A Topological Distance Measure between Multi-Fields for Classification and Analysis of Shapes and Data

Let $\mathbb{M}$ be a triangulated mesh of a compact $d$-manifold $\mathcal{M}$. A multi-field on $\mathbb{M}$ with $r$ component scalar fields is defined as a continuous map $\mathbf{f}=(f_{1},f_{2},...,f_{r}):\mathbb{M}\rightarrow\mathbb{R}^{r}$. For a value $\mathbf{c}\in\mathbb{R}^{r}$, its inverse $\mathbf{f}^{-1}(\mathbf{c})$ is called a fiber. Each connected component of a fiber is a fiber-component 2004-Saeki-Topology-of-Singular-Fibers ; 2014-Saeki-Visualizing-Multivariate-Data . In particular, for a scalar field $f:\mathbb{M}\rightarrow\mathbb{R}$, the inverse of $f$ corresponding to a point in the range is called a level set and each connected component of a level set is called a contour. The Reeb space of $\mathbf{f}$, denoted by $\mathcal{RS}_{\mathbf{f}}$, is the quotient space by contracting each fiber-component to a point 2008-Edelsbrunner-ReebSpaces . In particular, the Reeb space of a scalar field $f:\mathbb{M}\rightarrow\mathbb{R}$ is known as the Reeb graph $\mathcal{RG}_{f}$, which is obtained by contracting each contour to a point and $\tilde{f}:\mathbb{M}\rightarrow\mathcal{RG}_{f}$ is the corresponding quotient map. 2017-Saeki-Theory-of-Singular-Fibers . Carr et al.2014-Carr-JCN proposed the Joint Contour Net (JCN) data-structure, which is a quantized approximation of the Reeb space. For construction of the JCN of $\mathbf{f}$ , the range of $\mathbf{f}$ is subdivided into a finite set of $r$-dimensional intervals. For each interval, the inverse of $\mathbf{f}$ is a quantized fiber and each connected component of a quantized fiber is a quantized fiber-component or joint contour. The JCN is a graph where a node corresponds to a joint contour and an edge between two nodes corresponds to the adjacency of the corresponding joint contours in the domain.

A multi-dimensional Reeb graph (MDRG) of a multi-field $\mathbf{f}=(f_{1},...,f_{r}):\mathbb{M}\rightarrow\mathbb{R}^{r}$, proposed by Chattopadhyay et al.2014-Chattopadhyay-ExtractingJacobiStructures ; 2016-Chattopadhyay-MultivariateTopologySimplification , is the decomposition of the Reeb space (or a JCN) into a series of Reeb graphs in different dimensions. To construct the MDRG of $\mathbf{f}$, the Reeb graph $\mathcal{RG}_{f_{1}}$ of the function $f_{1}$ is computed. Each point $p\in\mathcal{RG}_{f_{1}}$ represents a contour $C_{p}$ of $f_{1}$. For every $p\in\mathcal{RG}_{f_{1}}$ a Reeb graph corresponding to the field $f_{2}$ is computed by restricting $f_{2}$ on $C_{p}$. This procedure is repeated by computing the Reeb graphs for the function $f_{i}$ by restricting $f_{i}$ on the contours corresponding to points in the Reeb graphs of $f_{i-1}$, where $2\leq i\leq r$. In practice, the MDRG is constructed from a quantized approximation of the Reeb Space (JCN). Therefore, similar to the JCN, the MDRG also depends on the subdivision of the range. Each node in a Reeb graph of the MDRG corresponds to a quantized contour and the adjacency between quantized contours is represented by an edge in the Reeb graph. Figure 1 shows the JCN and MDRG corresponding to a bivariate field. In the current paper, we compute persistence diagrams corresponding to Reeb graphs in the MDRG and propose a distance to compare the collection of persistence diagrams of two MDRGs based on the bottleneck distance between Reeb graphs 2014-Bauer-DistanceBetweenReebGraphs .

In this paper, we give a brief introduction to the notion of persistence diagrams and refer the readers to 2010-Edelsbrunner-book ; 2007-Zomorodian-ComputingPersistentHomology for further details on persistent homology.

