A Topological Similarity Measure between Multi-Field Data using Multi-Resolution Reeb Spaces
AA Topological Similarity Measure betweenMulti-Field Data using Multi-Resolution ReebSpaces
Tripti Agarwal ∗ Yashwanth Ramamurthi † Amit Chattopadhyay ‡ Abstract
Searching topological similarity between a pair of shapes or data is an importantproblem in data analysis and visualization. The problem of computing similaritymeasures using scalar topology has been studied extensively and proven useful inshape and data matching. Even though multi-field (or multivariate) topology-basedtechniques reveal richer topological features, research on computing similaritymeasures using multi-field topology is still in its infancy. In the current paper,we propose a novel similarity measure between two piecewise-linear multi-fieldsbased on their multi-resolution Reeb spaces - a newly developed data-structurethat captures the topology of a multi-field. Overall, our method consists of twosteps: (i) building a multi-resolution Reeb space corresponding to each of themulti-fields and (ii) proposing a similarity measure for a list of matching pairs (ofnodes), obtained by comparing the multi-resolution Reeb spaces. We demonstratean application of the proposed similarity measure by detecting the nuclear scissionpoint in a time-varying multi-field data from computational physics.
Keywords:
Topological Data Analysis, Multi-Field, Reeb Space, Multi-Resolution,JCN, Similarity Measure
Similarity measures using scalar topology have demonstrated significant applicationsin shape matching, classification of bio-molecular or protein structures, symmetrydetection, periodicity analysis in time-dependant flows, and so on [10, 19, 18, 14].The design of the similarity algorithms are mostly based on scalar topological data-structures, viz. contour tree, Reeb graph, merge tree, Morse-Smale complex, extremumgraph etc. ∗ International Institute of Information Technology, Bangalore, India. [email protected] † International Institute of Information Technology, Bangalore, India. [email protected] ‡ International Institute of Information Technology, Bangalore, India. [email protected] a r X i v : . [ c s . C G ] A ug ince multi-field topology is richer than scalar topology, using multi-field topologyone is expected to design more precise similarity measures for better classification ofshapes and data. However, this requires generalizations of the existing data-structuresfor scalar topology to capture multi-field topology. Towards this, in the current paper,we contribute as follows:• We introduce a novel Multi-resolution Reeb Space (MRS) data-structure to cap-ture the multi-field topology by generalizing the Multi-resolution Reeb Graph(MRG) by Hilaga et al. [10].• Then, we propose a similarity measure between two MRS structures by generaliz-ing the similarity measure for scalar fields by Hilaga et al. and Zhang et al.[10, 19].• Finally, we show the effectiveness of our method by detecting the nuclear scissionpoint from our similarity plots for the time-varying multi-field Fermium-256 atomdataset as in [8].The next section discusses the related works on topological similarity measures.Section 3 provides the necessary background to understand our method. Section 4and Section 5 describe our algorithms for computing an MRS and computing theproposed similarity measure between two MRSs, respectively. Finally, Section 6 showsan application of our method and concludes with a summary. Topological similarity and distance measures between scalar fields have been studiedextensively. Beketayev et al.[3] propose an interleaving distance as a distance betweenmerge trees. Bauer et al.[2] propose a stable functional distortion metric for computingdistance between two Reeb graphs. Saikia et al. [14] develop an extended branchdecomposition graph (eBDG) and identify the repeating topological structure in ascalar data. Thomas et al.[18] propose a multiscale symmetry detection techniqueusing contour clustering. Other work has been done to find similarity between scalarfields by proposing a distance metric between merge trees [3]. Saikia et al.[15] propose ahistogram feature descriptor to differentiate between subtrees of a merge tree. Narayananet al.[11] present a distance measure to compare scalar fields using extremum graphs.Sridharamurthy et al.[16] present an edit distance based method between merge treesfor feature visualization in time-varying scalar field data. Among the multi-resolutiontechniques, the most important ones are by Hilaga et al.[10] - a similarity measurebased on multi-resolution Reeb graphs (MRG) and by Zhang et al.[19] - topologymatching using multi-resolution dual contour trees. The current paper generalizesthese techniques for multi-fields.However, research on computing topological similarity measures for multi-fields isstill at a nascent stage. Recently, Agarwal et al.[1] have proposed a distance metric be-tween two multi-fields based on their fiber-component distributions of and demonstrateits usefulness over scalar-topology. It is worth mentioning few important data-structuresfor capturing multi-field topology. Carr et al.[4] develop a joint contour net (JCN) fora quantized approximation of the Reeb space. Duke et al.[8] successfully apply the2CN to visualize nuclear scission features in multi-field density data. Chattopadhyayet al.[5, 6] propose a hierarchical multi-dimensional Reeb graph (MDRG) structureequivalent to the Reeb space. These data-structures are useful for the development ofthe current algorithm.
