Uncertainty Visualization of Critical Points of 2D Scalar Fields for Parametric and Nonparametric Probabilistic Models

Abstract

This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.

Full text

Uncertainty Visualization of Critical Points of 2D Scalar Fields for
Parametric and Nonparametric Probabilistic Models
Tushar M. Athawale , Zhe Wang , David Pugmire , Kenneth Moreland , Qian Gong , Scott Klasky ,
Chris R. Johnson , and Paul Rosen
Fig. 1: Parametric (a-c) vs. nonparametric (d) models for uncertainty visualization of local minima of the magnitude of velocity field
vectors in the Red Sea ensemble simulations that correspond to eddy-like features. Images (e-f) show two randomly picked ensemble
members with critical points rendered in yellow. The proposed nonparametric models show new high-probability features (larger
spheres in red and orange) in green boxes that are not highlighted in parametric models. The results of nonparametric models can be
trusted more because of the ability of nonparametric models to capture the shapes of multimodel and skewed distributions [8,53],
unlike the restricted shape assumptions for the parametric models. The pink box shows a feature that is revealed by the multivariate
Gaussian and histogram models but not by others. The cyan boxes show high-probability critical points in the multivariate Gaussian
model that are not displayed as prominently in other models. The white boxes show two critical points that consistently have high
probability across all models. All visualizations are computed efficiently and visualized in ParaView [2] using VTK-m [42] as a backend.
Abstract—This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty
in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar
fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression),
however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored,
given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty
in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies
to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software.
We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we
derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods.
Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g.,
uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability
computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration
of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.
Index Terms—Topology, uncertainty, critical points, probabilistic analysis
1 INTRODUCTION
Topological data analysis (TDA) is increasingly used in scientific visu-
alizations because of its ability to concisely convey position and scale
of important data features in complex datasets. The application of TDA
can be found in diverse domains, including combustion science [12],
• Tushar M. Athawale, Zhe Wang, Qian Gong, Scott Klasky, Kenneth
Moreland, and David Pugmire are with the Oak Ridge National Laboratory.
E-mail:{athawaletm, wangz, pugmire, morelandkd, gongq, klasky
}@ornl.gov
• Chris R. Johnson and Paul Rosen are with the University of Utah E-mail:
{crj, prosen}@sci.utah.edu
Manuscript received xx xxx. 201x; accepted xx xxx. 201x. Date of Publication
xx xxx. 201x; date of current version xx xxx. 201x. For information on
obtaining reprints of this article, please send e-mail to: reprints@ieee.org.
Digital Object Identifier: xx.xxxx/TVCG.201x.xxxxxxx
molecular dynamics [43], and hydrodynamics [44]. Critical points are
fundamental topological descriptors of scalar fields and form the basis
of many topological visualization techniques, including persistent dia-
grams [17], contour trees [14,27], and Morse complexes [18]. Critical
points denote the domain position where the gradient of a field vanishes
(see technical details in Sec. 3). Uncertainty inherent in data arising
from instrument/simulation/model errors [13], however, creates uncer-
tainty regarding critical point positions. Ignoring uncertainty in critical
This manuscript has been authored by UT-Battelle, LLC under Contract No.
DE-AC05-00OR22725 with the U.S. Department of Energy. The publisher, by
accepting the article for publication, acknowledges that the U.S. Government
retains a non-exclusive, paid up, irrevocable, world-wide license to publish
or reproduce the published form of the manuscript, or allow others to do so,
for U.S. Government purposes. The DOE will provide public access to these
results in accordance with the DOE Public Access Plan (
http://energy.gov/
downloads/doe-public- access-plan).
arXiv:2407.18015v1 [cs.GR] 25 Jul 2024

point positions, therefore, can lead to errors in topological visualiza-
tions and analysis. Thus, it is necessary to quantify and visually convey
uncertainty in critical points to prevent misinformation. In this paper,
we study uncertainty in critical points as a function of uncertainty in
the underlying data modeled with probability distributions.
A significant number of studies have investigated uncertainty in
critical points arising from uncertain scalar fields. Petz et al. [49] and
Liebmann and Scheuermann [38] modeled noise in data with multi-
variate Gaussian distribution to derive critical point probabilities. The
former work utilized Monte Carlo (MC) sampling of the original space,
and the latter work utilized MC sampling of informative subspaces
(referred to as patch sampling in their work) for deriving critical point
probabilities. Günther et al. [25] investigated spatial bounds in the
domain where at least one critical point is guaranteed to exist for the
uniform noise assumption. Mihai and Westermann [41] derived confi-
dence intervals for gradient field and Hessian to visualize likely critical
point positions and their type in the domain. Vietinghoff et al. [63]
derived the critical point probability using the Bayesian inference and
derived confidence intervals [61]. Recently, Vietinghoff et al. devel-
oped a novel mathematical framework [62] that quantified uncertainty
in critical points by analyzing the variation in manifestation of the same
critical points occurring across realizations of the ensemble.
Inspired by these advances, we propose a novel closed-form theoreti-
cal framework for uncertainty quantification and visualization of critical
point uncertainty. In particular, we analytically derive the critical point
probability per grid position in a regular grid for independent para-
metric and nonparametric noise models with finite support. Although
the previously proposed multivariate Gaussian noise models [38,49]
can better handle the noise correlation compared to independent noise
models, they have two shortcomings. First, they resort to the MC sam-
pling approach for deriving the critical point probability, which can
be computationally expensive depending on the number of samples
and grid resolution, and they converge slowly to the true answer. Sec-
ond, because of the restricted shape assumptions for the (parametric)
Gaussian distributions, they can be less robust to the data outliers than
the nonparametric noise models [6,53]. We address these two short-
comings by developing a closed-form formulation and algorithm that
enhance the efficiency of results compared to MC sampling. Further,
we showcase how the nonparametric models can provide more robust
results compared to multivariate Gaussian and other parametric models.
Another important challenge associated with uncertainty visualiza-
tion algorithms is the additional computational costs they bear, which
prevents their integration with production-level software, e.g, VisIt [15]
and ParaView [2]. The problem of the added cost is amplified for expen-
sive MC sampling methods (e.g., in the case of multivariate Gaussian
models). It is, therefore, important to research techniques that facilitate
the integration of uncertainty visualization with production-level tools
to make them usable and accessible to a broader scientific community.
Recently, Wang et al. [65] provided a parallel and platform-portable
implementation of isosurface uncertainty visualization using the VTK-
m library [42]. They also showcased the integration of their VTK-m
implementation with ParaView. The Topology Toolkit (TTK) by Tierny
et al. [59] provided efficient implementation of mandatory critical
points [25] that is usable in ParaView. Motivated by these works, we
present a VTK-m parallel implementation of our closed-form critical
point probability computations that is integrable with ParaView.
To summarize, our contributions are threefold. First, we propose a
theoretical framework for uncertainty computation of critical points in
uncertain 2D scalar fields. In particular, we propose an algorithm for de-
riving local minimum, local maximum, and saddle probability in closed
form when data uncertainty is modeled as independent parametric (uni-
form, Epanechnikov) and nonparametric (histogram) distributions with
finite support. Second, we evaluate our algorithms by demonstrating
their enhanced accuracy and comparing their performance with the
conventional MC models. We showcase the increased robustness of our
proposed nonparametric models to data outliers compared to parametric
models through results on a synthetic dataset. We present the utility of
our methods through experiments on real datasets. Lastly, we imple-
ment our algorithms using the VTK-m library to present accelerated
computation of critical point uncertainty and demonstrate their usability
in ParaView for broader community access.