Let $f$ be a continuous real-valued function defined on $\mathbb{M}$. For a value $a\in\mathbb{R}$, the sublevel set $\mathbb{M}_{\leq a}$ consists of the points in $\mathbb{M}$ with $f$-value less than or equal to $a$, i.e. $\mathbb{M}_{\leq a}=f^{-1}(-\infty,a]$. For $a\leq b$, the $l$-th homology groups of the sublevel sets $\mathbb{M}_{\leq a}$ and $\mathbb{M}_{\leq b}$ are connected by the inclusion map $f^{a,b}_{l}:H_{l}(\mathbb{M}_{\leq a})\rightarrow H_{l}(\mathbb{M}_{\leq b})$. A value $a\in\mathbb{R}$ is a homological critical value of $f$ if $\exists l\in\mathbb{Z}^{*}$ such that $f^{a-\delta,a+\delta}_{l}$ is not an isomorphism for all sufficiently small $\delta>0$, where $\mathbb{Z}^{*}$ is the set of non-negative integers. We assume that $f$ is tame, i.e, the number of homological critical values of $f$ is finite and the homology groups $H_{l}(\mathbb{M}_{\leq a})$ are finite-dimensional $\forall l\in\mathbb{Z}^{*}$. Here, $a$ is any homological critical value of $f$. Let $a_{0}<a_{1}<...<a_{N}$ be the homological critical values of $f$. In this paper, we consider homology with coefficients in $\mathbb{Z}_{2}$, which is the group of integers modulo $2$. Therefore, $H_{l}(\mathbb{M}_{\leq_{a_{i}}})$ is a vector space for $0\leq i\leq N$. We have the following sequence of vector spaces,

where the homomorphisms $f_{l}^{a_{i},a_{i+1}}$ are induced by the inclusions $\mathbb{M}_{\leq a_{i}}\subseteq\mathbb{M}_{\leq a_{i+1}}$. A homology class $\gamma$ is born at $a$ if $\gamma\in H_{l}(\mathbb{M}_{a})$ but $\gamma\notin\text{Im }f_{l}^{a-\delta,a}$ for $0<\delta\leq b-a$. Further, a class $\gamma$ which is born at $a$ dies at $b$ if $f_{l}^{a,b-\delta}(\gamma)\notin\text{Im }f_{l}^{a-\delta,b-\delta}$ for any $\delta>0$, but $f_{l}^{a,b}(\gamma)\in\text{Im }f_{l}^{a-\delta,b}$. Such birth and death events are recorded by persistent homology. The $l$-th ordinary persistence diagram is a multiset of points in $\mathbb{\overline{R}}^{2}$, where $\mathbb{\overline{R}}=\mathbb{R}\cup\{-\infty,+\infty\}$. Each point $(a,b)$ in the $l$th ordinary persistence diagram corresponds to a $l$-homology class which is born at $a$ and dies at $b$. The multiplicity of a point $(a,b)$ with $a\leq b$ is defined in terms of the ranks of the homomorphism $f_{l}^{a,b}$ and the points along the diagonal have infinite multiplicity.

In general, not all homology classes die during the sequence in equation (1). Such homology classes are is said to be essential. An essential homology class of dimension $l$ is denoted by a point $(a_{i},\infty)$ in the $l$th ordinary persistence diagram, where $a_{i}$ is the critical value corresponding to the birth of the homology class. By appending a sequence of relative homology groups to equation (1), we obtain the following sequence:

where $\mathbb{M}_{\geq a_{i}}$ denotes the super-level set of $f$, $\mathbb{M}_{\geq a_{i}}=f^{-1}[a_{i},\infty)$ and $H_{l}(\mathbb{M},\mathbb{M}_{\geq a_{i}})$ is a relative homology group 2010-Edelsbrunner-book . Essential homology classes are created in the ordinary part and are destroyed in the relative part of the sequence in equation (2). The birth and death of essential homology classes are encoded by the extended persistence diagram. An essential homology class of dimension $l$ which is born at $H_{l}(\mathbb{M}_{\leq a_{i}})$ and dies at $H_{l}(\mathbb{M},\mathbb{M}_{\geq a_{j}})$ is denoted by the point $(a_{i},a_{j})$ in the $l$th extended persistence diagram.

The bottleneck distance between persistence diagrams $X$ and $Y$ is defined as follows:

where $\eta$ ranges over bijections between $X$ and $Y$. For each point $(a,b)$ in $X$, we add its nearest diagonal point $(\frac{a+b}{2},\frac{a+b}{2})$ in $Y$ and vice-versa. The number of points in $X$ and $Y$ are now equal. To compute the bottleneck distance, a bijection $\eta:X\rightarrow Y$ is constructed such that $\underset{x\in X}{\max}\|x-\eta(x)\|_{\infty}$ is minimized.

