BBaby Morse Theory in Data Analysis
Caren Marzban , ∗ and Ulvi Yurtsever , Applied Physics Laboratory, and Department of Statistics,University of Washington, Seattle, WA 98195, MathSense Analytics, 1273 Sunny Oaks Circle, Altadena, CA 91001 Hearne Institute for Theoretical Physics,Louisiana State University, Baton Rouge, LA 70803 ∗ Corresponding Author: [email protected] a r X i v : . [ s t a t . O T ] M a y bstract A methodology is proposed for inferring the topology underlying point clouddata. The approach employs basic elements of Morse Theory, and is capableof producing not only a point estimate of various topological quantities (e.g.,genus), but it can also assesses their sampling uncertainty in a probabilisticfashion. Several examples of point cloud data in three dimensions are utilizedto demonstrate how the method yields interval estimates for the topology ofthe data as a 2-dimensional surface embedded in R .Key Words: Topology, Morse theory, persistence, inference, signal detection. Introduction
There are many sources of high-dimensional data that are inherently structured butwhere the structure is difficult to conceptualize. The motivation to organize, asso-ciate, and connect multi-dimensional data in order to qualitatively understand itsglobal content has recently led to the development of new tools inspired by topo-logical methods of mathematics [5, 18, 20, 22]. The applications of topological dataanalysis methods include dimensionality reduction [16], computer vision [20], andshape discovery [1]. In most of these applications, the data is point cloud data , i.e.,the coordinates of points in some space. Such data arise naturally in LIDAR (LightDetection and Ranging) [12], image reconstruction [13], and in the geosciences [21].In addition, point-cloud data in multidimensional Euclidean space can arise fromnonlinear transforms of other kinds of data such as time series [10].Consider, for example, a cloud of points in 3-dimensional Euclidean space. Thecloud of points may be confined mostly to the surface of a 2-dimensional sphere; orto the surfaces of multiple disconnected spheres. The number of such spheres is anexample of a topological quantity, in contrast to the specific shape of the spheres (e.g.round vs. squashed) which is a geometrical quantity. Another example of a topologicalquantity is the number of handles; a sphere has none, but the surface of a doughnut hasone. A sphere and a doughnut are topologically distinct surfaces in that one cannotbe transformed to the other without cutting and gluing operations. The number ofhandles, known as genus , is important because it turns out any 2-dimensional compact3urface can be constructed by gluing handles onto a sphere [15]. Said differently, thegenus is a defining characteristic of the topology of a 2-dimensional compact surface.As a final example, compare the surface of a ball with that of a coffee mug; the formerhas genus 0, while the latter has genus 1. This type of topological information canbe useful in correctly identifying underlying structures in point cloud data.Whereas the human eye is capable of inferring such structures, one often requiresa method for performing that task objectively. For instance, the high dimensionalityof the data may not allow visualization in 3 dimensions. Even in 3 dimensions, itmay be that the topological structure must be inferred in a streaming environment,where a human operator cannot visually inspect every situation one at a time. Fi-nally, there may be situations wherein the existence of an underlying structure is notunambiguously evident even to a human expert. In such a situation, an algorithmcapable of assigning probabilities to the various topological structures can be usefulfor decision making [14].Inferring the various disconnected components of any structure can be done viaa class of statistical methods generally known as cluster analysis [9]. Some clusteranalysis methods are also naturally capable of assigning probabilities to the differentnumber of components/clusters. However, such methods are incapable of inferringhigher-order topological structures. For instance, no clustering algorithm can identifythe number of handles (i.e. genus) of a 2-dimensional surface underlying point clouddata. It is that task which is addressed in this paper. The method also producesprobabilities for the various possible genus values.4wo main methodologies of topological data analysis have been discussed so farin the literature: One is based on the idea of persistence, and the other on dis-cretized approaches to Morse Theory. This paper discusses a