From 1e816e47d07c7625075a089806a1551e0588e8d9 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 1 Oct 2026 16:15:14 +0200 Subject: [PATCH] [VC] Start fourier transforms --- README.md | 2 +- .../03_fourier/00_intro.tex | 23 +++++++++ .../parts/00_computer-vision/main.tex | 7 +++ core/vc/cheat-sheet-jh/vc-cheat-sheet.pdf | Bin 292497 -> 305765 bytes core/vc/summary/vc-summary.tex | 45 ------------------ 5 files changed, 31 insertions(+), 46 deletions(-) create mode 100644 core/vc/cheat-sheet-jh/parts/00_computer-vision/03_fourier/00_intro.tex delete mode 100644 core/vc/summary/vc-summary.tex diff --git a/README.md b/README.md index c9ab571..39a90f2 100644 --- a/README.md +++ b/README.md @@ -66,7 +66,7 @@ If you find mistakes, please open an issue or fix them yourself and open a PR. - [Cheat-Sheet (EN)](./core/cd/cheat-sheet-jh/cd-cheat-sheet.pdf) Author: Janis Hutz #### Visual Computing -- [Summary (EN)](./core/vc/summary/vc-summary.pdf) Author: Janis Hutz (NOTE: This is likely *not* to happen) +There won't be a summary for this subject, only a Cheat-Sheet. Provided script is good. - [Cheat-Sheet (EN)](./core/vc/cheat-sheet-jh/vc-cheat-sheet.pdf) Author: Janis Hutz #### Human Computer Interaction 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 new file mode 100644 index 0000000..0ca2b71 --- /dev/null +++ b/core/vc/cheat-sheet-jh/parts/00_computer-vision/03_fourier/00_intro.tex @@ -0,0 +1,23 @@ +\shortdefinition[Sinusoid] $f(x) = A \cdot \sin(2 \pi ux + \psi)$ with $\psi$ phase shift, equiv: $A \cos(\psi) \sin(2\pi ux) + A \sin(\psi)\cos(2\pi ux)$ + +\shortdefinition[Complex Exponential] $e^{i \theta} = \cos(\theta) + i \sin(\theta)$, complex conj. $e^{-i \theta} = \cos(\theta) - i \sin(\theta)$. For sinusoid: $\theta = 2\pi ux$. + +\shortdefinition[Cont. Time F. Ser] Sig as sum of sinusoids, $f(x) = \sum_{k = -\8}^{\8} c_k e^{i 2\pi kx \div T}$, with $T$ the period and $c_k$ fourier coeff. + +\shortdefinition[Cont. Time F. Transf] $F(u) = \sum_{\R} f(x)e^{-2\pi ux} \dx x$, $f(x) = \sum_{\R} F(u) e^{i 2\pi ux} \dx u$. $F(u)$ is \textit{typ} complex valued, + +\shortdefinition[Ampl. Spec] $A(u) = ||F(u)||$ + +\shortdefinition[Phase Spec] $\phi(u) = \arg F(u) = \arctan\left( \frac{\mathfrak{I}(F(u))}{\mathfrak{R}(F(u))} \right)$ + +\shortdefinition[DFT] used for analyzing discrete data. Applies base change to values via matrix multiplication. +More general: +\[ + F(k) = \sum_{n = 0}^{N - 1} x_n e^{-i 2\pi kn \div N} \text{ for } k \in \{ 0, 1, \ldots, N - 1 \} +\] + +For image: +\[ + F(u, v) = \sum_{x = 0}^{N - 1}\sum_{y = 0}^{M - 1} I(x, y) \cdot e^{-i2\pi\left( \frac{ux}{N} + \frac{vy}{M} \right)} +\] +% TODO: Formula to compute the matrix diff --git a/core/vc/cheat-sheet-jh/parts/00_computer-vision/main.tex b/core/vc/cheat-sheet-jh/parts/00_computer-vision/main.tex index af79ec2..9b09754 100644 --- a/core/vc/cheat-sheet-jh/parts/00_computer-vision/main.tex +++ b/core/vc/cheat-sheet-jh/parts/00_computer-vision/main.tex @@ -7,4 +7,11 @@ \subsection{Edge Detection} \input{parts/00_computer-vision/02_edge-detection.tex} + +\subsection{Fourier Transform} +\input{parts/00_computer-vision/03_fourier/00_intro.tex} +% \input{parts/00_computer-vision/03_fourier/} + + + % \input{parts/00_computer-vision/} diff --git a/core/vc/cheat-sheet-jh/vc-cheat-sheet.pdf 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a/core/vc/summary/vc-summary.tex b/core/vc/summary/vc-summary.tex deleted file mode 100644 index a44ac0a..0000000 --- a/core/vc/summary/vc-summary.tex +++ /dev/null @@ -1,45 +0,0 @@ -\documentclass{article} - -\input{../../../helpers.tex} - -\usepackage{lmodern} -\setFontType{sans} - -\setup{Visual Computing} - -\begin{document} -\startDocument - -\vspace{1cm} -\begin{center} - \includegraphics[width=0.5\linewidth]{~/projects/latex/assets/logo.jpg} -\end{center} - - -\vspace{2cm} -\begin{center} - \begin{Large} - \quote{A funny quote by a professor} - \end{Large} - - \hspace{3cm} - Prof. Dr. Professor Name, YEAR -\end{center} - -\vspace{1.5cm} -\begin{center} - HS2026, ETHZ - - \begin{Large} - Summary of the Script and Lectures - \end{Large} -\end{center} - - -\newpage -\printtoc{Aquamarine} - - - - - -\end{document}