In this section, we describe the necessary background to understand the proposedsimilarity measure between MRSs. More precisely, we briefly highlight the importanttools for capturing the scalar and multi-field topology.
Piecewise-Linear Multi-Fields.
Most of the data in scientific visualization comesas a discrete set of real values at every vertex (grid-point) of a mesh in a volumetricdomain. Let us consider the data domain as a compact m -dimensional manifold M and let M be a triangulation (mesh) of M whose vertices contain the data values. Let V ( M ) = { v , v , . . . , v p } be the set of vertices of M . An n -dimensional multi-field datacan be described by a vertex map ˆ f = ( ˆ f , ˆ f , . . . , ˆ f n ) ∶ V ( M ) → R n which maps eachvertex to a n -tuple of scalar values. From this discrete map we define a piecewise-linear(PL) multi-field f = ( f , f , . . . , f n ) ∶ M → R n as f ( x ) = ∑ pi = α i ˆ f ( v i ) where x ∈ σ (asimplex of M ) has a unique convex combination of its vertices that can be expressed as x = ∑ pi = α i v i with α i ≥ ∑ pi = α i =
1. We note, f is continuous and the restrictionof f over each simplex of M is linear. In the current paper, we consider the PL multi-field f ∶ M → R n with m ≥ n ≥ n = f ∶ M → R isa PL scalar field. Reeb Space and Reeb Graph.
Given a PL multi-field f ∶ M → R n and a range value c ∈ R n , the inverse image f − ( c ) = { x ∈ M ∶ f ( x ) = c } is called a fiber and a connectedcomponent of the fiber is called a fiber-component [12, 13]. Each fiber-component is anequivalence class obtained by an equivalence relation ∼ on M : x ∼ y ⇔ x , y ∈ M , f ( x ) = f ( y ) = h and x , y belong to the same connected component of f − ( h ) . This equivalencerelation partitions M into the set of all equivalence classes or fiber-components. Thespace W f formed by the set of equivalence classes along with the topology induced by aquotient map q f ∶ M → W f is called the Reeb space [9]. The quotient map q f maps eachpoint of M to its equivalence class. Geometrically, under some regularity conditions, W f is an n -dimensional polyhedron.In particular, for a PL scalar field f ∶ M → R , the fiber f − ( c ) = { x ∈ M ∶ f ( x ) = c } is known as a level set of the isovalue c ∈ R and a connected component of the level setis called a contour instead of fiber-component. Under some regularity conditions, theReeb space of the scalar field f ∶ M → R is a 1-dimensional CW-complex or a graphstructure, known as the Reeb graph and is denoted by RG f . Therefore, RG f consists ofa set of nodes and arcs , each arc connecting two of the nodes. Each point of the Reebgraph corresponds to a contour. In particular, the nodes of the Reeb graph correspondto the contours passing through the critical points [7] of f and the arcs connecting thenodes represent the contours which pass through the regular points (not critical!) of f .3 ulti-Resolution Reeb Graph. A multi-resolution Reeb graph of a PL scalar field h ∶ M → R , proposed by Hilaga et al. [10], is a data-structure that computes a finiteseries of Reeb graphs at various levels of data resolutions. In practice, each Reeb graphis obtained by subdividing the data range into a set of Q = k , ( k = , , . . . , N − ) levels of resolution by a dyadic subdivision (as shown in Figure 1). The domain M ispartitioned into fat (or quantized) contours accordingly and then the Reeb graph at thatresolution is obtained by constructing the adjacency graph of the fat contours [10]. TheReeb graphs in an MRG satisfy the following properties: 1. a parent-child relationshipis maintained between the nodes of the adjacent Reeb graphs at consecutive levels, 2.by repeating the process of subdivision, when the levels of resolution goes to infinitythe MRG converges to the actual Reeb graph of h and 3. a Reeb graph at a particularresolution contains all the information about the coarser resolution Reeb graphs. Figure1 shows the MRG of the height field of a standing double torus with legs, using 4resolutions. n n1 n n n n n2 n3 n n n6n5n4 n n n n r r r r r r r r r r r r r r r Figure 1: An MRG of the height function h of a standing double torus with legs. Thefigure shows the MRG with four Reeb graphs at four different resolutions - coarser tofiner Reeb graphs are shown from the left to right. Joint Contour Net.
