Transcript Document

Wavelet-based Coding
And its application in JPEG2000
Monia Ghobadi
CSC561 final project
[email protected]
Introduction
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Signal decomposition
Fourier Transform
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Frequency domain 
Temporal domain 
Time information?
What is wavelet transform?
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Wavelet transform decomposes a signal into a
set of basis functions (wavelets)
Wavelets are obtained from a single
prototype wavelet Ψ(t) called mother wavelet
by dilations and shifting:
1
t b
 a ,b (t ) 
(
)
a
a
where a is the scaling parameter and b is the shifting
parameter
What are wavelets

Wavelets are functions defined over a
finite interval and having an average
value of zero.
Haar wavelet
Haar Wavelet Transform
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Example: Haar Wavelet
0
1
Scaling
Function
h( n)  [
1 1
,
]
2 2
0
1
Wavelet
g ( n)  [
1
1
,
]
2
2
Haar Wavelet Transform
1.
2.
3.
4.
5.
6.
Find the average of each pair of samples.
Find the difference between the average and the samples.
Fill the first half of the array with averages.
Normalize
Fill the second half of the array with differences.
Repeat the process on the first half of the array.
1
3
5
7
1. Iteration
2
6
-1
-1
2. Iteration
4
-2
-1
-1
Signal
1.
2.
3.
4.
5.
6.
1+3 / 2 = 2
1 - 2 = -1
Insert
Normalize
Insert
Repeat
Haar Wavelet Transform
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 ]
Haar Basis
Lenna
Haar Basis
2D Mexican Hat wavelet
Time domain
 ( x, y )  ( x  y  2)e
2
2
1
 ( x2  y2 )
2
Frequency domain
( w1, w2)  2 (w1  w2 )e
2
2
1
 ( w12  w 2 2 )
2
2D Mexican Hat wavelet (Movie)
low frequency  high frequency
<Time Domain Wavelet>
<Fourier Domain Wavelet>
Scale = 38
Scale =2
Scale =1
Wavelet Transform
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Continuous Wavelet Transform (CWT)
Discrete Wavelet Transform (DWT)
Continuous Wavelet Transform
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continuous wavelet transform (CWT) of
1D signal is defined as
Wa f  (b)   f ( x )  a ,b ( x ) dx
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the a,b is computed from the mother
wavelet by translation and dilation
1  x  b
 a ,b ( x ) 


a  a 
Discrete Wavelet Transform
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CWT cannot be directly applied to analyze
discrete signals
Wa f  (b)   f ( x )  a ,b ( x ) dx
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CWT equation can be discretised by
restraining a and b to a discrete lattice
transform should be non-redundant, complete
and constitute multiresolution representation
of the discrete signal
Discrete Wavelet Transform
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Discrete wavelets
j
 j ,k  a0 2 (a0j t  k ),

j, k  Z .
In reality, we often choose
a0  2.
Discrete Wavelet Transform
In the discrete signal case we compute the
Discrete Wavelet Transform by successive low
pass and high pass filtering of the discrete
time-domain signal. This is called the Mallat
algorithm or Mallat-tree decomposition.
Pyramidal Wavelet
Decomposition
Wavelet Decomposition
The decomposition process can be iterated, with
successive approximations being decomposed in turn,
so that one signal is broken down into many lowerresolution components. This is called the wavelet
decomposition tree.
Lenna Image
Source: http://sipi.usc.edu/database/
Lenna DWT
Lenna DWT DC Level Shifted +70
Restored Image
Can you tell which is the original and which is the
restored image after removal of the lower right?
DWT for Image Compression
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Block Diagram
2D Discrete
Wavelet
Transform
Quantization
10
10
20
20
30
30
40
40
50
50
60
60
20
40
60
20
40
60
Entropy
Coding
2D discrete wavelet transform (1D
DWT applied alternatively to vertical
and horizontal direction line by line )
converts images into “sub-bands”
Upper left is the DC coefficient
Lower right are higher frequency
sub-bands.
DWT for Image Compression
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Image Decomposition
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Scale 1
LL1
HL1
LH1
HH1
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4 subbands: LL1, HL1, LH1, HH1
Each coeff.  a 2*2 area in the original image
Low frequencies: 0     / 2
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High frequencies:  / 2  
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DWT for Image Compression
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Image
Decomposition
LL2
HL2
HL1
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•
•
•
Scale 2
LH2
HH2
LL , HL , LH , HH
2
2
2
2
4 subbands:
Each coeff.  a
2*2 area in scale 1
image
Low Frequency: 0    / 4
High frequencies:  / 4   / 2
LH1
HH1
DWT for Image Compression
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Image Decomposition
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LL3
Parent
Children
Descendants:
corresponding coeff. at
finer scales
Ancestors: corresponding
coeff. at coarser scales
HL3
HL2
LH3
HH3
LH2
HL1
HH2
LH1
HH1
DWT for Image Compression
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Image Decomposition
Feature 1:
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Energy distribution similar to
other TC: Concentrated in low
frequencies
Feature 2:
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LL3
Spatial self-similarity across
subbands
HL3
HL2
LH3
HH3
LH2
HL1
HH2
LH1
HH1
The scanning order of the subbands
for encoding the significance map.
JPEG2000
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JPEG2000 (J2K) is an emerging
standard for image compression
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Achieves state-of-the-art low bit rate
compression and has a rate distortion
advantage over the original JPEG.
Allows to extract various sub-images from
a single compressed image codestream,
the so called “Compress Once, Decompress
Many Ways”.
ISO/IEC JTC 29/WG1 Security Working
Setup in 2002
JPEG 2000
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Not only better efficiency, but also more
functionality
Superior low bit-rate performance
Lossless and lossy compression
Multiple resolution
Range of interest(ROI)
JPEG2000
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Can be both lossless and lossy
Improves image quality
Uses a layered file structure :
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File structure flexibility:
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Progressive transmission
Progressive rendering
Could use for a variety of applications
Many functionalities
Why another standard?
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Low bit-rate compression
Lossless and lossy compression
Large images
Single decompression architecture
Transmission in noisy environments
Computer generated imaginary
“Compress Once, Decompress
Many Ways”
A Single Original
Codestream
By resolutions
By layers
Region of Interest
Components
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Each image is
decomposed into
one or more
components,
such as R, G, B.
Denote
components as Ci,
i = 1, 2, …, nC.
JPEG2000 Encoder
Block Diagram
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Key Technologies:
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Discrete Wavelet Transform (DWT)
Embedded Block Coding with Optimized
Truncation (EBCOT)
Rate Control
2-D Discrete
Wavelet
Transform
transform
Quantization
quantize
EBCOT Tier-1
Encoder
(CF + AE)
coding
EBCOT
Tier-2
Encoder
Resolution & ResolutionIncrements
J2K uses 2-D Discrete Wavelet
Transformation (DWT)
1-level DWT
Resolution and ResolutionIncrements
1-level DWT
2-level DWT
Discrete Wavelet Transform
LL2
HL2
LH2
HH2
LH1
HL1
HH1
Layers & Layer-Increments
L0
{L0, L1}
{L0, L1, L2}
All layerincrements
JPEG2000
v.s. JPEG
low bit-rate performance
JPEG2K - Quality Scalability
Improve decoding quality as receiving
more bits:
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Spatial Scalability
Multi-resolution decoding from one bitstream:
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ROI (range of interest)