Mathematical Methods of Image Processing (MA 490), Winter Quarter, 1999-2000
Homework 4 – due Monday, January 10, 2000
Instructions. As always, you may work with other people when you are thinking and talking
about the homework problems, but the problems must be written up individually.
1) First build the DFT Fn matrix for an arbitrary n: Then for n = 2; 4; 8; :::; 28 (larger if you
want to), compare the number of ‡ops from computing Fn v with C n2 (here let v = [1 : n]):
Plot the ‡ops vs n and try to …nd a good value for C (put both plots on the same set of
axes and use *’s or +’s rather than connecting the points with line segments). Then do the
same thing with f f t(v) and Cn ln(n): Note the di¤erence in ‡ops for larger values of n:
2) Reproduce Figures 2.25-2.28 on page 37. Also compute mD:
3) Recall that Ek is a column of the matrix on page 3 of the handout. Let Ck and Sk be the
real and imaginary parts of Ek : Use the fact that the Ek ’s are pairwise orthogonal to show
that the the family of vectors 1; Ck ; and Sk are also orthogonal (here 1 represents the vector
of all 1’s; ie 1 = E0 ): This is a little trickier than it might seem. First look at an example. I
think that you may need to use the fact that En¡k is the conjugate of Ek :
4) Exercise 2.10, page 39 (the ?? refers to the chart on page 24)
5) Exercise 2.15, page 39
6) Exercise 2.18, page 40
7) Exercise 2.24, page 40