# What are zernike polynomials and their use

Discussion in 'Camera Building, Repairs & Modification' started by Mustafa Umut Sarac, Nov 23, 2007.

1. ### Mustafa Umut SaracSubscriber

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Hello ,

I found several web pages about zernike polynomials . What are they and what are their use ? I researched to change aberrations of a photograph and simulate lens on photographs. I couldnt understand many things described.
Do this math way help me to rerender a photograph ?
This is not analog but lens design is depends on computers.

Best ,

Mustafa Umut Sarac

2. ### Bruce SchultzMember

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I'm not sure, but I bet they're good fried in batter with lots of cayenne pepper.

3. ### Dan FrommMember

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Mustafa, have you considered going back to school? The usual cure for not knowing the math involved in something is to learn it, and courses often help.

4. ### OleModeratorStaff MemberModerator

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Taking liberties and cutting corners:

Zernike polynomials are a set of mathematical functions that when added up give the solution to the form of the wavefront from a lens.

Functions of this sort, the best known being Fourier and Laplace, allow transforming an arbitrarily complex formula into a set of simpler formulas. Very simplistically: the fourier formulas for a sound are what are shown in the bouncing-bar equalizer display of an MP3 player. Finding how something (say a loudspeaker) will respond to a complex signal (say a snare drum) is very hard to calculate. But how a loudspeaker will respond to a pure tone is easy to calculate. So if one takes the the sound of a snare-drum and finds the set of pure tones that make up its sound then the response of the speaker to a snare drum will be the same as the adding together the loudspeaker's response to each of the pure tones. It is a lot easier to calculate the response of a thousand types of loudspeaker than build and measure a thousand types.

Ditto with lenses. A lens, after all, just applies a mathematical formula to the path of a ray of light.

The MTF response of a lens is the fourier transform of a lens' performance: the MTF graph is like the lens' equalizer graph.

There are many 'transforms', and some transforms are better at representing
some physical processes than others. Zernike functions are good at representing the effect of lenses on light.

The functions are called 'orthogonal' because they don't interact. Like measuring a box in the three orthogonal axis of geometry class: changing the the height does nothing to the depth or width.

The orthogonality is what lets one take the response to the simpler signals and add them up at the end to find the response to a complex signal. What the lens does to one Zernike polynomial has nothing to do with what it does to any other Zernike polynomial.

In sum -- the whole thing is a mathematical 'trick' to make calculations easier, similar to the 'casting out nines' trick of accountants.

6. ### Mustafa Umut SaracSubscriber

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I liked orthogonal concept. I read somewhere lens design is difficult because every single change on a aberration calculation effects the others.
i think zernike concept is useful because of that.
zernike got 1953 nobel if i m not wrong , 54 years and we dont know nearly anything about it.

7. ### tim_wallsMember

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As a general rule, I think it is dangerous to extrapolate the state of all human knowledge from one's own ignorance.

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