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\acmConference{De-cluttering Scatterplots with Integral Images}{2026}{Linz}

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\title{De-cluttering Scatterplots with Integral Images}

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\author{Lukas Heiligenbrunner}
\email{k12104785@students.jku.at}
\affiliation{%
  \institution{Johannes Kepler University Linz}
  \city{Linz}
  \state{Upper Austria}
  \country{Austria}
  \postcode{4020}
}

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\begin{abstract}
  Visualizing large multidimensional datasets using classical scatterplots often results in severe overplotting and visual clutter, which obscure data density and hinder the analysis of underlying structures.
  This paper presents a summarization of a novel density-equalizing technique proposed by Rave, Molchanov, and Linsen, designed to de-clutter scatterplots through a smooth, iterative transformation of the visual domain.
  The core algorithm utilizes integral images (summed-area tables) to compute a global regularization mapping that redistributes data samples into a nearly uniform configuration.
  Unlike previous displacement methods that may alter local sample ordering or require expensive collision detection, this approach guarantees the preservation of neighborhood relations and operates deterministically.
\end{abstract}

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%% Keywords. The author(s) should pick words that accurately describe
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\keywords{Scatterplots, Overplotting, Integral Images, Density Equalization}

%\received{20 February 2007}
%\received[revised]{12 March 2009}
%\received[accepted]{5 June 2009}

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\section{Introduction}
Scatterplots remain one of the most effective and widely utilized methods for visualizing multidimensional data, allowing analysts to relate pairs of data dimensions to reveal clusters, trends, and outliers.
However, as dataset sizes increase, classical scatterplots suffer significantly from scalability issues.
When the number of data samples grows, rendering them on a screen with fixed resolution inevitably leads to occlusion and overplotting.
This visual clutter negatively impacts the data analysis process by making it difficult to estimate sample density in crowded regions and restricting access to individual data points, thereby impeding detailed exploration.

To alleviate these issues, visualization research has traditionally employed three main strategies: appearance modification, data reduction, and spatial transformation.
Appearance modification, such as adjusting sample transparency (opacity), is a common technique to improve the visibility of local density.
While effective for density estimation, it does not resolve the overlap of interaction targets, leaving individual samples inaccessible.
Data reduction techniques, such as down-sampling, reduce the number of rendered elements but inevitably discard information, altering the representation of the underlying phenomenon.

The third category, spatial transformation, involves distorting the visualization domain to utilize screen space more efficiently.
This paper focuses on this domain, specifically addressing the limitations of existing deformation techniques.
Many prior spatial distortion methods rely on local collision detection or force-directed layouts, which can be computationally expensive (often $\mathcal{O}(n^2)$ or $\mathcal{O}(n^3)$) and may fail to preserve essential neighborhood relations.
Furthermore, previous attempts to use Integral Images (InIms) for smooth deformation, such as the work by Molchanov and Linsen, lacked stability and failed to converge to a uniform distribution in general cases.

The following summary examines the methodologies and contributions detailed in `De-cluttering Scatterplots with Integral Images` by Hennes Rave, Vladimir Molchanov, and Lars Linsen.
This work proposes a novel, deterministic algorithm for de-cluttering scatterplots using a corrected, stable regularization mapping based on Integral Images.
Unlike methods that rely on local collision handling, this approach evaluates the global density distribution to compute a smooth transformation.
This ensures that sample neighborhood relations are preserved without the need for expensive collision checks, enabling the processing of large datasets at interactive rates.
The authors present a parallel GPU-based implementation for fast computation and introduce visual encodings—such as deformed grids and density textures—to help users interpret the spatial distortions applied to the data~\cite{Rave_2025}.

\section{Method}
The core of the de-cluttering technique is a deterministic, iterative algorithm that uses global density information to redistribute data samples.
The method consists of three primary stages: constructing a density field, applying a smooth global deformation, and optimizing the process for real-time performance on a GPU.

\subsection{Density Field Construction}
To represent the distribution of $n$ data samples $z_i$ within the scatterplot domain, the algorithm first generates a smooth scalar-valued density function $d_r(x,y)$.
This is achieved by summing contributions from individual samples using a smooth radial basis function, such as a 2D Gaussian kernel with a dilation parameter $r$.
To ensure numerical stability, particularly in `empty` regions where the mapping might otherwise become singular, a global constant $d_0$ is added to the density field:

\begin{equation}
  \label{eq:dfc}
  d(i,j) = d_r(i,j) + D_{0}
\end{equation}

This constant $d_0$ is typically set to the average number of samples per pixel, representing the theoretical density of a perfectly uniform distribution~\cite{Rave_2025}.

\subsection{Smooth Global Deformation}
The transformation utilizes Integral Images (InIms), which provide a pixel-centered description of the global density distribution.
While prior work by Molchanov and Linsen proposed a global mapping $t(x,y;d)$, it failed to remain an identity transformation for constant density textures, making it unsuitable for iterative equalization.

This work introduces a corrected transformation $t(x,y)$ that subtracts the `defect` mapping of a constant texture $d_0$ from the original formula:

\begin{equation}
  \label{eq:sgd}
  t(x,y) = (x,y) + t(x,y; d) - t(x,y; d_0)
\end{equation}

By ensuring that $t(x,y)=(x,y)$ when the density is already uniform $(d=d0)$, the algorithm can be applied iteratively to converge toward a nearly uniform state.
This smooth deformation preserves essential neighborhood relations and the local ordering of data points without requiring expensive collision detection~\cite{Rave_2025}.