2 RE LATE D WORK
The research in uncertainty visualization dates back to early 2000 when
Pang [47] and Johnson [32,33] recognized the need for quantifying and
depicting uncertainty in visualization. Since then, multiple advances
in uncertainty visualization have been documented in multiple survey
reports, including those by Brodlie et al. [13], Potter et al. [52], and
Kamal et al. [35]. Uncertainty visualization specific to ensemble data
and topology was discussed in recent survey reports by Hazarika et
al. [64] and Yan et al. [69], respectively.
Multiple new techniques have been derived to portray uncertainty in
scalar, vector, tensor, and multivariate data. Uncertainty visualization
of scalar field data covers a range of algorithms, including isosurfaces
of univariate data [8,21,28,51,65], multivariate surfaces [5,55], direct
volume rendering [6,16,39,57], topological merge trees [70], contour
trees [68], persistence diagrams [60], and Morse complexes [7]. Al-
though not to the extent of scalar fields, vector-field uncertainty has
been explored to gain insight into important data features, including
critical point uncertainty [45,46], streamlines [20,30], and Finite-Time
Lyapunov Exponents [26]. A few techniques have been developed to
compute and visualize uncertainty in tensor field data captured with
high angular resolution diffusion imaging (HARDI) [31] and diffusion
tensor imaging (DTI) [58]. These prior contributions mainly included
Monte Carlo sampling, Bayesian statistics, closed-form solutions, em-
pirical ensemble analysis, and low-dimensional embedding techniques
to understand uncertainty in data features. The methods proposed in
this paper model uncertain 2D scalar data as a probabilistic field and
derive closed-form solutions to understand critical point uncertainty.
A variety of noise models have been previously explored to visualize
uncertainty in scientific data. Statistically independent Gaussian [50]
and uniform [4,25] distributions have been used to model uncertainty
in scalar fields and study their impact on features, such as level-sets and
critical points. This work was later extended to multivariate Gaussian
noise models [11,28,36,38,51] to capture the correlation among uncer-
tain data and avoid overestimation of feature probabilities in the final
visualization caused by the data independence assumption. Nonpara-
metric models (e.g., histograms, kernel density estimation, Gaussian
mixture models) have been used to show enhancements in visualization
quality over parametric models for various visualization techniques,
including level-sets [8,53], direct volume rendering [6,39], and fiber
surfaces [5], because of their higher robustness to outliers. Nonparmaet-
ric models, however, possess extra computational cost compared to
parametric models. Recently, copula-based models have been explored
to capture the correlation between both parametric and nonparametric
models for uncertainty visualization [29]. In this paper, we present our
methods by modeling data uncertainty with independent parametric
and nonparametric noise models with finite support.
Effective presentation of uncertainty is another important research
challenge. Visual attribute-mapping techniques, e.g., colormap-
ping [54] and point movement [24] proportional to uncertainty, have
been proposed to convey uncertainty in 3D surfaces. We utilize the ele-
vation map technique proposed by Petz et al. [49] to render critical point
uncertainty for the climate dataset in Sec. 5. The novel use of glyphs has
been previously proposed for conveying uncertainty in vector [66] and
tensor [34] field data. We utilize sphere glyphs to show critical point
uncertainty for the Red Sea dataset in Fig. 1. Animation techniques
have been proposed for volume rendering [40]. Effective quantification
and visualization of uncertainty for 2D and high-dimensional data still
remains a big challenge for the visualization community.
3 BACKGR OU ND A ND PRO BL EM SE TT IN G
We briefly define critical points. Let
M⊂R2
be a 2D domain with a
boundary discretized as a regular grid (we further ignore the boundary
condition for most of our discussion). Let
f:M→R
be a function;
∇f
denotes its gradient. A point
p∈M
is considered critical if
∇f(p) = 0
;
otherwise it is regular. We will assume that all critical points are non-
degenerate, i.e.,
f
is a Morse function. A critical point is categorized

into three types, local minimum (
lmin
), local maximum (
lmax
), and
saddle (
ls
). In particular, if
f(p)
is smaller than the function value of
all of its neighbors, then the point
p
is a local minimum. Similarly,
if
f(p)
is greater than the function value of all of its neighbors, then
the point
p
is a local maximum. If
f(p)
is smaller than the function
value of one neighbor and greater than the one for the next neighbor in
alternating fashion with the neighbors visited sequentially in clockwise
or counterclockwise manner, then the point pis a saddle.
Fig. 2: Depiction of a problem setting. The probability distributions at
a grid vertex
p
and its neighbors
e,n,w
, and
s
represent uncertainty in
data. Our aim is to compute the probability of point
p
to be critical when
distributions are represented with parametric and nonparametic models.
Uncertainty in data, however, creates uncertainty regarding whether
a point is critical or not. In this paper, we calculate the probability
for a domain point to be critical when uncertainty in data is modeled
as probability distributions. Specifically, the methods in this paper
consider uncertain data at four neighbors of a domain point along the
two coordinate axes directions in a regular grid to compute the proba-
bility of the domain point to be critical. Although several applications
consider six- or eight-pixel neighborhoods depending on the domain
triangulation for critical point visualization, we plan to research these
cases with a higher number of neighbors in the future.
Fig. 2 depicts the problem setting, which is also used to introduce
notation. Let
p
be a point for which we want to compute the probabil-
ity of it being critical. Let
X1∼PdfX1(x1)
denote a random variable
with parametric or nonparametric noise distribution
PdfX1
over the sup-
port
x1∈[a1,b1]
at a point
p
. Let
e,n,w
, and
s
be the four neighbors
of a point
p
in the east, north, west, and south directions with ran-
dom variables
X2∼PdfX2(x2)
,
X3∼PdfX3(x3)
,
X4∼PdfX4(x4)
, and
X5∼PdfX5(x5)
, respectively, that denote uncertainty in data. For each
random variable
Xi
,
xi∈[ai,bi]
with
ai<bi
. Our work presents all
derivations for the independent noise models with noise distribution
over a finite support, i.e., the random variables
Xi
with
i∈ {1,...,5}
are assumed to be independent and bounded by a finite support
[ai,bi]
.
As the local data are not always independent in real datasets and identi-
fying bounds
[ai,bi]
can be challenging considering noisy data acqui-
sition processes [13], we discuss the ramifications of our independent
noise and finite support assumptions in Sec. 6. Because of the data
independence assumption, the joint probability density
Pr joint
of the
random variables is the product of their individual probability densities,
i.e.,
Pdf joint =∏i=5
i=1PdfXi(xi)
. Let
dx=∏i=5
i=1dxi
. Given these data,
our goal is to find the probability of point
p
being a local minimum
Pr(p=lmin), local maximum Pr(p=lmax ), and saddle Pr(p=ls).
4 METHODS
We describe the mathematical formulation and our algorithm for critical
point probability computation in closed form for independent paramet-
ric and nonparametric noise models with finite support.
4.1 Critical Point Probability (Two-Pixel Neighborhood)
For simplicity, we describe our derivations and approach for computing
the probability of point
p
to be critical for 1D uncertain scalar fields.
Our methods for the 1D case generalize to uncertain 2D scalar fields,
as described in Sec. 4.2. For the 1D case, we consider only the two
neighbors with random variables
X2
and
X3
of a 1D point
p
with its
associated random variable
X1
. The rest of the problem settings are
similar to the 2D case described earlier in Sec. 3.
