Dimension-independent simplification and refinement of Morse complexes
Highlights
► We define simplification operators on Morse complexes in arbitrary dimensions. ► Simplification operators reduce the incidence relation on Morse complexes, and reduce the number of cells in Morse–Smale complexes at each step. ► We define the inverse refinement operators. ► Simplification and refinement operators form a complete set of basis operators for modifying Morse complexes. ► Simplification and refinement operators form a basis for building a multi-resolution representation of Morse complexes.
Introduction
Representing topological information extracted from discrete scalar fields becomes more challenging as more applications, such as analysis and visualization of terrain models, shape and volume data sets, and time-varying volume data sets see increasing amounts of data. Morse theory offers a natural and intuitive way of analyzing the structure of a scalar field as well as of compactly representing the scalar field through a decomposition of its domain into meaningful regions associated with the critical points of the field.
A discrete scalar field f is defined by its values at a finite set V of points on a manifold M in . The discretization of M is often obtained through a simplicial mesh (such as a triangle, or a tetrahedral mesh), or through a regular grid formed by square cells in 2D, or by hexahedral cells in 3D. This geometry-based description provides an accurate representation of a scalar field f, but it fails in capturing compactly its topological structure, defined by critical points and integral lines of f. Beside being compact, a topological description supports also a knowledge-based approach to analyze, visualize and understand the scalar field behavior (in space and time), as required, for instance, in visual data mining applications.
The descending Morse complex is composed of cells defined by the integral lines of f with the same destination. Dually, the ascending Morse complex is composed of cells defined by the integral lines with the same origin. The Morse–Smale complex describes the subdivision of M into cells determined by integral lines with the same origin and destination [40]. These subdivisions have been recognized as convenient representations for analyzing the topology of M, and the behavior of f over M.
Structural problems in Morse and Morse–Smale complexes, like over-segmentation in the presence of noise, or efficiency issues arising because of the very large size of the input data sets, can be faced and solved by defining simplification operators on those complexes and on their topological representations. Morse and Morse–Smale complexes can be simplified by cancelling critical points in pairs [35]. Cancellation eliminates two critical points of f, two cells in the Morse complexes, and two vertices in the Morse–Smale complexes. Surprisingly, a cancellation may increase the incidence relation on the Morse complexes, and the number of cells in the Morse–Smale ones.
We define two dual dimension-independent simplification operators, which do not have these undesirable properties. The two simplification operators are defined directly on the Morse complexes. We have introduced these operators in [13], where we have shown their effect on a graph-based representation of the two Morse complexes. Here, we describe the effect of these operators on the Morse complexes in detail. We define the inverse refinement operators to the simplification operators, and we describe in detail their effect on the Morse complexes. We show that simplification and refinements operators are valid, i.e., that the result of an application of a feasible simplification or refinement operator on a Morse complex is a Morse complex.
The combination of the simplification and refinement operators defines a minimally complete set of operators for creating and updating Morse decompositions. We prove this result formally by interpreting these operators as Euler operators, i.e., as operators, which affect a constant number of cells in the Euler–Poincaré formula on the ascending and descending Morse complexes. As a consequence, any operator for creating and modifying Morse complexes can be expressed as a suitable sequence of our simplification and refinement operators. In particular, we consider a macro-operator defined in [24] that consists of a 1-saddle-2-saddle cancellation followed by cancellations involving extrema, and we show that this macro-operator can be easily expressed as a sequence of a subset of our operators. The operators we define can be used to generate a multi-resolution model for Morse complexes. The basic ingredients of such model are refinement operators and a suitable dependency relation defined on them.
In summary, contributions of this work include
- •
a set of simplification operators on the Morse complexes, which:
- –
reduce the incidence relation on the Morse complexes, and the number of cells in the Morse–Smale complexes,
- –
can be seen as merging or collapsing of cells in the Morse complexes, and
- –
- •
a set of inverse refinement operators on the Morse complexes.
The simplification and refinement operators
- •
are defined in arbitrary dimension,
- •
maintain the topological validity condition expressed by the Euler–Poincaré formula,
- •
form a minimally complete set of operators for creating and updating Morse complexes, so that any macro-operator can be expressed as a suitable combination of operators in this set, and
- •
form a basis for a new general multi-resolution model of the Morse complexes.
The remainder of the paper is organized as follows. In Section 2, we review some basic notions on cell complexes and on Morse theory. In Section 3, we discuss related work. In Section 4, we investigate cancellation of critical points of a Morse function f, and its effect on the related Morse and Morse–Smale complexes in arbitrary dimensions. In Section 5, we define two dual simplification operators that we call removal and contraction, and we describe how ascending and descending Morse complexes are affected by these operators. In Section 6, we define the inverse refinement operators, and we describe how these operators affect the two dual Morse complexes. In Section 7, we give a proof of the validity of the simplification and refinement operators, and we show that they form a basis for the set of operators that modify Morse complexes in a topologically consistent manner. In Section 8, we explain the relationship between cancellation and removal and contraction operators in 3D, and we show how a 1-saddle-2-saddle macro-operator can be expressed through our operators. In Section 9, we draw some concluding remarks and we briefly discuss a multi-resolution model based on the simplification and refinement operators we introduced.
