Segmenting a surface mesh into simple pieces for further processing is a fundamental problem in many mesh processing applications such as texture mapping [levy2002least], collision detection [li2001decomposing], skeletonization [biasotti2003overview] and three-dimensional shape retrieval [zuckerberger2002polyhedral]. In surface matching, authors in [li2009surface] showed the use of a particular type of mesh segmentation called a pants decomposition, see Figure 1. It derives its name from its constituent piece termed a pair of pants, which, up to topology, is a genus zero orientable surface with boundary components. Besides surface matching, pants decomposition has found applications in surface classification and indexing [jin2009computing], and consistent mesh parametrization [kwok2012constructing]. This type of segmentation can be viewed as a common base domain segmentation [kwok2012constructing], where one partitions the mesh into parts with a common property such as having the same topology. Authors in [li2009surface] presented a pants decomposition algorithm based on the computations of certain basis cycles in the first homology group of the input surface. Other pants decomposition related algorithms proposed in the graphics literature can be found in [zhang2012optimizing] where the authors enumerates different classes of pants decompositions. However, these methods rely on computing certain curves on the surface called Handle and Tunnel loops [dey2007computing, dey2013efficient]. Computing such curves is expensive and more importantly requires the surface to be embedded in and not have any boundary. Morse theory allows our algorithms to get rid of these two constraints. Moreover, pants decomposition relies on finding a collection curves on the surface with certain topological properties. Finding these curves and manipulating them is a difficult problem. Morse theory allows us to avoid defining these curves explicitly by realizing them implicitly as level curves of a Morse function.
Many segmentation algorithms have been proposed in the graphics literature includes [mangan1999partitioning, shamir2008survey, li2009surface, chen2009benchmark, shlafman2002metamorphosis, attene2006hierarchical, yu2015geometry, li2016geometry]. One reason for the variety of segmentation algorithms suggested in the literature is that there is no one universal good algorithm that suits all applications. The techniques used in the algorithms are related to other areas in computer graphics such as image segmentation [reynolds1995robust, tomasi1998bilateral]
and machine learning[karypis1998software, cover1967nearest]. For good surveys on various segmentation algorithms see [attene2006mesh, shamir2008survey].
In this work we propose two algorithms based on Morse theory for computing a pants decomposition of an input surface mesh. The Morse theory connects the geometry and topology of manifolds via the critical values/points of a specific class of real-valued functions called Morse functions. Intuitively, given such a function on a manifold , Morse theory studies the topological changes of the level sets of . These level sets change in topology only at the critical values. Consequently, the space in between two consecutive critical levels becomes a product space of a fiber with an interval. This fact is used to decompose a surface into pants and cylinders, and the latter parts are glued inductively into pants. The second algorithm exploits a reduced structure called Reeb graph [reeb1946points] derived from a function on the surface. The Reeb graph of is a quotient space derived from and . For a Morse function, all vertices in the Reeb graph have valence except the extrema. We exploit this property to compute a pair of pant for every degree-3 vertex of the Reeb graph and treat the extrema specially.
Morse theory, in its original form, assumes smooth settings [milnor1963morse]. For operating on surface meshes, we need a piecewise linear (PL) version of the theory. We make this transition using the PL Morse theory proposed by [critical1967]. We show how one can compute a function on a surface mesh satisfying the specific property that our algorithms require. Furthermore, we extend our basic algorithms to the case when the surface has boundaries, or when the function allows degenerate critical points such as monkey saddles.
Our experiments show that both of our algorithms run much faster in practice than the algorithm of Li et al. while the Reeb-graph based algorithm performs the best. Furthermore, our tests show that both of the algorithms presented here are robust to noise.
2. Morse Theory and Handle Decomposition for Surfaces
Let be a compact smooth surface, and let , where , be a closed interval. Let be a smooth function defined on . A point is called a critical point of if the differential is zero. A value in is called a critical value of is contains a critical point of . A point in is called a regular point if it is not a critical point. Similarly, a value is called regular if it is not critical. The inverse function theorem implies that for every regular value in the level set is a -manifold, i.e., is a disjoint union of simple closed curves. A critical point is called non-degenerate if the matrix of the second partial derivatives of , called the Hessian matrix, is non-singular.
