# Filtering fun4

Posted by Steve Eddins,

Today I want to take the test pattern I created last time and subject it to a variety of frequency-based filters.

In this post I'll be using a variety of frequency design and frequency-response visualization tools in MATLAB, Signal Processing Toolbox, and Image Processing Toolbox, including fir1, ftrans2, freqspace, fwind1, and freqz2. I won't be giving much explanation of these filter design techniques. Perhaps in the future I'll explore filter design here in more detail. For now, though, I really just want to have some fun with this test pattern.

[x,y] = meshgrid(-200:200);
r = hypot(x,y);
km = 0.8*pi;
rm = 200;
w = rm/10;
term1 = sin( (km * r.^2) / (2 * rm) );
term2 = 0.5*tanh((rm - r)/w) + 0.5;
g = (term1 .* term2 + 1) / 2;

imshow(g, 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')
title('Test pattern')

Example 1: Design a one-dimensional lowpass filter and apply it in both directions.

b = fir1(30,0.3);
h = b' * b;
g1 = imfilter(g, h);

imshow(g1, 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')

Example 2: Design a one-dimensional highpass filter and apply it in both directions.

b2 = fir1(30, 0.3, 'high');
h2 = b2' * b2;
g2 = imfilter(g, h2);

imshow(g2, [], 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')

Example 3: Design a one-dimensional bandpass filter and apply it on both directions.

b3 = fir1(30, [0.3 0.4]);
h3 = b3' * b3;
g3 = imfilter(g, h3);

imshow(g3, [], 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')

Example 4: Apply the one-dimensional bandpass filter from the previous step in only one direction.

g4 = imfilter(g, b3);

imshow(g4, [], 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')

Example 5: Use ftrans2 to turn the one-dimensional bandpass filter from the previous example into a two-dimensional bandpass filter that is approximately circularly symmetric.

h5 = ftrans2(b3);
g5 = imfilter(g, h5);

imshow(g5, [], 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')

Example 6: Use fwind1 to design a two-dimensional filter that has a wedge-shaped frequency response.

[f1, f2] = freqspace(101, 'meshgrid');
theta = cart2pol(f1, f2);
Hd = ((theta >= pi/6) & (theta < pi/3)) | ...
((theta >= -5*pi/6) & (theta < -2*pi/3));
mesh(f1, f2, double(Hd))
title('Ideal frequency response')
h6 = fwind1(Hd, hamming(31));
freqz2(h6)
title('Designed filter frequency response')
g6 = imfilter(g, h6);

imshow(g6, [], 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')

Example 7: Compare the use of an averaging filter and a Gaussian filter. This example is inspired by Cris Luengo's comment from last week.

g7 = imfilter(g, fspecial('average', 9));
imshow(g7, [], 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')
title('13-by-13 averaging filter')

You can see that some very high frequencies remain in the output. The Gaussian filter performs much better:

g9 = imfilter(g, fspecial('gaussian', 9, 2));
imshow(g9, [], 'XData', [-.8 .8], 'YData', [-.8 .8])
axis on
xlabel('Normalized frequency'), ylabel('Normalized frequency')
title('13-by-13 Gaussian filter')

OK, I'm going to give this test pattern a rest for now. Phew!

Get the MATLAB code

Published with MATLAB® 7.12

Vitaly Boyko replied on : 1 of 4

Hi Steve, it looks nice. However, what is about creating a digital image processing toolbox with basic statistics and signal processing functions embedded into it. For example the ClusterTrack script from http://lccb.hms.harvard.edu/software.html
also requires statistics and signal processing toolboxes, which obviously should not cost USD 2’000 extra. I guess that Marketing Staff is primarily responsible for generating “vast” profits. What is about being Customer Friendly in this respect? Many thanks in advance. Vitaly

Steve replied on : 2 of 4

Vitaly—As it says on the upper right of this page, I write this blog about “image processing concepts, algorithm implementations, and MATLAB.” In general, I only allow comments that are relevant to the particular topic of the post. For this post, for example, I could imagine following up on questions regarding filters, filter design, visualization in the frequency domain, applications of filtering, etc. I regard off-topic comments as being unfriendly to other readers of this blog, and I don’t normally allow them.

Carlos Ricci replied on : 3 of 4

Steve, I enjoyed your post. Your radial chirp test function has some interesting properties. In IE9 on Windows 7, when you hover over the toolbar you get a thumbnail image of whatever that window happens to be displaying. When you do this with your test image, you get a surprise!

I believe the pattern is indeed from aliasing, as the thumbnail must be just a straight sub-sampling of the underlying window image. But it’s an interesting phenomenon nonetheless… how aliasing can make the high-frequency contours curve back into copies of dc. I suppose that is how it must be for all the boundary conditions to be satisfied, but still, knowing that doesn’t make it any less fascinating to look at!

Wayne Bretl replied on : 4 of 4

For Carlos’ information, the “radial chirp function” has a name – it is called a “zone plate.” It is the most common test signal for illustrating two-dimensional frequency response.

A moving or pulsing zone plate is sometimes used in studies of three-dimensional (horizontal, vertical, temporal) response of imaging systems.