From e7573d82299440b6d22ba9325cf5caf5cb0f2c17 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 2 Oct 2026 08:00:33 +0200 Subject: [PATCH] [VC] Finish fourier transform sec --- .../03_fourier/00_intro.tex | 27 ++++++++++++++++-- core/vc/cheat-sheet-jh/vc-cheat-sheet.pdf | Bin 305765 -> 311286 bytes core/vc/cheat-sheet-jh/vc-cheat-sheet.tex | 1 + 3 files changed, 26 insertions(+), 2 deletions(-) diff --git a/core/vc/cheat-sheet-jh/parts/00_computer-vision/03_fourier/00_intro.tex b/core/vc/cheat-sheet-jh/parts/00_computer-vision/03_fourier/00_intro.tex index 0ca2b71..73db45e 100644 --- a/core/vc/cheat-sheet-jh/parts/00_computer-vision/03_fourier/00_intro.tex +++ b/core/vc/cheat-sheet-jh/parts/00_computer-vision/03_fourier/00_intro.tex @@ -15,9 +15,32 @@ More general: \[ F(k) = \sum_{n = 0}^{N - 1} x_n e^{-i 2\pi kn \div N} \text{ for } k \in \{ 0, 1, \ldots, N - 1 \} \] +% TODO: Formula to compute the matrix -For image: +For images at location $(u, v)$, or $F = Uf$, with $U$ Fourier Mat.: \[ F(u, v) = \sum_{x = 0}^{N - 1}\sum_{y = 0}^{M - 1} I(x, y) \cdot e^{-i2\pi\left( \frac{ux}{N} + \frac{vy}{M} \right)} \] -% TODO: Formula to compute the matrix +For continuous time, we get: +\[ + \cF[g(x, y)](u, v) = \int \int_{\R^2} g(x, y) e^{-2i\pi (ux + vy)} \dx x \dx y +\] +Inverted by negating exp and swapping $g$ and $F$ (or $F$ and $I$) + +\shorttheorem[Convolution] $\cF[f * g] = F \cdot G$, $\cF[f \cdot g] = F * G$, with $F, G$ the fourier transforms + +\shortremark[Filtering] Can be achieved: $h * f = \cF^{-1}[H \cdot F]$ + +\shortremark[Dirac Delta] $\delta(x) = \begin{cases} + \8 & \text{ if }x = 0 \\ + 0 & \text{otherwise} + \end{cases}$, +following prop for sampling to alleviate issue of $0$ everywhere after fourier: +\[ + \int_{-\8}^{\8} f \cdot \delta(x - a) \dx = f(a) +\] + +\shortremark[Restoration] To restore image, inverse kernel $\tilde{h}(x) = \cF^{-1}\left[ \frac{1}{H(x)} \right]$, with $H(x) = \cF[h(x)]$ +and $h(x)$ degradation filter. + +\shortexample 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\renewcommand{\theoremShortNamingEN}{Thm} +\renewcommand{\exampleShortNamingEN}{Ex} \renewcommand{\descriptorNameDisplay}[1]{\textbf{#1}} \setupCheatSheet[0.5cm]{Visual Computing}