The definition of a Morse function is motivated mainly by the following Lemma.
(Morse Lemma) Let be a smooth surface, be a smooth function and be a non-degenerate critical point of . We can choose a chart around such that takes exactly one of the following three forms:
The index of a critical point of , denoted by
, is defined to be the number of negative eigenvalues of its Hessian matrix. Since the Hessian of a scalar function on smooth surface is asymmetric matrix, then the index takes the values or . One can see that on a non-degenerate critical point of index the function takes a minimum value, on a non-degenerate critical point of index , the graph of the function looks like a saddle and on a non-degenerate critical point of index the function takes a maximum value. See Figure 2
If all the critical points of are non-degenerate and all critical points have distinct values, then is called a Morse function. If the surface has boundary, then we also require two other conditions : (1) and (2) the critical points of lie in the interior of .
2.1. Handle Decomposition of a Surface
Our first algorithm uses the attachment of cylinders called handles at
the critical points. We need a few definitions first.
Let be a smooth surface and let be a Morse function defined on . Define the set
Let , , be two reals. Define
When it is clear from the context, we will drop from the notation and use simply and to refer to the previous two sets. Morse theory studies the topological changes of as varies. The following is well-known [matsumoto2002introduction].
Let be a smooth function on a smooth surface . For two reals , , if has no critical values in the interval , then the surfaces and are diffeomorphic.
The previous theorem says that the topology of the surface does not change as passes through regular values. In the following we use to denote the interval . The end points of are given by . Given a Morse function on M, the following theorem gives s precise description for the change that occurs in the topology of as passes through a critical value.
Let be Morse function. Let be a critical point of index and be its corresponding critical value. Let be chosen small enough so that has no critical values in the interval .
If , then is diffeomorphic to the disjoint union of and a -dimensional disk .
If , then can be obtained from by attaching a -handle. This means that can be obtained by gluing a rectangular strip to the boundary of along .
If , then can be obtained by capping off the surface with a disk . This means that is obtained by gluing a disk along its boundary to one of the boundary components of .
2.2. Morse Theory for Triangulated Surfaces
Morse theory was extended to triangulated surfaces (-manifolds) by Banchoff [critical1967]. Let be a triangulated surface, and let be a piece-wise linear (PL) continuous function on . The link of a vertex is defined as the set of all vertices that shares an edge with. The upper link of is defined as
and the lower link is defined by
and mixed link
Using the link definitions we can classify the interior vertices of. The boundary vertices are treated separately as we will require to have identical values on them. An interior vertex is regular if , is a maximum with index if , is a minimum with index if , and is a saddle with index and multiplicity if . See Figure 3. A vertex is a simple critical point if it is either minimum, or maximum, or a saddle with multiplicity . A PL function on a closed triangulated surface is PL Morse if all its vertices are either PL regular or simple PL critical and have distinct function values. If the mesh has a non-empty boundary then we also require (1) , and (2) the critical vertices of lie in the interior of .
2.3. Reeb Graph
Let be a surface possibly with boundary , and let be continuous. Define the equivalence relation on by if and only if and and belong to the same connected component of the level set . The set with the standard quotient topology is called Reeb graph of . When is smooth and Morse, or PL Morse, every vertex of the arises from a critical point of or a boundary component. Every maximum or minimum of gives rise to a degree -node of . Since is Morse, every boundary component also gives rise to a degree -node and every saddle of gives rise to degree -node. If is an embedded surface without boundary then can be recovered up to a homeomorphism from as the boundary of an oriented -dimensional regular neighborhood of the graph . See Figure 4 for an example of a Reeb graph.
One reason that we require the Morse function to assume constant values on boundaries is that it makes it consistent with the definition of the Reeb graph we give here. Note that each boundary component maps exactly to one point on the Reeb graph.
The definition of Reeb graph goes back to G. Reeb [reeb1946points]. It was first introduced to computer graphics in [sato1994matrix]. Reeb graph has found applications in shape understanding [attene2003shape], quadrangulation [hetroy2003topological], surface understanding [biasotti2000extended], segmentation [werghi2006functional], parametrization [patane2004graph, zhang2005feature], animation [kanongchaiyos2000articulated] and many other applications. Reeb graphs algorithms can be found in many papers such as [shinagawa1991constructing, cole2003loops, pascucci2007robust, doraiswamy2009efficient]. The most efficient algorithms in terms of time complexity are due to [Salman12, HWW10].
