Pre-class Exercise
• Recall from Lecture 8, the Fourier transform of the Laplacian
operator is
∂2 f ∂2 f
∇ f ( x, y ) = 2 + 2
∂x
∂y
2
⇔ − (2π ) 2 (u 2 + v 2 ) F (u , v)
• So the equivalent Fourier domain filter for the Laplacian filter is
H (u , v) = −(2π ) 2 (u 2 + v 2 )
• Using Matlab, plot the magnitude of H(u,v) as a function of (u,v).
M = 100;
N = 100;
[u,v] = meshgrid(-M/2:M/2, -N/2:N/2);
% This is the true Fourier transform of a Laplacian.
H1 = ...
% Fill in here the equation for H(u,v)
figure, imshow(abs(H1), []), impixelinfo
figure, surf(abs(H1)), colormap jet;
1
Colorado School of Mines
Department of Electrical Engineering and Computer Science
Pre-class Exercise (continued)
• Now look at the equivalent Fourier domain filter for the digital
approximation to the Laplacian; ie
0 1 0


h = 1 − 4 1
0 1 0


Pad h with zeros so that it is the same size as the (u,v) space used
previously. Then take the discrete Fourier transform (using
Matlab’s fft2) and plot the magnitude of H(u,v).
Compare the two plots. They should be very similar at low
frequencies, but differ quite a bit at large frequencies.
2
Colorado School of Mines
Department of Electrical Engineering and Computer Science
A note about units
• You may have noticed that the scale of the two plots is
quite a bit different.
• This is because of the difference between the units of
measure for the spatial domain and the frequency
domain.
• As explained on slide 10 of lecture 8.1, unit increments
of u actually correspond to increments in frequency of
1/(M∆x).
– If we use ∆x=1, then we should actually use
u = 1/M, 2/M, 3/M, … M/M
– instead of
u = 1,2,3, … M
3
Colorado School of Mines
Department of Electrical Engineering and Computer Science