4.1.1 Local Minimum Probability
The probability of point
p
being a local minimum,
Pr(p=lmin)
, can be
computed by integrating the joint probability
Pr joint
over its support
where random variable
X1
is simultaneously smaller than all neigh-
boring random variables (i.e.,
X2
and
X3
). Mathematically, the local
minimum probability can be represented as follows:
Pr(p=lmin) = Pr[(X1<X2)and (X1<X3)]
=Zx1=bmin
x1=a1Zx2=b2
x2=max(x1,a2)Zx3=b3
x3=max(x1,a3)(Pdf joint )dx,
where bmin =min(b1,b2,b3)
(1)
Equation (1) represents the core integration formula for the computation
of local minimum probability at a domain position
p
. We describe our
approach for the computation of local minimum probability in three
parts. (1) We explain the integral limits and piecewise simplifications of
the core integration formula in Eq. (1). (2) We describe our piecewise
integration approach to efficiently compute the formula in Eq. (1).
(3) We show a running illustration of our piecewise integration approach
to compute the local minimum probability.
Limits
a1
and
bmin
of the outer integral in Eq. (1):The outer
integral of Eq. (1) indicates the portion of data range of a random vari-
able
X1
(i.e.,
[a1,b1]
) that can result in point
p
being a local minimum.
In particular, the portion
[a1,bmin]
(with
a1<bmin
) of random variable
X1
can result in point
p
being a local minimum, where
bmin
denotes the
minimum among
b1,b2
, and
b3
. In contrast, the data range
[bmin,b1]
for
bmin =b1
cannot result in a point
p
as a local minimum
(lmin)
because
it will be always greater than the random variables
X2
or
X3
depending
on if
bmin =b2
or
bmin =b3
, respectively. Thus, mathematically, for
any value
x1≥bmin
,
Pr(p=lmin) = 0
. In the case
bmin <a1
, then
Pr(p=lmin) = 0
because there is at least one random variable between
X2and X3that will be always smaller than X1.
Limits
max(x1,ai)
and
bi
of the inner integrals in Eq. (1):The
two inner integrals in Eq. (1) integrate the joint distribution
Pdf joint
over its support where random variables
X2
and
X3
are simultaneously
greater than
x1∈[a1,bmin]
in the outer integral. The inner integral
lower limits are
max(x1,ai)
for
i∈ {2,3}
. The maximum is taken
because the support of a random variable
Xi
is restricted to
[ai,bi]
.
Thus, for
x1<ai
in any inner integral, the entire support
[ai,bi]
with
i∈ {2,3}
will always be greater than
x1
, and the inner integral does not
depend on the value of
x1
. In contrast, for
x1>ai
, the inner integration
depends on the value of
x1
because
x1
assumes values in support of
distributions. It is guaranteed that the upper limit of inner integrals in
Eq. (1) is greater than their respective lower limit, i.e.,
bi≥max(x1,ai)
,
for two reasons. First, for any random variable
Xi
, we assume
ai<bi
.
Second, the maximum value of
x1
is equal to
bmin =min(b1,b2,b3)
based on the outer integral (see the previous paragraph), and it cannot
exceed the upper limits
b2
and
b3
in inner integrals. Depending upon
whether the
max(x1,ai)
is equal to
x1
or
ai
, the integral in Eq. (1) can be
simplified and computed differently, which necessitates the evaluation
of the integral in Eq. (1) in a piecewise manner, as described next.
Piecewise simplification of Eq. (1):The core integration formula
in Eq. (1) can be simplified differently for different subsets of the range
of the outer integral (i.e.,
[a1,bmin]
). We refer to each subset as a
piece
P
. For a piece
P
, where
x1<a2
and
x1<a3
, the inner integrals
in Eq. (1) attain the range
x2∈[a2,b2]
and
x3∈[a3,b3]
. In other words,
the inner integrals do not depend on
x1
when
x1<a2
and
x1<a3
.
Thus, the integration over the entire support of random variables
X2
and
X3
simplifies Eq. (1) to the integration over a marginal distribution
of X1for the piece P, i.e., Rx1∈PPdfX1(x1)dx1.
For a piece
P
, where
x1∈[a2,b2]
and
x1<a3
, the inner inte-
grals in Eq. (1) attain the range
x2∈[x1,b2]
and
x3∈[a3,b3]
. In
this case, only the first inner integral related to the range of random

Fig. 3: Illustration of our three-step approach for critical point uncertainty computation for the 1D case. (1) Determine the range of random variable
X1
for which critical point probability is nonzero (i.e.,
[a1,bmin =b2]
) and determine its pieces. The range for which critical point probability is zero
is shown in brown in (a). Each new start point
ai∈[a1,bmin]
creates a new piece. Here,
a3
results in two pieces
P
1= [a1,a3]
and
P
2= [a3,bmin]
. (2)
Compute integration for piece
P
1
(i.e.,
IP
1
) depending on which intervals overlap it. Since the interval
[a2,b2]
overlaps with
P
1
,
IP
1
corresponds to
integral over a joint distribution of
X1
and
X2
, as depicted in (b). (3) Update integrals for next pieces depending on observed start points (e.g., inclusion
of random variable
X3
in the integral
IP
2
in (c) based on the start point
a3
) and sum all piecewise integrals to compute the local minimum probability.
variable
X2
depends upon
x1
. Thus, Eq. (1) simplifies as the inte-
gration over the joint distribution of
X1
and
X2
for the piece
P
, i.e.,
Rx1∈PPdfX1(x1)PdfX2(x2)dx2dx1
. In summary, various pieces of the
outer integral in Eq. (1) can be simplified differently based on whether
the inner integrals depend on x1or not.
Approach for computing local minimum probability: The com-
putation of Eq. (1) depends on the ordering of start points
ai
(i.e.,
a1,a2
,
and
a3
) and
bmin
. Thus, in 1D case, there are
4! =24
permutations of
ai
and
bmin
. These permutations increase fast for 2D/high-dimensional
cases. We, therefore, devise an efficient algorithm that computes the
piecewise integrals on the fly depending on the observed permutation
of aiand bmin without needing to go through all permutations.
We now describe our approach for the computation of the local
minimum probability at a domain position
p
[i.e.,
Pr(p=lmin)
]. Our
approach comprises three main tasks. Task 1: Determination of pieces
P
i
of the outer integral in Eq. (1) needed for performing piecewise
integration. Initially, we compute
bmin
. If
bmin <a1
, then there are
no pieces and
Pr(p=lmin) = 0
. If
bmin >a1
, then we sort intervals
representing uncertain data ranges (i.e.,
[ai,bi]
) based on their start
points
ai
and keep them in the array named
Isorted
. We note the index
of the interval
[a1,b1]
in
Isorted
(referred to as
ida1
) and the index of
interval in
Isorted
from the end that contains
bmin
(referred to as
idbmin
),
as
a1
and
bmin
constitute limits of the outer integral in Eq. (1). Any start
points
ai
contained in the range of indices
ida1
and
idbmin
determine the
pieces
P
i
for integration. This process generalizes to any ordering of
ai
to determine the pieces piof the outer integral in Eq. (1).
Next, we compute the integration for piece
P
1
denoted as
IP
1
.Task
2: Integration over piece
P
1
of the outer integral range
[
[
[a
a
a1
1
1,
,
,b
b
bmin]
]
]
.
