Homework 1, EE 528 Fall 2009
For Matlab problems, use as the image, I(m,n), either a chessboard image or a fence image.

1. Problems 4.9, 4.11a, 4.12 of AK Jain. Also verify 4.11a using MATLAB. A continuous function can be simulated by choosing x = [0:1/(100 zeta_s): 5/zeta_0]  (using a very small sampling interval).
2. Redo Example 4.2 with  (a) f(x,y) = cos 2pi(x/4a + y/8a) + cos 2pi(x/128a + y/128a), (b) f(x,y) = cos 2pi(x/4a + y/8a) + cos 2pi(4x/a + 4y/a).
   
3. Matlab assignment:
   
1. Demonstrate Moire effect for a DC source (constant grey scale image) using MATLAB. Note what I said in class wasn't exactly correct. You will need to decimate the image first and then interpolate using different size averaging filters.
   
2. Decimation and Interpolation (DO NOT use standard Matlab functions)
   
• First decimate without applying the low pass filter (Decimate in two steps: first do zeroing in the original image and then downsample). See the effect of aliasing in the decimated image. Start with decimating by a factor of 2 and keep going until you "see" the effect of aliasing.
  
• Let the chosen decimation factor is D. Now apply a low pass filter (say a 3x3 or a 5x5 averaging filter) to I(m,n) and then decimate by D. Keep increasing size of the averaging filter until the effect of aliasing is "negligible".
• DFT: See all of the above in the frequency domain by taking a DFT of a row of the image.
  
• Details for decimation by 2 (similarly do for any factor D)
  
• Take the image I(m,n), Set every alternate row and column to zero. Call this I_Z(m,n). Now downsample to get I_D(m,n) which is half the size.
  
• Compute DFT of I, I_Z and I_D  for either whole image or for a piece (or just 1D DFT) where you can "see" aliasing most easily
• Low pass filter the original image and repeat the above steps
• Do the same thing for different decimation rates D and for different
• Interpolation: Take the decimated image I_D(m,n), add D-1 zeros to along rows and columns to get an original size image I_U(m,n). Interpolate the upsampled image (try zero-order hold, linear or cubic interpolation) to get the reconstructed image I_hat(m,n). Do this both for the decimation of the original image and decimation of the LPF'ed image. Notice that the effect of aliasing will not go away by interpolation. To get a quantitative measure, you may want to look at the mean squared error between the reconstructed and original image in both cases. Note for the LPF'ed case, the original image will be the LPF'ed one.  Compute DFT again.
4. For one row of the chessboard image, write out equations for every image you got after decimation and after interpolation, i.e. do the entire thing on paper. One row of the chessboard can be written out as a difference of two unit step functions: u(x) - 2u(x-20)