

shift in discrete steps. DWT forms an orthonormal basis of L2(R). Multiresolution analysis provides the tool to adapt signal resolution to only r

elevant details for a particular task.

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Image Compression Standards

JPEG (Joint Photographic Experts Group) an image compression standard accepted as an international standard in 1992. a lossy image compression method by using DCT

Useful when image contents change relatively slowly human less to notice loss of very high spatial frequency component Visual acuity is much greater for gray than for color.

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Main Steps in JPEG Image Compression

Transform RGB to YIQ or YUV and subsample color. DCT on image blocks. Quantization. Zig-zag ordering and run-length encoding. Entropy coding.

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JPEG Image Compression

DCT on image blocks Each image is divided into 8X8 blocks. 2D DCT is applied to each block image f(i,j), with output being the DCT co

efficients F(u,v) for each block.

Quantization F(u,v) represents a DCT coefficient, Q(u,v) is a “quantization matrix" entr

y, and ^ F(u,v) represents the quantized DCT coefficients which JPEG will use in the succeeding entropy coding

Zig-zag ordering and run-length encoding RLC on AC coefficients make to hit a long run of zeros:

a zig-zag scan used to turn the 8X8 matrixinto a 64-vector

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JPEG2000 Standard

To provide a better rate-distortion tradeoff and improved subjective image quality.

To provide additional functionalities lacking in JPEG standard. addresses the following JPEG problems:

Lossless and Lossy Compression Low Bit-rate Compression large Images Single Decompression Architecture Transmission in Noisy Environments Progressive Transmission Region of Interest Coding Computer Generated Imagery Compound Documents

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Properties of JPEG2000 Image Compression

Uses Embedded Block Coding with Optimized Truncation (EBCOT) algorithm which partitions each subband LL, LH, HL, HH produced by the wavelet transform into small blocks called “code blocks".

A separate scalable bitstream is generated for each code block => improved error resilience.

Code block structure of EBCOT.

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Region of Interest Coding in JPEG2000

Particular regions of the image may contain important information, thus should be coded with better quality than others.

A scaling-based method (MXSHIFT) to scale up thecoefficients in the ROI so that they are placed intohigher bitplanes.

Region of interest (ROI) coding of an image using a circularly shaped ROI. (a) 0.4 bpp, (b) 0.5 bpp, (c) 0.6bpp, and (d) 0.7 bpp.