A Joint Contour Net (JCN) of a PL multi-field f = ( f , . . . , f n ) ∶ M → R n , proposed by Carr et al. [4], is a quantized approximation of the Reeb spaceusing a chosen number of quantization levels (or levels of resolution). The data rangeis first quantized or subdivided into Q = q × q × . . . × q n levels of quantization (here,range of f i is quantized into q i levels, i = , , . . . , n ), similar as MRG. Thus the range of f is discretized into a finite set, such as a subset of Z n . In this case, for a quantized rangevalue h ∈ Z n , instead of a fiber we obtain a quantized fiber or joint level set , denotedas ˜ f − ( h ) = { x ∈ M ∶ round ( f ( x )) = h } where the round function is applied on eachcomponent. Quantized fibers are not always connected. A connected component ofa quantized fiber is called a quantized fiber-component or joint contour . The jointcontour net is an adjacency graph of the joint contours or quantized fiber-components.Figure 2(a) shows an example of JCN corresponding to a simulated bivariate dataset. Inour method, to compute a multi-resolution Reeb space corresponding to a multi-field,we compute joint contour nets at different resolutions.4 a) (b) (4,4) (3,4) (2,4) (3,4) (4,4) (0,2) (1,2)(2,1) (1,1) (2,1)(1,3) (2,3)(2,3)(1,2) 4,0)(3,0)(2,0)(3,0)(4,0)(3,1)(2,2)(3,3) (3,3)(2,2)(3,1) f f (c) Figure 2: (a) A PL bivariate data over a 2D mesh, (b) JCN at 4 × Multi Dimensional Reeb Graph.
A Multi Dimensional Reeb Graph (MDRG) of aPL multi-field f ∶ M → R n , proposed by Chattopadhyay et al.[5, 6], is a hierarchicaldecomposition of the Reeb space (or the corresponding joint contour net) into a set ofReeb graphs in different dimensions. In particular, to construct the MDRG for a PLbivariate field f = ( f , f ) ∶ M → R , we first compute the Reeb graph RG f of the field f (in the first dimension). Now each point p ∈ RG f corresponds to a contour C p of f . We restrict function f (second dimension) on C p and define the restricted function ̃ f p = f ∣ C p . Then for the second dimension, we compute Reeb graphs RG ̃ f p for eachof these restricted functions ̃ f p . Thus MDRG of f , denoted by MDRG f , can be definedas: MDRG f = {( p , p ) ∶ p ∈ RG f , p ∈ RG ̃ f p } . The definition can be extendedfor any PL multi-field f ∶ M → R n with m ≥ n >
2. Fig. 2(b) shows an example of aquantized Reeb space or JCN for a PL bivariate field (in Fig. 2(a)) and Fig. 2(c) showsits MDRG.
In this section, we develop a new multi-resolution Reeb space structure that capturesthe topology of a PL multi-field data at different resolutions, similar as MRG of a PLscalar field [10]. The Reeb space at a particular resolution is approximated by the JCN.The idea is to develop a series of JCNs at various resolutions.