Let $f$ be a real-valued continuous function defined on $\mathbb{M}$. The Reeb graph of $f$, denoted by $\mathcal{RG}_{f}$, is the quotient space of contours of $f$. The function $f$ induces a real-valued function $\bar{f}:\mathcal{RG}_{f}\rightarrow\mathbb{R}$, which takes each point in $p\in\mathcal{RG}_{f}$ to the value of $f$ corresponding to its contour $\tilde{f}^{-1}(p)$ in the domain, $f=\bar{f}\circ\tilde{f}$.

The persistence of topological features of $\mathcal{RG}_{f}$ are encoded in the persistence diagrams $\mathrm{Dg}_{0}(\mathcal{RG}_{f})$, $\mathrm{Dg}_{0}(\mathcal{RG}_{-f})$, $\mathrm{ExDg}_{0}(\mathcal{RG}_{f})$ and $\mathrm{ExDg}_{1}(\mathcal{RG}_{f})$. The $0$-dimensional persistent homological features corresponding to the sub-level set and super-level set filtrations of $f$ are captured in $\mathrm{Dg}_{0}(\mathcal{RG}_{f})$ and $\mathrm{Dg}_{0}(\mathcal{RG}_{-f})$ respectively. $\mathrm{ExDg}_{0}(\mathcal{RG}_{f})$ encodes the range of $f$ and $\mathrm{ExDg}_{1}(\mathcal{RG}_{f})$ captures the $1$-cycles or loops in $\mathcal{RG}_{f}$.

To compute the persistence diagrams of $\mathcal{RG}_{f}$, we require $\mathcal{RG}_{f}$ to be the Reeb graph of a Morse function. If $f$ is a Morse function, i.e. all its critical points are non-degenerate and are at different levels, then the critical nodes of $\mathcal{RG}_{f}$ have distinct values of $\bar{f}$ and belong to one of the following five-types: (i) a minimum (with down-degree $=0$, up-degree $=1$), (ii) a maximum (with up-degree $=0$, down-degree $=1$), (iii) a down-fork (with down-degree $=2$, up-degree $=1$) and (iv) a up-fork (with up-degree $=2$, down-degree $=1$). A regular node has up-degree = $1$ and down-degree $=1$. A down-fork (similarly, up-fork) node is called an essential down-fork node when it contributes to a loop (cycle) of the Reeb graph. Otherwise it is called an ordinary down-fork node. To ensure that $\mathcal{RG}_{f}$ is the Reeb graph of a Morse function, we first eliminate degenerate critical nodes in $\mathcal{RG}_{f}$ by breaking them into non-degenerate critical nodes 2019-Tu-PropogateAndPair . After eliminating degenerate critical nodes, we ensure that the critical nodes of $\bar{f}$ are at different levels. If two critical nodes are at the same level, then the value of one of the nodes is increased/decreased by a small value $\epsilon$. After removing degenerate critical nodes and ensuring that critical nodes are at different levels, $\mathcal{RG}_{f}$ becomes the Reeb graph of a Morse function.

The points in $\mathrm{Dg}_{0}(f)$ are computed by pairing ordinary down-forks with minima, ordinary up-forks with maxima and the global minimum with global maximum. Let $u$ be an ordinary down-fork of $\mathcal{RG}_{f}$. The two lower branches of $u$ correspond to two different components $C_{1}$ and $C_{2}$ in $(\mathcal{RG}_{f})_{\leq\bar{f}(u)}$. Let $x_{1}$ and $x_{2}$ be the global minimum of $C_{1}$ and $C_{2}$ respectively. Let us assume that $\bar{f}(x_{1})<\bar{f}(x_{2})$. Then a $0$-dimensional homology class is born at $x_{2}$ and dies at $u$. $x_{2}$ is paired with $u$ and point $(\bar{f}(x_{2}),\bar{f}(u))$ is created in $\mathrm{Dg}_{0}(\mathcal{RG}_{f})$ (see Figure 2(b)). A symmetric procedure is applied on $\mathcal{RG}_{-f}$ to obtain points in $\mathrm{Dg}_{0}(\mathcal{RG}_{-f})$ by pairing up-forks with maxima. Let $x,y\in\mathcal{RG}_{f}$ correspond to the global minimum and maximum of $\bar{f}$ respectively. We pair $x$ with $y$, giving rise to the point $(\bar{f}(x),\bar{f}(y))$ in $\mathrm{ExDg}_{0}(\mathcal{RG}_{f})$ (see Figure 2(c)).

