Home / Problem Set 11 / Cats

Cats

The questions below are due on Monday May 03, 2021; 10:00:00 PM.
 
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Note that this exercise is intended to be solved by hand, without the use of computation.

Ben Bitdiddle took a photograph of his cat, but he only saved the associated DFT coefficients X[k_r, k_c], rather than saving the original image. However, he knows the original 77 \times 51 image looked like this:

Ben tries several different methods of recovering the original image based on X[k_r, k_c]. For each of the methods described below, indicate which of the images (labeled A-T) would have resulted from that approach. In these images, grey colors represent positive values (black represents 0, white represents 1), and red represents negative values (black represents 0, bright red represents -1).

For all parts, assume that r=0, c=0 corresponds to the upper-left corner of the image (rows increase downward, and columns increase to the right).

Approaches

  1. Applying the inverse DFT to the real part of X.
  2. Applying the inverse DFT to the imaginary part of X.
  3. Applying the inverse DFT to j times the imaginary part of X.
  4. Applying the inverse DFT to X after setting X[0,0] = 0.
  5. Applying the inverse DFT to X after setting X[25, 38] = 0.
  6. Applying the inverse DFT to X after subtracting 1\over {51\times 77} from every value.
  7. Applying the inverse DFT to X after multiplying every value by e^{j\pi}.
  8. Applying the inverse DFT to X after multiplying every value except X[0,0] by e^{j\pi}.
  9. Applying the inverse DFT to X after negating the phase of every value.

Images

 

A B C D E
F G H I J
K L M N O
P Q R S T

Question

Which image matches each process? Enter a sequence of letters in the box below. For example, entering GDIC... would imply that G matches with process 1, D matches with process 2, I with 3, C with 4, ...

If an approach would have led to an image with nonzero imaginary components (and thus would have resulted in a Python error when trying to save the image), enter X for that approach.

Matching letters: