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

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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.

A <strong>Scale</strong> <strong>Space</strong> <strong>Based</strong> <strong>Persistence</strong> <strong>Measure</strong><br />

<strong>for</strong> <strong>Critical</strong> <strong>Po<strong>in</strong>ts</strong> <strong>in</strong> <strong>2D</strong> Scalar Fields<br />

Jan Re<strong>in</strong><strong>in</strong>ghaus, Natallia Kotava, David Günther, Jens Kasten,<br />

Hans Hagen, Senior Member, IEEE, and Ingrid Hotz<br />

Abstract— This paper <strong>in</strong>troduces a novel importance measure <strong>for</strong> critical po<strong>in</strong>ts <strong>in</strong> <strong>2D</strong> scalar fields. This measure is based on a<br />

comb<strong>in</strong>ation of the deep structure of the scale space with the well-known concept of homological persistence. We enhance the noise<br />

robust persistence measure by implicitly tak<strong>in</strong>g the hill-, ridge- and outlier-like spatial extent of maxima and m<strong>in</strong>ima <strong>in</strong>to account. This<br />

allows <strong>for</strong> the dist<strong>in</strong>ction between different types of extrema based on their persistence at multiple scales. Our importance measure<br />

can be computed efficiently <strong>in</strong> an out-of-core sett<strong>in</strong>g. To demonstrate the practical relevance of our method we apply it to a synthetic<br />

and a real-world data set and evaluate its per<strong>for</strong>mance and scalability.<br />

Index Terms—<strong>Scale</strong> space, persistence, discrete Morse theory<br />

1 INTRODUCTION<br />

Computer assisted analysis of two-dimensional scalar data has become<br />

an essential tool <strong>in</strong> scientific research. To deal with the grow<strong>in</strong>g<br />

amount of data, feature extraction methods are frequently employed<br />

<strong>in</strong> applications like medical imag<strong>in</strong>g, geosciences and computational<br />

fluid dynamics. The features can often be described by the extremal<br />

po<strong>in</strong>ts of the scalar field or its derived quantities. Due to the <strong>in</strong>tr<strong>in</strong>sic<br />

uncerta<strong>in</strong>ty <strong>in</strong> measurements and the f<strong>in</strong>ite precision of numerical<br />

simulations these k<strong>in</strong>ds of data usually conta<strong>in</strong> noise. This noise <strong>in</strong>troduces<br />

a lot of spurious local extrema, which complicates automatic<br />

analysis. One is there<strong>for</strong>e <strong>in</strong>terested <strong>in</strong> methods that allow to discrim<strong>in</strong>ate<br />

dom<strong>in</strong>ant from spurious critical po<strong>in</strong>ts.<br />

One such method is homological persistence, <strong>in</strong>troduced by Edelsbrunner<br />

et al. [14]. This method assigns an importance measure to<br />

the critical po<strong>in</strong>ts of a scalar-valued function. The measure is based<br />

on a certa<strong>in</strong> pair<strong>in</strong>g of critical po<strong>in</strong>ts. Loosely speak<strong>in</strong>g, this pair<strong>in</strong>g<br />

is def<strong>in</strong>ed by the changes of the topology of the sublevel sets of<br />

the function. The persistence of a critical po<strong>in</strong>t is then given by the<br />

difference between the function values of the po<strong>in</strong>t and its assigned<br />

neighbor. Due to its noise robustness persistence has become popular<br />

<strong>in</strong> data analysis. One important property of persistence is its <strong>in</strong>variance<br />

with respect to de<strong>for</strong>mations of the doma<strong>in</strong>. While this can be a useful<br />

property <strong>for</strong> certa<strong>in</strong> applications it also implies that this importance<br />

measure does not take <strong>in</strong>to account the spatial extent of a critical po<strong>in</strong>t<br />

(see Figure 8b). It is thereby extremely sensitive to critical po<strong>in</strong>ts <strong>in</strong>duced<br />

by outliers <strong>in</strong> the data (see Figure 9d).<br />

Another importance measure <strong>for</strong> critical po<strong>in</strong>ts, based on the deep<br />

structure of the scale space, was presented by L<strong>in</strong>deberg [28]. <strong>Scale</strong><br />

space is a well-known concept <strong>in</strong> the area of computer vision. It is a<br />

one-parameter family of images, obta<strong>in</strong>ed by cumulative smooth<strong>in</strong>g of<br />

the <strong>in</strong>itial function. Consider<strong>in</strong>g the scale space as a time-dependent<br />

function, one can track the critical po<strong>in</strong>ts of the <strong>in</strong>itial function through<br />

multiple scales. S<strong>in</strong>ce every function turns <strong>in</strong>to a constant function<br />

with <strong>in</strong>creas<strong>in</strong>g scale, all critical po<strong>in</strong>ts disappear eventually. L<strong>in</strong>deberg<br />

there<strong>for</strong>e def<strong>in</strong>ed the importance of a critical po<strong>in</strong>t by its life time<br />

