Lab: Fourier Transforms (2)

CSC 295 - Computer Vision - Weinman

Summary:

In this lab you will examine the filtering properties of some common kernels in the frequency domain.

Due:

Wed 2/24

Deliverables

• The Matlab script used to make your comparisons and generate all figures
• (5 points) Comparison of Gaussian and its FFT (A.7)
• (10 points) FFT of cameraman and Gaussian (B.3, B.4)
• (5 points) Product of transforms (B.6)
• (5 points) Frequency observations (B.7)
• (10 points) Inverted product (B.9)
• (10 points) Inverted product observations (B.10)
• (5 points) LoG kernel (C.2)
• (5 points) LoG FFT magnitude (C.5)
• (5 points) LoG frequency observations (C.6).
• (5 points) LoG filter responses (C.9)
• (5 points) LoG kernel detection predictions and observations (C.3, C.10)
• (5 points) Box filter FFT magnitude (D.5)
• (10 points) Box filter observations and commentary (D.6, D.7)
• (10 points) Professionalism of write-up

Exercises

A. Gaussian Fourier Transform

1. The command linspace gives a vector of N linearly spaced numbers between an upper and lower bound. We can combine this with meshgrid to generate a domain for creating and plotting functions. Create a 256×256 domain over -20 to 20 as follows.

   [xx,yy] = meshgrid( linspace(-20,20,256), linspace(-20,20,256);
2. Using its functional form,
   

   g(x,y)=exp(-(x2+y2)/2),

   create a matrix containing a 2-dimensional Gaussian with a variance of 1 using the domains for x and y you created above.
3. Normalize your gaussian kernel matrix so that its contents sum to 1.
4. Use mesh to plot your kernel in a new figure window.
5. Use fft2 to take the Fourier transform of your Gaussian kernel.
6. Use mesh to plot the magnitude of the kernel's Fourier transform (don't forget that you need to do fftshift on the result before you visualize it.)
7. How do the two mesh plots compare?

B. Filtering in the Frequency Domain

1. Load the cameraman image and convert it to doubles.

   X = im2double(imread('cameraman.tif'));
2. Take the Fourier transform of the cameraman.
3. Use imshow to display the (shifted and log compressed) magnitude of the cameraman transform. Save this image using print.
4. Use imshow to display the (shifted and log compressed) magnitude of the Gaussian kernel transform. Save this image using print.
5. Take the componentwise product of the cameraman and Gaussian kernel transforms.
6. Use imshow to display the (shifted and log compressed) magnitude of the product. Save this image using print.
7. What effect does this product have on the low and high frequencies, respectively?
8. Take the inverse Fourier transform with ifft2. You'll likely need to extract the real component of the result (using the command real), and you may also need to do an ifftshift on the result to get the quadrants back in the right place.
9. Use imshow to display the result. Save this "filtered" image.
10. Does the operation have the intended result? How do the borders compare to a version created using conv2?

C. Other Filters

1. Using the functional form given above, create a matrix containing the 2-D LoG with the same x and y domains used above.
2. Display your LoG kernel using both imshow and mesh. Save one of these images.
3. As a filter, what types of structures would you expect the LoG to detect?
4. Take the Fourier transform of the LoG.
5. Use imshow to display the (shifted and log compressed) magnitude of the LoG's transform. Save this image using print.
6. What effect do you expect the LoG to have on the low, mid, and high frequencies?
7. Take the componentwise product of the cameraman and LoG kernel transforms.
8. Take the inverse Fourier transform with ifft2. You'll likely need to extract only the real component (using the command real), and you may also need to do an ifftshift on the result to get the quadrants back in the right place.
9. Use imshow to display the result (the LoG is not an averaging filter, so you'll have to relax the bounds on the display). Save this image using print.
10. Where are the responses the strongest? Does it match your predictions?

D. Box Filter

1. Create a 256×256 kernel of zeros.
2. Makes the first 10 rows and columns of your kernel a uniform box filter. Don't forget to normalize!
3. Take the Fourier transform of your box filter kernel.
4. Use imshow to display the (shifted and log compressed) magnitude of the box filter's transform. You may wish to add false color using a color map to more easily see the differences in magnitudes. The command

   colormap(jet)

   renders the grayscale values as indices into a colormap that ranges from cool to warm colors.
5. Save your image using print.
6. Is the result "isotropic?" (That is, is it the same in all directions?) How does its elimination of high frequencies compare to the Gaussian?
7. Why might we avoid smoothing with the box filter before downsampling?