The integration for piece
P
1
depends on the intervals that started before
a1
because the inner integral in Eq. (1) depends on
x1
for a random
variable
Xi
(with
i∈2,3
) started before
a1
, as max(
x1
,
ai
) is equal to
x1
. Task 2, therefore, corresponds to finding the intervals that started
before
ida1
, which also determines the upper limits of inner integrals
for piece P
1depending on observed order of intervals.
The computation of the integral in Eq. (1) for piece
P
1
(as well as
any arbitrary piece
P
i
) simplifies to one of the three types of integration
formulae, which we call integration templates. Generally, the simpli-
fication depends on the number of intervals overlapping a piece
P
i
, as
explained earlier. If there is no overlap with a piece, then the integral
in Eq. (1) simplifies to the integration of the probability distribution
of random variable
X1
(Template 1) over a piece. If only one random
variable (
X2
or
X3
) is overlapping with a piece, then the integral in
Eq. (1) simplifies to the integral over the joint distribution of
X1
and a
random variable corresponding to the overlapping interval (Template
2). If both random variables
X2
and
X3
are overlapping with a piece,
then the integral in Eq. (1) corresponds to the integral over the joint
distribution of all random variables (Template 3).
Having determined the integration for piece
P
1
, we compute inte-
gration for successive pieces. Task 3: Integration over piece P
iwith
i>1
.Essentially, each new start point
ai
observed between the outer
integral limits
a1
and
bmin
of Eq. (1) creates a new piece. Generally
speaking, each new start point
ai
results in a different simplification of
Eq. (1) because
max(x1,ai)
in Eq. (1) becomes equal to
x1
at each new
start point. Thus, encountering a new start point adds one inner integral
in a simplified form compared to the piece before encountering a new
start point. Finally, the integration of all pieces is summed to compute
the local minimum probability at a point p, i.e., Pr(p=lmin ).
Illustration of Local Minimum Probability Computation: Fig. 3
illustrates our method for computing the local minimum probability
for a domain point
p
. As shown for the example in Fig. 3,
a2<a1<
a3<b2<b1<b3
. Initially, we determine the range of a random
variable
X1
that can result in point
p
being a local minimum. As
shown in Fig. 3a, each value in the range
[x1=a1,x1= (bmin =b2)]
has a nonzero probability of being simultaneously smaller than the
neighboring random variables (i.e.,
X2
and
X3
). On the contrary, the
range
[x1= (bmin =b2),x1=b1]
is always greater than the random
variable
X2
, and therefore, cannot result in point
p
as a local minimum.
In Task 1, we determine the pieces
P
i
of the range
[x1=a1,x1=
(bmin =b2)]
. As depicted in Fig. 3, the array
Isorted
has intervals
ordered by
ai
, where
a2<a1<a3
. For this
Isorted
,
ida1=2
and
idbmin =3
. Since
a3
is a start point in interval indexed by
idbmin
, it
divides the outer integral range
[a1,bmin]
in Eq. (1) into two pieces
(depicted in blue and orange in Fig. 3).
In Task 2, we determine the simplification of the formula in Eq. (1)
for piece
P
1
. The simplification for piece
P
1= [a1,a3]
in Fig. 3 (denoted
by blue) depends on the number of intervals that started before
a1
. As
observed in Fig. 3, the interval
[a2,b2]
starts before
a1
. Since
x1<a3
in
piece
P
1
, the formula in Eq. (1) integrates random variable
X3
over its
entire support and simplifies to the integration over the joint distribution
of random variables X1and X2shown in Fig. 3b.
In Task 3, we determine the simplification of the formula in Eq. (1)
for successive pieces formed by each new start point
ai∈[a1,bmin]
. In
Fig. 3, the start point
a3∈[a1,bmin]
results in a new piece
P
2
shown
in orange. All inner integrals for piece
P
2
stay the same as piece
P
1
except for one newly added inner integral with limits
[x1,b3]
, as shown
in Fig. 3c, because
x1=max(x1,a3)
for piece
P
2
, unlike the piece
P
1
in
which
a3=max(x1,a3)
. Thus, we make such updates to inner integrals
for each new piece corresponding to a new start point ai∈[a1,bmin].
Time complexity: Our presented approach for local minimum
probability computation initially sorts all the intervals based on start
points
ai
with
i∈1,2,3
and
bmin
to determine pieces for integration
(Task 1), which is a constant time operation. Task 2 and Task 3 comprise
a single loop, which runs a maximum of three times corresponding to
the three entries of a sorted interval array
Isorted
. Each loop iteration
computes the integral template (simplification of the formula in Eq. (1))
on the fly depending on the observed data in array
Isorted
in constant
time. The algorithm is, therefore, linear time complexity with the
number of input intervals (here, three) and extremely efficient.

4.1.2 Local Maximum Probability
Having derived a probabilistic framework for computation of local min-
imum probability (Sec. 4.1.1), the computation of the local maximum
probability
Pr(p=lmax)
at a domain point
p
is fairly straightforward.
Computation of the local maximum probability corresponds to comput-
ing
Pr[(X1>X2)and (X1>X3)]
for the 1D case, which is equivalent
to computing
Pr[(−X1<−X2)and (−X1<−X3)]
. This negation for-
mat is equivalent to Eq. (1). Thus, we create negated random variables
X′
1=−X1
,
X′
2=−X2
, and
X′
3=−X3
. We then apply our proposed local
minimum probability computation algorithm (Sec. 4.1.1) to these new
random variables X′
ifor computing the local maximum probability.
4.1.3 Saddle Probability
The probability of point
p
being a saddle,
Pr(p=ls)
, can be computed
by integrating the joint probability
Pdf joint
over its support where the
random variable
X1
is simultaneously smaller than
X2
and greater than
X3
(and the other way around). Mathematically, the saddle probability
can be represented as follows:
Pr(p=ls) =(t1=Pr[(X1<X2)and (X1>X3)])+
(t2=Pr[(X1>X2)and (X1<X3)])
The term t1in the equation above can be written as follows:
t1=Pr[(X1<X2)and (X1>X3)]
=Zx1=balt
min
x1=aalt
max Zx2=b2
x2=max(x1,a2)Zx3=min(x1,b3)
x3=a3
(Pdf joint )dx,
where aalt
max =max(a1,a3),balt
min =min(b1,b2)
(2)
Equation (2) represents the core integration formula for the computation
of the saddle probability at a domain point
p
. We derive our closed-
form computations and algorithm only for the term
t1=Pr[(X1<
X2)and (X1>X3)]. The term t2can be computed by creating negated
random variables
X′
i=−Xi
with
i∈ {1,2,3}
and plugging
X′
i
in place
of Xiinto the derivation for term t1.
The algorithm to compute the saddle probability is similar to the
three tasks of the local minimum probability computation in Sec. 4.1.1.
In Task 1, the algorithm first determines the data range
[aalt
max,balt
min]
that
qualifies for point
p
being a saddle (similar to the range determination
for local minimum probability computation, illustrated in Fig. 3a).
The algorithm then divides this range into pieces
P
i
depending on the
ordering of points
[aalt
max,balt
min,a2,b3]
that correspond to the limits of
integrals in Eq. (2). In Task 2, the integral of piece
P
1
is computed
depending on how
a2
and
b3
are ordered with respect to the lower
limit
aalt
max
. In Task 3, the integrals of successive pieces are computed
depending on the order in which
a2
and
b3
appear until the upper
limit
balt
min
. All integrals fit into one of the templates that represent
simplification of the core integration in Eq. (2). All piecewise integrals
are finally added to compute the term t1.
