Wavelet Transform

• Continuous wavelet transform (CWT):-

• Where b is translation (shift), a is varying time scale.

Disadvantage of CWT

• It provide an over sampling of the original wave form

For recovery all of the coefficients will be required and the computational effort will be excessive , so we will use Discrete Wavelet Transform (DWT).

Discrete Wavelet Transform• DWT based analysis is best

described in terms of filter banks.• The filters used must achieve

perfect reconstruction condition

Discrete Wavelet Transform

2-D Discrete Wavelet Transform• In the 2D case, the 1D analysis filter

bank is first applied to the columns of the image and then applied to the rows (or vice versa)

2-D Discrete Wavelet Transform

(a )Original image )b( Image Decomposition

DWT image analysis presenting )a( the original image and )b( its decomposition into the first level

Application of 2-D Discrete Wavelet Transform in De-noising Images

Wavelet Image De-noising

Choice of a wavelet and number of levels or scales for the

decomposition

Computation of the forward wavelet transform of the noisy image

Estimation of a threshold

Choice of a thresholding rule and application of the threshold to

the detail coefficients .

Here we used soft thresholding.

Application of the inverse transform (wavelet reconstruction )

using the modified )thresholded( coefficients

Filters Used

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Normalized Frequency ) rad/sample(

Ma

gn

itu

de

)d

B(

Magnitude Response )dB(

Hlp Lowpass Analysis filterHhp Highpass Analysis filterGlp Lowpass Synthesis filterGhp Highpass Synthesis filter

the filter used is FIR filter banks, with order 25 and pass band edge 0.3 rad/sampleand stopband-ripple 0.0001

Thresholding

Soft thresholding deletes the coefficients under the threshold, but scales the ones that are left. The general soft shrinkage rule is defined by:

( ) (| ( ) | ) | ( ) |( )

0 | ( ) |h

sign c k c k if c kc k

if c k

•The soft threshold is continuous function, but we lose some high-frequency information, so that reduce the accuracy of reconstructed signal and blur the edge of signal.

Original knee image Noisy image De-noised image

Results

Threshold value

RMSERelative RMSE

SNR (dB)

277.66970.0002017145.385

Original Image Noisy ImageDe-noised Image

Original brain image Noisy image De-noised image

0 5 10 15 20 25 30 35 40 45 508

10

12

14

16

18

20RMS error V.S. Threshold Pt.

Threshold pt.

RM

S e

rror

0 5 10 15 20 25 30 35 40 45 503

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8x 10

-4 Relative RMS error V.S. Threshold Pt.

Threshold pt.

Rel

ativ

e R

MS

err

or

0 5 10 15 20 25 30 35 40 45 5020

22

24

26

28

30

32

34

36SNR(dB) V.S. Threshold Pt.

Threshold pt.

SN

R(d

B)

Threshold value

RMSERelative RMSE

SNR (dB)

198.99760.000321635.984

Original Shoulder image Noisy image De-noised image

0 5 10 15 20 25 30 35 40 45 506

8

10

12

14

16

18

20RMS error V.S. Threshold Pt.

Threshold pt.

RM

S e

rror

Threshold value

RMSERelative RMSE

SNR (dB)

2810.5480.0002179149.992

Morphological Segmentation Based

On Watersheds

Morphology BasicsMorphology Basics

• Inputs are images but outputs are attributes extracted from those images.

• Erosion and Dilation

Erosion Dilation

Morphological Segmentation by Watersheds• The concept of watersheds is based

on visualizing an image in three dimensions: two spatial coordinates versus intensity

Algorithm

Visualize the image f)x,y( as a topographic surface, with both

Valleys and mountains.

Assume that there is a hole in each minima and the surface

is immersed into a lake

The water will enter through the holes at the minima

and flood the surface.

To avoid the water coming from two different minima to meet, a Dam is build whenever there would be a merge of the water.

The only thing visible of the surface would be the damThese dam walls are called

the watershed lines.

Dam Construction• Implemented by dilation ,

under two conditions: • The dilation has to be

constrained to q (the center of the structuring element can be located only at points in q during dilation),

• The dilation cannot be performed on points that would cause the sets being dilated to merge (become a single connected component).

Results

)a( )b(

)c( )d(

Segmentation of an MR image of the knee . (a) MRI of the knee, (b) watershed lines, (c) binary

image of the selected object, and (d) segmented bone

)a( )b(

)c( )d(