Wavelet-based Coding And its application in JPEG2000 Monia Ghobadi CSC561 final project

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

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

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. Normalize 5. Fill the second half of the array with differences. 6. Repeat the process on the first half of the array Iteration 2. Iteration / 2 = = -1 3.Insert 4.Normalize 5.Insert 6.Repeat Signal

Haar Wavelet Transform Signal Iteration Signal [ ] Signal recreated from 2 coefficients [ ]

Haar Basis Lenna Haar Basis

2D Mexican Hat wavelet Time domainFrequency domain

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

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

Continuous Wavelet transform Grossman-Morlet Wavelets Daubechies Wavelets Gabor-Malvar Wavelets

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

Discrete Wavelet Transform Discrete wavelets In reality, we often choose

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:

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 Block Diagram 2D Discrete Wavelet Transform Quantization 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 Image Decomposition Scale 1 4 subbands: Each coeff. a 2*2 area in the original image Low frequencies: High frequencies: LL 1 HL 1 LH 1 HH 1

DWT for Image Compression Image Decomposition Scale 2 4 subbands: Each coeff. a 2*2 area in scale 1 image Low Frequency: High frequencies: HL 1 LH 1 HH 1 HH 2 LH 2 HL 2 LL 2

DWT for Image Compression Image Decomposition Parent Children Descendants: corresponding coeff. at finer scales Ancestors: corresponding coeff. at coarser scales HL 1 LH 1 HH 1 HH 2 LH 2 HL 2 HL 3 LL 3 LH 3 HH 3

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 HL 1 LH 1 HH 1 HH 2 LH 2 HL 2 HL 3 LL 3 LH 3 HH 3 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

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 C i, i = 1, 2, …, n C.

JPEG2000 Encoder Block Diagram Key Technologies: Discrete Wavelet Transform (DWT) Embedded Block Coding with Optimized Truncation (EBCOT) transformquantize 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 LL 2 HL 2 LH 2 HH 2 HL 1 LH 1 HH 1

Layers & Layer-Increments L0L0 {L 0, L 1 } {L 0, L 1, L 2 } 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)