Let f = ( f , f , . . . , f n ) ∶ M → R n be a PL multi-field and JCN f ( q ) denote the JCNof f with parameter-vector q = ( q , q , . . . , q n ) , each parameter q i being the levels ofresolution of the component field f i for i ∈ { , , . . . , n } . Thus, using q , we obtain anapproximated Reeb space JCN f ( q ) with Q = q × q × . . . × q n levels of resolution ofthe multi-field f . Now to construct a multi-resolution Reeb space, for simplicity, weconsider a finite sequence { q ( k ) = ( k , k , . . . , k ) ∶ k = , , . . . , N − } of N (≥ ) f . Corresponding to this sequence of resolutions, weobtain a sequence { JCN f ( q ( k ) ) ∶ k = , , . . . , N − } of N approximated Reeb spacesthat defines a Multi-resolution Reeb Space (MRS) of f with N levels of resolution and isdenoted by MRS f , N . Moreover, JCN f ( q ( k ) ) or JCN f , k (in short) for k = , , . . . , N − f , k and JCN f , k + . For example, in Figure 3, node n ofthe JCN in (d) is the parent of the nodes { n , n , n , n , n , n } of the JCN in (e).(P2) Note that when the levels of resolution goes to infinity the JCN graph convergesto the Reeb space of f , i.e. JCN f , k converges to the Reeb space W f as k → ∞ (see [6] for details). In otherwords, we can say the multi-resolution Reeb spaceMRS f , N converges to W f as the levels of resolution N tends to ∞ .(P3) A JCN of a certain resolution in MRS f , N implicitly contains all the informationof the JCNs of coarser resolutions, i.e. JCN f , k contains all the information ofJCN f , , JCN f , , . . . , JCN f , k − . Once a JCN of a certain resolution is constructed,the coarser resolution JCN can be constructed by grouping the adjacent nodes insame coarser range interval, as shown in Figure 3. Consider the parameter vector q ( N − ) = ( N − , N − , . . . , N − ) of (dyadic) levels ofresolutions corresponding to PL multi-field f = ( f , f , . . . , f n ) where N (≥ ) is aninteger. Then the JCN of f , using q ( N − ) , can be constructed using the algorithmdescribed by Carr et al. [4] and is denoted by JCN f ( q ( N − ) ) or JCN f , N − . To constructthe multi-resolution Reeb space MRS f , N , we start with JCN f , N − and construct itscoarser resolution Reeb spaces, sequentially, by merging each pair of consecutiverange intervals and grouping the adjacent nodes in the corresponding intervals. Thisis similar to the construction of MRG by Hilaga et al.[10]. Algorithm 1 outlinesthe method for constructing a coarser resolution Reeb space JCN f , k − from the finerresolution Reeb space JCN f , k . Let R f = [ f min1 , f max1 ] × [ f min2 , f max2 ] × . . . × [ f min n , f max n ] be the n -dimensional range interval of the multi-field f where f min i = min x ∈ M f i ( x ) and f max i = max x ∈ M f i ( x ) for i = , , . . . , n . The range [ f min i , f max i ] of the component field f i (for i = , , . . . , n ) is subdivided into q = k ( k ∈ N ∶ set of natural numbers) dyadiclevels of resolution or sub-intervals: r ( i ) = [ x ( i ) , x ( i ) ) , r ( i ) = [ x ( i ) , x ( i ) ) , . . . , r ( i ) q − =[ x ( i ) q − , x ( i ) q ] , where x ( i ) = f min i and x ( i ) q = f max i ( i = , , . . . , n ). Thus the range R f issubdivided into Q = q × q × . . . × q ( n times) dyadic levels of resolution or sub-intervals,denoted by r i i ... i n = r ( ) i × r ( ) i × . . . × r ( n ) i n (where i , i , . . . , i n = , , . . . , q − f , k − (where k ≥