4.2 Critical Point Probability (Four-Pixel Neighborhood)
Our methods for the two-pixel neighborhood case (Sec. 4.1) generalize
to the four-pixel neighborhood case (depicted in Fig. 2) straightfor-
wardly. Here, we briefly discuss a few new specifics related to the
four-pixel neighborhood case. The detailed algorithms and illustrations
are provided in the supplementary material.
Local Minimum Probability: The core integration formula for the
computation of local minimum probability at a domain position
p
in
the case of a four-pixel neighborhood takes a form similar to Eq. (1).
Pr(p=lmin)
=Pr[(X1<X2)and (X1<X3)and (X1<X4)and (X1<X5)]
=Zx1=bmin
x1=a1Zx2=b2
x2=max(x1,a2)···Zx5=b5
x5=max(x1,a5)(Pdf joint )dx,
where bmin =min(b1,b2,b3,b4,b5)
(3)
Computation of the integral in Eq. (3) has a workflow similar to the
two-pixel neighborhood case. Initially, pieces of the range
[a1,bmin =
min(b1...b5)]
are determined based on the ordering of integral limits,
i.e.,
a1,a2,a3,a4,a5
, and
bmin
. Based on these six points of interest,
there are
6! =720
possible permutations. We compute piecewise sim-
plification of Eq. (3) (templates) on the fly depending on the observed
ordering of the six points without needing to go through all permuta-
tions. Finally, all piecewise integrals are summed to compute the local
minimum probability.
Local Maximum Probability: Similar to the two-pixel neighbor-
hood case, the local maximum probability in the four-pixel neighbor-
hood case is computed by negating the random variables followed by
the application of our algorithm for computing the local minimum
probability.
Saddle Probability The core integration formula for computing
probability of point
p
being a saddle,
Pr(p=ls)
in the case of four-pixel
neighborhood is mathematically represented as follows:
Pr(p=ls)
= (t1=Pr[(X1<X2)and (X1>X3)and (X1<X4)and (X1>X5)])+
(t2=Pr[(X1>X2)and (X1<X3)and (X1>X4)and (X1<X5)])
The term
t1
in the equation above takes a form similar to the two-pixel
neighborhood case in Eq. (2) and can be written as follows:
t1=Pr[(X1<X2)and (X1>X3)and (X1<X4)and (X1>X5)]
=Zx1=balt
min
x1=aalt
max Zx2=b2
x2=max(x1,a2)Zx3=min(x1,b3)
x3=a3Zx4=b4
x4=max(x1,a4)Zx5=min(x1,b5)
x5=a5
...
(Pdf joint )dx, where aalt
max =max(a1,a3,a5),balt
min =min(b1,b2,b4)
(4)
Again, the saddle probability can be efficiently computed using
piecewise integration with piece limits determined by ordering of the
integral limits
aalt
max
,
balt
min
,
a2
,
a4
,
b3
,
b5
in Eq. (4). We compute piece-
wise simplification of Eq. (4) (templates) on the fly depending on the
observed ordering of the six points and sum them up to compute the
saddle probability, similar to the two-pixel neighborhood case.
4.3 Parametric Noise Models
In this section, we derive critical point probability computations for the
uniform and Epanechnikov distributions. The derivation for the uniform
noise model acts as a building block for our histogram model derivation
in Sec. 4.4.1. The uniform distribution is given by
PdfX(x) = 1
b−a
,
where
x∈[a,b]
. Although we do not have a closed-form solution for
the Gaussian noise model, similar to the Gaussian, the Epanechnikov
model gives more weight to the data mean, has a bell-like shape, and is
smoother than the uniform noise distribution (see Fig. 4). The Epanech-
nikov distribution, therefore, can yield better results than the uniform
noise model, as presented later in Sec. 5.2. The Epanechnikov distribu-
tion is given by
PdfX(x) = 3
2∗(b−a)[1−(x−m
w)2]
, where
m= (a+b)/2
,
w= (b−a)/2
, and
x∈[a,b]
. These distribution formulae can be
plugged into the integration formulae for critical point probability com-
putations (i.e., Eq. (3) and Eq. (4)) and their simplifications (templates)
to compute results for these two types of distributions.
Fig. 4: Epanechnikov distribution (b) gives more weight to the mean
unlike the uniform distribution (a), and hence, can provide enhanced
visualization compared to the uniform noise model.

The derivation of integral templates for the uniform and Epanech-
nikov kernel can be cumbersome because of high-order functions result-
ing from integrals. For example, since the Epanechnikov is an order-2
kernel, the integration templates can result in formulae of order-15
functions because of the five integrals in the case of four-pixel neigh-
borhood without simplification (Sec. 4.2). Thus, we use the Wolfram
Alpha software [67] for deriving the integral templates
1
. Note that
these high-order functions may produce numerical instabilities for large
or fractional data values. Thus, the dataset range needs to be properly
scaled to ensure stable computations with the Epanechnikov kernel.
4.4 Nonparametric Noise Models
For nonparametric models, we do not assume a particular distribution
type for data. Instead, we derive a histogram with a user-specified
number of bins at each grid vertex to derive critical point probabilities.
Our derivations for the parametric models act as building blocks for
derivations of nonparametric models. We propose closed-form and
semianalytical (mix of MC and closed-form) solutions.
4.4.1 Closed-Form Formulation
Each histogram can be considered as a weighted combination of
nonoverlapping uniform distributions. Mathematically, the
pdfX(x)
using a histogram can be written as
pdfX(x) = wi∑i=h
i=1Kb(x−xi)
,
where
∑i=h
i=1wi=1
,
Kb
denotes a uniform kernel
K
with width
b
,
xi
denotes the bin center, and
h
denotes the number of histogram bins,
and
x∈[x1−b/2,xh+b/2]
. Thus, our derivation and algorithm for the
uniform distribution (Sec. 4.3) can be leveraged for critical point proba-
bility computation with the histogram noise model. Let
Kbi
,
Kbj
,
Kbk
,
Kbl
, and
Kbm
denote the uniform kernels of histograms for random vari-
ables
X1
,
X2
,
X3
,
X4
, and
X5
, respectively. The critical point probability
can be computed by going through all combinations of the uniform
kernels across five random variables and summing the critical point
probability for each possible combination weighted by its probability.
Mathematically, the local minimum probability at a grid point
p
can be
computed as follows:
Pr(p=lmin) = w
i=h
∑
i=1
···
m=h
∑
m=1
Pr(p=lmin)i,j,k,l,m,(5)
where
Pr(p=lmin)i,j,k,l,m
denotes the probability of each kernel
Kbi
of random variable
X1
being simultaneously smaller than kernels
Kbj,...,Kbm
of random variables
X2,...,X5
, respectively, weighted
by the probability of choosing kernels denoted by
w=wiwjwkwlwm
.
We verify the correctness of our formulation through quantitative and
qualitative evaluation in our results.
