1. Even Discrete Cosine Transform The Chinese University of Hong Kong
Math 3360: Mathematical Imaging
Lecture 11:
More about DFT &
Even Discrete Cosine Transform
Prof. Ronald Lok Ming Lui Department of Mathematics,
The Chinese University of Hong Kong

2. Recap: Properties of DFT

3. Recap: Algorithm of FFT

4. Elementary images of DFT decomposition
Recap

5. Elementary images of DFT decomposition
Recap

6. Real example Truncate smallest X% Fourier coefficients
Truncate 70% of Fourier coefficients
Truncate 50% of Fourier coefficients
Truncate 30% of Fourier coefficients
Truncate 10% of Fourier coefficients

7. Real example Truncate smallest X% Fourier coefficients Difference
Image:
Original minus Compressed image
Truncate ~63% of Fourier coefficients
Original

8. Real example Truncate smallest X% Fourier coefficients Difference
Image:
Original minus Compressed image
Truncate 10% of Fourier coefficients

9. Real example Truncate smallest X% Fourier coefficients
Truncate 30% of Fourier coefficients

10. Real example Truncate smallest X% Fourier coefficients
Truncate 50% of Fourier coefficients

11. Real example Truncate smallest X% Fourier coefficients
Truncate 70% of Fourier coefficients

12. Even Discrete Cosine Transform
1D and 2D Even Discrete Cosine Transform:
1D:
2D:
For details, please refer to Lecture Note Chapter 2

13. Inverse Even Discrete Cosine Transform
Image Decomposition

14. Elementary images of EDCT decomposition

15. Reconstruction w/ EDCT 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…

16. Other similar transforms
Odd Discrete Cosine Transform:
Even Discrete Sine Transform:
Odd Discrete Sine Transform:
All of them have explicit formula for their inverses.
(For details, please refer to Lecture Note Chapter 2)

17. Other similar transforms
Inverse Odd Discrete Cosine Transform:
Inverse Even Discrete Sine Transform:
Inverse Odd Discrete Sine Transform:
(For details, please refer to Lecture Note Chapter 2)

18. Elementary images of ODCT decomposition

19. Reconstruction w/ ODCT 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…

20. Elementary images of EDST decomposition

21. Reconstruction w/ EDST 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…

22. Elementary images of ODST decomposition

23. Reconstruction w/ ODST 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…

24. Comparison of errors
The flower example:

25. More example on DCT decomposition

26. More example on DCT decomposition
Original image
DCT

27. More example on DCT decomposition
Original image
DCT

28. More example on DCT decomposition
Original image
DCT

29. More example on DCT decomposition

30. More example on DCT decomposition

31. More example on DCT decomposition

32. More example on DCT decomposition

33. More example on DCT decomposition

34. More example on DCT decomposition