1. Math 3360: Mathematical Imaging
Lecture 6:
Haar, Walsh & Discrete Fourier Transform
Prof. Ronald Lok Ming Lui Department of Mathematics,
The Chinese University of Hong Kong

2. Recap: Haar transform Definition of Haar functions:
For details, please refer to Supplementary note 3
Definition of Haar functions:

3. Recap: Haar transform Definition of Haar transforms:
For details, please refer to Supplementary note 3
Definition of Haar transforms:
(N = power of 2)

4. Recap: Walsh transform
For details, please refer to Supplementary note 3
Definition of Walsh functions:

5. Recap: Walsh transform
For details, please refer to Supplementary note 3
Definition of Walsh transforms:
(N = power of 2)

6. Haar transform elementary images
Haar transform basis image. White = positive; Black = negative; Grey = 0

7. Walsh transform elementary images
Walsh transform basis image. White = positive; Black = negative; Grey = 0

8. Reconstruction using Haar decomposition
= using 1x1 elementary images (first 1 row and first 1 column elementary images;
= using 2x2 elementary images (first 2 rows and first 2 column elementary images…
And so on…

9. Error under Haar decomposition

10. Reconstruction using Walsh decomposition
= using 1x1 elementary images (first 1 row and first 1 column elementary images;
= using 2x2 elementary images (first 2 rows and first 2 column elementary images…
And so on…

11. Error under Walsh decomposition