new approach to discreteMorse Theory, illustrated by analyzing simple examples of point-cloud data in three-dimensional Euclidean space. The alternative methodology, based on persistence,utilizes the idea of capturing topological features in data by analyzing continuousstructures which are associated to data points as a function of a varying scale param-eter that measures, roughly, how coarsely the data points are assumed to sample anunderlying topological manifold.To briefly illustrate the idea of persistence, assume a given data set D consistingof a sampling of a smooth submanifold M ⊂ R n of n -dimensional Euclidean space.Evolution has moulded the human perception system with the ability to reconstructgeometric information from two-dimensional projections; but this capability is onlyuseful in dimension n = 3. For submanifolds embedded in higher-dimensional Eu-clidean space R n ( n ≥ M ⊂ R n cannot be readout from visual inspection of two-dimensional projections. One topological invariantthat is immensely useful in ascertaining the “global configuration” of the surface M ,and therefore the true global nature of the “model” M of the given data D , is thesequence of homology groups H k ( M ), further described below. As mentioned previ-ously, some statistical methodologies like clustering can be viewed as methods for theextraction of homological information. Persistence is a general method to extract ho-mology information from a given data set. The natural question persistence attempts5o answer is: how can one compute H k ( M ) from only the knowledge of the discretesample of points D ⊂ M ? The strategy persistence uses to answer this question isthe following: Fix a distance parameter (cid:15) >
0, and build a simplicial complex C (cid:15) byjoining a m -simplex whenever m + 1 data points in D are mutually within distance (cid:15) of each other. In this way, when (cid:15) is sufficiently small (but not too small) and D ⊂ M is a sufficiently dense sampling of the submanifold M , the complex C (cid:15) is guaranteedto have the same homotopy type (and thus the same sequence of homology groups) as M . For a given fixed data set D , however, if (cid:15) is chosen smaller than a threshold value C (cid:15) becomes a discrete set (with uninteresting topology), and, similarly, if (cid:15) is largerthan some other threshold value C (cid:15) becomes a single giant simplex whose topologygives no information about M . The true topology of M is reflected in the simplex C (cid:15) only when (cid:15) ranges in an “optimal” interval between these two thresholds. Theidea of persistence is to inspect the variation in the topology of C (cid:15) as (cid:15) varies, andidentify the largest interval in which the topology is “persistent” as a function of (cid:15) .This persistent topology is then the statistical estimate of the “true” topology of thedata D .While persistence relies on sophisticated constructions derived from algebraictopology, Morse Theory supplies the set of tools for an alternative approach to topo-logical data analysis [4, 6, 7]. The latter provides a framework conducive to statisticalanalysis, because a probabilistic estimate of the topology follows naturally. There ex-ists a large body of knowledge on the applications of Morse Theory [17]. Althoughsome of these works are quite complex and sophisticated, to the knowledge of the6resent authors, some of the simpler applications have not appeared in data analysiscircles. In this paper, a few synthetic examples of 3-dimensional point cloud data areutilized to illustrate these simple applications of Morse Theory.Morse Theory, in its simplest form, can be thought of as a set of topologicalconstraints which must be satisfied by a surface, if/when some function on the surfaceis known. For example, consider a circle (i.e., a 1-dimensional, compact surface) in 3dimensions, oriented along the conventional z-axis. Also, consider the height functionon such a circle; it is a function defined on the circle which produces the height ofevery point on the circle from the x-y plain. Such a function has two critical points,at the bottom and at the top of the circle, where its derivative is zero. As shownin the next section, these critical points of the height function restrict the topologyof the surface over which the function is defined. More specifically, it is shown herethat by computing the height function and its critical points for point cloud data,Morse Theory allows one to infer the genus of the underlying surface. Moreover,resampling [8, 11] is employed to compute the empirical sampling distribution