4 ms·

That’s a really interesting way of phrasing it I’d never fully considered: of course you are right, a physical lens is just performing predictable mathematical

That’s a really interesting way of phrasing it I’d never fully considered: of course you are right, a physical lens is just performing predictable mathematical transformations on the light passing through it. Thank you, that was something of a revelation in understanding how these cameras actually work.

joshu 4y ago
More importantly, focusing a lens is one of many possible transformations. Look up coded apertures. so much is possible.
weinzierl 4y ago
"just performing predictable mathematical transformations" Just linear algebra, to be precise. A lens is just a matrix and rays are simply vectors. Combining lenses is merely matrix multiplication and complex optical systems can be represented by the result, which is a plain simple matrix again. The Eigenvalues tell you interesting properties of your optical system. I've always enjoyed this part of physics because it is so simple and elegant, yet so powerful. If you look closely linear algebra pops up in many places.
- ddalex 4y ago
  Going on a metaphysical tangent, it is a bit weird that A LOT of physical processes can be modelled with linear algebra, and don't require something way more advanced ...
  - fhars 4y ago
    That is actually deliberate, we tend to organize everyday life and the devices we build around processes that are easy to model, that is why many things look like linear processes and harmonic oscillators (the first two term of the Taylor expansion of the actual behavior). We change the type of spring we use if the current one starts to wear out early under normal operational conditions.
    - IIAOPSW 4y ago
      Let me rephrase on his behalf. Isn't it curious just how unreasonably capable our math is at expressing those physical laws of the universe to which we did not invent.
      - jamiek88 4y ago
        I don’t know, seems a bit Anthropic Puddle to me.
  - Someone 4y ago
    I think it’s more that we developed linear algebra that made us approximate physical processes by linear models. If you look close enough, almost nothing is linear. For example, we happily draw a linear scale on a mercury thermometer, even though we know that not to be correct, even if we incorrectly assume that the coefficient of expansion of mercury is independent of its temperature. Also, try explaining why the conversions between Celsius, Fahrenheit and Kelvin are all linear (answer for Kelvin and Celsius: they technically only are since 2019, when we redefined them (https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_ba...). I think the Fahrenheit scale was redefined as a shift on the Kelvin one around the same time)
- fhars 4y ago
  Well, sort of... there goes quite a bit of materials science and engineering into making and shaping the glass of a lens to make it behave like a matrix (i.e. to extend the range in which this linear approximation to the actual behavior of a blob of glass is valid).
  - gumby 4y ago
    It sure does. But if you can replace many of the lenses with computation it's a big win. If you can compensate for some of the inherent aberration in a lens design, or specific manufacturing variation in a lens, its an even bigger win. In other words, one does not obviate the other.
  - gnramires 4y ago
    Blobs of glass are actually overwhelmingly linear :) Non-linear effects in optics are generally negligible unless you go to very high energies (scattering, like raman scattering) or special materials (like fluorescent ones). You need a high powered laser or certain materials for those regimes. That said, the geometric functions (i.e. the positions of rays w.r.t. other objects) are probably non-linear in the object position parameters indeed, and intensities certainly are (example: moving an object in front of a screen by X amount changes the illumination in front of the screen non-linearly): it's important to keep in mind what we mean by linearity (it's in terms of light field intensities for a fixed geometry scene) -- the scene are the transfer function coefficients. What really makes things complicated in real life is (1) the presence of noise; (2) imperfections (and unknowns) in your physical/geometric apparatus. Even if you know the system perfectly, noise generally disallows reverting (or easily reverting) transfer functions, i.e. undoing blurs and arbitrarily refocusing images with simple detectors. The imperfections and even lack of rigidity of lenses and your system add even more difficulty. That's why making a simple computational lens is not so easy.
dreamcompiler 4y ago
Lenses compute 2D Fourier transforms. And they do it really, really fast. https://en.wikipedia.org/wiki/Fourier_optics#Fourier_transforming_property_of_lenses https://en.wikipedia.org/wiki/Fourier_optics#Fourier_transfo...

