Wavelet-based CodingAnd its application in JPEG2000

Monia GhobadiCSC561 final projectmonia@cs.uvic.ca

Introduction

Signal decomposition Fourier Transform

Frequency domain Temporal domain Time information?

What is wavelet transform? 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:

where a is the scaling parameter and b is the shifting parameter

)(1

)(, a

bt

atba

What are wavelets

Haar wavelet

Wavelets are functions defined over a finite interval and having an average value  of zero.

Haar Wavelet Transform

Example: Haar Wavelet

1 0 0 1

Scaling Function Wavelet

]2

1,

2

1[)( nh ]

2

1,

2

1[)( ng

Haar Wavelet Transform1. 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.

1357

1. Iteration

2. Iteration

1. 1+3 / 2 = 22. 1 - 2 = -13. Insert4. Normalize5. Insert6. Repeat

Signal

-1

-1-1

-1

6

-2

2

4

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 ]

Haar Basis

Lenna Haar Basis

2D Mexican Hat wavelet

)(2

122

22

)2(),(yx

eyxyx

Time domain)21(

2

122

22

)21(2)2,1(ww

ewwww

Frequency domain

2D Mexican Hat wavelet (Movie)low frequency high frequency

<Time Domain Wavelet> <Fourier Domain Wavelet>

Scale = 38

Scale =2

Scale =1

Wavelet Transform

Continuous Wavelet Transform (CWT)

Discrete Wavelet Transform (DWT)

Continuous Wavelet Transform

continuous wavelet transform (CWT) of 1D signal is defined as

the a,b is computed from the mother wavelet by translation and dilation

dxxxfbfW baa )()()( ,

abx

axba

1)(,

Discrete Wavelet Transform

CWT cannot be directly applied to analyze discrete signals

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

dxxxfbfW baa )()()( ,

Discrete Wavelet Transform

Discrete wavelets

In reality, we often choose

),( 02

0, ktaa jj

kj ., Zkj

.20 a

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.

Discrete Wavelet Transform

Pyramidal Wavelet Decomposition

The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower-resolution components. This is called the wavelet decomposition tree.

Wavelet Decomposition

Lenna Image

Source: http://sipi.usc.edu/database/

Lenna DWT

Lenna DWT DC Level Shifted +70

Restored ImageCan you tell which is the original and which is the restored image after removal of the lower right?

DWT for Image Compression

Block Diagram

2D DiscreteWaveletTransform

QuantizationEntropyCoding

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 frequencysub-bands.

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 HL1

LH1 HH1

1111 ,,, HHLHHLLL

2/0

2/

DWT for Image Compression

Image Decomposition Scale 2 4 subbands:• Each coeff. a

2*2 area in scale 1 image

• Low Frequency: • High frequencies:

HL1

LH1 HH1

HH2LH2

HL2LL2

2,

2,

2,

2HHLHHLLL

4/0 2/4/

DWT for Image Compression Image Decomposition

Parent Children Descendants:

corresponding coeff. at finer scales

Ancestors: corresponding coeff. at coarser scales

HL1

LH1 HH1

HH2LH2

HL2

HL3LL3

LH3 HH3

DWT for Image Compression

Image Decomposition Feature 1:

Energy distribution similar to other TC: Concentrated in low frequencies

Feature 2: Spatial self-similarity

across subbands

HL1

LH1 HH1

HH2LH2

HL2

HL3LL3

LH3 HH3

The scanning order of the subbands for encoding the significance map.

JPEG2000 (J2K) is an emerging standard for image compression 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

JPEG2000

JPEG 2000

Not only better efficiency, but also more functionality

Superior low bit-rate performance Lossless and lossy compression Multiple resolution Range of interest(ROI)

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

Why another standard?

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

Each image is decomposed into one or more components, such as R, G, B.

Denote components as Ci, i = 1, 2, …, nC.

JPEG2000 EncoderBlock Diagram

Key Technologies: Discrete Wavelet Transform (DWT) Embedded Block Coding with

Optimized Truncation (EBCOT)

QuantizationEBCOT Tier-1

Encoder(CF + AE)

EBCOTTier-2

Encoder

Rate Control

2-D DiscreteWavelet

Transform

transform quantize coding

Resolution & Resolution-Increments

1-level DWT

J2K uses 2-D Discrete Wavelet Transformation (DWT)

Resolution and Resolution-Increments

2-level DWT

1-level DWT

Discrete Wavelet Transform

LL2 HL2

LH2 HH2HL1

LH1 HH1

Layers & Layer-Increments

L0 {L0, L1} {L0, L1, L2}

All layer-increments

JPEG2000 v.s. JPEG

low bit-rate performance

JPEG2K - Quality Scalability

Improve decoding quality as receiving more bits:

Spatial Scalability

Multi-resolution decoding from one bit-stream:

ROI (range of interest)