Project Plan: A Guide To Convolution In Action | Modeling and Experimental Tools with Prof. Magnes

Description of project

Convolution is the theory behind interpreting the data presented. When physicists use optical tools for their experiments they need to understand whether the optics they are using will transmit the proper information. In practice one is limited to the material of the filters, as the wave fronts get distorted by the material they pass through. In order to be able to work with what you have, physicists model their possible optics layout before purchasing the optics. To retrieve the desired information physicists model the convoluted wave front, and then prepare a good deconvolution mechanism that in practice should yield the necessary data. This project is an attempt to understand convolution and deconvolution of electromagnetic waves through optical filters. I will be exploring the theory behind convolution, demonstrate examples of convolution, attempt deconvolution of a convoluted distributions, and then hopefully show a real world example of actual optical filters.

Theory

Convolution is the mixture of functions that individually have known Fourier transforms. A Fourier transform is the transformation of a square-integrable function $f(x)$ from one domain to another (5). It is defined as such:

$f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }\!\Phi(k)e^{ikx}dk \qquad \Phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^{\infty }\!f(x)e^{-ikx}dx$

Both the function and the Fourier transform describe the same system though under different but related spaces (See: Plancherel’s Theorem). The advantage is if you have a machine that measures time but you are trying the measure the system as a function of frequency, then you can measure the system with that machine and simply take the Fourier transform of your data. Unfortunately, the deviations of the Fourier pairs are bounded by the uncertainty principle. A function with a narrow spread will yield a transformation with a wider spread, and vice versa.

$\sigma^2_\Psi\sigma^2_\Omega\geq\left(\frac{1}{2i}\langle[\hat{\Psi},\hat{\Omega}]\rangle\right)^2$

where $\hat{\Psi}$ and $\hat{\Omega}$ are two arbitrary operators, and $\sigma_{i}$ is the standard deviation.

The definition of a convolution of two functions is defined as follows:

$f_{1}(x)\ast f_{2}(x)=\int_{-\infty }^{\infty }\!f_{1}(x')f_{2}(x-x')\mathrm{d}x' \\ \\ \Phi_{1}(k)\ast \Phi_{2}(k)=\int_{-\infty }^{\infty }\!\Phi_{1}(k')\Phi_{2}(k-k')\mathrm{d}k'$

The convolution theorem then states that the Fourier transform of a convolution is the product of the Fourier transforms to a factor of $\sqrt{2\pi}$ . Similarly, the Fourier transform of a product of functions is the convolution of the Fourier transforms (5):

${\widehat_{f_{1}\ast f_{2}}} (x)= \sqrt{2\pi}\;{\Phi_1}(k)\cdot {\Phi_2}(k) \qquad {\widehat_{f_{1}\cdot f_{2}}} (x)= \frac{1}{\sqrt{2\pi}}\;{\Phi_1}(k)\ast {\Phi_2}(k)$

The implication of this theorem is if any arbitrary curve can be expressed as a product of functions with given Fourier counterparts, then it can undergo deconvolution to yield the desired data as represented by known functions (3).

$f_{1}(x)\ast f_{2}(x)\rightleftharpoons \Phi_{1}(k)\cdot \Phi_{2}(k)$

Timeline

All work will either be done on my computer or at the sci-vis lab.

Week 1: I will become familiar with Professor Magnes’s MATLAB script and reconstruct it in Mathematica. I will use the information provided for me in the texts to create new examples of various convoluted distributions. I will also provide an introduction to the theory that pertains to my project.

Week 2: By now I should be familiar enough with Mathematica to display some cool examples. I will attempt to provide an example of real world convolution of two optical filters from thorlabs. In practice convolution of two actual filters is not as easy as adding two equations. I will need to figure out how to generally apply my ideal examples to sets of data.

Week 3: I will now attempt to demonstrate the deconvolution of a distribution. I will show this by using one of my previous examples. In theory I should be able to demonstrate proper deconvolution of two functions.

Week 4: Now that I have demonstrated deconvolution, I should be able to demonstrate deconvolution of the data I had previously showed in convolution form. This would be very tricky but hopefully I should be able to succeed.

Week 5: I may be setting the bar up too high, so possibly some of my work will take more time than expected. I should be finished with everything now. Here I will tweak my project, and conclude my demonstration.

Note: I may stray from my timeline, but that is only because I got caught up in an interesting phenomenon. Should I decide to alter my direction, I will update my timeline accordingly.

Resources

Will be updated if necessary

(1) Griffiths, David J. Introduction to Electrodynamics. Upper Saddle River, NJ: Prentics Hall, 1999. Print.

(2) Hecht, Eugene. Optik. München: Oldenbourg, 2001. Print.

(3) James, J. F. A Student’s Guide to Fourier Transforms: With Applications in Physics and Engineering. Cambridge: Cambridge UP, 2011. Print.

(4) Pedrotti, Frank L., Leno S. Pedrotti, and Leno Matthew. Pedrotti. Introduction to Optics. Upper Saddle River, NJ: Pearson Prentice Hall, 2007. Print.

(5) Sadun, Lorenzo Adlai. Applied Linear Algebra: The Decoupling Principle. Providence, RI: American Mathematical Society, 2008. Print.

Question to the reader: Is the verb of deconvolution: deconvolve or deconvolute?

1 thought on “Project Plan: A Guide To Convolution In Action”

Jenny Magnes April 18, 2014 at 9:53 am

I like your explanation of convolution. It might be beneficial to mention that the “machine” (instrument) itself is performing a FT transform. For example, a grating FT the light into its frequency domain. Thinking of the process that way might make conceptual sense.

It might be good for you to have someone else proofread your post to avoid typos.

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