Title: Introduction to Wavelet Transform
1Introduction to Wavelet Transform
2comment
- Giving some helpful Pre-knowledge for reading
other paper about wavelets in the future. - This presentation will focus on
- Brief introductions for basic wavelets theory
- The implement of Discrete Wavelet transform
3Outline
- Basic Wavelets Theory
- Discrete Wavelet transform
- Wavelets in Mpeg-4 and Jpeg-2000
- Discussion
4Outline
- Basic Wavelets Theory
- Discrete Wavelet transform
- Wavelets in Mpeg-4 and Jpeg-2000
- Discussion
5Basic Wavelets Theory
- Wavelet transform(WT), like Fourier transform
(FT), is a powerful mathematical tool for many
problem in science and engineering. - FT the basis function are sines and
cosines. - WT a hierarchical set of wavelet functions.
6Basic Wavelets Theory
- Wavelet
- little wave or localized wave
- Oscillatory
- Little decay quick to zero
- No DC component
7Basic Wavelets Theory
- Mother wavelet, is a continuous function
which has two properties. - The function integrates to zero
- Its square integrable or, equivalently, has
finite energy
8Basic Wavelets Theory
- Wavelets are dilations and translations of the
mother wavelet. - By parameters(a,b), the is dilated by a
factor a and translated by factor b
9Basic Wavelets Theory
- For many applications a restricted set is
- used and (j,kintegers)
-
- wavelet at scale j with shift k
10Basic Wavelets Theory
- The simplest wavelet is the Haar mother wavelet
which defined on 0,1)
11Basic Wavelets Theory
- Wavelets at different j and k
- j0 j1
12Basic Wavelets Theory
- Any admissible functions can be expressed
as - The coefficients
13Basic Wavelets Theory
- However, the Haar wavelets of scales
- j 0 are not enough to represent all
functions (like constant function). - A father wavelets is used to remedied the
deficiency
14Basic Wavelets Theory
- In the Haar case, the father wavelet is defined
to be - The father wavelets is sometimes named the
scaling function. - Constant functions can be represented easily by a
multiple of the father wavelets.
15Basic Wavelets Theory
- A function can be represented more accurately
below - In connect with filter banks, it is thehigh-pass
filterthat leads to the low-pass
filter lead to scaling function
16Outline
- Basic Wavelets Theory
- Discrete Wavelet transform
- Wavelets in Mpeg-4 and Jpeg-2000
- Discussion
17Discrete Wavelet transform
- In discrete wavelet transform(DWT), the DWT turns
a data sequence into a set of discrete wavelets
coefficients. - The DWT consists of three main components
- Low-pass filter
- High-pass filter
- Sampling operator
18Discrete Wavelet transform
Analysis
Synthesis
19Discrete Wavelet transform
- Why need downsample the signal?
- h0 low-pass filter h1 high-pass filter
- Yy0y1h0x h1x (without downsampling)
- The volume of data Y is the double of data X
(without downsampling).
20Discrete Wavelet transform
- Downsample is represented by the symbol ,and
this operation is not invertible.
21Discrete Wavelet transform
- In order to perfect reconstruction(PR) the
original signal, the filters must have some
properties - (1)
- (2)
22Discrete Wavelet transform
- Example
- Input signal
- Use(5,3)filter bank
23Discrete Wavelet transform
- Example
- After filter
- h0
- h1
- Downsample and combine two data together
24Discrete Wavelet transform
25 26Discrete Wavelet transform
27Outline
- Basic Wavelets Theory
- Discrete Wavelet transform
- Wavelets in Mpeg-4 and Jpeg-2000
- Discussion
28Wavelets in Mpeg-4 and Jpeg-2000
- Mpeg-4
- Support the coding of still textures in the
visual texture coding mode of MPEG-4. - Jpeg-2000
- Compress different types of still images with
different characteristics.
29Wavelets in Mpeg-4 and Jpeg-2000
- Mpeg-4(Visual texture coding)
- Efficient compression
- Arbitrarily shape coding
- Spatial and quality scalability
- Error robustness
- tiling
- Jpeg-2000(Types of still image)
- Types of still image
- Lossless and lossy compression
- Spatial and quality scalability
- Error resilient coding
- Region of interest coding
- Sequential build up and tiling
- Random code-stream access and processing
30Wavelets in Mpeg-4 and Jpeg-2000
31Discussion
- The main advantages or properties of Wavelets
- Easy to implement because of recursive process
- Achieve the multiresolusion analysis(MRA) concept
both in time and frequency intuitively - Lower bit-rate and higher performance for image
compression - PR concept lead to lossless compression
-