Hybridoma Technology ( Production , Purification , and Application )
3.Wavelet Transform(Backup slide-3)
1. Wavelet Transformation
Department of Computer Science And Engineering
Shahjalal University of Science and Technology
Nashid Alam
Registration No: 2012321028
annanya_cse@yahoo.co.uk
Masters -2 Presentation
(Backup Slides# 3)
Courtesy :
Prof. Fred Hamprecht
Heidelberg University
Source:
https://www.youtube.com/watch?v=DGUuJweHamQ
3. Wavelet
Working with wavelet:
1. Convolve the signal with wavelet filter(h/g)
2. Store the results in coefficients/frequency response
(Result in number is put in the boxes)
3. Coefficients/frequency response:
- The representation of the signal in the new domain.
Properties:
• Maximum frequency depends on the length of the signal.
• Recursive partitioning of the lowest band in subjective to the application.
Details in upcoming slides
4. Good temper resolution in high frequencies
Good frequency resolution in low pass band
OBTAION:
Wavelet
A high pass filter
Temper resolution : A vertical high-resolution
Frequency resolution : The sample frequency divided by the number of samples
O/P of Low Pass Filter High Pass Filter = A Band Pass Result
5. 1.A length 8 signal
3.Convolve the signal with
the high pass filter
2.Split/divide the signal in two parts
Wavelet
7. • For perfect low pass filter
• Leave everything intact in 0 (zero)
Spectrodensity of the signal at this point
Unit cell
Unit cell is shrunk by half(1/2)
Wavelet
No information loss due to shrinking
First partitioning of lower and higher frequency band
8. Wavelet
Spectrodensity of the signal at this point
For perfect low pass filter For perfect high pass filter
This works even not for perfect high pass/low pass filter
17. Wavelet
Positive half
of
the frequency axis
Details at level 2
Details at level 3
Details
at
level 1
Approximation
Good frequency resolution in low pass band
23. Wavelet
Want to understand
The effect of this label
Have to perform
convolution
Understand The effect of each this label
24. Wavelet
Graph 01: Transfer functions of the wavelet transforms
Works for Signals more then 8 samples
23= 8, Sample=8, level=3.
Level 1
details
Level 2
details
Level 3
details
Level 4
details
Level 5
details
Transfer functions of
Approximation:
The low pass
result
That we keep at
the end
25. Wavelet
Graph 01: Transfer functions of the wavelet transforms
Level
details
+approximation= 1
Property of wavelet
26. Wavelet
Approximation is a sinc
- A perfect low pass filter
sincA-sincB
A=A frequency
B=A frequency
-A perfect bandpass filter
27. Wavelet
Signal with
more than
eight samples
Scenario:
Temper resolution : A vertical high-resolution
Frequency resolution : The sample frequency divided by the number of samples
Temper resolution>
Frequency resolution
Increasing
frequency resolution
Decreases
temporal resolution.
29. Discrete Wavelet Transform(DWT)
Requires a wavelet ,Ψ(t), such that:
- It scales and shifts
from an orthonormal basis
of the square integral function.
)2/)2((
2
1
)(, jt
j
t n
j
nj
Scale Shift
Denote Wavelet
j and n both are integer
nmjlmlnj ., ,,
To offer an orthonormal basis:)(, tnj
Orthonormal basis: A vector space basis for the space it spans.
.
.
39. Backward transformation of Wavelets
Opposite of forward transformation
Mirror the forward transformation on the right hand side
Replace the down-sampling by up-sampling.
Signal
Wavelet
transform
of the Signal
Wavelet
transform
of
the Signal
Signal
49. Use Separable Transform
2D Wavelet Transform
Four region:
Blue= Diagonal Details at label 1
Green=Horizontal Details at label 1
Purple=vertical details at label 1
Yellow= Approximation at Label 1
(Low pass in both x and y direction)
50. Use Separable Transform
2D Wavelet Transform
Doing the above steps recursively:
Take the current approximation
51. Use Separable Transform
2D Wavelet Transform
Doing the above steps recursively:
1. Take the current approximation
2. And further split it up
52. Use Separable Transform
2D Wavelet Transform
Doing the above steps recursively:
1. Take the current approximation
2. And further split it up
53. Use Separable Transform
2D Wavelet Transform
New
approximation
Doing the above steps recursively:
1. Take the current approximation
2. And further split it up
3. Getting new approximation
54. Use Separable Transform
2D Wavelet Transform
Diagonal Details
Horizontal Details
vertical details
Approximation
(can be further
decomposed)
In summary
60. Use Separable Transform
2D Wavelet Transform
More
precise
Visualization
Original image:
Gray square on a
Black Background
Diagonal Details
Horizontal Details
(row by row)
Vertical details
(column by column)
70. CT vs. DWT
DWT Target Goal:
1.Applying a DWT to decompose a digital mammogram into different subbands.
2.The low-pass wavelet band is removed (set to zero) and
the remaining coefficients are enhanced.
3.The inverse wavelet transform is applied to recover
the enhanced mammogram containing microcalcifications [7].
7. Wang T. C and Karayiannis N. B.: Detection of Microcalcifications in Digital Mammograms Using Wavelets, IEEE
Transaction on Medical Imaging, vol. 17, no. 4, (1989) pp. 498-509
The results obtained by the Contourlet Transformation (CT)
are compared with
The well-known method based on the discrete wavelet transform