Quantum circuit for the fast Fourier transform(link.springer.com)
link.springer.com
Quantum circuit for the fast Fourier transform
https://link.springer.com/article/10.1007/s11128-020-02776-5
12 comments
Ears
A cochlea? Yes, that does occur naturally, and it does work to allow us to hear different frequencies of sound...
So, now the next question is, could we prove that a cochlea implements a Fourier Transformation? Note that this Fourier Transform isn't applicable to all frequencies, that is, our ears cannot hear vibrations beyond a certain high pitch, so if a Fourier Transformation could be proved, we could almost think of this as a Fourier Transformation inside of a specific range of frequencies...
But anyway, an interesting observation!
So, now the next question is, could we prove that a cochlea implements a Fourier Transformation? Note that this Fourier Transform isn't applicable to all frequencies, that is, our ears cannot hear vibrations beyond a certain high pitch, so if a Fourier Transformation could be proved, we could almost think of this as a Fourier Transformation inside of a specific range of frequencies...
But anyway, an interesting observation!
We know how it works, and it's a simple version of a Fourier transform. The cochlea has hairs of different lengths that vibrate according to the contribution of the sine wave that corresponds to the resonant frequency of that hair's length. By the time neurons get involved at all, it's already a signal of the frequency domain over time.
Great question!
Prisms and diffraction pattern gratings split light according to the frequency of the photons (colours) in the input. That's a Fourier transform in a certain sense, and you'd use that to make a spectral analyzer. But what about the Fourier transform of an image, according to the frequency components of objects in the image? Like fence posts at a regular spacing, and such?
Well you might be excited to learn that optical lenses actually compute the Fourier transform of their input image:
http://web.mit.edu/2.710/Fall06/2.710-wk10-a-sl.pdf (the ideal thin lens as a Fourier transform engine, PDF)
Prisms and diffraction pattern gratings split light according to the frequency of the photons (colours) in the input. That's a Fourier transform in a certain sense, and you'd use that to make a spectral analyzer. But what about the Fourier transform of an image, according to the frequency components of objects in the image? Like fence posts at a regular spacing, and such?
Well you might be excited to learn that optical lenses actually compute the Fourier transform of their input image:
http://web.mit.edu/2.710/Fall06/2.710-wk10-a-sl.pdf (the ideal thin lens as a Fourier transform engine, PDF)
Great link, thank you very much!
Also (since I didn't know what a "thin lens" was prior to reading this (I never studied Optics)) I Googled it, and found the Wikipedia page for it:
https://en.wikipedia.org/wiki/Thin_lens
>"In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are sometimes called thick lenses.
The thin lens approximation ignores optical effects due to the thickness of lenses and simplifies ray tracing calculations."
Which makes me wonder if there's a link between Fourier Transformations and Ray Tracing, and if so, how does it express itself?
Although, that's probably going seriously off-topic for this HN article! <g>
Anyway, thanks again for the link/info!
Also (since I didn't know what a "thin lens" was prior to reading this (I never studied Optics)) I Googled it, and found the Wikipedia page for it:
https://en.wikipedia.org/wiki/Thin_lens
>"In optics, a thin lens is a lens with a thickness (distance along the optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is not negligible are sometimes called thick lenses.
The thin lens approximation ignores optical effects due to the thickness of lenses and simplifies ray tracing calculations."
Which makes me wonder if there's a link between Fourier Transformations and Ray Tracing, and if so, how does it express itself?
Although, that's probably going seriously off-topic for this HN article! <g>
Anyway, thanks again for the link/info!
Your average water droplet acts as a dispersive element (a prism). Dispersive elements can act as temporal Fourier transforms. Lenses act as spatial Fourier transforms. Going further, any mechanical device that exhibits resonance can be claimed to act as a Fourier transform (but that starts sounding meaninglessly vague).
> then is there anything self-existing in Nature which could be used to (or does in actuality) implement a Fourier Transform?
The lens in the human eye is continuously doing Fourier transforms via the lens:
https://en.wikipedia.org/wiki/Fourier_optics#Fourier_transfo...
The lens in the human eye is continuously doing Fourier transforms via the lens:
https://en.wikipedia.org/wiki/Fourier_optics#Fourier_transfo...
Don't think prism and FT are analogous. Fourier transform is a symmetric operation between dual expressions of the same thing - frequency form and amplitude form.
What effect do yall think this could have on AI? There is literature describing performance advantages of using Fourier transform space for convolutional neural networks
For those interested there is a cool relationship between a convolution and a Fourier transform:
https://en.wikipedia.org/wiki/Convolution_theorem
The fourier transform is a linear operator (implementable as a dense NN layer) so I wouldn't be surprised if some NNs learn it or an approximation during training.
If a Fourier Transform can be computed from something in which Quantum Bits are implemented by some underlying physical system, then is there anything self-existing in Nature which could be used to (or does in actuality) implement a Fourier Transform?
Now, I don't know the answer to that.
Something not occuring naturally in Nature is a Prism, and a Prism splits light, and that's sort of, kind of analogous to what a Fourier Transform does, although, perhaps there's a limited range of frequencies at which the effect is applicable...
So now the next question is, is there anything in Nature, that is, naturally occurring, which acts like a Prism does?
Although, perhaps we're not looking for something that necessarily splits light, perhaps we're looking for something that splits other electromagnetic radiation, that is, waves of some sort...
Also, maybe there's some kind of naturally occuring shape of rock that could be used to "split" ocean waves into one or more component parts... while perhaps not a true Fourier Transform, it might be a step in the right direction for conceptualization purposes...
Anyway, just thinking aloud... It would be interesting to know all of the systems, both natural and man-made, which are capable of Fourier Transforms -- or even just splitting waves of different sorts into component parts...