optical convolution processor

so i've been sifting through my old photos and uncovered some images which, having been unfiled, didn't find it's way into the lab reports last semester. this one in particular is for the optical convolution processor under the session on fourier optics.

Equipment

HeNe Laser
Lenses
Filters
Pinhole
CCD

Set-up

errata: "image" → should be "object". that is, the image plane is the plane with the ccd.

Analysis & Discussion

An optical convolution processor, or 4f processor, is an optical setup which enables real time implementation on the mathematical cross-correlation and convolution methods. It states that the convolution of two images (f(x,y) and h(x,y))



can be expressed as a simple product of the fourier transforms of both images:



Given a certain image, we first would like to view its fourier transform. Using the setup shown above, we obtain the following images at the specified planes:

object mask placed on the object plane.

image at fourier plane: "A" decomposed in it's spatial frequencies.

image at CCD without filter at fourier plane

if a low pass filter, like this, is placed at the fourier plane, we can clearly see that light corresponding to the high frequencies are filtered out. hence, in the image plane we see the image below.

low pass filter

image at CCD with a low-pass filter at fourier plane

on the other hand, placing a high-pass filter on the fourier plane generates the following image:

image at CCD with a high-pass filter at fourier plane

when a pinhole is set in the fourier transform plane, intuitively we would expect it to give a better, clearer output of the image at the observation plane. this is not the case that was observed, however. since the lenses that make up the system are imperfect, then their imperfection adds to the noise which is contained in the high frequencies. decreasing the aperture removes the ambient noise hence we can see the image become clearer. further decreasing the aperture removes more higher frequencies hence the image becomes more blurred at the edges.

these are simple examples of spatial filtering. spatial filtering is a technique for filtering out certain spatial frequencies usually in order to improve input image caused by scattering by defects or particles in the air. when focusing a beam, the image of the source composed of low frequencies is concentrated at the centre while higher order frequencies, noise, are focused further away.by employing a pinhole at the fourier transform plane, the higher order frequencies will be filtered out thus, theoretically, giving a clean spatial profile at the output.

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