A Scale Space Based Persistence Measure for Critical Points in 2D ...

A Scale Space Based Persistence Measure
for Critical Points in 2D Scalar Fields
Jan Reininghaus, Natallia Kotava, David Günther, Jens Kasten,
Hans Hagen, Senior Member, IEEE, and Ingrid Hotz
Abstract— This paper introduces a novel importance measure for critical points in 2D scalar fields. This measure is based on a
combination of the deep structure of the scale space with the well-known concept of homological persistence. We enhance the noise
robust persistence measure by implicitly taking the hill-, ridge- and outlier-like spatial extent of maxima and minima into account. This
allows for the distinction between different types of extrema based on their persistence at multiple scales. Our importance measure
can be computed efficiently in an out-of-core setting. To demonstrate the practical relevance of our method we apply it to a synthetic
and a real-world data set and evaluate its performance and scalability.
Index Terms—Scale space, persistence, discrete Morse theory
1 INTRODUCTION
Computer assisted analysis of two-dimensional scalar data has become
an essential tool in scientific research. To deal with the growing
amount of data, feature extraction methods are frequently employed
in applications like medical imaging, geosciences and computational
fluid dynamics. The features can often be described by the extremal
points of the scalar field or its derived quantities. Due to the intrinsic
uncertainty in measurements and the finite precision of numerical
simulations these kinds of data usually contain noise. This noise introduces
a lot of spurious local extrema, which complicates automatic
analysis. One is therefore interested in methods that allow to discriminate
dominant from spurious critical points.
One such method is homological persistence, introduced by Edelsbrunner
et al. [14]. This method assigns an importance measure to
the critical points of a scalar-valued function. The measure is based
on a certain pairing of critical points. Loosely speaking, this pairing
is defined by the changes of the topology of the sublevel sets of
the function. The persistence of a critical point is then given by the
difference between the function values of the point and its assigned
neighbor. Due to its noise robustness persistence has become popular
in data analysis. One important property of persistence is its invariance
with respect to deformations of the domain. While this can be a useful
property for certain applications it also implies that this importance
measure does not take into account the spatial extent of a critical point
(see Figure 8b). It is thereby extremely sensitive to critical points induced
by outliers in the data (see Figure 9d).
Another importance measure for critical points, based on the deep
structure of the scale space, was presented by Lindeberg [28]. Scale
space is a well-known concept in the area of computer vision. It is a
one-parameter family of images, obtained by cumulative smoothing of
the initial function. Considering the scale space as a time-dependent
function, one can track the critical points of the initial function through
multiple scales. Since every function turns into a constant function
with increasing scale, all critical points disappear eventually. Lindeberg
therefore defined the importance of a critical point by its life time
• Jan Reininghaus, David Günther, Jens Kasten and Ingrid Hotz are with
Zuse Institute Berlin, Germany, E-mail: {reininghaus, guenther, kasten,
hotz}@zib.de.
• Natallia Kotava and Hans Hagen are with the University of
Kaiserslautern, Germany, E-mail: kotava@rhrk.uni-kl.de,
hagen@informatik.uni-kl.de.
Manuscript received 31 March 2011; accepted 1 August 2011; posted online
23 October 2011; mailed on 14 October 2011.
For information on obtaining reprints of this article, please send
email to: tvcg@computer.org.
in the scale space. It is essential to have a very stable tracking of the
critical points in scale space for this method to work effectively. When
a coarsely sampled data set contains noise the critical lines may be interrupted
which severely affects the importance value of the critical
points (see Figure 9e). There are also data sets whose critical points
have an infinite lifetime (see Figure 7b).
In this paper we propose a new importance measure for critical
points, which builds upon the above two methods. The basic idea
is to accumulate the homological persistence value of a critical point
through its evolution in the scale space. We will show that our importance
measure has the following essential properties:
1. it is able to effectively deal with data containing outliers,
2. it assigns an importance value to each extremal point of the original
input data - no preprocessing is necessary,
3. it contains only the sampling of the scale space as computational
parameter - all other parts of the algorithm are parameter-free,
4. it is applicable to large data sets due to low memory requirements
and practical running times.
We introduce the underlying mathematical notions of our method
in Section 3 and present the method in detail in Section 4. In Section
5 we demonstrate the above mentioned properties on a synthetic and a
real world dataset and conclude the paper in Section 6.
2 RELATED WORK
In this section we give an overview of the previous work related to
our method. In particular, we discuss the research done in the areas of
scale space theory, discrete Morse theory and homological persistence.
Scale Space Theory was first described by Iijima [20], as discovered
by Weickert et al. [36]. Later it was independently proposed by
Witkin [39] and Koenderink [24]. As mentioned in Section 1, the
scale space of a function is a family of images generated by a smoothing
operator. This operator has a rigorous axiomatic basis described
in the early works of Iijima. In R n the unique operator which satisfies
all axioms is a convolution with the Gaussian kernel. This scale
space is usually referred to as linear scale space. By relaxing some
axioms nonlinear scale spaces can be defined [31]. Recently Duits et
al. [12] proved that a one-parameter family of scale spaces proposed by
Pauwels et al. [30] fulfills all basic scale space axioms. This family is
usually referred to as α scale space. Koenderink [24] proposed to investigate
the evolution of critical points of the initial function through
its scale space, called deep structure, see Lindeberg [28] for an extensive
introduction. In contrast to our approach, the deep structure is
usually defined in a continuous setting and is extracted using numerical
methods.

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