1. Lecture 7: Sampling Review of 2D Fourier Theory We view f(x,y) as a linear combination of complex exponentials that represent plane waves. F(u,v) describes the weighting of each wave. has a frequency and a direction The wave

2. Review of 2D Fourier theory, continued. F(u,v) can be plotted as real and imaginary images, or as magnitude and phase. |F(u,v)| = And Phase(F(u,v) = arctan(Im(F(u,v)/Re(F(u,v)) Just as in the 1D case, pairs of exponentials make cosines or sines. We can view F(u,v) as

3. Review of 2D Fourier theory, continued: properties - Similar to 1D properties Let Then 1. Linearity 2. Scaling or Magnification 3. Shift 4. Convolution also let then,

4. Review of 2D Fourier theory, continued: properties If g(x,y) can be expressed as g x (x)g y (y), the F {g(x,y)} =

5. Relation between 1-D and 2-D Fourier Transforms Rearranging the Fourier Integral, x y F(u) y Taking the integrals along x gives, Taking the integrals of along y gives F(u,v)

6. Sampling Model We use the comb function and scale it for the sampling interval X.

7. Spectrum of Sampled Signal Prior to sampling, After sampling,... Multiplication in one domain becomes convolution in the other,

8. Spectrum of Sampled Signal: Band limiting and Aliasing If G(u) is band limited to u c, (cutoff frequency) G(u) = 0 for |u| > u c. ucuc To avoid overlap (aliasing), ucuc Nyquist Condition: Sampling rate must be greater than twice the highest frequency component.

9. Spectrum of Sampled Signal: Restoration of Original Signal Can we restore g(x) from the sampled frequency-domain signal? ucuc Yes, using the Interpolation Filter

10. Spectrum of Sampled Signal: Restoration of Original Signal(2) From previous page, g(x) is restored from a combination of sinc functions. Each is weighted and shifted according to its corresponding sampling point.

11. Visualizing Sinc Interpolation Original functionSampled function Weighted and shifted sincs for 3 sample points shown by black arrows Each row shows convolution of shifted sinc with a sampled point. Sum lines along vertical direction to get output.

12. Original functions and output a) Continuous waveform b) Sampled waveform c) Sinc interpolation of sampled waveform ( sum of vertical lines in lower left plot from previous slide. a)b)

13. Two Dimensional Sampling

14. 2D Sampling Goal: Represent a 2D function by a finite set of points. - particularly useful to analysis w/ computer operations. Points are sampled every X in x, every Y in y. How will the sampled function appear in the spatial frequency domain?

15. Two Dimensional Sampling: Sampled function in freq. domain How will the sampled function appear in the spatial frequency domain? Since The result: Replicated G(u,v), or “islands” every 1/X in u, and 1/Y in v.

16. Example Let g(x,y) =  (x/16  y/16) be a continuous function Here we show its continuous transform G(u,v) Now sampling the function gives the following in the space domain

17. v u Fourier Representation of a Sampled Image 1/X 1/Y Sampling the image in the space domain causes replication in the frequency domain

18. Two Dimensional Sampling: Restoration of original function will filter out unwanted islands. Let’s consider this in the image domain.

19. Two Dimensional Sampling: Restoration of original function(2) Each sample serves as a weighting for a 2D sinc function. Nyquist/Shannon Theory: We must sample at twice the highest frequency in x and in y to reconstruct the original signal. (No frequency components in original signal can be or)

20. Two Dimensional Sampling: Example 80 mm Field of View (FOV) 256 pixels Sampling interval = 80/256 =.3125 mm/pixel Sampling rate = 1/sampling interval = 3.2 cycles/mm or pixels/mm Unaliased for ± 1.6 cycles/mm or line pairs/mm