We are interested in the persistent features which are born and die within the range of $\bar{f}$. In Figure 2(b), the point $(1,\infty)\in\mathrm{Dg}_{0}(\mathcal{RG}_{f})$ corresponds to the unique homology class born at the global minimum of $f$ and persists throughout the sub-level set filtration. However, the point $(1,12)\in\mathrm{ExDg}_{0}(\mathcal{RG}_{f})$ encodes both the minimum and maximum of $\bar{f}$ (see Figure 2(c)). Therefore, $\mathrm{Dg}_{0}(\mathcal{RG}_{f})$ is combined with $\mathrm{ExDg}_{0}(\mathcal{RG}_{f})$ to obtain $\mathrm{PD}_{0}(\mathcal{RG}_{f})$. This persistence diagram consists of the points in $\mathrm{Dg}_{0}(\mathcal{RG}_{f})$ except the point with infinite persistence $(1,\infty)$, instead the point $(1,12)$ in $\mathrm{ExDg}_{0}(\mathcal{RG}_{f})$ is included. Thus $\mathrm{PD}_{0}(\mathcal{RG}_{f}):=\mathrm{Dg}_{0}(\mathcal{RG}_{f})\cup\mathrm{ExDg}_{0}(\mathcal{RG}_{f})\setminus\{(1,\infty)\}$ (see Figure 2(d)).

To compute the points in the $1$st extended persistence diagram $\mathrm{ExDg}_{1}(\mathcal{RG}_{f})$, we pair essential down-forks with essential up-forks. Let $u$ be an essential down-fork of $\mathcal{RG}_{f}$. Of all the cycles born at $u$, let $\gamma$ be a cycle having the largest minimum value of $\bar{f}$. Let $v\in\gamma$ be the node corresponding to the minimum on $\gamma$. It can be seen that $v$ is an essential up-fork 2006-Agarwal-ExtremeElevation . $u$ is paired with $v$ giving rise to the point $(\bar{f}(u),\bar{f}(v))$ in $\mathrm{ExDg}_{1}(f)$ (see Figure 2(e)). In this paper, we compute the persistence diagrams $\mathrm{PD}_{0}$ and $\mathrm{ExDg}_{1}$ of the Reeb graphs in an MDRG.

Next, we provide a brief description of the Laplace Beltrami operator and show its utility in computing descriptors of a shape.

Let $\mathcal{M}$ be a Riemannian $2$-manifold in $\mathbb{R}^{3}$. For a twice-differentiable function $f:\mathcal{M}\rightarrow\mathbb{R}$, the Laplace Beltrami (LB) operator $\Delta_{\mathcal{M}}$ is defined as follows:

where grad $f$ is the gradient of $f$ and div is the divergence on the manifold 1984-chavel-eigenvalues . Let $\psi$ be a diffeormorphism from an open set $U\subset\mathcal{M}$ to $\mathbb{R}^{2}$ with

where the matrix $G$ is a Riemannian tensor on $\mathcal{M}$ and $|G|$ is the determinant of $G$. The LB operator can now be written as follows 1997-Rosenberg-Laplacian :

The Riemannian metric $g$ determines the intrinsic properties of $\mathcal{M}$, which are independent of the embedding of $\mathcal{M}$. Further, since the definition of $g$ is based on inner products which are rotation invariant, $g$ is also invariant to rotations of $\mathcal{M}$.

Let $\mathbf{f}=(f_{1},f_{2}),\,\mathbf{g}=(g_{1},g_{2})$ be two piecewise linear bivariate fields defined on $\mathbb{M}$ and let $\mathbb{MR}_{\mathbf{f}}$ and $\mathbb{MR}_{\mathbf{g}}$ be the corresponding MDRGs. To compute the proposed distance between $\mathbb{MR}_{\mathbf{f}}$ and $\mathbb{MR}_{\mathbf{g}}$, we compute a sum of bottleneck distances between the component Reeb graphs in each dimension of the MDRGs by considering all possible bijections between the component Reeb graphs.

