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Well, you can model pixels as you wish and you can translate coordinates as you wish, as long as you are consistent with your assumptions. Modelling pixels as p

Well, you can model pixels as you wish and you can translate coordinates as you wish, as long as you are consistent with your assumptions. Modelling pixels as points (i.e., delta distributions) instead of squares or other continuous distributions makes everything more prone to aliasing artifacts and is arguably less physically justified, but can sometimes be the right thing.

mark-r 6y ago
Did you even read the link I gave? Sampling theory relies on treating samples as sizeless points rather than rectangles, and aliasing is completely predicted by sampling theory.
- giomasce 6y ago
  Maybe we're just using different words. To me the "shape" of a pixel is the "shape" of the sampling filter you're using, and assuming that I stay by my first comment: you can choose whatever distribution you want (provided that you can couple it with the function space you're choosing for the non-sampled images, but let's avoid technicalities). Unless I am mistaken, this aspect is not discussed in that essay, only lightly touched. They speak more about reconstruction. In other words, let's say a non-sampled (black and white, for simplicity) image is a function [0,1]^2 -> [0,1]. You have to choose function space, like C^0, L^1, L^2, L^\infty, there are many. Whatever function space you choose, it will probably be infinite-dimensional, do you want to sample it to reduce it to a finite dimension and be able to represent it in a computer. Usually you do that by convolving it with a kernel. Using a delta kernel (i.e., taking the value of the function at the location of a pixel) is choice, but not the only one, and certainly not the one that models physical processes like a camera or a scanner (as the essay itself discusses). Once you have chosen your sampling filter, you can process to choose your reconstruction filter, and again you have a lot of freedom. One property that you will reasonably want to retain is that the reconstruction filter is a right inverse of the sampling filter, i.e., if you take a sampled image, reconstruct the non-sampled image and then sample it again, you will probably want to get the same image you started with. But being the sampling filter a map from an infinite-dimensional space to a finite-dimensional one, it has a lot of right inverses, and you can choose. Depending on these choices you get different aliasing and interpolation results. My point is that there is no hardcoded predefined choice: depending on your models you will have different outcomes, and you have to choose wisely depending on what you want. This is probably not far from what that essay says, but then I don't see why its conclusion is that a pixel is a sizeless point: it can be whatever you want.

mark-r 6y ago

Did you even read the link I gave? Sampling theory relies on treating samples as sizeless points rather than rectangles, and aliasing is completely predicted by sampling theory.

giomasce 6y ago
Maybe we're just using different words. To me the "shape" of a pixel is the "shape" of the sampling filter you're using, and assuming that I stay by my first comment: you can choose whatever distribution you want (provided that you can couple it with the function space you're choosing for the non-sampled images, but let's avoid technicalities). Unless I am mistaken, this aspect is not discussed in that essay, only lightly touched. They speak more about reconstruction. In other words, let's say a non-sampled (black and white, for simplicity) image is a function [0,1]^2 -> [0,1]. You have to choose function space, like C^0, L^1, L^2, L^\infty, there are many. Whatever function space you choose, it will probably be infinite-dimensional, do you want to sample it to reduce it to a finite dimension and be able to represent it in a computer. Usually you do that by convolving it with a kernel. Using a delta kernel (i.e., taking the value of the function at the location of a pixel) is choice, but not the only one, and certainly not the one that models physical processes like a camera or a scanner (as the essay itself discusses). Once you have chosen your sampling filter, you can process to choose your reconstruction filter, and again you have a lot of freedom. One property that you will reasonably want to retain is that the reconstruction filter is a right inverse of the sampling filter, i.e., if you take a sampled image, reconstruct the non-sampled image and then sample it again, you will probably want to get the same image you started with. But being the sampling filter a map from an infinite-dimensional space to a finite-dimensional one, it has a lot of right inverses, and you can choose. Depending on these choices you get different aliasing and interpolation results. My point is that there is no hardcoded predefined choice: depending on your models you will have different outcomes, and you have to choose wisely depending on what you want. This is probably not far from what that essay says, but then I don't see why its conclusion is that a pixel is a sizeless point: it can be whatever you want.