forked from lukas/Seminar_in_AI
403 lines
13 KiB
TeX
403 lines
13 KiB
TeX
\documentclass[usenames,dvipsnames]{beamer}
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%----------------------------------------------------------------------------------------
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% Struktur und Pointer Referat
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% 20.04.2020
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%----------------------------------------------------------------------------------------
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\usetheme[nofirafonts]{focus}
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\usepackage[utf8]{inputenc}
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\usepackage{booktabs}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{amsfonts}
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\usepackage{bbm}
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\usepackage{hyperref}
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\usepackage{graphicx}
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\usepackage{xcolor}
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\usepackage{mathtools}
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\RequirePackage[T1]{fontenc}
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\PassOptionsToPackage{sfdefault}{FiraSans}
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\RequirePackage{FiraSans}
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\RequirePackage{FiraMono}
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% Farbdefinitionen
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\definecolor{backgroundcoloreq}{RGB}{180,140,0}
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\definecolor{codegreen}{rgb}{0,0.6,0}
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\definecolor{codegray}{rgb}{0.5,0.5,0.5}
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\definecolor{codepurple}{rgb}{0.58,0,0.82}
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\definecolor{codeorange}{RGB}{190,100,0}
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% we wanna use default caleographic alphabet
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\DeclareMathAlphabet{\mathcal}{OMS}{cmbrs}{m}{n}
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%----------------------------------------------------------------------------------------
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% TITLE SLIDE
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%----------------------------------------------------------------------------------------
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\title{De-Cluttering Scatterplots}
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\subtitle{with Integral Images}
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\author{Lukas Heiligenbrunner}
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\date{\today}
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%------------------------------------------------
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\begin{document}
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%------------------------------------------------
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\begin{frame}
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\maketitle
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\end{frame}
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%----------------------------------------------------------------------------------------
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% SECTION 1: INTRODUCTION
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%----------------------------------------------------------------------------------------
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\section{Introduction}
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\begin{frame}{Goal of the Paper}
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\begin{itemize}
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\item Scatterplots are fundamental for exploring multidimensional data
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\item But: with large datasets they suffer from \textbf{overplotting}
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\item Dense regions obscure structure, samples become inaccessible
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\item Goal:
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\begin{itemize}
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\item Reduce clutter
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\item Preserve neighborhood relations
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\item Achieve uniform sample distribution
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\item Maintain interpretability
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\end{itemize}
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\end{itemize}
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\end{frame}
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%----------------------------------------------------------------------------------------
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% SECTION 2: PROBLEM
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%----------------------------------------------------------------------------------------
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\section{Problem: Overplotting}
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\begin{frame}{Why Scatterplots Clutter}
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\begin{itemize}
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\item Modern datasets: millions of samples
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\item Pixel resolution fixed → many samples map to the same pixel
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\item Consequences:
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\begin{itemize}
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\item Occlusion of clusters + outliers
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\item Loss of density information
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\item Hard to select individual items
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\item Misleading visual perception
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\end{itemize}
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\item A method is needed to \textbf{declutter} without losing structure
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\end{itemize}
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\end{frame}
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\begin{frame}
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\centering
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\includegraphics[scale=0.8]{rsc/overplotting}
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\footnotesize\text{Source: https://statisticsglobe.com/avoid-overplotting-r}
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\end{frame}
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\begin{frame}{Limitations of Traditional Approaches}
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\begin{itemize}
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\item Transparency-based methods
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\begin{itemize}
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\item Improve density perception
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\item But still lose individual sample visibility
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\end{itemize}
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\item Down-sampling
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\begin{itemize}
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\item Removes data → not acceptable for analysis
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\end{itemize}
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\item Local spatial distortions
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\begin{itemize}
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\item Risk of collisions
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\item Often non-monotonic mappings
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\end{itemize}
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\item Need a \textbf{global}, \textbf{smooth}, \textbf{monotonic}, \textbf{collision-free} method
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\end{itemize}
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\end{frame}
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%----------------------------------------------------------------------------------------
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% SECTION 3: BACKGROUND
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%----------------------------------------------------------------------------------------
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\section{Background: Density Fields \& Integral Images}
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\begin{frame}{Density Estimation}