3. Pants Decomposition
Let be a compact, orientable, and connected surface. We say that is of type if is of genus and has a boundary components. A pair of pants is a surface of type . See Figure 5.
A pants decomposition of is a finite collection of unordered pairwise disjoint simple closed curves embedded in with the property that the closure of each connected component of is a pair of pants. Two pants decompositions of are equivalent if they are isotopic. See Figure 19 for an example of two non-isotopic pants decompositions of a genus- surface.
Let be a compact orientable surface and connected surface of type . The Euler characteristic of M, denoted by is defined as . Every connected, compact and orientable surface with , genus and boundary components admits a pants decomposition with simple closed curves and the number of complementary components is .
4. Handle-Based Pants Decomposition
In this section we use handles given by a Morse function to design an algorithm for decomposing a surface with into a collection of surfaces of type . Our algorithm works for arbitrary surface with and with or without a boundary. However, in order to guarantee the correctness of our algorithm we must choose a function with certain properties. Ideally, this function should be PL Morse with these properties when is a surface mesh. We cannot always guarantee that the function is PL Morse, but we can compute one that satisfies all required properties except the simplicity of the saddles. We first describe the algorithm assuming a PL Morse function and later mention how to handle the exceptions of degenerate saddles.
4.1. Orientable Surfaces With and No Boundary
In this section we give an algorithm to compute a pants decomposition of a triangulated surface with genus without boundary. Let be a compact connected orientable surface with genus without boundary and let be a PL Morse function on . Suppose that are the critical values for ordered in an ascending order. Let be the corresponding critical points of . Choose a real number small enough so that for each there are no critical values for on the interval except . Finally we assume that function has exactly one minimum and exactly one maximum. It is clear from the choice of the function that one of the points and is the global maximum and one of them is the global minimum. Since multiplying any Morse function on a surface by changes its critical points of index to critical points of index and vise versa, we can always choose our Morse function so that is the global minimum and is the global maximum. It should be noted that such a function can be constructed in practice and we will talk about the construction of such functions later. We need the following lemma for the correctness of our algorithm.
Let be a compact connected orientable surface with a Morse function chosen as specified above then is homeomorphic to a surface of type or a surface of type .
The choice of the scalar function implies immediately that for each we have . Moreover, by construction we have and . By Theorem 2.3 we conclude that is diffeomorphic to a disk and is diffeomorphic to surface of type .
Moreover, by Theorem 2.3, when passes through the surface is obtained from by gluing a rectangular strip to the boundary of along . Up to a homeomophism, there are two possible ways of gluing the rectangular strip to the boundary of along . See Figure 7. We either glue this rectangular strip to the same boundary component of to obtain a surface of type or we glue each side of the strip on one of the boundary components to obtain a surface of type . See Figure 8.
Using similar argument one can prove that the surface is either of type or a surface of type .
Lemma 4.1 holds in the case when the function is PL Morse on a triangulated surface. However, the case when the PL scalar function has saddle points with multiplicity larger than or equal to needs special treatment and Lemma 4.1 is no longer valid. We deal with such cases in section 4.3.
Compute the critical points of and put them in an ascending order. Let be the sequence of ordered critical points of and let be their corresponding critical values. Note that by our choice of the scalar function .
For each let . After reindexing, we define the set . In other words, the set is a set of ordered regular values for such that there is exactly one critical value for the function in the intervals for . See Figure 9 for an example.
Cut the surface along the level sets for all . Note again that the values are all regular values.
By Lemma 4.1 the surface is either of type or of type . If is of type , then we trace a loop from the saddle point to the minimum point and cut the surface along that loop. We explain later a method of tracing this loop. See Figure 10. Otherwise, if the surface is of type we do not need to do anything. In either cases, we denote the resulting surface of type by .
Consider the manifold with boundary . This surface is a finite disjoint union of one surface of type and multiple surfaces of type . Attach every surface of type to . Note that this gluing does not change the homeomorphism type of . See Figure 11 (b).
