Spatial and Temporal Data Mining V. Megalooikonomou Data Compression.

Spatial and Temporal Data Mining

V. Megalooikonomou

Data Compression

General Overview

Data Compression Run Length Coding Huffman Coding PCM, DPCM, ADPCM Quantization

Scalar quantization Vector quantization

Image Compression Video Compression

Data Compression

Why data compression? Storing or transmitting multimedia data requires large space or

bandwidth The size of one hour 44K sample/sec 16-bit stereo (two

channels) audio is 3600x44000x2x2= 633.6MB, which can be recorded on one CD (650 MB). MP3 compression can reduce this number by factor of 10

The size of a 500x500 color image is 750KB without compression (JPEG can reduced this by a factor of 10 to 20)

The size of one minute real-time, full size, color video clip is 60x30x640x480x3= 1.659GB. A two-hour movie requires 200GB. MPEG2 compression can bring this number down to 4.7 GB (DVD)

Data Compression

Run length coding

Example: A scanline of a binary image is 00000 00000 00000

00000 00010 00000 00000 01000 00000 00000 Total of 50 bits However, strings of consecutive 0’s or 1’s can be

represented more efficiently 0(23) 1(1) 0(12) 1(1) 0(13) If the counts can be represented using 5 bits, then we

can reduce the amount of data to 5+5*5=30 bits. A compression ratio of 40%

Huffman coding

The basic idea behind Huffman coding algorithm is to assign shorter codewords to more frequently used symbols

Example: let there be 4 letters in language “A”, “B”, “S”, “Z”

To uniquely encode each letter, we need two bits: A- 00 B-01 S-10 Z-11 A message “AAABSAAAAZ” is encoded with 20

bits Now how about assign A- 0, B-100, S-101, Z-11 The same message can be encoded using 15 bits

Huffman coding

Given a set of N symbols S={si, i=1,…N} with probabilities of occurrence Pi, i=1,…N, find the optimal encoding of the symbol to achieve the minimum transmission rate (bits/symbol)

Algorithm: Each symbol is a leaf node in a tree Combining the two symbols or composite symbols with the

least probabilities to form a new parent composite symbol, which has the combined probabilities. Assign a bit 0 and 1 to the two links

Continue this process until all symbols are merged into one root node (bottom up)

For each symbol, the sequence of the 0s and 1s from the root node to the symbol is the codeword

Pulse code modulation

The process of digitizing audio signal is called pulse code modulation Sampling the analog waveform at a minimum rate Each sample is quantized using a fixed number of

bits To reduce the amount of data, we can

Reduce the sampling rate Reduce the number of bits per sample

Differential Pulse Code Modulation (DPCM)

Encode the changes between consecutive samples Example

The value of the differences between samples are much smaller than those of the original samples. Less bits are used to encode the signal (e.g., 7 bits instead of 8 bits)

For decoding the difference is added to the previous sample to obtain the value of the current sample. Lossless coding is achieved

Adaptive Differential Pulse Code Modulation (ADPCM)

One observation is that a small difference between samples happens more often than large changes

An Entropy coding method such the Huffman coding scheme can be used to encode the difference for additional efficiency The probabilities of occurrence of different differences are

first obtained using a large data base Huffman coding is used to determine the codeword for each

difference The codeword is fixed and made available to decoders

Linear Predictive Coding (LPC)

In DPCM, the value of the current sample is guessed based on the previous sample. Can a better prediction be made ?