1) we merge theadjacent levels of JCN f , k (in pairs) and obtain the coarser sub-intervals as R i i ... i n = merge ( r ( ) i , r ( ) i + ) × merge ( r ( ) i , r ( ) i + ) × . . . × merge ( r ( n ) i n , r ( n ) i n + ) (for i , i , . . . , i n = , , . . . , q − f , k − reduces to ˜ Q = q × q × . . . × q .6 /0 2/02/0 1/0 2/11/11/1 0/10/2 1/21/22/2 2/23/32/31/32/3 3/02/13/3 n n n n (4,4) (4,4)(4,0) (4,0) f f (4,4) (4,4)(2,4)(4,0) (4,0)(2,0)(2,2) (2,2)(0,2) f f (4,4) (3,4) (2,4) (3,4) (4,4) (0,2) (1,2)(2,1) (1,1) (2,1)(1,3) (2,3)(2,3)(1,2) 4,0)(3,0)(2,0)(3,0)(4,0)(3,1)(2,2)(3,3) (3,3)(2,2)(3,1) f f (4,0) (a) (b) (c)(f)(e)(d) n n n n n n n n n Figure 3: Multi-resolution Reeb Space corresponding to a PL bivariate data: (ring,height). (a) Each component field is quantized into one level, and the correspondingJCN in (d) consists of only one node n . (b) Each component field is quantized into twolevels and the corresponding JCN is shown is (e). (c) Each component field is quantizedinto four levels, and the corresponding JCN is shown in (f). Parent-child relationshipsbetween the nodes of JCNs in consecutive resolutions are shown, e.g. n is parent of { n , n , n , n , n , n } .We construct a Union-Find structure UF [17] from the adjacency of the nodes of JCN f , k with ranges in R i i ... i n . Each connected component of UF becomes a node of the coarserReeb space JCN f , k − and the adjacencies of these new nodes are determined by theadjacencies of the components in JCN f , k . 7 lgorithm 1 CreateCoarserReebSpace
Input:
JCN f , k Output:
JCN f , k − for i = q − do for i = q − do ⋱ for i n = q − do % RANGE MERGING FOR COARSER RESOLUTION R i i ... i n = merge ( r ( ) i , r ( ) i + )× merge ( r ( ) i , r ( ) i + )× . . . × merge ( r ( n ) i n , r ( n ) i n + ) Create Union-Find Structure UF for the nodes of JCN f , k in range R i i ... i n % CREATING NODES OF COARSER JCN for each component C j in UF do Create a node n C j in JCN f , k − Map node-ids and field-values of the finer JCN
Set node n C j as the parent for the nodes in C j end for end for end for ⋰ end for % ADDING EDGES IN COARSER JCN for each edge e e in JCN f , k do if e , e ∈ components C j ≠ C l and f ( e ) ≠ f ( e ) then Add edge e ( n C j , n C l ) in JCN f , k − if not already present end if end for return JCN f , k − Figure 3 illustrates an MRS for a simple bivariate data in a 2D box domain with 3levels of resolution. The construction of the MRS starts with the construction of theReeb space with Q = × levels at the finest resolution. To obtain a quantitative similarity measure between two multi-resolution Reeb spaces,we associate several attributes to the nodes of the MRS that quantify different topologicaland geometrical properties of the multi-field data, similar as in [10, 19]. The attribute setcorresponding to a node n , denoted by ̃ n , is defined as ̃ n = {V( n ) , R( n ) , B ( n ) , D( n )} where V( n ) = Volume ( n ) Total Volume ( M ) is the normalized volume of the node n , R( n ) = measure ( range ( n )) measure ( range ( f )) is the normalized range, B ( n ) is the number of components of the joint level set cor-responding to n and D( n ) is the degree of n in the corresponding JCN.8 A Similarity Measure between MRSs
Let MRS f , N and MRS g , N be two multi-resolution Reeb spaces with same levels ofresolution (here, N ) corresponding to two PL multi-fields f and g , respectively. Ourmethod of computing the similarity measure between two MRSs has two steps: 1. Cre-ating a list of matching pairs from the nodes of the respective MRSs and 2. Computingthe similarity measure between the MRSs by defining a similarity between the nodesof a matched pair. We describe these steps in the following subsections.Figure 4: Label propagation is demonstrated for a matching pair in two MDRGs. For abivariate field, two lists of labels need to be maintained. To create the list of matching pairs between two multi-resolution Reeb spaces, denotedby MPAIR, we search from the coarser to the finer resolution Reeb spaces. Nodes m ∈ MRS f , N and n ∈ MRS g , N form a matching pair ( m , n ) ∈ MPAIR if they satisfy thefollowing matching rules (generalizing the rules in [10, 19]):(i) m ∈ MRS f , N and n ∈ MRS g , N do not belong to any other matched pair,(ii) m and n belong to the Reeb space of same resolution and both have the samerange level,(iii) Parent