For each consider the manifold with boundary . This surface is again a finite disjoint union of one surface of type and multiple surfaces of type . Attach every surface of type to . See Figure 11 (c).
The remaining part is either of type or of type . If the surface is of type then trace a loop from the saddle point to the point and cut the surface along this loop to obtain a pant. Otherwise, if the surface is of type we do not need to do anything.
Note that the algorithm constructs a collection of pants inductively. We start by having the first pair of pants in step and then we go to the next level which is a finite disjoint union of a single pant and some topological cylinders. We attach the cylinders to the previous pant and then we go to the next level and repeat the same process.
We should clarify here what we mean by tracing a loop from the saddle to the minimum point we mentioned in step . Let the simple saddle point and be the unique minimum point for the PL scalar function on specified in our algorithm above. The point is a simple saddle, hence the set can be decomposed into two disjoint connected components and . Pick the vertex in such that for all . The vertex is chosen similarly. A descending path from a regular vertex , denoted by , is defined to be a finite sequence of vertices on such that is an edge on for , , and is a minimum. A loop connecting and can be computed as the concatenation of , , , and . The loop in step is computed analogously by considering an ascending path.
4.2. Orientable Surfaces With and Non-Empty Boundary
In this section we present an algorithm to decompose a surface with with boundary components. The algorithm follows almost similarly as before. The main difference is that we need to take care of the choice of the Morse function. We have two cases, The surface has exactly one boundary component The surface has more than one boundary component.
Let be a compact connected orientable surface with and one boundary component . We pick a point on the surface and construct a Morse function that satisfies the following conditions :
and for all in .
The point is a global maximum.
The function does not have any critical point of index or except for .
The pants decomposition algorithm for a surface with and one boundary component algorithm goes now as follows :
Compute the critical points of and put them in an ascending order. Let be the ordered critical points of and let be the corresponding critical values. By our choice of the Morse function, we have for all and .
For each let . Define the set . In other words, the set is a set of ordered regular values for such that there is exactly one critical value in the interval . See Figure 12 for an example.
Cut the surface along the level sets for all .
Consider the manifold . By our choice of the Morse function is a of type .
The rest of the pants decomposition algorithm for a surface is similar to steps , and of the algorithm in section 4.1.
Now we discuss the final case. Suppose that has boundary components where . Here we also need to construct a Morse function that serves our purpose. We need a Morse function that satisfies the following :
We choose one of the boundary component, say , and we construct such that and for all .
The function does not have any critical point of index or .
As we did earlier, let be the ordered critical points of and let be the corresponding critical values of . Define the values for all . By our choice of the Morse function, the manifold is homeomorphic to a finite disjoint union of one surface of type and multiple surfaces of type . In particular is a pant. The pants decomposition algorithm now is similar to the previous algorithm except that we do not trace any loop in this case since all connected components after cutting along the regular values of are pants or cylinders. See Figure 13 for an example.
Figure 14 shows multiple examples of pants decomposition of surfaces using this algorithm.
4.3. Dealing with Degenerate Cases
In this section we deal with the case when the index of a critical point point is and has multiplicity . We formulate our arguments in terms of Reeb graph for clarity.
If the point is a degenerate saddle point then the Reeb graph obtained from the quotient space of appears as in Figure 15.
In this case the surface is of type where and we can decompose this surface into pants as described earlier.
The point is a simple saddle and the point is a degenerate one. In this case the Reeb graph obtained from the surface has two possibilities and they both appear in Figure 16.
In the case when the Reeb graph appears as in left hand side of Figure 16 then is homeomorphic to a surface of type where . This surface can be decomposed into pants as we described earlier. The case when the Reeb graph appears as in the right hand side of Figure 16 then the is surface of type where . We can cut this surface along the simple saddle using descending path by tracing a loop from the simple saddle point the unique minimum using descending path as we described earlier into a sphere with boundary components. The latter can be also decomposed into pants as explained in earlier sections. Note that this case is an extension of the simple case we considered in Lemma 4.1.
