2003 Wavelet Transform

8/8/2019 2003 Wavelet Transform

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Wavelet TransformWavelet TransformGroup 820Group 820


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System ArchitectureSystem Architecture

Haar Transform

EZW

Arithmetic Coding


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Haar Wavelet TransformHaar Wavelet Transform

To calculate the Haar transform of an array of samples:

1. Find the average of each pair of samples.

2. Find the difference between the average and the samples.

3. Fill the first half of the array with averages.

4. Normalize

5. Fill the second half of the array with differences.6. Recurse - repeat the process on the first half of the array.

(Note: The array length should be a power of two)

11 33 55 77

1. Iteration

2. Iteration

1. 1+3 / 2 = 22. 1 - 2 = -1

3. Insert

4. Normalize

5. Insert

6. Recurse

Signal

-1

-1-1

-1

6

-2

2

4

Haar Transform

EZW

Arithmetic Coding


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Haar Wavelet TransformHaar Wavelet Transform

Signal 1

3

5

7

4

-2

-1

-1

2. Iteration

Signal

[ 1 3 5 7 ]

Signal recreated from 2 coefficients

[ 2 2 6 6 ]

Haar Transform

EZW

Arithmetic Coding


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Haar BasisHaar Basis

Lisa Haar Basis

Haar Transform

EZW

Arithmetic Coding


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Haar Transform

EZW

Arithmetic Coding


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EZWEZW Embedded Zerotree of Wavelet CoefficientsEmbedded Zerotree of Wavelet Coefficients

Haar Transform

EZW

Arithmetic Coding

Developed by Shapiro in 93 Enabled wavelet compression to compete with JPEG

Builds on two observations:

1. Large coefficients are most important (contains most

information) and should be stored first.

2. Magnitude of coefficients decrease as one moves

from lower frequency subbands to higher frequencies.

E: Embedded. Progressive algorithm

Z: Zerotree. Quadtree data structure central to algorithm

W: Wavelet. Designet specifically for wavelet compression


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Why it worksWhy it worksHaar Transform

EZW

Arithmetic Coding


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How it worksHow it worksHaar Transform

EZW

Arithmetic Coding

threshold = 2^(floor(lg(max(c)))

do {

dominant pass (encode PS,NS,ZT,IZ)

subordinate pass (refinement of PS and NS)

t /= 2

} while (PSNR < T1 && BITRATE < T2);

Decode algorithm similar

EBCOT used in JPEG2000 uses similar

algorithm


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Haar Transform

EZW

Arithmetic Coding


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Arithmetic CodingArithmetic CodingHaar Transform

EZW

Arithmetic Coding

A symbol with the probability of 0.4 should ideally be

encoded with 1.32 bits.

Arithmetic Coding assigns one long code to entire stream!


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Arithmetic CodingArithmetic Coding Encoding stepEncoding step Haar Transform

EZW

Arithmetic Coding

Divide the interval of [0;1) into subintervals based on theprobabilities of the individual symbols

Based on the symbol read from the input stream, select the

corresponding interval

Divide this interval further into subinterval still based on

probabilities of the symbols

Repeat the last two steps until the end of input stream


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Arithmetic CodingArithmetic Coding Example of the encodingExample of the encoding Haar Transform

EZW

Arithmetic Coding

Input: ABAABB

p(A)=0.5 and p(B) = 0.5

As interval [0;0.5)

Bs interval [0.5;1)

Upper and lower determines current interval

A1: lower = 0, upper = 0.5

B1: lower = 0.25, upper = 0.5

..

B3: lower = 0.34375, upper = 0.375


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Arithmetic CodingArithmetic Coding Example of the encodingExample of the encoding Haar Transform

EZW

Arithmetic Coding

Final interval: [0.34375;0.375)

We choose 0.36 and throw away 0. -> 36 is a code for the

string ABAABB

=6 bit


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Arithmetic CodingArithmetic Coding DecodingDecoding Haar Transform

EZW

Arithmetic Coding

From the code the initial interval is determined

For next symbol (Repeat until done):

Subtract lower limit of previous

Divide by width of subinterval


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ResultResult

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2003 Wavelet Transform

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