• Jan Re<strong>in</strong><strong>in</strong>ghaus, David Günther, Jens Kasten and Ingrid Hotz are with<br />

Zuse Institute Berl<strong>in</strong>, Germany, E-mail: {re<strong>in</strong><strong>in</strong>ghaus, guenther, kasten,<br />

hotz}@zib.de.<br />

• Natallia Kotava and Hans Hagen are with the University of<br />

Kaiserslautern, Germany, E-mail: kotava@rhrk.uni-kl.de,<br />

hagen@<strong>in</strong><strong>for</strong>matik.uni-kl.de.<br />

Manuscript received 31 March 2011; accepted 1 August 2011; posted onl<strong>in</strong>e<br />

23 October 2011; mailed on 14 October 2011.<br />

For <strong>in</strong><strong>for</strong>mation on obta<strong>in</strong><strong>in</strong>g repr<strong>in</strong>ts of this article, please send<br />

email to: tvcg@computer.org.<br />

<strong>in</strong> the scale space. It is essential to have a very stable track<strong>in</strong>g of the<br />

critical po<strong>in</strong>ts <strong>in</strong> scale space <strong>for</strong> this method to work effectively. When<br />

a coarsely sampled data set conta<strong>in</strong>s noise the critical l<strong>in</strong>es may be <strong>in</strong>terrupted<br />

which severely affects the importance value of the critical<br />

po<strong>in</strong>ts (see Figure 9e). There are also data sets whose critical po<strong>in</strong>ts<br />

have an <strong>in</strong>f<strong>in</strong>ite lifetime (see Figure 7b).<br />

In this paper we propose a new importance measure <strong>for</strong> critical<br />

po<strong>in</strong>ts, which builds upon the above two methods. The basic idea<br />

is to accumulate the homological persistence value of a critical po<strong>in</strong>t<br />

through its evolution <strong>in</strong> the scale space. We will show that our importance<br />

measure has the follow<strong>in</strong>g essential properties:<br />

1. it is able to effectively deal with data conta<strong>in</strong><strong>in</strong>g outliers,<br />

2. it assigns an importance value to each extremal po<strong>in</strong>t of the orig<strong>in</strong>al<br />

<strong>in</strong>put data - no preprocess<strong>in</strong>g is necessary,<br />

3. it conta<strong>in</strong>s only the sampl<strong>in</strong>g of the scale space as computational<br />

parameter - all other parts of the algorithm are parameter-free,<br />

4. it is applicable to large data sets due to low memory requirements<br />

and practical runn<strong>in</strong>g times.<br />

We <strong>in</strong>troduce the underly<strong>in</strong>g mathematical notions of our method<br />

<strong>in</strong> Section 3 and present the method <strong>in</strong> detail <strong>in</strong> Section 4. In Section<br />

5 we demonstrate the above mentioned properties on a synthetic and a<br />

real world dataset and conclude the paper <strong>in</strong> Section 6.<br />

2 RELATED WORK<br />

In this section we give an overview of the previous work related to<br />

our method. In particular, we discuss the research done <strong>in</strong> the areas of<br />

scale space theory, discrete Morse theory and homological persistence.<br />

<strong>Scale</strong> <strong>Space</strong> Theory was first described by Iijima [20], as discovered<br />

by Weickert et al. [36]. Later it was <strong>in</strong>dependently proposed by<br />

Witk<strong>in</strong> [39] and Koender<strong>in</strong>k [24]. As mentioned <strong>in</strong> Section 1, the<br />

scale space of a function is a family of images generated by a smooth<strong>in</strong>g<br />

operator. This operator has a rigorous axiomatic basis described<br />

<strong>in</strong> the early works of Iijima. In R n the unique operator which satisfies<br />

all axioms is a convolution with the Gaussian kernel. This scale<br />

space is usually referred to as l<strong>in</strong>ear scale space. By relax<strong>in</strong>g some<br />

axioms nonl<strong>in</strong>ear scale spaces can be def<strong>in</strong>ed [31]. Recently Duits et<br />

al. [12] proved that a one-parameter family of scale spaces proposed by<br />

Pauwels et al. [30] fulfills all basic scale space axioms. This family is<br />

usually referred to as α scale space. Koender<strong>in</strong>k [24] proposed to <strong>in</strong>vestigate<br />

the evolution of critical po<strong>in</strong>ts of the <strong>in</strong>itial function through<br />

its scale space, called deep structure, see L<strong>in</strong>deberg [28] <strong>for</strong> an extensive<br />

<strong>in</strong>troduction. In contrast to our approach, the deep structure is<br />

usually def<strong>in</strong>ed <strong>in</strong> a cont<strong>in</strong>uous sett<strong>in</strong>g and is extracted us<strong>in</strong>g numerical<br />

methods.


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