Wavelet Transform

1
Wavelet Transform

• Group 820

2
System Architecture
Haar Transform
EZW
Arithmetic Coding
3
Haar Wavelet Transform
Haar Transform
EZW
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)
Arithmetic Coding

• 13 / 2 2
• 1 - 2 -1
• Insert
• Normalize
• Insert
• Recurse

Signal
1. Iteration 2. Iteration
-1
-1
6
2
-1
-1
-2
4
4
Haar Wavelet Transform
Haar Transform
EZW
Arithmetic Coding
Signal
1 3 5 7
Signal
1 3 5 7
Signal recreated from 2 coefficients
2. Iteration
4 -2 -1 -1
2 2 6 6
5
Haar Basis
Haar Transform
EZW
Arithmetic Coding
Lisa
Haar Basis
6
System Architecture
Haar Transform
EZW
Arithmetic Coding
7
Encoding the Coefficients
Haar Transform
EZW

• How do you encode the coefficients effectively?
• Various ad hoc methods
• Store 0 if coefficient is below some threshold
• Store coefficient if gt threshold as (x,y,c)
• Suboptimal approach. Bad compression.

Arithmetic Coding
8
EZW Embedded Zerotree of Wavelet Coefficients
Haar Transform

• Developed by Shapiro in 93
• Enabled wavelet compression to compete with JPEG
• Builds on two observations
• Large coefficients are most important (contains
  most information) and should be stored first.
• 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

EZW
Arithmetic Coding
9
Why it works
Haar Transform
EZW
Arithmetic Coding
10
How it works
Haar Transform
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 lt
T1 BITRATE lt T2)
EZW
Arithmetic Coding

• Decode algorithm similar
• EBCOT used in JPEG2000 uses similar algorithm

11
System Architecture
Haar Transform
EZW
Arithmetic Coding
12
Arithmetic Coding
Haar Transform
EZW

• 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!

Arithmetic Coding
13
Arithmetic Coding Encoding step
Haar Transform
EZW

• Divide the interval of 01) into subintervals
  based on the probabilities 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

Arithmetic Coding
14
Arithmetic Coding Example of the encoding
Haar Transform
EZW

• Input ABAABB
• p(A)0.5 and p(B) 0.5
• As interval 00.5)
• Bs interval 0.51)
• 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

Arithmetic Coding
15
Arithmetic Coding Example of the encoding
Haar Transform
EZW

• Final interval 0.343750.375)
• We choose 0.36 and throw away 0. -gt 36 is a code
  for the string ABAABB
• 6 bit

Arithmetic Coding
16
Arithmetic Coding Decoding
Haar Transform
EZW

• From the code the initial interval is determined
• For next symbol (Repeat until done)
• Subtract lower limit of previous
• Divide by width of subinterval

Arithmetic Coding
17
Result
48
13
5