joshu 4y ago

More importantly, focusing a lens is one of many possible transformations. Look up coded apertures. so much is possible.

weinzierl 4y ago

"just performing predictable mathematical transformations" Just linear algebra, to be precise. A lens is just a matrix and rays are simply vectors. Combining lenses is merely matrix multiplication and complex optical systems can be represented by the result, which is a plain simple matrix again. The Eigenvalues tell you interesting properties of your optical system. I've always enjoyed this part of physics because it is so simple and elegant, yet so powerful. If you look closely linear algebra pops up in many places.

ddalex 4y ago
Going on a metaphysical tangent, it is a bit weird that A LOT of physical processes can be modelled with linear algebra, and don't require something way more advanced ...
- fhars 4y ago
  That is actually deliberate, we tend to organize everyday life and the devices we build around processes that are easy to model, that is why many things look like linear processes and harmonic oscillators (the first two term of the Taylor expansion of the actual behavior). We change the type of spring we use if the current one starts to wear out early under normal operational conditions.
  - IIAOPSW 4y ago
    Let me rephrase on his behalf. Isn't it curious just how unreasonably capable our math is at expressing those physical laws of the universe to which we did not invent.
    - jamiek88 4y ago
      I don’t know, seems a bit Anthropic Puddle to me.
- Someone 4y ago
  I think it’s more that we developed linear algebra that made us approximate physical processes by linear models. If you look close enough, almost nothing is linear. For example, we happily draw a linear scale on a mercury thermometer, even though we know that not to be correct, even if we incorrectly assume that the coefficient of expansion of mercury is independent of its temperature. Also, try explaining why the conversions between Celsius, Fahrenheit and Kelvin are all linear (answer for Kelvin and Celsius: they technically only are since 2019, when we redefined them (https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_base_units https://en.wikipedia.org/wiki/2019_redefinition_of_the_SI_ba...). I think the Fahrenheit scale was redefined as a shift on the Kelvin one around the same time)
fhars 4y ago
Well, sort of... there goes quite a bit of materials science and engineering into making and shaping the glass of a lens to make it behave like a matrix (i.e. to extend the range in which this linear approximation to the actual behavior of a blob of glass is valid).
- gumby 4y ago
  It sure does. But if you can replace many of the lenses with computation it's a big win. If you can compensate for some of the inherent aberration in a lens design, or specific manufacturing variation in a lens, its an even bigger win. In other words, one does not obviate the other.
- gnramires 4y ago
  Blobs of glass are actually overwhelmingly linear :) Non-linear effects in optics are generally negligible unless you go to very high energies (scattering, like raman scattering) or special materials (like fluorescent ones). You need a high powered laser or certain materials for those regimes. That said, the geometric functions (i.e. the positions of rays w.r.t. other objects) are probably non-linear in the object position parameters indeed, and intensities certainly are (example: moving an object in front of a screen by X amount changes the illumination in front of the screen non-linearly): it's important to keep in mind what we mean by linearity (it's in terms of light field intensities for a fixed geometry scene) -- the scene are the transfer function coefficients. What really makes things complicated in real life is (1) the presence of noise; (2) imperfections (and unknowns) in your physical/geometric apparatus. Even if you know the system perfectly, noise generally disallows reverting (or easily reverting) transfer functions, i.e. undoing blurs and arbitrarily refocusing images with simple detectors. The imperfections and even lack of rigidity of lenses and your system add even more difficulty. That's why making a simple computational lens is not so easy.

dreamcompiler 4y ago

Lenses compute 2D Fourier transforms. And they do it really, really fast. https://en.wikipedia.org/wiki/Fourier_optics#Fourier_transforming_property_of_lenses https://en.wikipedia.org/wiki/Fourier_optics#Fourier_transfo...