Nonparametric models can increase the robustness of visualization
to outliers [6,8,53] because they do not assume any particular shape
of the distribution. We show the increased robustness of visualizations
under uncertainty with the proposed nonparametric models compared
to parametric models in the results, Sec. 5. However, the increased
quality of nonparametric models comes at the cost of an increased
number of computations. The time complexity of computing the local
minimum probability is
O(h5)
, as observed from Eq. (5), where
h
is
the number of histogram bins. Our histograms method in Eq. (5) can
be extended to a more general kernel density estimation [48], in which
each noise sample is assigned a kernel. However, the time complexity
can grow sharply with an increase in the sample count, and KDE can
quickly become impractical for use in visualization. Thus, we restrict
our nonparametric methods to histograms with a user-specified number
of bins. To accelerate the performance of nonparametric methods, we
propose two solutions. First, we present a more efficient semianalytical
approach (Sec. 4.4.2), which provides an approximate but reliable
solution at a greater speed. Second, we accelerate our critical point
uncertainty computation using a parallel implementation (Sec. 4.5).
1
All integral templates and code are available at
https://github.com/
tusharathawale/UCV/tree/exp_critical_point_noplugin.
4.4.2 Semianalytical Solution
We propose a mix of MC and closed-form formulation to get approxi-
mate but reliable solutions at a greater speed. For our method, we draw
c
samples from
pdfX1(x1)
at a point
p
. For each sample, the probability
of that sample being critical can be computed in closed form. Specifi-
cally, we compute the probability of a sample being smaller (or greater)
than neighboring random variables in closed form. For example, in
order to compute
Pr(sample <X2)
, we first find the histogram bin of
PdfX2(x2)
to which the sample belongs. We then integrate the bin den-
sity for values greater than the sample (similar to a uniform distribution)
and sum it with the integration of densities of subsequent bins. Note
that integration of subsequent bins is precomputed efficiently using the
prefix sum (scan) method and, therefore, the integration computation
time does not depend on the number of bins. Let
Pr2...Pr5
denote
Pr(sample <X2)...Pr(sample <X5)
, respectively. Because of the in-
dependent assumption, the local minimum probability for the sample
corresponds to a product
Prsampl e =Pr2Pr3Pr4Pr5
. Saddle and local
maximum probabilities for the sample can be computed in a similar
manner. Let
Prtotal
denote the sum of
Prsampl e
for all
c
samples. Thus,
the probability of the point
p
being critical can be found as the ratio
of
Prtotal
and
c
. The computational complexity of this semianalytical
method is proportional to the number of samples (i.e.,
c
) drawn from a
single distribution
PdfX1(x1)
. This method, therefore, provides faster
results compared to the exponential closed-form solution in Sec. 4.4.1.
4.5 Integration with VTK-m and ParaView
Motivated by Wang et al.’s work [65], we integrate the critical point
uncertainty code with the VTK-m software [42]. Since the computation
of critical point probability depends only on the local neighbors and
is independent of data at other pixels, it is embarrassingly parallel.
We therefore implemented our code using VTK-m. The advantage
of VTK-m is that it allows optimized access to neighbors, and it is
platform portable. With our VTK-m implementation, we showcase
significant speed-up in critical point uncertainty computation on various
architectures, including AMD, NVIDIA, and Intel processors. Further,
we export our VTK-m code as a plugin for the use in ParaView software.
With our plugin, the uncertainty of critical points can be visualized
in ParaView in near-real time and can be combined with the other
ParaView filters for better analysis of uncertainty.
5 RE SULTS AN D DISCUSSIONS
5.1 Validation and Performance of Proposed Algorithms
We validate the correctness of our proposed closed-form computations
and algorithms through qualitative and quantitative comparison with
respect to the conventional MC sampling approach. We also demon-
strate the performance improvements of the proposed methods over
MC sampling. Figure 5 demonstrates the correctness and enhanced per-
formance of our algorithms through experiments on a synthetic Ackley
function [1] sampled on a uniform grid. In particular, the uncertain
data is synthetically generated by injecting random uniform noise at
each grid position to produce an ensemble of
50
members. At each
grid vertex, the minimum and maximum values are computed from the
ensemble to estimate the range of a uniform distribution. The local
minimum probability
Pr(p=lmin)
and saddle probability
Pr(p=ls)
are then computed at each grid vertex
p
using the MC sampling method
and our proposed algorithms. The results are visualized in Fig. 5.
Columns Fig. 5a-b show the results of MC sampling. Column
Fig. 5c shows the results obtained with our closed-form formulation and
algorithms. Column Fig. 5d visualizes the difference image between
columns Fig. 5c and column Fig. 5b. Column Fig. 5e visualizes the
convergence of MC solutions to our closed-form solution by plotting
the root mean squared error (RMSE) between the MC and closed-form
solutions. The white isocontours (isovalues
0.1
and
0.01
for local
minimum and saddle probability results, respectively) shown in Fig. 5a-
c enclose the possible critical point positions. As observed in Fig. 5a-c,
the isocontour structure in MC results converges to our closed-form
solutions as we increase the number of MC samples from
100
to
2000
.

Fig. 5: The qualitative and quantitative proof of correctness and enhanced performance of our proposed closed-form computations (column c) with
the MC sampling approach (columns a and b) as the baseline. The results are shown for the uniform noise model. The solution obtained with 2000
MC samples converges to closed-form computations with our algorithms (see the difference images in column d and convergence curves in column
e), thereby confirming their correctness. Our methods provide
64×
and
119×
speed-up with respect to the MC sampling approach with
2000
samples.
In MC sampling, uniform distributions are sampled to estimate the
local minimum/saddle probability. As observed in Fig. 5a, drawing
100
samples per grid vertex not only causes more computation time but
also results in a lower accuracy compared to our closed-form results in
Fig. 5c. As we increase the number of MC samples to
2000
in Fig. 5b,
the accuracy increases (i.e., results converge), but the computational
performance reduces. Specifically, compared to the MC sampling
method with
2000
samples, our closed-form solution provides a
64×
speed-up for local minimum probability and
119×
speed-up for the
saddle probability computation with a comparable accuracy. The tim-
ings reported are for the Python serial implementation on a quad-core
Intel I7 processor. Note that the MC sampling approach for a saddle
takes on average more time compared to MC sampling for the local
minimum because of the computation of two terms,
t1
and
t2
, for a
saddle (see Sec. 4.2). The RMSE and maximum probability difference
(
Error_max
) between the MC and our closed-form solutions shown in
Fig. 5d and a convergence curve in Fig. 5e confirm the correctness of
our derivations and algorithms. Similar convergence curves and results
for the Epanechnikov and histogram (closed-form and semianalytical)
models are reported in the supplement.
5.2 Comparison of Parametric Vs. Nonparametric Models
We present a comparison of parametric and nonparametric models in
terms of accuracy and performance for computation of critical point
probability. Here, we show the results for the synthetic data modeled
as a Gaussian mixture model, which is a common data model used in
scientific and topological analysis [10,60,70]. In particular, we generate
the ensemble of 50 members, in which each member comprises a
mixture of two Gaussians. As shown in Fig. 6, 40 members correspond
to a Gaussian mixture, in which peaks (yellow) are oriented in NW-SE
direction. These
40
members are considered as the ground truth, and the
Fig. 6: Gaussian mixture data for computation of critical point probability.
variation across them corresponds to small noise randomly added. The
remaining
10
members are a
90◦
rotated version of the first
40
members
oriented in NE-SW direction. These rotated members represent the
noisy data or outliers. In other words, the peaks represented by these
rotated members are not the true peaks.
We apply parametric (independent uniform, Epanechnikov, Gaus-
sian, and multivariate Gaussian) and nonparametric (histogram with
5
bins) noise models to the Gaussian mixture ensemble (Fig. 6) for
visualization of the local maximum probability (i.e.,
Pr(p=lmax)
).
