csc461 monia wavelet

8/13/2019 Csc461 Monia Wavelet

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Wavelet-based CodingAnd its application in JPEG2000

Monia Ghobadi

CSC561 final project

[email protected]


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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 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.



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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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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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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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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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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)

csc461 monia wavelet

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