Csc461 Monia Wavelet
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Transcript of Csc461 Monia Wavelet
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Wavelet-based CodingAnd its application in JPEG2000
Monia Ghobadi
CSC561 final project
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Introduction Signal decomposition
Fourier Transform
Frequency domain
Temporal domain Time information?
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What is wavelet transform? Wavelet transform decomposes a signal into
a set of basis functions (wavelets)
Wavelets are obtained from a singleprototype wavelet (t) called motherwaveletby dilationsand shifting:
where ais the scaling parameter and bis the shiftingparameter
)(1
)(,a
bt
a
tba
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What are wavelets
Haar wavelet
Wavelets are functions defined over afinite interval and having an average
value of zero.
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Haar Wavelet Transform Example: Haar Wavelet
100 1
ScalingFunction Wavelet
]2
1,
2
1[)( nh ]
2
1,
2
1[)( ng
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Haar Wavelet Transform
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. Normalize5. Fill the second half of the array with differences.6. Repeat the process on the first half of the array.
1 3 5 7
1. Iteration
2. Iteration
1. 1+3 / 2 = 2
2. 1 - 2 = -13. Insert
4. Normalize
5. Insert
6. Repeat
Signal
-1
-1-1
-1
6
-2
2
4
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Haar 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 ]
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Haar Basis
Lenna Haar Basis
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2
D Mexican Hat wavelet
)(2
1
22
22
)2(),(yx
eyxyx
Time domain
)21(2
1
22
22
)21(2)2,1( wwewwww
Frequency domain
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2D Mexican Hat wavelet (Movie)
low frequency
high frequency
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Scale = 38
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Scale =2
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Scale =1
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Wavelet Transform
Continuous Wavelet Transform (CWT)
Discrete Wavelet Transform (DWT)
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Continuous Wavelet
Transform continuous wavelet transform (CWT) of
1D signal is defined as
the a,bis computed from the mother
waveletby translation and dilation
dxxxfbfW baa )()()( ,
a
bx
axba
1)(,
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Discrete Wavelet Transform
CWT cannot be directly applied to analyzediscrete signals
CWT equation can be discretised byrestraining aand bto a discrete lattice
transform should be non-redundant,complete and constitute multiresolutionrepresentation of the discrete signal
dxxxfbfW baa )()()( ,
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Discrete Wavelet Transform
Discrete wavelets
In reality, we often choose
),(0
2
0,
ktaa jj
kj
., Zkj
.20a
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In the discrete signal case we compute theDiscrete Wavelet Transform by successive low
pass and high pass filtering of the discretetime-domain signal. This is called the Mallatalgorithm or Mallat-tree decomposition.
Discrete Wavelet Transform
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Pyramidal WaveletDecomposition
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The decomposition process can be iterated, withsuccessive approximations being decomposed in turn,so that one signal is broken down into many lower-
resolution components. This is called the waveletdecomposition tree.
Wavelet Decomposition
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Lenna Image
Source: http://sipi.usc.edu/database/
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Lenna DWT
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Lenna DWT DC Level Shifted +70
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Restored Image
Can you tell which is the original and which is the
restored image after removal of the lower right?
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DWT for Image Compression
Block Diagram
2D DiscreteWavelet
Transform
QuantizationEntropy
Coding
20 40 60
10
20
30
40
50
60
20 40 60
10
20
30
40
50
60
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.
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DWT for Image Compression
Image Decomposition
Scale 1
4 subbands:
Each coeff. a 2*2 area in the original image
Low frequencies:
High frequencies:
LL1 HL
1
LH1 HH
1
1111 ,,, HHLHHLLL
2/0
2/
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DWT for Image Compression
ImageDecomposition
Scale 2 4 subbands:
Each coeff. a
2*2 area in scale 1image
Low Frequency:
High frequencies:
HL1
LH1 HH
1
HH2LH2
HL2LL2
2,
2,
2,
2 HHLHHLLL
4/0
2/4/
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DWT for Image Compression
Image Decomposition Parent
Children
Descendants:corresponding coeff. atfiner scales
Ancestors: corresponding
coeff. at coarser scales
HL1
LH1 HH
1
HH2LH2
HL2
HL3
LL3
LH3 HH
3
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DWT for Image Compression Image Decomposition
Feature 1:
Energy distribution similar toother TC: Concentrated in low
frequencies Feature 2:
Spatial self-similarity across
subbands
HL1
LH1 HH1
HH2LH2
HL2
HL3LL3
LH3 HH
3
The scanning order of the subbands
for encoding the significance map.
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JPEG2000 (J2K) is an emergingstandard for image compression
Achieves state-of-the-art low bit ratecompression and has a rate distortionadvantage over the original JPEG.
Allows to extract various sub-images from
a single compressed image codestream,the so called Compress Once, DecompressMany Ways.
ISO/IEC JTC 29/WG1Security Working
Setup in 2002
JPEG2000
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JPEG 2000
Not only better efficiency, but also morefunctionality
Superior low bit-rate performance
Lossless and lossy compression
Multiple resolution
Range of interest(ROI)
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JPEG2000
Can be both lossless and lossy
Improves image quality
Uses a layered file structure :
Progressive transmission
Progressive rendering
File structure flexibility:
Could use for a variety of applications
Many functionalities
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Why another standard?
Low bit-rate compression
Lossless and lossy compression Large images
Single decompression architecture
Transmission in noisy environments
Computer generated imaginary
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Compress Once, DecompressMany Ways
A Single OriginalCodestream
By resolutionsBy layers
Region of Interest
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Components
Each image is
decomposed intoone or morecomponents,such as R, G, B.
Denotecomponents as Ci,i = 1, 2, , nC.
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JPEG2000 EncoderBlock Diagram
Key Technologies:
Discrete Wavelet Transform (DWT)
Embedded Block Coding with OptimizedTruncation (EBCOT)
Quantization
EBCOT Tier-1
Encoder
(CF + AE)
EBCOT
Tier-2
Encoder
Rate Control
2-D Discrete
Wavelet
Transform
transform quantize coding
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Resolution & Resolution-Increments
1-level DWT
J2K uses 2-D Discrete Wavelet
Transformation (DWT)
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Resolution and Resolution-Increments
2-level DWT
1-level DWT
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Discrete Wavelet Transform
LL 2 HL 2
LH2 HH2HL 1
LH1
HH1
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Layers & Layer-Increments
L0
{L0
, L1} {L
0, L
1, L
2}
All layer-increments
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JPEG2000v.s. JPEG
low b it-rate perform ance
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JPEG2K - Quality Scalability
Improve decoding quality as receivingmore bits:
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Spatial Scalability
Multi-resolution decoding from one bit-stream:
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ROI (range of interest)