Assignment 11a: Image Processing

by Mike Gleicher on November 28, 2014

This assignment is “due” on Wednesday, December 3. However, you don’t need to turn it in. Just make sure you can do these kinds of problems – the exam is coming up soon, and these kinds of problems make for great exam questions (even if we don’t have time for an image processing project).

Question 1:

Be sure you can do discrete convolutions!

For example, if:
f = [ 1 2 1 3 1 4 1 5 1 6 2 5 3 4 3 ]
g = 1/3 [1 1 1]
h = 1/6 [1 2 3]  (remember: convolutions reverse)

Make sure you can compute f*g, and f*h.

Make sure you know how to deal with the boundaries. (zero pad, repeat ends, …)

Question 2:

Make sure you can reconstruct a signal given the discrete signal and the sampling kernel.

Consider reconstructing the signal from the following samples (the first sample was at t=0):
f = [ 1 2 1 3 1 2 2 ]

Compute the value of the reconstructed signal at t=1.5, t=2, and t=3.25 with the following reconstruction filters (your answer for each should be 3 numbers).

2.A The unit box (g(t) = 1 if -.5 < t <= .5, 0 otherwise)
    Note: in the book, this is the continuous case and r=1/2
2.B The unit tent (g(t) = (1-t) if -1 < t <= 0, t if 0 < t < 1, and 0 otherwise)
    Note: in the book, this is the tent filter of r=1/2

Question 3:

Make sure you understand the idea of resampling by pre-filtering.

Consider resampling the following signal:
   [ 0 0 4 4 0 0 4 0 4 0 4 0 0 0 4 0 0 0 ]
   using the pre-filtering kernel 1/4 [1 2 1]

   3.A If you resample the signal at half the sampling rate, what result would you get?

   3.B If you made a small change in how you sampled in 2.A (say, chose even instead of odd values), would the results be very much different? What does this say about the adequacy of the kernel for doing this resampling?

   3.C If you resample the signal at 1/3rd the samping rate (pick every third sample), what result would you get?

   3.D If you made a small change in how you sampled in 2.C (say, shifted the samples a little), could the results be very much different? What does this say about the adequacy of the kernel for doing this resampling?

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