1. Part I: Image Transforms DIGITAL IMAGE PROCESSING

2. 1-D SIGNAL TRANSFORM GENERAL FORM Scalar form Matrix form

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

4. 1-D INVERSE SIGNAL TRANSFORM GENERAL 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) TRANSFORM GENERAL FORM

10. 2-D IMAGE TRANSFORM SPECIFIC 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. Example Removing undesirable frequencies remove high frequencies reconstructed signal frequenciesnoisy 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) before scalingafter scaling original image

19. Amplitude and Log of the Amplitude

21. Original and Amplitude

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

23. How can we compute ? How can we compute ? 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. Translation in spatial domain: Translation in frequency domain: DFT PROPERTIES: TRANSLATION

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

30. no translation after translation

31. DFT PROPERTIES: ROTATION

32. DFT PROPERTIES ADDITION-MULTIPLICATION

33. DFT PROPERTIES: SCALE

34. DFT PROPERTIES: AVERAGE

35. Original Image Fourier Amplitude Fourier Phase

36. 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

37. 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!)

38. Magnitude and Phase of DFT

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

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

42. Reconstruction from phase of one image and amplitude of the other