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\begin{itemize}
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\item Given samples $z_i = (x_i, y_i)$
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\item Build smoothed density:
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\[
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d_r(x,y) = \sum_{p=1}^n \varphi_r(x-x_p, y-y_p)
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\]
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\item Typically Gaussian kernel
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\item Add global constant $d_0$ for stability:
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\[
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d(i,j) = d_r(i,j) + d_0
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\]
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\item Ensures no empty regions → avoids singular mappings
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\end{itemize}
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\end{frame}
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\begin{frame}{Integral Images (InIms)}
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\begin{itemize}
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\item Integral images compute cumulative sums over regions
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\item Four standard tables:
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\[
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\alpha,\beta,\gamma,\delta
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\]
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\item Four tilted (45°) tables:
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\[
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\alpha_t, \beta_t, \gamma_t, \delta_t
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\]
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\item Each encodes global density distribution
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\item Key advantage:
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\begin{itemize}
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\item Displacements depend on \textbf{global density}, not local neighborhood
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\item Avoids collisions
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\end{itemize}
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\end{itemize}
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\end{frame}
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%----------------------------------------------------------------------------------------
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% SECTION 4: METHOD
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%----------------------------------------------------------------------------------------
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\section{Density-Equalizing Mapping}
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\begin{frame}{Original Mapping (Molchanov \& Linsen)}
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\begin{itemize}
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\item Prior work defined mapping:
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\[
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t(x,y; d) = \frac{
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\alpha q_1 + \beta q_2 + \gamma q_3 + \delta q_4
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+ \alpha_t (x,1) + \beta_t (1,y) + \gamma_t (x,0) + \delta_t (0,y)
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}{2C}
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\]
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\item But:
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\begin{itemize}
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\item Not identity for uniform density
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\item Iteration unstable
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\item Does not converge to equalized distribution
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Corrected Mapping (This Paper)}
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\begin{itemize}
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\item Compute deformation for true density $d$
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\item Compute deformation for constant density $d_0$
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\item Subtract:
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\[
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t(x,y) = (x,y) + t(x,y; d) - t(x,y; d_0)
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\]
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\item This ensures:
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\begin{itemize}
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\item Identity for uniform density
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\item Smooth monotonic deformation
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\item Progressive convergence to equalization
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\item No overlap of regions
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Iterative Algorithm Overview}
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\begin{enumerate}
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\item Rasterize and smooth density
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\item Compute integral images
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\item Compute corrected deformation $t(x,y)$
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\item Apply bi-linear interpolation to sample positions
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\item Iterate until:
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\begin{itemize}
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\item Time budget reached
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\item Uniformity threshold reached
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\end{itemize}
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\end{enumerate}
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\end{frame}
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\begin{frame}
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\centering
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\begin{figure}
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\centering
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\begin{minipage}{0.4\textwidth}
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\centering
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\includegraphics[width=\textwidth]{rsc/2408.06513v1_page_7_1}
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\vspace{4pt}
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\footnotesize MNIST Dataset (UMAP)
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\end{minipage}
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\begin{minipage}{0.15\textwidth}
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\centering
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$\Longrightarrow$
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\end{minipage}
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\begin{minipage}{0.4\textwidth}
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\centering
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\includegraphics[width=\textwidth]{rsc/2408.06513v1_page_7_2}
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\vspace{4pt}
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\footnotesize Visual encoding of the density-equalizing transform
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\end{minipage}
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\label{fig:figure}
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\end{figure}
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\end{frame}
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%----------------------------------------------------------------------------------------
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% SECTION 6: VISUAL ENCODING
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%----------------------------------------------------------------------------------------
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\section{Visual Encoding of Deformation}
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\begin{frame}{Problem After Deformation}
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\begin{itemize}
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\item After equalization:
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\begin{itemize}
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\item Local densities lost
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\item Cluster shapes distorted
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\item Distances no longer meaningful
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\end{itemize}
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\item Need additional encodings to preserve structure
