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docs: fix broken relative links #632
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| Original file line number | Diff line number | Diff line change |
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@@ -11,7 +11,7 @@ Radon transform: Fourier-slice diagonalization and pseudo-inversion | |
| - Submitter: Kim Morrison | ||
| - Notes: The Fourier slice theorem diagonalizes the Radon transform (1D Fourier of a projection = a 2D-Fourier slice), and the transform has a left inverse on Schwartz functions. Trusted helpers radon, fourier1, fourier2 (non-holes). Mathlib has the 1D/2D Fourier transforms and Schwartz space but no Radon transform, Fourier slice theorem, or filtered back-projection. The pseudo-inverse is stated existentially (the explicit filtered-back-projection form would need the Hilbert transform / Riesz potential). Candidate from §100 of the Knill survey. | ||
| - Source: J. Radon (1917); R. N. Bracewell (Fourier slice theorem, 1956). Knill, *Some fundamental theorems in mathematics*, §100. | ||
| - Informal solution: Fourier slice theorem: writing the Radon projection R f(p,θ) and taking its 1D Fourier transform in p, interchange integrals (Fubini) and change variables so the line-integral-then-Fourier becomes the 2D Fourier transform of f restricted to the line through the origin at angle θ: F₁[Rf(·,θ)](k) = F₂[f](k cos θ, k sin θ). This diagonalizes R (it becomes a slice/multiplication operator). Pseudo-inversion: the slice identity plus 2D Fourier inversion on Schwartz functions makes R injective on 𝓢, so a left inverse exists; the canonical one is filtered back-projection u ↦ backproject(Hilbert-filter(u)). Mathlib lacks the slice theorem and the filter. | ||
| - Informal solution: Fourier slice theorem: writing the Radon projection R f(p,θ) and taking its 1D Fourier transform in p, interchange integrals (Fubini) and change variables so the line-integral-then-Fourier becomes the 2D Fourier transform of f restricted to the line through the origin at angle θ: F₁\[Rf(·,θ)\](k) = F₂[f](k cos θ, k sin θ). This diagonalizes R (it becomes a slice/multiplication operator). Pseudo-inversion: the slice identity plus 2D Fourier inversion on Schwartz functions makes R injective on 𝓢, so a left inverse exists; the canonical one is filtered back-projection u ↦ backproject(Hilbert-filter(u)). Mathlib lacks the slice theorem and the filter. | ||
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Author
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This was an interesting one, basically we need the escape |
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| Do not modify `Challenge.lean` or `Solution.lean`. Those files are part of the | ||
| trusted benchmark and fixed by the repository. | ||
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Should we consider pinning links in this doc to a particular commit SHA, rather than linking to the
masterbranch? The code could evolve over time and become stale, or the file could be deleted, etc.If yes, I"m not sure which commit SHA to use (maybe just the latest one...?)