# Filtering fun 4

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

## 4 CommentsOldest to Newest

**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

**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.

**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!

**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.

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