of thegenus, which in turn allows for a probabilistic assessment of topology.The contributions of this work are twofold. First, it is shown that Morse Theorycan be employed to infer the topology of the manifold underlying point cloud data. Inthe examples, which are point clouds in R , the manifolds are 2-dimensional surfacesand their topology is uniquely set by one integer: the genus. Second, we point out thatthe genus (and more generally, all algebraic topological invariants of the data) mustbe treated as a random variable when inferred from data. A resampling method is7mployed to compute the empirical sampling distribution of the genus, which in turn,conveys its sampling variability. As such, one can predict the underlying topology ina probabilistic fashion. To demonstrate the methodology, four compact surfaces are considered (Figure 1).The choice of these specific examples is based on the desire to have nontrivial, realistic,topology, but also sufficiently simple topology to allow for a lucid presentation. Thetop/right panel in Figure 1 is topologically a sphere. However, two “dimples” areintroduced in order to generate more critical points for the height function, renderingthe problem less trivial. The top/left panel shows the next nontrivial example, namelya torus. These two surfaces have genus 0 and 1, respectively. The next example(lower/left panel) is a genus 1 surface, but with “dimples,” again for the purpose ofhaving a more complex height function. The final example (bottom/left) is a 2-torus,i.e., a genus 2 surface. Recall that the goal is to infer these values of genus from data.The particular embeddings/shapes of the surfaces shown in Figure 1 are em-ployed in the remainder of the article. Other embeddings/orientations lead to differ-ent height functions; alternatively, functions other than the height coordinate can beused to assess the topology. The discussion section addresses the effect of changing8he embedding for the purpose of obtaining more precise (less variable) estimates ofthe genus.Point cloud data are simulated by adding a zero-mean random Gaussian variableto the height function of the four surfaces. The variance of this variable controlsthe level of noise in the data. Naturally, small values generally lead to accurateand precise estimates of genus. Said differently, the inferred value of genus is thecorrect one, and the uncertainty of the estimate is small. Although larger values ofthe variance are associated with less precise estimates of genus, for sufficiently largevalues the estimates become inaccurate as well, in the sense that the most likely genusinferred from data is the wrong genus altogether. An analysis of the sensitivity ofthe method to noise level is sufficiently complex to be relegated to a separate article(reported later). The complexity of that analysis arises because the effect of noiselevel is confounded with the relative size of the various loops around the handles. Forexample, even with low noise levels, if one of the tori in the 2-torus is much smallerthan the other, then the method is likely to imply that the underlying surface hasgenus one. For the present work, suffice it to say that the standard deviation of thenoise is fixed at 0.1. Loosely speaking, given that the radius of the small loop in thetorus example is 4 (grid lengths), a standard deviation of 0.1 corresponds to a signalto noise ratio of about 40.As shown in the next section, Morse Theory can place bounds on the genus ofa surface from knowledge of the critical points of a function defined on the surface.Specifically, what is required is the number of minima, saddle points, and maxima.9here exist numerous standard methods for finding critical points of a function, butin this article a relatively simple approach is adopted for the sake of clarity.
Although the height function is a standard function on a surface [2, 17], the functionadopted in this article is the area of the surface up to some height h , denoted S ( h ).The area function is closely related to the height function, but is more natural whendealing with data. First, the height function for data is more noisy than the areafunction, because the latter is inherently an integral. Second, critical points of theheight function correspond to points in S ( h ) where the derivative S (cid:48) ( h ) is discontinu-ous. The more robust nature, and the presence of “kinks” in the area function makeit a natural choice to use for identifying the critical points in the height function.Given that S ( h ) is computed from data, it is a random variable. In other words,every realization