# Jähne test pattern – take 35

Posted by Steve Eddins,

Earlier today I told you that I was feeling a little dense because I couldn't figure out the right parameters to use in the tanh term of this test pattern:

(This is equation 10.63 in Practical Handbook on Image Processing for Scientific Applications by Bernd Jahne.)

I'm grateful to reader Alex H for quickly enlightening me. He described as an approximation to a step function, where a is the location of the step and w is the width of the transition. That was very helpful.

I've also realized that I misinterpreted the meaning of . In the book this is described as the "maximum radius of the pattern," and I assumed this would be fixed as the distance from the center of the square image to one of its corners. But now I realize that the author intended for this to be an adjustable parameter. That is, one can set so that the maximum instantaneous frequency is reached closer to the center than at the image corners.

For example, I can set the parameters so that the maximum instantaneous frequency of is reached in the center of the image edges, and then the tapering function prevents aliasing artifacts from appearing as you move out to the corners. The book's figure 10.23 is based on maximum instantaneous frequency of (a period of 2.5 samples) reached at the edges, so I'll use that.

[x,y] = meshgrid(-200:200);
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;
imshow(g,[])

Finally, a result that looks like the figure in the book!

Thanks again, Alex.

Get the MATLAB code

Published with MATLAB® 7.12

### Note

Mo replied on : 1 of 5

Something really weird – if I quickly scroll up and down, the image (a) seems to be modulated by a sinusoidal manifold that runs along the direction of motion and, better (b) seems to have smaller ‘ripples’, specially at the cardinal points.

The original scan doesn’t seem to have much of the first and definitely not the second – in fact if you look from different angles, you can see the ‘cardinal ripples’ in the final image, and lighter ripples symmetrically organized.

What’s going on?! (If it’s just me, I’m off to get an MR)

Dave S. replied on : 2 of 5

Thanks Steve, one definitely can’t stare at this pattern too long :)

It looks like an MTF could be calculated using a radial slice or average — is there anything in the Jahne book that leads in this direction?

Steve replied on : 3 of 5

Dave—Yes, MTF measurement using this pattern is discussion in a section called “Test Pattern for OTF Measurements.” You can find scanned pages of this section of the book’s second edition at Google Books. Search for “MTF”.

Xin replied on : 4 of 5

The pattern is also referred to as Fresnel zone plate in the holography field. A feature of this pattern is that the frequency is increasing or decreasing with the space. As for the pattern, an accurate Fourier transform result is difficult to obtain.

Sepide replied on : 5 of 5

How can I create a Fresnel zone plate with finite number of rings?