Part I: Image Transforms

1. Part I: Image Transforms DIGITAL IMAGE PROCESSING

2. 1-D SIGNAL TRANSFORMGENERAL FORM • Scalar form • Matrix form

3. 1-D SIGNAL TRANSFORM cont.REMEMBER THE 1-D DFT!!! • General form • DFT

4. 1-D INVERSE SIGNAL TRANSFORMGENERAL FORM • Scalar form • Matrix form

5. 1-D INVERSE SIGNAL TRANSFORM cont.REMEMBER THE 1-D DFT!!! • General form • DFT

6. 1-D UNITARY TRANSFORM • Matrix form

7. SIGNAL RECONSTRUCTION

8. IMAGE TRANSFORMS • Many times, image processing tasks are best performed in a domain other than the spatial domain. • Key steps: (1) Transform the image (2) Carry the task(s) in the transformed domain. (3) Apply inverse transform to return to the spatial domain.

9. 2-D (IMAGE) TRANSFORMGENERAL FORM

10. 2-D IMAGE TRANSFORMSPECIFIC FORMS • Separable • Symmetric

11. Separable and Symmetric • Separable, Symmetric and Unitary

12. ENERGY PRESERVATION • 1-D • 2-D

13. ENERGY COMPACTION Most of the energy of the original data concentrated in only a few of the significant transform coefficients; remaining coefficients are near zero.

14. Why is Fourier Transform Useful? • Easier to remove undesirable frequencies. • Faster to perform certain operations in the frequency domain than in the spatial domain. • The transform is independent of the signal.

15. ExampleRemoving undesirable frequencies noisy signal frequencies remove high frequencies reconstructed signal

16. How do frequencies show up in an image? • Low frequencies correspond to slowly varying information (e.g., continuous surface). • High frequencies correspond to quickly varying information (e.g., edges) Original Image Low-passed

17. 2-D DISCRETE FOURIER TRANSFORM

18. Visualizing DFT • Typically, we visualize • The dynamic range of is typically very large • Apply stretching: ( is constant) original image before scaling after scaling

19. Amplitude and Log of the Amplitude

20. Amplitude and Log of the Amplitude

21. Original and Amplitude

22. DFT PROPERTIES: SEPARABILITY Rewrite as follows: If we set: Then:

23. DFT PROPERTIES: SEPARABILITY • How can we compute ? • How can we compute ?

24. DFT PROPERTIES: SEPARABILITY

25. DFT PROPERTIES: PERIODICITY The DFT and its inverse are periodic with period N

26. DFT PROPERTIES: SYMMETRY If is real, then

27. DFT PROPERTIES: TRANSLATION • Translation in spatial domain: • Translation in frequency domain:

28. DFT PROPERTIES: TRANSLATION Warning: to show a full period, we need to translate the origin of the transform at

29. DFT PROPERTIES: TRANSLATION

30. DFT PROPERTIES: TRANSLATION no translation after translation

31. DFT PROPERTIES: ROTATION

32. DFT PROPERTIESADDITION-MULTIPLICATION

33. DFT PROPERTIES: AVERAGE

34. Original Image Fourier Amplitude Fourier Phase

35. Magnitude and Phase of DFT • What is more important? • Hint: use inverse DFT to reconstruct the image using magnitude or phase only information magnitude phase

36. Magnitude and Phase of DFT Reconstructed image using magnitude only (i.e., magnitude determines the contribution of each component!) Reconstructed image using phase only (i.e., phase determines which components are present!)

37. Magnitude and Phase of DFT

38. Original Image-Fourier AmplitudeKeep Part of the Amplitude Around the Origin and Reconstruct Original Image (LOW PASS filtering)

39. Keep Part of the Amplitude Far from the Origin and Reconstruct Original Image (HIGH PASS filtering)

41. Reconstruction fromphase of one image and amplitude of the other

43. Reconstruction fromphase of one image and amplitude of the other