Wavelet Transform

1. Wavelet Transform • Continuous wavelet transform (CWT):- • Where b is translation (shift), a is varying time scale.

2. 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).

3. Discrete Wavelet Transform • DWT based analysis is best described in terms of filter banks. • The filters used must achieve perfect reconstruction condition

4. Discrete Wavelet Transform

5. 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)

6. 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

7. Estimation of a threshold Wavelet Image De-noising Choice of a thresholding rule and application of the threshold to the detail coefficients. Here we used soft thresholding. Choice of a wavelet and number of levels or scales for the decomposition Application of the inverse transform (wavelet reconstruction) using the modified (thresholded) coefficients Computation of the forward wavelet transform of the noisy image Application of 2-D Discrete Wavelet Transform in De-noising Images

8. Filters Used the filter used is FIR filter banks, with order 25 and pass band edge 0.3 rad/sample and stopband-ripple 0.0001

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

10. Results Original knee image Noisy image De-noised image

12. Original brain image Noisy image De-noised image

14. De-noised image Original Shoulder image Noisy image

16. Morphological Segmentation Based On Watersheds

17. Morphology Basics • Inputs are images but outputs are attributes extracted from those images. • Erosion and Dilation Dilation Erosion

18. Morphological Segmentation by Watersheds • The concept of watersheds is based on visualizing an image in three dimensions: two spatial coordinates versus intensity

19. Algorithm Visualize the image f(x,y) as a topographic surface, with both Valleys and mountains. To avoid the water coming from two different minima to meet, a Dam is build whenever there would be a merge of the water. Assume that there is a hole in each minima and the surface is immersed into a lake The only thing visible of the surface would be the dam These dam walls are called the watershed lines . The water will enter through the holes at the minima and flood the surface.

20. 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).

21. Results 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