1. From Diffraction Patterns to
Lecture 6
From Diffraction Patterns to
Image Analysis

2. Recap…. Outline of Lecture 6
Diffraction and convolution
Spatial frequencies and image analysis
Outline of Lecture 6
The diffraction grating
2D Fourier transforms
Image processing via (spatial) frequency analysis

3. No lecture next Monday (March 19)!
Problems classes will run at usual times.
Matlab – B11 (Monday and Friday), B15 (Thursday)

4. The diffraction grating (Problems Class 5)
An (infinite) diffraction grating has a transmission function which
looks like:
We saw earlier how the double slit transmission function could be represented as a convolution of two functions. The grating transmission function can be treated similarly. ?
The transmission function above can be represented as the convolution of two functions. Sketch them.

5. The diffraction grating
The ‘train’ of delta functions is known as a Dirac ‘comb’ (or a Shah function).
…whose Fourier transform is another Dirac comb:
where:

6. The diffraction grating
has Fourier transform: ?
At what value of k is the first zero in G(k) located? ?
Sketch the Fourier transform (i.e. the diffraction pattern) of the transmission function for the infinite diffraction grating.
See Problems Class 5

7. The diffraction grating
Now, what happens if we want to consider a real diffraction grating (i.e. one that is not infinite in extent)? ?
The slits in the infinite grating above are spaced by an amount L. Imagine that we want to determine the Fourier transform of a grating which is 50L wide. How do we convert the transmission function for the infinite grating into that for a real grating which is 50L wide?

8. The diffraction grating

9. Reciprocal space and spatial frequencies
Just as we can build up a complex waveform from a variety of sinusoids of different amplitudes and phases, so too can we generate an image from a Fourier integral.

10. Image Processing via Fourier Analysis
Extracting information from an image
Enhancing an image
James Bouwer, UCSD

11. Patterns: Here, There, and Everywhere
Is there a characteristic spatial frequency…?

12. Fourier and Fingerprints
1cm
1cm
0=black
255=white
shades
of grey 2 D a r y ( 5 6 )
Find average spacing between ridges
2. Remove noise

13. Fourier and Fingerprints
Average spacing between ridges
(a) Manually: 9 r i d g e s 4 : 8 m R p a c n 5 3
1cm

14. Fourier and Fingerprints
(b) Automatically:
Simplifying observations:
- Ridges are “locally parallel”

15. Fourier and Fingerprints: 2D cosine waves
2. Locally parallel regions are similar to 2D cosine waves
Plot 3D surface
2D cos wave as image

16. A recap of FT of 1D cosine wave( x ) = A c o s k + Á f ( x ) = A c o s k + Á 1 2 e i B u t F T f e i k x g = p 2 ( ) + e i Á ( k ) F = p 2 A F ( k ) A p = 2 e i Á

17. 2D cosine wave ¸ y y µ = d i r e c t o n o f t r a v e l x x µ y x ( ;( ; ) [ k = 2 ] f A c o s + Á x = c o s + y i n f ( ; ) A [ k Á ]

18. 2D Fourier Transform F T : ( k ; ) = 1 2 ¼ R f e i d F ( k ; ) = 1 p 2x ; y ) = 1 2 R f e i d F ( k x ; y ) = 1 p 2 R f e i d = F T y f x ( ; ) g I f ( x ; y ) = X Y F k T g [ s e p a r b l ] = F T y f Y ( ) x X g = F T x f X ( ) g y Y

19. Fourier Transform of 2D cosine wave( x ; y ) = A c o s [ k + i n Á ] = A c o s [ k ( x + y ) Á ] = A 2 h e i [ k ( x c + y s ) Á ] = A 2 e i Á k c x s y + m l a r X ( ) Y F ( k x ; y ) = A 2 e i Á p c s + m l a r F ( k x ; y ) = A e i Á c o s n + k y x A e i Á k k s i n c o
Fourier Transform of 2D cosine wave

20. k y F ( k ; ) k j F ( x ; y ) [ c o m p u t e r ] k x f ( x ; y ) ¹ ¸B 2 A A
2D FT k x A j F ( x ; y ) [ c o m p u t e r ] A B C k 2 x f ( x ; y ) : 5 4 m [ c f . p r e v i o u s a l 3 ]

21. Nanoparticle networks on silicon
2D FFT and radially averaged FFT for 20 x 20 mm2 image
Fourier analysis
Intensity (a.u.)
k (nm-1)
2 mm
P. Moriarty, MDR Taylor, and M. Brust
Phys. Rev. Lett (2002)
Peak at k ~ 7 x 10-3 nm-1 (i.e. l ~ 880 nm) but otherwise structure factor is featureless - no orientational order

22. ) Key points F e a t u r o f n i m g v s l h - c ¸ d µ P k p ¼ 2 = , .x . P e a k c o r s p n d t m h i g : f b u . w [ ] l

23. High frequency “noise”
2. Removing noise
Top right corner of filtered fingerprint
Top right corner of fingerprint FT
High frequency “noise”
Low
frequency
“noise”
2D FT of whole fingerprint
Band pass
filter

24. Original
Band-pass filtered

25. Things to think about: W h a t d o e s p k ( ; ) = r n i c l ? ® w u mx ; y ) = r n i c l ? w u m v g f I s t h e r i n g o f w d l y b c a u p m ? I f t h e r i d g s w m o a n , c u l v 2 D F T ?

26. 2D Images and 2D Fourier Transforms
Consider an aperture: ?
f(x,y) in this case can be broken down into two functions f(x) and f(y). Sketch those functions.

27. 2D Images and 2D Fourier Transforms
So, for a square aperture we have two sinc functions,
one along kx and one along ky
Figures taken from Optics, Hecht (Addison-Wesley, 2nd Ed. 1987)

28. 2D Images and 2D Fourier Transforms?
Which area of the diffraction pattern is associated with low spatial frequencies? With high spatial frequencies?

29. 2D Images and 2D Fourier Transforms
Aperture function (2 slits)
2 slit pattern ?
What is the effect on the image if we only pass the spatial frequencies within the circle shown?

30. 2D Images and 2D Fourier Transforms?
What is the effect on the image if we block the
spatial frequencies within the circle shown?

31. Niamh’s Fourier transform (modulus2)
Complex images: Fourier transforming and spatial filtering
Niamh
Niamh’s Fourier transform (modulus2)

32. Complex images: Fourier transforming and spatial filtering

33. Complex images: Fourier transforming and spatial filtering
Optical computer