Comment by whatshisface
5 years ago
>That "infinitely many overlaps" I was talking about happens with lenses too and is essentially a convolution where you convolve the image with itself (actually many different perspectives of itself if I am correct). Which is just the Fourier transform.
That statement is a bit muddled, let me unpack it. The infinitely many overlaps thing can be expressed mathematically as the convolution of the image with a function that's one where the aperture is open, and zero where it's not. The thing about the Fourier transform is actually related to a different phenomenon. When the slit is really small, you start getting diffraction effects. The diffraction bands are approximately the Fourier transform of the slit function. However that is not significant unless the slit is extraordinarily tiny.
Indeed! We like to think of light as photons, traveling in rays in straight lines. That's geometric optics and works great for most purposes in photography.
Physical optics takes into account the wave nature of light. This becomes important when the size of the lens becomes small (eg pinholes) ... there's diffraction around edges and pixels receive contributions from many points in space.
Geometric optics lets you model using ray-tracing, reflections, and Snell's law refraction.
Physical optics uses tools such as Fourier transforms, convolutions, and sinc functions.
Understanding a simple lens system? Geometric optics is your friend. Building an astronomical telescope? Check into physical optics.
Thanks for weighing in. My kid and I are really enjoying The CuCoo's Egg!
I read that a few times when I was younger. It's such a great story, and I've always been meaning to revisit it as an adult. When my kids get a little older I'm sure we'll enjoy it.
The diffraction bands are approximately the Fourier transform of the slit function.
Is this correct from a physics standpoint? Feynman had a lot to say about light in his book (QED https://www.amazon.com/QED-Strange-Theory-Light-Matter/dp/06...) and never framed it this way. At one point he remarked that the explanations in the book were related to diffraction, and the explanation there was very different from the Fourier transform.
In short, yes. An ideal image source incident on a positive optical lens produces its spatial Fourier transform at the lense's focal point. This is easiest to see with a lasersl backlighting a transparency, since the light is collimated and monochromatic. The transparency produces diffraction at its edges, which causes the effect. Actually, you'd also see the spatial Fourier transform at infinity if you took away the lens. The result of this is that you can do cool spatial frequency filtering effects at the focal point, then convert it back into an image with another lens. Laser systems that require high precision will use such a setup to remove high-frequency components and pass just the collimated light.
Pinhole lenses create visible diffraction effects pretty fast. Actually, even normal lenses cause diffraction at small ~f/11-f/22 apertures.
I suspect the GP comment is referring in context to the use of the Fourier transform to efficiently implement convolution.