The case when is a degenerate saddle where . In this case , for a sufficiently , is a finite disjoint union of multiple surfaces of type and one surface of of type where . When we have this, we attach the cylinders just as before to previous pants and we apply Morse function-based algorithm again on the single remaining sphere with boundary components to decompose it into pants.
We left out two cases, namely the cases when is degenerate and the case when is simple saddle and is a degenerate one. These two cases can be dealt with as cases (1) and (2) above.
5. Reeb Graph-Based Pants Decomposition
In this section we give pants decomposition algorithm using the Reeb graph of a Morse function. This algorithm is implicit in the work of [hatcher1999pants]. We extend this algorithm to handle degenerate cases that show up in practice. The Reeb graph decomposition algorithm has many advantages over the previous algorithm. The main advantage of this algorithm lies in the fact that it does not put any restriction on the choice of the Morse function. This allows us to choose a scalar function with better geometric properties. The second advantage is that choosing a cutting circle on a surface is much more flexible when using the structure of the Reeb graph than choosing the inverse image of a regular value of a Morse function.
Let be a compact orientable (triangulated) surface, possibly with boundary, such that . Let be an arbitrary (PL) Morse function of . The Reeb graph-based pants decomposition algorithm of the surface and the Morse function goes as follows:
Compute the Reeb graph of .
There are two types of -valence nodes on the graph , the -valence nodes that are a result of collapsing the boundary components of and the -valence nodes that are coming from critical points of of index and . We consider the graph obtained by taking the deformation retract of the graph that leaves the edges coming from the boundary component without retraction. This step can be done by iteratively deleting the edges one of whose nodes has valence until there are no more such edges except the ones which have -valence nodes originating from boundary components. See Figure 17 step for an example of such retraction.
We remove all the nodes on the graph of valency and we combine the two edges that meet at such a node into one edge. We also denote the graph obtained from this step by . Note that this graph is trivalent by construction.
We select an interior point on every edge of the graph provided that neither one of the two nodes defining that edge has valency . Note that the selection of the these point on the graph corresponds to partitioning the graph into a collection of small graphs each one of them is a vertex connected by small three arcs. See step in Figure 17. Each one of these small graphs corresponds to a pair of pants on the surface .
Each choice of an interior point induces a choice of a simple closed curve on the original surface . The collection of all curves obtained in this way defines a pants decomposition of the surface .
Figure 18 shows an application of this algorithm on some surfaces.
5.1. Dealing with degenerate cases
In the case when some critical points of index have multiplicity , we proceed as described in steps through . However, the final result of the decomposition will no longer be a collection of pants but rather it will contain some surfaces of type where . For each single surface of these surfaces we apply one of the pants decomposition algorithms again in order to decompose it into a collection of pants.
6. Choosing an Appropriate Scalar Function
We need to construct a function that suits the algorithms that we have presented. For the algorithm given in section 4.1 we need a scalar function that has one global minimum and one global maximum. This can be done by solving a Laplace equation on a mesh with Dirichlet boundary condition. A scalar function that satisfies the Laplace equation is called harmonic. We are seeking here is a scalar function which satisfies the Laplace equation subject to the Dirichlet boundary conditions for all . Here is a set of constrained vertices and are known scalar values providing the boundary conditions. This system has a unique solution provided . Furthermore, the solution for such a system has an important property that it has no local extrema other than the constrained vertices. This property of harmonic functions is usually called the maximum principle [rosenberg1997laplacian]. Designing such a function is possible in practice. Recall that on triangulated mesh the standard discretization for the Laplacian operator at a vertex is given by :
where is a scalar weight assigned to the edge such that . Choosing the weights such that for all edges guarantees the solution of the Laplace equation has no local extrema other than at constrained vertices [floater2003mean]. These conditions are satisfied by the mean value weights:
where the angles and are the angles on either sides of the edge at the vertex . Mean value weights are used to approximate harmonic map and they have the advantage that they are always non-negative which prevents any introduction of extrema on non-constrained vertices in the solution of the Laplace equation specified above. On the other hand, the cotangent weights may become negative in presence of oblique triangles and this can produce local extrema on non-constrained vertices. In this context see also [wardetzky2007discrete] for various discretizations of the Laplace-Beltrami operator and their properties. In our Morse function-based algorithm for pants decomposition we needed a scalar function that has precisely one global minimum and one global maximum. Hence, the constrained vertices is chosen to have exactly two vertices such that . This choice will guarantee that the solution has a single minimum at and a single maximum at . Hence, every other critical point for must be a saddle point which is a requirement for 4.1. Similarly, a harmonic scalar function can be used to obtain the pants decomposition algorithm 4.2 for a surface with multiple boundary components.
