1. 1D and 2D signal processing
Convolution
1D and 2D signal processing

2. Convolution Theorem Let F and H be the Fourier transforms of f and h
For big filters, can be faster to convert to fourier domain, then multiply, then convert back
Let F and H be the Fourier transforms of f and h
Convolution in the spatial (image) domain is equivalent to multiplication in the frequency (Fourier) domain

3. Symmetric theorem:
Convolution in the frequency domain is equivalent to multiplication in the spatial domain

4. 2-D Convolution thm f(x,y) h(x,y) g(x,y) *  |F(sx,sy)| |H(sx,sy)|
|G(sx,sy)|

5. Consider the delta function

6. Time-shift delta

7. Sample the input (it’s a convolution!)

8. What is the spectrum?

9. Fourier Coefficients

10. CTFT

11. Euler’s identity

12. Sine-cos Rep

13. Harmonic Analysis

14. Convolution=time-shift&multi

15. Convolution Thm multiplication in the time domain =
convolution in the frequency domain

16. Sample

17. Spectrum reproduced
spectrum to be reproduced at intervals

18. Summary

19. Example of 1D convoln

20. 2D Convolution

21. 2D Convolution

22. Region of Support
The region of support is defined as that area of the .kernel which is non-zero
linear convolution:=signal has infinite extent but kernel has finite support
If function has finite region of support we have compact support

23. Real images have finite region of support
But we treat them as periodic and infinite!
We repeat kernels so that they have the same period as the images.
We call this cyclic convolution.

24. Convolution in 2D

25. Avoid the Mod op

26. What is wrong with avoiding the mod op?
How do I find the center of the kernel?

27. Cyclic Convolution

28. Implementing Convolution
for(int y = 0; y < height; y++) {
for(int x = 0; x < width; x++) {
sum = 0.0;
for(int v = -vc; v <= vc; v++)
for(int u = -uc; u <= uc; u++)
sum += f[cx(x-u) ][cy(y-v)] * k[ u+uc][v+vc];
if (sum < 0) sum = 0;
if (sum > 255) sum = 255;
h[x][y] = (short)sum; }

29. What happens to the image if you ignore the wrap?

30. Cyclic Convolution keeps the edges

31. Can you think of a better way to implement convolution?
Keep the edges!
Don’t use the mod operation.
How about growing the image by the size of the kernel*2?

32. Convolution is slow, how can I speed it up?
JAI!
FFT!?
Other ideas?

33. But JAI…
Eats my edges!
Devised j2d Border
Implement that using MDI!

34. For HW
Integrate the BorderFrame in j2d.border into the MDI interface so that you can apply the BorderPanel to the input image.
Use the RunSpinnerSlider to dynamic alter the parameters.
Update the image dynamically
Apply the change with an apply button.