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\end{itemize}
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\end{frame}
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\begin{frame}{Three Proposed Encodings}
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\begin{itemize}
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\item \textbf{Deformed grid lines}
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\begin{itemize}
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\item Show local expansion / contraction
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\end{itemize}
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\item \textbf{Background density texture}
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\begin{itemize}
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\item Shows cluster cores after deformation
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\end{itemize}
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\item \textbf{Contour lines}
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\begin{itemize}
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\item Reveal subcluster structure
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Three Proposed Encodings II}
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\centering
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\begin{figure}
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\centering
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\begin{minipage}{0.3\textwidth}
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\centering
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\includegraphics[width=\textwidth]{rsc/2408.06513v1_page_7_2}
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\vspace{4pt}
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\footnotesize Deformed grid lines
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\end{minipage}
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\begin{minipage}{0.3\textwidth}
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\centering
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\includegraphics[width=\textwidth]{rsc/2408.06513v1_page_7_3}
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\vspace{4pt}
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\footnotesize Background density texture
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\end{minipage}
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\begin{minipage}{0.3\textwidth}
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\centering
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\includegraphics[width=\textwidth]{rsc/2408.06513v1_page_7_4}
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\vspace{4pt}
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\footnotesize Contour lines
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\end{minipage}
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\label{fig:figure2}
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\end{figure}
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\end{frame}
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%----------------------------------------------------------------------------------------
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% SECTION 5: IMPLEMENTATION
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%----------------------------------------------------------------------------------------
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\section{GPU Implementation}
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\begin{frame}{Efficient GPU Computation}
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\begin{itemize}
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\item All major steps implemented on GPU:
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\begin{itemize}
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\item Density accumulation
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\item Gaussian smoothing
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\item Integral image computation
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\end{itemize}
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\item Fast multi-pass reduction for InIms
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\item Complexity:
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\[
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O(n + m)
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\]
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where $m = 2^k \times 2^k$ is texture resolution
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\item Achieves interactive rates for millions of samples
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\end{itemize}
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\end{frame}
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%----------------------------------------------------------------------------------------
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% SECTION 7: RESULTS
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%----------------------------------------------------------------------------------------
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\section{Results}
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\begin{frame}{Performance}
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\begin{itemize}
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\item Runs at interactive frame rates:
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\begin{itemize}
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\item e.g. 4M samples in $\approx 28$ ms per iteration
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\end{itemize}
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\item Standard deviation of samples/bin decreases monotonically
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\item Overplotting fraction also decreases monotonically
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\end{itemize}
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\centering
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\includegraphics[scale=0.4]{rsc/results}
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\end{frame}
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% --- THE END
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\begin{frame}[focus]
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Thanks for your Attention!
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\end{frame}
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%----------------------------------------------------------------------------------------
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% CLOSING/SUPPLEMENTARY SLIDES
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%----------------------------------------------------------------------------------------
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\appendix
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\begin{frame}{Sources}
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\nocite{*} % Display all references regardless of if they were cited
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\bibliography{sources}
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\bibliographystyle{plain}
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\end{frame}
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\section{Backup}\label{sec:backup}
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\begin{frame}{User Study}
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\begin{itemize}
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\item 25 participants, 3 tasks:
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\begin{enumerate}
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\item Estimate cluster size
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\item Sort clusters by size
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\item Select clusters (lasso)
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\end{enumerate}
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\item Findings:
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\begin{itemize}
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\item Size estimation (T1): regularized significantly better
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\item Sorting (T2): regularized significantly better
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\item Cluster selection (T3):
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\begin{itemize}
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\item Grid encoding: worst
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\item Background texture: better
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\item Original scatterplot: best
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\end{itemize}
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\end{itemize}
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\end{itemize}
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\end{frame}
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\end{document}
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