Yes! For example, we can use the previous two samples to predict the current one

LPC is more general than DPCM. It exploits the correlation between multiple consecutive samples

General Overview




Quantization

One-dimensional quantizer interval endpoints and levels indicated on a horizontal line

Quantization is the discretization of a continuous-alphabet source (signal)

X: original value, X’: codeword, Q: quantizer Distortion:

d(X,X’): a measure of overall quality degradation due to Q Mean Squared Error (MSE): E[(X-X’)2]

Scalar Quantization

One-dimensional quantizer interval endpoints and levels indicated on a horizontal line

Approximates a source symbol by its closest representative from a codebook

An N-point scalar quantizer Q is a mapping: Q: R C where R is the real line and C={y1, y2,…, yN} R is the

codebook of size N Q(x)=D(E(x)) where E: R I is the encoder and D: IC is the decoder I={1,2,…,N}

Vector Quantization

VQ: a generalization of scalar quantization to quantization of a vector

VQ is superior to scalar quantization. Why?

Vector Quantization

VQ: a generalization of scalar quantization to quantization of a vector

VQ is superior to scalar quantization. Why? Exploits linear and non-linear dependence that exists among the

components of a vector VQ is superior even when the components of the random vector

are statistically independent of each other. How? A vector quantizer Q of dimension k and size N is a

mapping from a vector (or a “point” in Rk), into a finite set C={y1, y2,…, yN}, yiRk, the codebook of size N

Q: RkC It partitions Rk into N regions or cells, Ri for iJ{1,2,

…,N} Ri={x Rk: Q(x)=yi}

Motivation for Vector Quantization

Scalar quantization

Plot two successive values as a single vector (x(1), x(2))

Scalar quantization

Vector quantization

Scalar vs Vector Quantization

Vector Quantization - Design

The goal is to find: A codebook (decoder) – representation

levels A partition rule (encoder) – decision levels

To maximize an overall measure of performance

VQ Design – Optimality Conditions

Nearest Neighbor Condition

For a given codebook, C, the optimal regions {Ri: i=1,…,N} satisfy the condition:

Ri {x: d(x,yi) d(x,yj); j}

That is Q(x)=yi only if d(x,yi) d(x,yj) j

VQ Design – Optimality Conditions

Centroid Condition

For given partition regions {Ri:i=1,…,N} the optimal codewords satisfy the condition:

yi = cent(Ri) For the SE measure, the centroid of a set R is the

arithmetic average:

for R={xi : i = 1,…,|R|}

R

iix

RRcent

1

1)(

VQ Design – The Generalized Lloyd Algorithm (GLA)

It produces a locally optimal codebook from a training sequence, T. It starts with an initial codebook and iteratively improves it.

Lloyd Iteration Given a codebook Cm={yi} generate an improved

codebook Cm+1 as follows: Partition T into cells Rj using the Nearest Neighbor Condition:

Ri={x : d(x,yi) d(x,yi); ji} Using the Centroid Condition compute the centroids of the cells

just found to obtain the new codebook, Cm+1={cent(Ri)}

Compute the average distortion for Cm+1 , Dm+1. If the fractional drop (Dm- Dm+1) / Dm is below a certain threshold stop, else continue with mm+1


To solve the initial codebook generation problem a partition split mechanism is used. How?


To solve the initial codebook generation problem a partition split mechanism is used. How?

Starts with a codebook containing only one codeword (which one?) In each repetition and before the application of the Lloyd iteration, it doubles the number of codewords from the previous iteration

See: http://www.data-compression.com/vq.html#animation

General Overview




Image Compression

From the 1D case, we observe that data compression can be achieved by exploiting the correlation between samples

This idea is applicable to 2D signals as well. Instead of predicting sample values, we can use the so

called transformation method to obtain a more compact representation

Discrete Cosine Transform (DCT) DCT is the real part of the 2D Fourier Transform

Discrete Cosine Transform (DCT)

DCT

Inverse DCT

DCT transform of 2D Images

DCT Example

DCT of images can also be considered as the projection

of the original image into the DCT basis functions. Each basis function is in the form of

DCT transform of 2D Images

The basis functions for an 8x8 DCT

DCT compression of 2D Images

After DCT compression, only a few DCT coefficients have large values

We need to: Quantize the DCT coefficients Encode the position of the large coefficients Compress the value of the coefficients

Spatial and Temporal Data Mining V. Megalooikonomou Data Compression.

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