\subsection{Implementation and Visual Encodings}
To maintain interactivity with datasets containing millions of points, the authors developed a parallel GPU-based scheme using compute shaders to calculate InIms in linear O(n) time.
This multi-pass approach computes column and full integrals to determine new sample positions via bi-linear interpolation.
Because this spatial distortion alters original distances, the method incorporates critical visual cues to help users maintain context.
These include deformed regular grids to show area expansion, density background textures to highlight original cluster locations, and contour lines to define the boundaries of the original data structures~\cite{Rave_2025}.

\section{Results}
The effectiveness of the proposed density-equalization method was evaluated through a combination of algorithmic performance benchmarks, quantitative metrics for structure preservation, and a controlled user study.

\begin{figure}[htbp]
  \centering
  % First Row
  \begin{subfigure}[b]{0.2\textwidth}
    \centering
    \includegraphics[width=\textwidth]{../presentation/rsc/2408.06513v1_page_7_1}
    \caption{Original Scatterplot}
    \label{fig:original}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.2\textwidth}
    \centering
    \includegraphics[width=\textwidth]{../presentation/rsc/2408.06513v1_page_7_2}
    \caption{Density Estimation}
    \label{fig:density}
  \end{subfigure}

  \begin{subfigure}[b]{0.2\textwidth}
    \centering
    \includegraphics[width=\textwidth]{../presentation/rsc/2408.06513v1_page_7_3}
    \caption{Result with background texture}
    \label{fig:grid}
  \end{subfigure}
  \hfill
  \begin{subfigure}[b]{0.2\textwidth}
    \centering
    \includegraphics[width=\textwidth]{../presentation/rsc/2408.06513v1_page_7_4}
    \caption{Result with contour lines}
    \label{fig:final}
  \end{subfigure}

  \caption{MNIST Dataset with CMAP applied and then de-cluttered using InIms~\cite{Rave_2025}}\label{fig:figure}
  \label{fig:2x2grid}
\end{figure}

\subsection{Performance and Scalability}
A significant contribution of this work is the GPU-accelerated implementation of the integral image-based mapping.
By leveraging parallel compute shaders, the algorithm achieves linear time complexity, O(n), relative to the number of data points.
Benchmarks conducted on an NVIDIA GeForce RTX 3080 showed that the entire pipeline—including density estimation, InIm computation, and sample mapping—takes less than 10 milliseconds for datasets with up to 106 points.
This performance level enables seamless interactivity within visualization software, allowing users to adjust regularization parameters in real-time.

\subsection{Quantitative Comparison}
The method was compared against `Hagrid`, a state-of-the-art grid-based de-cluttering technique.
Using metrics such as the preservation of k-nearest neighbors (KNN) and spatial stress, the proposed integral image approach demonstrated superior stability.
Unlike Hagrid, which can introduce artifacts due to its discrete grid-based nature, the continuous transformation provided by this method ensures that the relative ordering of points remains consistent.
The results showed that while both methods successfully utilize screen space, the InIm-based approach maintains a higher correlation with the original local data structure.

\subsection{User Study Findings}
%A user study involving 20 participants was conducted to evaluate the practical utility of the regularized scatterplots.
%The study focused on two primary tasks: estimating the relative size of clusters and analyzing class distributions within overlapping regions.

%\begin{enumerate}
%\item \textbf{Accuracy:} Participants were significantly more accurate at estimating the number of samples within dense clusters when using the regularized view compared to the original scatterplot (even with transparency).
%\item \textbf{Confidence:} Users reported higher confidence levels when performing class-separation tasks in the equalized view, as the spatial expansion made individual color-coded classes more distinguishable.
%\item \textbf{Interpretation:} While the distorted view required the use of visual cues (like background textures and grids) to understand the original density, participants found these encodings intuitive and effective for maintaining context.
%\end{enumerate}

A controlled study involving 20 participants demonstrated that the regularized view significantly improves accuracy in estimating sample counts within dense clusters compared to traditional scatterplots.
Users reported higher confidence levels when performing class-separation tasks, as the spatial expansion made color-coded classes more distinguishable.
Although the distortion requires the use of visual aids like background textures and grids to maintain context, participants found these encodings intuitive and effective for interpreting the original data distribution.

\section{Conclusion}
This paper presented a robust and scalable visualization technique for de-cluttering dense scatterplots using density-equalizing transformations.
By correcting previous mapping formulas and utilizing GPU-accelerated integral images, the authors established a deterministic framework that transforms cluttered, overlapping data into a nearly uniform distribution while strictly preserving neighborhood relationships.

The integration of visual aids, such as deformed grids and density contours, successfully bridges the gap between the equalized layout and the original data distribution, ensuring that analysts do not lose spatial context.
The quantitative results and user study confirm that this approach significantly improves the readability of large datasets without sacrificing the performance required for interactive exploratory analysis.
Future work may explore extending this density-equalizing mapping to 3D scatterplots or integrating it into automated dimensionality reduction pipelines to better visualize manifold structures.

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