of the Gaussian about the surface will lead to a different value. Inorder to assess the variability of S ( h ) resampling is employed [8, 11]. Specifically, 100samples/realizations are drawn and the distribution of S ( h ), at prespecified values of h is generated. Each distribution is summarized with a boxplot and displayed for all h values as a means of displaying the functional dependence of S ( h ), as well as itsvariability, on h .Note that each sample/realization of data gives rise to a sequence of S ( h ) valuesat prespecified h values. As such, S ( h ) can be considered a stochastic time series.10dditionally, it is a monotonic, non-decreasing time series. This monotonic nature ofthe time series makes it difficult to identify its kinks (i.e., critical points of the heightfunction). A more useful quantity is the first derivative of S ( h ) with respect to h .Second derivatives are also useful, but here only the time series of the first derivatives, S (cid:48) ( h ), is examined. It is the critical points of the S (cid:48) ( h ) time series which are used inMorse Theory to infer topology. The sampling variability of S (cid:48) ( h ) is again assessedvia resampling, and displayed with boxplots.Figure 2 shows the above ideas for the specific example of a dimpled sphere. Thetop/left panel shows a vertical cross-section of the surface. Here the h values varyfrom the global minimum of the surface to its global maximum, in increments of 0.5.The data simulated about this surface are not shown, but boxplots summarizing thedistribution of the S ( h ) are shown in the top/right panel. Although the boxplots arerelatively small, and difficult to see, their medians are connected by a red line as avisual aid. Also difficult to see are the “kinks” in the red line at the critical points,marked by the blue horizontal lines. The first derivative (left/lower panel) bettershows both the kinks and the sampling variability. It is evident that some kind of akink exists at each of the critical points of S (cid:48) ( h ) (again, marked by the blue lines).The kinks can be generally classified into three types: an increasing step function, acusp (i.e., ∧ ), and a decreasing step function, respectively corresponding to minima,saddle points, and maxima. The second derivative of S ( h ) is also shown (lower/rightpanel), only to illustrate that it too carries information useful for identifying criticalpoints. However, it is not used in the present work.11he analogous figures for the torus example are shown in Figure 3. Again, it canbe seen that the kinks in the area function (and its derivatives) occur at the locationsof the critical points of S (cid:48) ( h ), and that the shape of the kinks in the first derivativeare of the same type as seen previously, namely step functions, and cusps. Similarresults are found for the dimpled torus and the 2-torus (not shown). Although there exist standard methods for finding critical points of a time series,most rely on some sort of time series modeling. The time series models, in turn,have numerous parameters which must be determined. Although there exist criteria(e.g., maximum likelihood) for estimating the best models, for the sake of clarity, avery simple approach is adopted here. The approach is based on template matching.Specifically, three templates are selected corresponding to the aforementioned threekinks observed in the series S (cid:48) ( h ), namely 1) an increasing step function for findinglocal minima in the time series, 2) a cusp function for finding the saddle points, and3) a decreasing step function for identifying local maxima in the series.By sliding each of the templates across the time series for S (cid:48) ( h ), and computing theresiduals, one obtains three additional time series. The left column in Figure 4 showsthese series for one realization of data about the dimpled sphere. The vertical linesare at the h values corresponding to the critical points. Given that these time seriesare of residuals, near-zero values indicate a close agreement between the template12nd the time series of S (cid:48) ( h ). It can be seen that the residuals corresponding to thefirst template (top/left panel in Figure 4) approach zero only at the location of thelocal minima. Similarly, the residuals for the second template (middle/left panel)are near zero only at the location of the saddle points. The final panel shows theresiduals for the last template, and the residuals are near zero only at the locationof the local maxima. To quantify the notion of “near-zero,” the histogram of theresiduals is examined (right column