p ( m ) of m and parent p ( n ) of n must have been matched, i.e. ( p ( m ) , p ( n )) is already a matching pair, except for the nodes in the coarsest resolution,(iv) m and n must be topologically consistent. Unlike computing consistency usingReeb graphs in [10], here we consider the MDRGs of the corresponding Reebspaces. That is, m and n should be in the same branches of the respectiveMDRGs as their matched siblings in all dimensions. In particular, for a bivariatefield, to satisfy the topological consistency in each dimension, two lists of labelsare maintained corresponding to each node. Once two nodes are matched andlabeled, their siblings in the same branch of the MDRG get the same labels (ineach dimension), as shown in Figure 4.9reating the list MPAIR of matching pairs between two MRSs is outlined in Algorithm2. Algorithm 2
CreatingMatchingPairs
Input:
Multi-resolution Reeb spaces MRS f , N , MRS g , N Output:
MPAIR - list of matched pairs for k = , , . . . , N − do Add all the nodes of JCN f , k to a priority queue Q where the priority of a nodeis set as its volume attribute. while Q is not empty do m ← Pop ( Q ) . Search for its best matching pair n ∈ JCN g , k , satisfying the matching rules (i)-(iv). if n is found then MPAIR ← Add ({( m , n ) , k }) end if end while end for return MPAIR
Following Zhang et al.[19], first we define a real-valued similarity function ϕ for eachmatched pair ( m , n ) ∈ MPAIR as: ϕ ( m , n ) = ω ̃ ϕ (V( m ) , V( n )) + ω ̃ ϕ (R( m ) , R( n ))+ (1) ω ̃ ϕ (B ( m ) , B ( n )) + ω ̃ ϕ (D( m ) , D ( n )) where the weights ω i satisfy 0 ≤ ω i ≤ i = , , , ∑ i = ω i = ̃ ϕ ∶ R + × R + → R + is defined as ̃ ϕ ( r , r ) = min ( r , r ) max ( r , r ) . Thus, we have 0 ≤ ϕ ( m , n ) ≤ ϕ ( m , m ) = ϕ ( n , n ) = Φ between two k -th resolution JCNs JCN f , k and JCN g , k , by the weighed sum of the similarities for all pairs ( m i , n i ) ∈ MPAIR with m i ∈ JCN f , k and n i ∈ JCN g , k , as Φ ( JCN f , k , JCN g , k ) = s ∑ i = V( m i ) + V( n i ) ϕ ( m i , n i ) (2)where s is the total number of such pairs. Finally, we define the similarity Φ betweentwo MRSs MRS f , N and MRS g , N as Φ ( MRS f , N , MRS g , N ) = N N − ∑ k = Φ ( JCN f , k , JCN g , k ) (3)We note, 0 ≤ Φ ( MRS f , N , MRS g , N ) ≤ Φ ( MRS f , N , MRS f , N ) = Implementation and Application
We implement our algorithm for computing the similarity between two multi-resolutionReeb spaces under the JCN implementation framework [4]. As an application of ourtool, we consider the time-varying Fermium-256 atom dataset as described by Dukeet al. [8]. The dataset is defined on a 19 × ×
19 sized grid and consists of proton and neutron densities for 40 regularly spaced time-steps. Figure 5 shows the similarityplots by pairwise comparison of the datasets at consecutive time-stamps - for 4 differentresolutions and 4 attributes. From the plots, we see a major topological event at site 26which is the nuclear scission , as described by Duke et al.[8].Figure 5:
Top-four rows:
Similarity plots for time-varying Fermium atom data. Eachrow shows the metric plots using multi-resolution Reeb spaces with 4 different attributes.Each column shows the metric plots using multi-resolution Reeb spaces of 4 differentresolutions.
Bottom-row:
Nucleus is visualized at Sites: 23 −
27, the split happens atSite 26 - corresponding to the ‘lowest’ dip in the plots.11
Conclusion
In this article, we propose a novel Reeb space based method for measuring the topo-logical similarity between two multi-field data. To compute the similarity measure, wedevelop a multi-resolution Reeb space data-structure which converges to the actual Reebspace as the levels of resolution goes to infinity. We show effectiveness of our methodin the application of detecting nuclear scission point in a time-varying multi-field data.
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