1. 4. DIGITAL IMAGE TRANSFORMS 4.1. Introduction
Digital image processing Digital image transforms
4. DIGITAL IMAGE TRANSFORMS
4.1. Introduction
4.2. Unitary orthogonal two-dimensional transforms
Separable unitary transforms
4.3. Properties of the unitary transforms
Energy conservation
Energy compaction; the variance of coefficients
De-correlation
Basis functions and basis images
4.4. Sinusoidal transforms
The 1-D discrete Fourier transform (1-D DFT)
Properties of the 1-D DFT
The 2-D discrete Fourier transform (2-D DFT)
Properties of the 2-D DFT
The discrete cosine transform (DCT)
The discrete sine transform (DST)
The Hartley transform
4.5. Rectangular transforms
The Hadamard transform = the Walsh transform
The Slant transform
The Haar transform
4.6. Eigenvectors-based transforms
The Karhunen-Loeve transform (KLT)
The fast KLT
The SVD
4.7. Image filtering in the transform domain
4.8. Conclusions

2. Digital image processing Digital image transforms

4. Digital image processing Digital image transforms

5. Digital image processing Digital image transforms

6. Digital image processing Digital image transforms

7. Digital image processing Digital image transforms

8.  Basis functions and basis images
Digital image processing Digital image transforms
 Basis functions and basis images
KLT
Haar
Walsh
Slant
DCT
Basis functions (basis vectors)
Basis images (e.g.): DCT, Haar, ….

9. Digital image processing Digital image transforms

10. Digital image processing Digital image transforms

11. Digital image processing Digital image transforms

12. Digital image processing Digital image transforms

13. Digital image processing Digital image transforms

14. Digital image processing Digital image transforms

15. Digital image processing Digital image transforms

16. Digital image processing Digital image transforms

17. Digital image processing Digital image transforms

18. Walsh-Hadamard transform
Digital image processing Digital image transforms
Basis vectors for the
Walsh-Hadamard transform

19. Original image Ordered Hadamard Non-ordered Hadamard
Digital image processing Digital image transforms
Original image
Ordered Hadamard
Non-ordered Hadamard

20. Digital image processing Digital image transforms

21. Digital image processing Digital image transforms

22. Digital image processing Digital image transforms
Applying the Haar transform at block level (e.g. 2×2 pixels blocks => Hr[2×2]):
Rearrange
coefficients:
Block
transform:
Applying the Haar transform at block level for a 4×4 pixels blocks => Hr[4×4]:
Rearrange
coefficients:
Block
transform:

23. Digital image processing Digital image transforms

24. Digital image processing Digital image transforms

25.  KLT (PCA) Eigenimages – examples: Facial image set
3 eigenimages and the individual variations on those components
Corresponding “eigenfaces”
Face aproximation, from rough to detailed, as more coefficients are added

26. Digital image processing Digital image transforms

27. DFT IDFT LPF 2-D DFT = sinc 2-D for the square + cst. (for noise)
Original image = (white square, grey background)
+ aditive noise
DFT
LPF 2-D
IDFT

28. Noisy image; periodic noise as vertical lines
The 2-D spectrum of the image and the filters applied:
In the regions corresponding to the vertical lines frequencies
Image restoration through filtering

29. Digital image processing Digital image transforms