in Figure 4). Specifically, any residual within onestandard deviation of zero is defined to be “near-zero.” This 1-standard-deviationvalue is displayed with the vertical line on the histograms in Figure 4.In short, sliding three templates across the time series of S (cid:48) ( h ), and examiningnear-zero values of the ensuing residuals correctly identifies the locations of the criticalpoints of S ( h ). This method for automatically identifying critical points of the heightfunction for data can be improved upon. And as mentioned previously, there existmore sophisticated methods for identifying critical points. However, that is not themain goal of the present work. The rudimentary method outlined here is sufficientto demonstrate the main goal of the work - that Morse Theory can be employed toestimate the topology underlying data, and to express the statistical uncertainty inthat estimate. 13 Morse Theory
The material presented in this subsection is only a small portion of Morse Theory,and so, has been called Baby Morse Theory [2, 3].Given a surface S , the Poincare polynomial is defined as P ( S ) = (cid:88) k b k t k , where − ≤ t ≤
1, is a quantity with no special meaning, and b k is the k th Bettinumber. For a 2-dimensional surface, k = 0 , ,
2. Intuitively, b is the number ofsimply-connected components of S , b is the number of noncontractable loops onthe surface, and b is the number of noncontractable surfaces. For example, for a2-sphere, P ( S ) = 1 + t , and for a torus, P ( S ) = 1 + 2 t + t . The 2 t term reflects thefact that there are two noncontractable loops on a torus - one around the “hole” ofthe doughnut, and another going around the “handle.” As another example, considera 2-torus for which P ( S ) = 1 + 4 t + t . It is important to point out that P ( S ) is atopological quantity in the sense that any 2-sphere (symmetric, squashed, dimpled,or otherwise) has P ( S ) = 1 + t . The same is true of the other examples considered;their Poincare polynomial is independent of their embedding/shape.Given a function f defined on a surface, the Morse polynomial is defined as M ( f ) = (cid:88) P i t n i ,P i denotes the critical points of f , and n i is the index of f at the i th critical point.The index is defined to be the number of non-decreasing directions for f . Unlike the14oincare polynomial, the Morse function is not a topological quantity. For example,consider the perfectly round 2-sphere. Then the height function has 2 critical points,with indices 0 and 2, corresponding to the South and North poles, respectively. Thisis so, because at the South pole there are no directions in which the height functiondecreases, while there are two such directions at the North pole. Then, for the heightfunction on this sphere one has M ( f ) = 1 + t . By contrast, a 2-sphere with dimplesin it (e.g., top/left panel in Figure 1) has 6 critical points with indices 0, 1, 2, 0, 1,2, respectively, moving up from the bottom of the figure. For this height function, M ( f ) = 2 + 2 t + 2 t . As another example, for the height function on the torus in thetop/right panel of Figure 1, one has M ( f ) = 1 + 2 t + t .Central to Morse Theory are the so-called Morse inequalities [2, 17]. They areexpressed in two forms - “weak” and “strong:” M ( f ) ≥ P ( S ) , M ( f ) − P ( S ) = (1 + t ) Q ( t ) , (1)where Q ( t ) is any polynomial in t with non-negative coefficients.In the above examples, note that for some functions one has M ( f ) = P ( S ).Such functions are called “perfect.” Intuitively, such a function tightly “hugs” thesurface. As such, the coefficients in the corresponding Morse function are equal tothe Betti numbers. As a result, knowledge of a perfect function is tantamount toprecise knowledge of the topology (technically, homology) of the underlying surface.For all non-perfect functions, the Morse inequalities provide only an upper bound onthe Betti numbers, and do not uniquely identify the topology.15he search for perfect functions is aided by the Lacunary principle [2]: If theproduct of all consecutive coefficients in M ( f ) is zero, then f is perfect. Anotheruseful corollary of the strong form of the inequalities follows upon considering t = − (cid:88) P i ( − n i = (cid:88) k b k ( − k . (2)This places a constraint on the allowed number of minima, saddle points, and maxima: n min − n saddle + n max = b − b + b . (3)And since in this article only surfaces with b = b = 1 are considered, then b = 2 − n min + n saddle − n max . (4)Finally, given that any 2-dimensional surface can be constructed by gluing tori toa sphere, it follows that b must