In the first dimension of the MDRGs, there is only one Reeb graph corresponding to each of the fields $f_{1}$ and $g_{1}$, denoted by $\mathcal{RG}_{f_{1}}$ and $\mathcal{RG}_{g_{1}}$, respectively. Let, $Range(f_{1})=[\underset{x\in\mathbb{M}}{\min}f_{1}(x),\underset{x\in\mathbb{M}}{\max}f_{1}(x)]$ and $Range(g_{1})=[\underset{x\in\mathbb{M}}{\min}g_{1}(x),\underset{x\in\mathbb{M}}{\max}g_{1}(x)]$ be the ranges of the functions $f_{1}$ and $g_{1}$. We define $R(f_{1},g_{1})=Range(f_{1})\cup Range(g_{1})$ as the union of the ranges of $f_{1}$ and $g_{1}$. Now for each $c\in R(f_{1},g_{1})$, in the second dimension, the MDRGs $\mathbb{MR}_{\mathbf{f}}$ and $\mathbb{MR}_{\mathbf{g}}$ may have multiple Reeb graphs corresponding to $f_{2}$ and $g_{2}$ restricted on $f_{1}^{-1}(c)$ and $g_{1}^{-1}(c)$, respectively. Let $\mathbb{S}_{f_{1}}^{c}=\{\mathcal{RG}_{\widetilde{f_{2}^{p}}}|\bar{f_{1}}(p)=c\}$ be the set of Reeb graphs from the second dimension of $\mathbb{MR}_{\mathbf{f}}$ and $\mathbb{S}_{g_{1}}^{c}=\{\mathcal{RG}_{\widetilde{g_{2}^{p}}}|\bar{g_{1}}(p)=c\}$ be the set of Reeb graphs from the second dimension of $\mathbb{MR}_{\mathbf{g}}$. We consider all possible bijections $\psi_{c}$ between $\mathbb{S}_{f_{1}}^{c}$ and $\mathbb{S}_{g_{1}}^{c}$. Then we define a distance between $\mathbb{MR}_{\mathbf{f}}$ and $\mathbb{MR}_{\mathbf{g}}$ as follows.

where $\mathrm{Dg}(\mathcal{RG})$ is a persistence diagram of the Reeb graph $\mathcal{RG}$, $\psi_{c}$ ranges over bijections between $\mathbb{S}_{f_{1}}^{c}$ and $\mathbb{S}_{g_{1}}^{c}$. If $Range(f_{1})\cap Range(g_{1})\neq\emptyset$, then $|R(f_{1},g_{1})|$ is the length of the interval $R(f_{1},g_{1})$, otherwise $|R(f_{1},g_{1})|=|Range(f_{1})|+|Range(g_{1})|$.

In the current paper, corresponding to each of the Reeb graph in an MDRG, we construct two persistence diagrams, namely $\mathrm{PD}_{0}$ and $\mathrm{ExDg}_{1}$, as discussed in Section 3.5. We compute the distance between MDRGs defined in equation (LABEL:eqn:bottleneck-distance-mdrgs) based on $\mathrm{PD}_{0}$ and $\mathrm{ExDg}_{1}$, which are denoted by $d_{\mathbb{MR}}^{0}(\mathbb{MR}_{\mathbf{f}},\mathbb{MR}_{\mathbf{g}})$ and $d_{\mathbb{MR}}^{1}(\mathbb{MR}_{\mathbf{f}},\mathbb{MR}_{\mathbf{g}})$, respectively. We consider the proposed distance measure between $\mathbb{MR}_{\mathbf{f}}$ and $\mathbb{MR}_{\mathbf{g}}$ as the weighted sum of $d_{\mathbb{MR}}^{0}\left(\mathbb{MR}_{\mathbf{f}},\mathbb{MR}_{\mathbf{g}}\right)$, $d_{\mathbb{MR}}^{0}\left(\mathbb{MR}_{-\mathbf{f}},\mathbb{MR}_{-\mathbf{g}}\right)$ and $d_{\mathbb{MR}}^{1}\left(\mathbb{MR}_{\mathbf{f}},\mathbb{MR}_{\mathbf{g}}\right)$.

where, $0\leq w_{0},w_{1},w_{2}\leq 1$ and $w_{0}+w_{1}+w_{2}=1$. Here, $d_{\mathbb{MR}}^{0}\left(\mathbb{MR}_{\mathbf{f}},\mathbb{MR}_{\mathbf{g}}\right)$ and $d_{\mathbb{MR}}^{0}\left(\mathbb{MR}_{-\mathbf{f}},\mathbb{MR}_{-\mathbf{g}}\right)$ respectively compute the distance between persistence diagrams of sub-level set and super-level set filtrations of the Reeb graphs in the MDRGs and $d_{\mathbb{MR}}^{1}\left(\mathbb{MR}_{\mathbf{f}},\mathbb{MR}_{\mathbf{g}}\right)$ is the distance between the $1$-st extended persistence diagrams of the Reeb graphs in $\mathbb{MR}_{\mathbf{f}}$ and $\mathbb{MR}_{\mathbf{g}}$ respectively. Next, we discuss how to compute the proposed distance measure for quantized fields.

In this sub-section, we discuss the properties of the proposed distance between MDRGs. Let $\mathbf{f}=(f_{1},f_{2}),\mathbf{g}=(g_{1},g_{2})$ be two bivariate fields. First, we show that $d_{T}(\mathbb{MR}_{\mathbf{f}},\mathbb{MR}_{\mathbf{g}})$ is a pseudo-metric.