# Continuous Scatterplots

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### Abstract

Scatterplots are well established means of visualizing discrete data values
with two data variables as a collection of discrete points. We aim at
generalizing the concept of scatterplots to the visualization of spatially
continuous input data by a continuous and dense plot. An example of a
continuous input field is data defined on an n-D spatial grid with
respective interpolation or reconstruction of in-between values. We propose a
rigorous, accurate, and generic mathematical model of continuous scatterplots
that considers an arbitrary density defined on an input field on an n-D
domain and that maps this density to m-D scatterplots. Special cases are
derived from this generic model and discussed in detail: scatterplots where the
n-D spatial domain and the m-D data-attribute domain have identical
dimension, 1-D scatterplots as a way to define continuous histograms, and 2-D
scatterplots of data on 3-D spatial grids. We show how continuous histograms
are related to traditional discrete histograms and to the histograms of
isosurface statistics. Based on the mathematical model of continuous
scatterplots, respective visualization algorithms are derived, in particular
for 2-D scatterplots of data from 3-D tetrahedral grids. For several
visualization tasks, we show the applicability of continuous scatterplots.
Since continuous scatterplots do not only sample data at grid points but
interpolate data values within cells, a dense and complete visualization of the
data set is achieved that scales well with increasing data set size. Especially
for irregular grids with varying cell size, improved results are obtained when
compared to conventional scatterplots. Therefore, continuous scatterplots are a
suitable extension of a statistics visualization technique to be applied to
typical data from scientific computation.

Index Terms - Scatterplot, histogram, continuous frequency plot, interpolation.

Last modified: Oct. 10th, 2008

by Sven Bachthaler