be even (including zero). The genus of a compactsurface is then found to be genus = b / . (5)For a sphere, a torus, and a 2-torus (e.g. in Figure 1), the genus is 0, 1, and 2,respectively. Intuitively, the genus counts the number of “holes” or “handles” in acompact surface. Inferring the genus is the main goal of the present work. Armed with a method to find the number of minima, saddle points, and maxima ofthe height function (section 2.3), one can then examine the distribution of each. The16op/left panel in Figure 5 shows the boxplots summarizing the three distributionsfor the dimpled sphere example. Recall that for this example, the correct number ofminima, saddle points, and maxima is 2. The median of the three boxplots is preciselyat 2, as well. The 1st and 3rd quartiles of the distribution (i.e., the bottom and topsides of the boxes) suggest an uncertainty of about ± The Morse inequalities are reviewed. It is shown that when specialized to the caseof a 2-dimensional surface embedded in 3 dimensions, they place severe constraintson the topology of the surface. Three examples are employed to show that all ofthe quantities appearing in the Morse inequalities can be estimated from point clouddata, thereby providing a statistical/probabilistic view of the topology of the surfaceunderlying the data. Empirical sampling distributions are produced for the varioustopological entities, all of which can then lead to traditional confidence intervals orhypothesis tests of the topological parameter of interest. Throughout the paper,an attempt is made to avoid complex mathematics (e.g., algebraic and differentialtopology, and homology), with the hope that the utility of Morse Theory in dataanalysis may be appreciated by a wider readership.As mentioned above, the sensitivity of the inferred quantities to noise level hasnot been examined here. The main reason is that the noise level and the physical size18f the structures underlying the point cloud data are confounded. This complicationis not unsurmountable; it simply calls for a more careful analysis wherein the size ofthe noise and the typical features in the data are both varied/controlled.The typical size of the features in the data also affects the uncertainty of theinferred topology. The empirical sampling distribution of the genus spreads out whenthe topological features are small relative to noise level. Although not shown here,we have found that this uncertainty depends on the orientation of the surface. Thisis expected, because the height function depends on the orientation. So, it is possibleto orient the surface in a way that would allow for more precise estimation of thecritical points. In other words, it is possible to add another step to the proposedmethod, wherein the variance of the distribution of genus is minimized across differ-ent orientations of the point cloud. Such a rotation can also be used to identify aperfect height function, in which case the Betti numbers can be computed precisely,as opposed to being only bounded at the top. This idea will be examined at a latertime.In the examples considered here the goal is to identify the genus of a 2-dimensionalcompact surface underlying 3-dimensional point cloud data. Several generalizationare possible. The dimensionality of the embedding space, or of the “surface” (embody-ing the underlying structure), can both be generalized. Of course, a single numberlike genus will no longer suffice to define the topology uniquely, but the set of Bettinumbers does. In other words, if the manifold of interest underlying the data hasdimension larger than 2, then more parameters need to be estimated. From a statis-19ical point of view, the consequence of this increase in the number of parameters isthat more data will be required to estimate the parameters with precision.The general formulation of Morse Theory does not require the underlying manifoldto be compact. There are also extensions of Morse Theory that allow for degener-ate critical points, as well as extensions to manifolds with boundary, and to Morsefunctions that take values in more general spaces than R (e.g., circle-valued MorseTheory where Morse functions are S -valued) [19]. The application of these morepowerful topological tools to data analysis is a fruitful frontier for exploration. Acknowledgements