The results in Fig. 7 demonstrate the quality and performance of vari-
ous noise models. Figure 7a visualizes the result for the independent
uniform noise model using our closed-form computational algorithm
(Sec. 4.2). At each pixel, the data range is computed based on the min-
imum and maximum data values observed across the ensemble. The
closed-form uniform noise model is the fastest, but shows all critical
points in NW-SE-NE-SW directions equally likely with moderate prob-
ability. This is not a desirable result, as the critical points in the NE-SW
directions correspond to noise or outliers (see Fig. 6) and should not
result in important features in a probabilistic visualization.
The quality of results is improved overall in the case of independent
Gaussian and Epanechnikov noise models, as observed in Fig. 7b and
Fig. 7c, respectively. For both models, we determine the distribution
range based on the sample mean and standard deviation per pixel across
the ensemble members. Since the Gaussian model does not have a
closed-form solution, we performed MC sampling with
1000
samples
to compute the local maximum probability, which is computationally
expensive compared to the uniform noise model. On the other hand, the
Gaussian noise model highlights the true critical points (NW-SE) better
with respect to local neighbors compared to the outlier critical points
(NE-SW). However, both true and outlier critical points get moderate
probability values assigned. Our proposed closed-form result for the
Epanechnikov model in Fig. 7c is similar to the result for the Gaussian
model in Fig. 7b, obtained in reduced time with about
2.58×
speed-up.
The Epanechnikov model enhances the results compared to the uniform
model because of its similar characteristics to the Gaussian model, i.e.,
greater mean weight, a bell-like shape, and more smoothness, compared
to the uniform model (see Sec. 4.3). The multivariate Gaussian model in
Fig. 7d from the previous work [38,49] prominently highlights both the
true and outlier peaks with high probabilities (bright yellow). Capturing
data correlation assigns high probability to the outlier critical points
and counter intuitively exhibits less robustness to outliers.
Figure 7e and Fig. 7f visualize the results for the independent non-

Fig. 7: A comparison of parametric vs. nonparametric noise models in terms of quality and performance of local maximum probability (i.e.,
Pr[p=lmax]
)
computation for the ensemble dataset shown in Fig. 6. Our closed-form uniform noise model (a) is the fastest, but it exhibits less accurate results in
that it equally highlights both the true and noisy peaks with moderate probability. The independent Gaussian noise (b) model without a closed-form
solution and the independent Epanechnikov model with a closed-form solution (c) highlight the true peaks better with respect to the local neighbors,
but with moderate probability and slightly more compute time. The multivariate Gaussian model (d) from previous work [38,49] highlights both
the true and noisy peaks with high probabilities (bright yellow). The proposed closed-form histogram method (e) exhibits the most robust result to
outliers clearly highlighting the true local maximum positions with high probability (bright yellow), but it is the slowest in performance. The proposed
semianalytical histogram method (f) reduces the time while maintaining sufficient accuracy when compared to (e).
parametric histogram model (Sec. 4.4) with
5
bins, which exhibit the
greatest robustness to outliers. Specifically, they clearly highlight the
true local maximum positions with a high probability (bright yellow)
and smooth out noisy peaks with a moderate probability (indicated by
the arrows in Fig. 7e-f). The closed-form nonparametric models, how-
ever, require more computations (see Sec. 4.4.1). The computational
time of the closed-form nonparametric solution is mitigated with the
semianalytical solution (see Sec. 4.4.2) while maintaining comparable
accuracy, as seen from Fig. 7e and Fig. 7f. All timing results reported in
Fig. 7 are again obtained with a serial Python implementation on a quad-
core Intel I7 processor. To be able to accommodate the high-quality
nonparametric solutions in the visualization systems, we accelerate
our algorithms using a C++ parallel implementation with VTK-m [42]
(Sec. 4.5), as presented for the real dataset results in the next section.
5.3 Real Datasets
Climate Data: Figure 8 visualizes the mean sea-level pressure
variable simulated from an earth and climate simulation model – Energy
Exascale Earth System Model (E3SM) [22] – with a grid resolution
of
0.25◦
. The result is visualized for a tropical region sampled on a
regular grid of size
240 ×960
, in which the local minima in relatively
low pressure regions (i.e., blue) indicate the potential of existence of
tropical cyclones. The dataset was compressed using an error-controlled
lossy compressor – MGARD [23] – under a relative L2 error bound (
eb
)
of
1×10−3
, which provided a compression ratio of
16.68
. We study
the uncertainty of critical points in data decompressed using MGARD.
Figure 8a visualizes the original mean sea-level pressure data col-
ormapped with magnitude. The critical points (i.e., local minima
lmin
)
are shown as the blue spheres extracted using TTK [59]. Figure 8b
visualizes the decompressed data and its critical points. As observed,
the number of critical points increases in certain regions due to the com-
pression errors, as illustrated by the inset view. There is, however, no
indication of how much uncertainty or probability of critical points is
in a decompressed field, which can lead to less reliable TDA. Figure 8c
visualizes critical points in a decompressed field along with uncertainty.
We utilize the value of error bound
eb
to derive critical point proba-
bilities. Specifically, at each grid position, the uncertain data range
is
[d′−eb
2,d′+eb
2]
, where
d′
denotes the decompressed value. Since
we do not have prior knowledge of the distribution over the uncertain
range, we model data uncertainty with the uniform distribution and
apply our proposed algorithms (Sec. 4.2).
In Fig. 8c, we visualize the derived local minimum probability using
a heightfield, an idea similar to the previous work by Petz et al. [49]. In
a heightmap, each grid point of interest is elevated and colormapped
proportional to the critical point probability. In Fig. 8c, the heightfield
tailored to critical points clearly indicates points with relatively high
probability. For the two inset views in Fig. 8c, the high probability
critical points (enclosed by the green dotted box) also appear in the
original data in Fig. 8a. In contrast, newly created critical points due
to compression errors have a low probability. Figure 8d visualizes the
heightfield for every grid pixel, in which the magenta regions (with
Pr(p=lmin)>0.2
) reflect a pattern of critical point positions that is
similar to the one in the original data (Fig. 8a). We additionally present
a quantitative evaluation for the dataset in the supplementary material.
We measure the performance and accuracy of the climate dataset
results through our VTK-m implementation. Since the VTK-m im-
plementation is platform portable, we run it on a serial processor and
AMD GPUs on the Oak Ridge National Laboratory’s Frontier super-
computer [3] and NVIDIA GPUs on National Energy Research Sci-
entific Computing Center’s Perlmutter supercomputer [37]. Using a
conventional MC solution with
1000
samples per grid point takes
4.94
seconds on a serial backend. The proposed algorithms for closed-form
computation (Sec. 4.2) compute the true solution in
0.012
seconds on
a serial backend, thereby providing a
411×
speed-up. Running the
VTK-m code on AMD GPU yields a closed-form result in
0.003
sec-
onds, which corresponds to a
1646×
speed-up compared to the MC
solution and a
4×
speed-up compared to the closed-form solution on a
serial backend. The NVIDIA GPU yields a closed-form result in
0.004
seconds, a performance close to the AMD GPU. We present additional
accuracy and performance results of our VTK-m code for the AMD
GPU in the supplementary material.
Oceanology Data: In our next experiment, we evaluate the quality
and performance of parametric and nonparametric noise models for
the Red Sea ensemble simulations [56]. The results are visualized in
Fig. 1. The dataset is downloaded from the 2020 IEEE SciVis contest
website. The ensemble comprises
20
members each with grid resolution
500 ×500
. Understanding eddy positions is crucial for oceanologists
to gain insight into energy and particle transport patterns in oceans.
