3 ms·

Coauthor of the paper here: this tangent does not apply to this post. The key thing about the actual work linked in this post is that it is pointing out and ad

by jrk 6y ago

Coauthor of the paper here: this tangent does not apply to this post. The key thing about the actual work linked in this post is that it is pointing out and addressing an important case where discontinuous computations (polygon coverage at edges) are in fact meaningfully differentiable.

gugagore 6y ago
Thanks for re-highlighting this point, and for doing it with a pun! While not directly related to the differentiability question, the idea that you can resolve spatial movement to well below the spatial resolution of pixels in an imager is relevant. It is possible to resolve displacements that are e.g. 0.02 pixels wide [1] by relying on the "natural" anti-aliasing effect of the point-spread function [2] that arises from 1. the optics (where even if lenses were free of any aberration, the aperture cannot be infinitely small nor can the lens diameter be infinitely large), 2. the pixels don't sample light at a point. This enables video motion amplification/magnification, which has been on the front page of HN before [3]. There, the rasterizer is a physical process that needs to be modeled. Here, the rasterizer is constructed to have the necessary property. Note that if you had a perfect model of the physical process, and if it had no noise (e.g. no photon noise and no Johnson noise---impossible), and if the pixel intensities were given to infinite precision (as opposed to being discretized by the imager bit depth), then you could resolve arbitrarily small displacements of point light sources. I wonder if that points more to a continuity question than a differentiability question... but I felt compelled to make the connection in any case. Returning back to your comment: > case where discontinuous computations (polygon coverage at edges) are in fact meaningfully differentiable It seems like the rasterizer ends up being a continuous computation, unless we disagree on what counts as the "rasterizer" (I'm including any anti-aliasing strategy, including one that underlyingly makes multiple calls to a discontinuous rasterizer.) [1] Sound-Induced Motions of Individual Cochlear Hair Bundles https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1304819/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1304819/ [2] https://en.wikipedia.org/wiki/Point_spread_function https://en.wikipedia.org/wiki/Point_spread_function [3] https://hn.algolia.com/?dateRange=all&page=0&prefix=false&query=Eulerian%20Video%20magnification&sort=byPopularity&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...

by jrk 6y ago

gugagore 6y ago

Thanks for re-highlighting this point, and for doing it with a pun! While not directly related to the differentiability question, the idea that you can resolve spatial movement to well below the spatial resolution of pixels in an imager is relevant. It is possible to resolve displacements that are e.g. 0.02 pixels wide [1] by relying on the "natural" anti-aliasing effect of the point-spread function [2] that arises from 1. the optics (where even if lenses were free of any aberration, the aperture cannot be infinitely small nor can the lens diameter be infinitely large), 2. the pixels don't sample light at a point. This enables video motion amplification/magnification, which has been on the front page of HN before [3]. There, the rasterizer is a physical process that needs to be modeled. Here, the rasterizer is constructed to have the necessary property. Note that if you had a perfect model of the physical process, and if it had no noise (e.g. no photon noise and no Johnson noise---impossible), and if the pixel intensities were given to infinite precision (as opposed to being discretized by the imager bit depth), then you could resolve arbitrarily small displacements of point light sources. I wonder if that points more to a continuity question than a differentiability question... but I felt compelled to make the connection in any case. Returning back to your comment: > case where discontinuous computations (polygon coverage at edges) are in fact meaningfully differentiable It seems like the rasterizer ends up being a continuous computation, unless we disagree on what counts as the "rasterizer" (I'm including any anti-aliasing strategy, including one that underlyingly makes multiple calls to a discontinuous rasterizer.) [1] Sound-Induced Motions of Individual Cochlear Hair Bundles https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1304819/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1304819/ [2] https://en.wikipedia.org/wiki/Point_spread_function https://en.wikipedia.org/wiki/Point_spread_function [3] https://hn.algolia.com/?dateRange=all&page=0&prefix=false&query=Eulerian%20Video%20magnification&sort=byPopularity&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...