Even though our second algorithm works on a generic Morse function, we choose a function that captures the geometry and the symmetry aspects of the mesh. This was not possible in the previous algorithm due to the restriction of the input function. Our choice for scalar functions were made based on the object itself. For organic objects like the human body we choose the Poisson field [dong2005harmonic]. On the other hand, for objects with some symmetry, we found that isometry invariant scalar functions [sun2009concise, kim2010mobius, wang2014spectral] and multi-source heat kernel maps [hajij2015constructing] give the best results.
7.1. Run-time comparison
We ran our experiments on a GHz AMD(R) A6-6300 with 10.0 Gb memory. We implement all the algorithms presented in Table 1 using C++ on a Windows platform. The algorithms were tested on different models and compared their run-time with a publicly available software running the algorithm of [li2009surface]. Table 1 shows that time comparison between these algorithms. The running times that are provided in the table includes the time needed for computing the scalar functions needed as an input for in out algorithms and exclude the time needed to compute the homology generators needed for the algorithm of [li2009surface]. We did not include the pants decomposition algorithm presented in [zhang2012optimizing] since it is not different fundamentally from the one in [li2009surface]. The main purpose of the framework in [li2009surface] is to enumerate different pants decomposition classes starting from an initial pants decomposition which is computed using the method given in [li2009surface]. The algorithms that we presented here perform much better than the algorithm in [li2009surface]. In particular, the Reeb graph algorithm gives us the best time efficiency.
|Vertices||Topology||Alg. time||Alg. time.||Alg. [li2009surface] time|
We mentioned in 6 that our choice of the scalar function relies on the geometry that we deal with. In Table 2 we list our choices for the scalar functions that we used in our experimentation in the Reeb graph-based pants decomposition algorithm. We also list in the same table other scalar functions that we found potentially useful and could be used on the same model to produce similar results. In our Handle-based pants decomposition algorithm all scalar fields are Harmonic fields by our remark in Section 6.
|Model||Our choice of the scalar field||Other appropriate scalar fields|
|David||Poisson fields||Harmonic functions|
|-Torus||Multi-source heat kernel maps||Isometry invariant fields|
|Vase||Harmonic functions||Poisson fields|
|Greek||Harmonic functions||Poisson fields|
|Topology||Isometry invariant fields||Poisson fields|
|Knotty||Poisson fields||Harmonic functions|
Figure 19 shows examples of the scalar functions used in the input for our algorithms. The Figure also shows the critical points of the scalar functions.
7.2. Imperfect input
Just to test how robust are our algorithms against noise, we ran our algorithms on surfaces corrupted with deliberate noise. We applied random displacements to the coordinates of the input mesh and tested the results. We applied noise amplitude up to of the bounding box length for the input mesh. Both algorithms maintained correct and meaningful outputs. See Figure 20 for some examples.
Morse theory is a powerful mathematical tool that uses local differential properties of a manifold to infer its global topological properties.
Using a PL version of the theory, we have given two algorithms to decompose a surface mesh into pants. The algorithms we proposed here remove the constrains required by earlier algorithms and the run time much faster than the existing state-of-the-art method.
The choice of a Morse function has an impact on the output of the pants decomposition algorithms that we propose here. We have proposed some scheme to compute functions suitable for our pants decomposition algorithms. These scalar functions satisfies certain desirable geometric properties. By the term desirable we mean one or more of the following properties
The isolines of the scalar function are shape-aware in the sense that they follow one of the principal directions of the surface.
The critical points of the scalar function coincide with feature or the symmetry points on the surface.
If the surface has some sort of symmetry then the scalar field also inherits the symmetry of the surface.
Minimal user input.
However, are there alternative schemes which give better pants with some desirable geometric properties? We intend to address this issue in future.