Therefore, we investigate the local minima positions that potentially
correspond to the eddy features. Since the original simulation data are
too noisy, we applied topological simplification [19] using TTK [59] to
each ensemble member as a denoising (preprocessing) step until each
ensemble member has approximately the same number of critical points.
This strategy corresponds to the persistence graph idea from the prior
work by Athawale et al. [7], which plots the number of local minima
against the persistence simplification level to decide the amount of
simplification. Having simplified the topology, we efficiently compute
and visualize critical point uncertainty in ParaView with our VTK-m
code as a backend, as documented in the supplement.
Fig. 1a-b visualize the results for the parametric noise models. We ap-
ply spherical glyphs in ParaView to help quickly identify the positions
with high probability. The points with a larger local minimum proba-
bility (
Pr(p=lmin)
), therefore, have a bigger radius and a red/orange
color in the sphere glyphs. The result of the multivariate Gaussian

Fig. 8: Critical point visualization for the climate dataset. (a) The colormapped original data with local minima shown as the blue spheres. (b)
Compression errors result in increased numbers of critical points for which no uncertainty is visualized. (c) Computation of critical point probability
with the uniform noise model and visualization of uncertainty through elevation proportional to probability. The critical points in the original data have
higher probabilities (tall mountain with magenta color enclosed by the green box). (d) The heightmap of probabilities for every domain position. High
probabilities (i.e., tall mountains with magenta peaks) are observed in similar regions as the critical points in the original data.
model from previous work by Petz et al. [49] and Liebmann et al. [38]
with 2000 MC samples is visualized in Fig. 1c. Capturing the cor-
relation using the multivariate Gaussian model significantly reduces
the probability of local minima in certain regions (blue regions) and
emphasizes fewer critical points. Fig. 1d visualizes the results for our
proposed nonparametric histogram model with four bins (Sec. 4.4.1).
The uniform, Epanechnikov, multivariate Gaussian, and histogram mod-
els took
0.094
,
0.102
,
0.167
and
0.145
seconds, respectively, on the
Frontier supercomputer’s AMD GPU. Fig. 1e-f visualize the critical
points (yellow) extracted from two random members of the ensemble.
Since we do not know the true critical points for the ensemble
data, we make a few interesting observations by comparing results of
different noise models (shown with the boxes in Fig. 1). The white
boxes indicate positions where the two local minimum positions are
consistently observed with high probability across all models, and
therefore, can be trusted. The green boxes show critical points that are
captured as high probability by the proposed nonparametric histogram
models, but not by other models. This result is interesting because
the histogram showed the highest resilience to outliers, and therefore,
more trustworthy results, in our synthetic experiments in Fig. 7. As
observed in individual ensemble members, critical points are also seen
in the areas marked by the green boxes. Similarly, the cyan boxes mark
critical points that are captured by the multivariate Gaussian model, but
not by any other models. This result necessitates further investigation
because the multivariate Gaussian model was less robust to outliers in
our synthetic experiment in Fig. 7. Lastly, the pink boxes show the
positions where the multivariate Gaussian and nonparametric models
agree, whereas the others do not agree.
6 CONCLUSION AND FUTURE WORK
In this paper, we study the propagation of uncertainty in critical points,
the fundamental topological descriptors of scalar field data. Our main
contribution of this paper is the creation of a novel efficient algorithms
that compute critical point probability in closed form for parametric
and nonparametric uncertainty with finite support. We demonstrate
the effectiveness of our algorithms through enhanced accuracy and
performance over classical MC sampling. We integrate our algorithms
with the VTK-m library [42] to further accelerate performance using
serial, AMD, and NVIDIA backends. We show seamless integration
of our VTK-m algorithms with ParaView [2] (see the supplement),
which is a key to making our algorithms accessible to a wide audience.
Our synthetic experiments show the greater resilience of our proposed
nonparametric models to outliers compared to parametric models, sim-
ilar to the prior studies [8,53]. We present the practical utility of our
techniques through application to climate and oceanology datasets.
A few limitations of this work are important and need to be ad-
dressed in the future. Currently, we assume that the data at each grid
point have uncertainty only over finite bounds. Although the finite
bounds assumption is generally true in practice, finding exact bounds
that are needed for our proposed algorithms can be nontrivial. In the
demonstrated results, we derived the upper and lower bounds for the
climate and Red Sea ensemble datasets at each grid point based on the
data captured by simulations. Thus, the quality of our results strongly
depends on how well the application can provide upper and lower
bounds for the values at a grid point. That said, one of the benefits of
the proposed nonparametric models is that they can mitigate the effects
of overestimation (or coarseness) of bounds resulting from the outliers
by assigning a lower weight or probability to outliers (see Sec. 4.4 and
the synthetic experiments in Sec. 5.2). In the frequent case of ensemble
simulations, however, a priori knowledge or experiments demonstrating
the robustness of lower and upper bounds provided by the ensembles
can help to further improve the trustworthiness of the results.
Another limitation of our work is that we assume neighboring data
points to be uncorrelated. This independent noise assumption can be
true in the case of measurement data but is not often true for the real
datasets or ensemble simulations encountered in practice [4,9,13,51].
Spatially close grid points are often strongly correlated in real datasets
because to reliably represent a smooth function, any reasonable grid
must have a spatial resolution finer than frequencies of relevance. Our
independent noise assumption, therefore, can lead to overestimation
of probabilities from ignoring the local correlation (as shown too in
the previous studies [9,51]). Considering spatial correlation, however,
has a caveat of higher sensitivity of results to outliers, as demonstrated
for the multivariate Gaussian model in our synthetic experiments in
Sec. 5.2. Even though the core integration formulae in Eq. (3) and
Eq. (4) are valid for the correlated data, their computation becomes
more challenging because of the complexity of accommodating linear
or nonlinear correlations. Thus, further research is needed to derive
closed-form solutions for critical point probability computation that
can accommodate spatial correlation and are robust to outliers.
Our methods are currently limited to critical points of uncertain 2D
fields based on four neighbors per grid point. In the future, we would
like to extend our work to more neighbors (e.g., six- or eight-pixel
neighborhood based on the triangulation) and 3D datasets, which can be
complex because of higher order of integration templates. We utilize the
sphere glyphs and heightmaps [49] for the visualization of critical point
probabilities. However, both methods can lead to occlusion and clutter.
Thus, further study is needed to evaluate the perceptual quality of sphere
glyph and heightmap methods and, possibly, derive new rendering
techniques for enhanced perception of uncertainties. Finally, we will
also investigate uncertainty in other topological visualizations based on
critical points, including contour trees and persistence diagrams.

ACK NOW LEDGMEN TS
This work was supported in part by the U.S. Department of En-
ergy (DOE) RAPIDS-2 SciDAC project under contract number DE-
AC0500OR22725, NSF III-2316496, the Intel OneAPI CoE, and the
DOE Ab-initio Visualization for Innovative Science (AIVIS) grant
2428225. This research used resources of the Oak Ridge Leadership
Computing Facility (OLCF), which is a DOE Office of Science User
Facility supported under Contract DE-AC05-00OR22725, and National
Energy Research Scientific Computing Center (NERSC), which is a
DOE National User Facility at the Berkeley Lab. We would also like to
thank the reviewers of this article for their valuable feedback.
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