Marina Meilˇa is acknowledged for valuable discussion. support for this project wasprovided by the National Geospatial-Intelligence Agency, award number HM1582-06-1-2035. [1] A. Adan, C. Cerrada, V. Feliu, Modeling Wave Set: Definition and Applicationof a New Topological Organization for 3D Object Modeling, Computer Visionand Image Understanding, 79 (2000) 281-307.[2] R. Bott, Morse theoretic aspects of Yang-Mills Theory, in Recent Developments20n Gauge Theories, Eds. G. ’tHooft, et al., Plenum Publishing, 1980, pp. 46-67.[3] R. Bott, Lectures on Morse Theory, Old and New, Bull. Amer. Math. Soc. (N.S.),7 (1982) 331-358.[4] P-T. Bremer, V. Pascucci, A Practical Approach to Two-Dimensional ScalarTopology, in Topology-based Methods in Visualization, H. Hauser, H. Hagen, H.Theisel (Eds), Springer-Verlag, Berlin, 2007. ISBN-13 978-3-540-70822-3.[5] G. Carlsson, Topology and Data, Bull. (New Series) Amer. Math. Soc., 46 (2009)255308.[6] F. Cazals, F. Chazal, T. Lewiner, Molecular Shape Analysis Based upon theMorse-Smale Complex and the Connolly Function. Proc. 19th ACM Symp. Com-putational Geometry (SoCG), 2003, pp. 351-360.[7] M. Connolly, Molecular Surfaces: A Review, Network Science, (1996) .[8] B. Efron, R. J. Tibshirani, An introduction to the bootstrap, Chapman & Hall,London, 1993.[9] B. S. Everitt, Cluster Analysis, Second Edition, Heinemann Educational Books.London, 1980.[10] R. Gilmore, M. Lefranc, The Topology of Chaos, Wiley-Interscience, New York,2002. ISBN 0-47 1-40816-6[11] P. I. Good, Introduction to Statistics Through Resampling Methods and R/S-PLUS, Wiley, 2005. ISBN 0-471-71575-12112] D. Grejner-Brzezinska, C. Toth, Deriving Vehicle Topology and Attribute Infor-mation Over Transportation Corridors From LIDAR Data. Proceedings of the59th Annual Meeting of The Institute of Navigation and CIGTF 22nd GuidanceTest Symposium, June 23 - 25, Albuquerque, NM., 2003, pp. 404-410.[13] M. R. Hajihashemi, M. El-Shenawee, Shape Reconstruction Using the Level SetMethod for Microwave Applications, Antennas and Wireless Propagation Letters,7 (2008) 92-96.[14] R. W. Katz, A.H. Murphy, Economic Value of Weather and Climate Forecasts,Cambridge University Press, Cambridge, 1997.[15] J. M. Lee, Introduction to Topological Manifolds, Springer-Verlag, New York,2000. ISBN 0-387-98759-2[16] J. A. Lee, M. Verleysen, Nonlinear dimensionality reduction, Information Scienceand Statistics, Springer, 2007.[17] L. I. Nicolaescu, An Invitation to Morse Theory, Springer Monograph, XIV,2010. ISBN 978-0-387-49509-5.[18] P. Niyogi, S. Smale, S. Weinberger, Finding the homology of submanifolds withhigh confidence from random samples, Combinatorial and Discrete Geometry, 39(2008) 419-441.[19] A. V. Pajitnov, Circle-valued Morse Theory, De Gruyter Studies in Mathematics32, 2006. ISBN 978-3-11-015807-6. 2220] V. Pascucci, X. Tricoche, H. Hagen, J. Tierny, Topological Methods in DataAnalysis and Visualization; Theory, Algorithms, and Applications. Springer,2010.[21] T. F. Wawrzyniec, L. D. McFadden, A. Ellwein, G. Meyer, L. Scuderi, J.McAuliffe, P. Fawcett, Chronotopographic analysis directly from point-clouddata: A method for detecting small, seasonal hillslope change, Black Mesa Es-carpment, NE Arizona, Geosphere, 3, (2007) 550-567. DOI: 10.1130/GES00110.1[22] A. Zomorodian, Topology for Computing, Cambridge Monographs on Appliedand Computational Mathematics (No. 16), 2005.23igure 1. Four example surfaces: The “dimpled sphere” in the top/left panel istopologically a sphere (genus = 0), in spite of the dimples. The top/right panel showsa surface with less trivial topology, namely a torus (genus = 1). Another example ofa genus-1 surface is shown in the lower/left panel; it is a skinny torus, with “dimples”at the top and the bottom. The last example is a 2-torus (genus = 2) shown in thelower/right panel. 24igure 2. a) A vertical cross-section of the dimpled sphere shown in Figure 1.The blue lines mark the height of the critical points. b) The dependence of the areafunction S ( h ) on the height h shown along the y axis. The blue horizontal lines markthe height of the critical points. c) The first derivative of S ( h ) with respect to h , andthe second derivative in panel d). 25igure 3. Same as Figure 2, but for the torus.26igure 4. Left column: The time series generated by sliding three template acrossthe time series of S (cid:48) ( h ) and computing a measure of the error/residual between thetime series and each template. Right column: The histogram of the three templateerrors. From top to bottom, the templates are the increasing setp function, the cusp,and the decreasing step function. 27) and computing a measure of the error/residual between thetime series and each template. Right column: The histogram of the three templateerrors. From top to bottom, the templates are the increasing setp function, the cusp,and the decreasing step function. 27