Walsh transform

This presentation is helpful for image transformation. This ppt is in a very simple manner.

1. Seminar Topic on Walsh Transform Presented to: Presented By: Dr. Amod Kumar Sachin Maithani Electronics and Communication ME(Regular) Engineering Department 202604 1

2. Table of Content 1.Introduction 2. Walsh Transform 3.Example of Walsh Transform 4. Walsh Transform Example in Matlab 4. Application of Walsh Transform 5.Refrences 2

3. INTRODUCTION Remember that the Fourier transform is based on trigonometric terms. The Walsh transform consists of basis functions whose values are only 1 and -1. They have the form of square waves. These functions can be implemented more efficiently in a digital environment than the exponential basis functions of the Fourier transform. 3

4. WALSH TRANSFORM Based on Hadamrad Transformation (HT) Walsh transform is just a sequence ordered hadamard transform. sequence means, the no. of sign changes in a row. non-sinusoidal, orthogonal transformation It is focus on the count the sign change in the (H matrix). WT can be used in many different applications, such as power spectrum analysis, filtering, processing speech and medical signals, multiplexing and coding in communications, characterizing non-linear signals, solving non-linear differential equations, and logical design and analysis. 4

5. 1-D Walsh Transform We define now the 1-D Walsh transform as follows: The above is equivalent to: The transform kernel values are obtained from:    N  1 N x    ( u )   (  1 ) n  1  i  W ( u )  b ( x ) b n  1  i 1 i f ( x ) 1 N  1 W ( u )  n  1  b i ( x ) b n  1  i ( u )  f ( x )(  1) i  1 N x  (  1 )       (  1 ) T ( u , x )  T ( x , u )  n  1 i  1 n  1  i  i n  1 b i ( x ) b n  1  i ( u ) N  i   b ( x ) b ( u ) 1 N 1 5

6. 2-D Walsh Transform We define now the 2-D Walsh transform as a straightforward extension of the 1-D transform: The above is equivalent to:     ( u )  b ( y ) b   ( v )   (  1 ) n  1  i  W ( u , v )  n  1  i b ( x ) b N  1 N  1 n  1  i 1 i i f ( x , y ) N x  y  1 N  1 N  1 W ( u , v )  n  1  ( b i ( x ) b n  1  i ( u )  b i ( y ) b n  1  i ( v ))   f ( x , y )(  1) i  1 N x  y  6

7. 1-D Inverse Walsh Transform Base on the last equation of the previous slide we can show that the Inverse Walsh transform is almost identical to the forward transform! The above is again equivalent to The array formed by the inverse Walsh matrix is identical to the one formed by the forward Walsh matrix apart from a multiplicative factor N.   ( u )   x  n  1  i  b ( x ) b n  1  i i  W ( u )   (  1) N  1 f ( x )  N  1 n  1  b i ( x ) b n  1  i ( u ) f ( x )   W ( u )(  1) i  1 x  7

8. 2-D Inverse Walsh Transform We define now the Inverse 2-D Walsh transform. It is identical to the forward 2-D Walsh transform! The above is equivalent to: ( u )  b ( y ) b   ( v )   n  1  i  n  1  i b ( x ) b n  1  i i i   W ( u , v )   (  1) x  y  f ( x , y )  N  1 N  1 N  1 N  1 f ( x , y )  n  1  ( b i ( x ) b n  1  i ( u )  b i ( x ) b n  1  i ( u ))   W ( u , v )(  1) i  1 x  y  8

9. Implementation of the 2-D Walsh Transform The 2-D Walsh transform is separable and symmetric. Therefore it can be implemented as a sequence of two 1-D Walsh transforms, in a fashion similar to that of the 2-D DFT. 9

10. H = 1 1 1 1 = For first row Walsh Matrix Formation 10

11. 1 - 1 1 -1 The second row =3 Third row Fourth row 1 1 - 1 -1 1 1 - 1 - 1 1 1 1 = 1 = 2 11 1 1 1

12. 12 The Walsh matrix build according to number of sign change 1 ) 1 D F = W . f 2 ) 2D F = W . f . W T = W . f . W 3) Symmetric

13. Example f = { 1, 2, 0, 3} Find WT F = = 13

14. f = Find WT 2D F = W . f . W Sym. 14 Example F =

15. F = F = 15

16. Walsh Transform Example in Matlab 16

17. Application of Walsh transform   Speech recognition medical and biological  image processing   digital holography 17

18. References:- Google Wikipedia You tube Walsh, J.L. (1923).  "A closed set of normal orthogonal functions“.   Amer. J. Math. 45 (1): 5–24.  doi : 10.2307/2387224.   JSTOR   2387224 .  S2CID   6131655 . 18

19. THANK YOU 19