Section snippets
Background notions
In this Section, we briefly review some basic notions on cell complexes (for more details on algebraic topology, see [34]). A survey on topological shape representations based on cell complexes is given in [15]. We then review the basic notions of Morse theory in the case of n-manifolds (for more details, see [35], [36]).
Related work
In this Section, we review related work on topological representation of scalar fields, focusing on algorithms that assume a discretization of the domain of the field as a manifold simplicial complex (a triangle or a tetrahedral mesh in 2D and in 3D, respectively). We will concentrate on three different research areas, which are relevant to the work presented here, namely: (i) discrete representations of Morse and Morse–Smale complexes, (ii) algorithms for computing an approximation of a Morse
Cancellation operator
In this section, we review the cancellation operator, which simplifies a Morse function f defined on a manifold M by eliminating its critical points in pairs [35]. We explain how this operator modifies the gradient vector field of f around the two cancelled critical points in the differentiable case, and how it affects the discrete representation of f given by Morse and Morse–Smale complexes.
Simplification operators
We define two dual dimension-independent simplification operators, that we call removal and contraction. We define these operators as a special case of a cancellation, by imposing additional constraints on the feasibility of a cancellation. These constraints guarantee that the two operators behave like a cancellation involving an extremum: they do not increase the incidence relation on the Morse complexes or the number of cells in the Morse–Smale ones, and they can be seen as merging of cells
Refinement operators
One of the aims of Morse theory is to relate the homotopy type of a manifold M to that of a cell complex with a cell of dimension i for each critical point of index i of a scalar function f defined over M. A cancellation operator decreases the number of critical points of f and provides a representation of the homotopy type of M through a cell complex with fewer cells. In Morse theory, an inverse operator of a cancellation in arbitrary dimensions has not been defined. Only in the
Validity and minimality of simplification and refinement operators
In Section 7.1, we show that each feasible simplification and refinement operator is valid, i.e., that applied on a combinatorial Morse complex Γd on M it produces a combinatorial Morse complex on M. In Section 7.2, we show that these operators form a basis for the set of operators that modify Morse complexes on M in a topologically consistent manner.
Cancellations and simplifications in the 3D case
In this Section, we discuss the relationship between cancellations and removals and contractions in the 3D case.
In 3D, there are two types of cancellation. The first type cancels a maximum and a 2-saddle, and it is the same as a removal of index 2, or dually it cancels a minimum and a 1-saddle, and it is the same as a contraction of index 1. Let us recall the effect of a cancellation of a maximum p and a 2-saddle q on a descending Morse complex, and on Morse–Smale complex. In a descending Morse
Concluding remarks
We have defined operators for simplification and refinement of Morse complexes in arbitrary dimension. We have shown their effect on the ascending and descending Morse complexes, we have shown that these operators are valid, and we have shown that they form a minimally complete set of basis operators for creating and modifying Morse complexes on a manifold M. We have proven this result by interpreting our operators as Euler operators, that is, as operators that affect a constant number of cells
Acknowledgments
This work has been partially supported by the National Science Foundation through Grant CCF-0541032.
References (45)
- et al.
Topology preserving data simplification with error bounds
Computers and Graphics
(1998) Morse theory for cell complexes
Advances in Mathematics
(1998)- et al.
Ascending and descending regions of a discrete Morse function
Computational Geometry: Theory and Applications
(2009) Topological models for boundary representation: a comparison with n-dimensional generalized maps
Computer Aided Design
(1991)- et al.
Topological volume skeletonization and its application to transfer function design
Graphical Models
(2004) Critical points and curvature for embedded polyhedral surfaces
American Mathematical Monthly
(1970)- B.G. Baumgart, A polyhedron representation for computer vision, in: Proceedings AFIPS National Computer Conference,...
- et al.
Describing shapes by geometrical–topological properties of real functions
ACM Computer Survey
(2008) - et al.
A multi-resolution data structure for two-dimensional Morse functions
- et al.
A topological hierarchy for functions on triangulated surfaces
Transactions on Visualization and Computer Graphics
(2004)
Maximizing adaptivity in hierarchical topological models
Representing geometric structures in D dimensions: topology and order
Discrete and Computational Geometry
Molecular shape analysis based upon the Morse–Smale complex and the Connolly function
Vector field editing and periodic orbit extraction using Morse decomposition
Transactions on Visualization and Computer Graphics
Cancellation of critical points in 2D and 3D Morse and Morse–Smale complexes
Modeling and simplifying Morse complexes in arbitrary dimensions
Topological analysis and characterization of discrete scalar fields
Multiresolution representation of shapes based on cell complexes
Algorithms in Combinatorial Geometry
Cited by (16)
Topologically-consistent simplification of discrete Morse complex
2015, Computers and Graphics (Pergamon)Citation Excerpt :In the literature, several strategies have been proposed for topologically simplifying the morphological representation of a dataset [12]. The problem of simplifying a Morse–Smale complex has been addressed in 2D [3,13,14], 3D [6,15] and in nD [16]. A common characteristic of all simplification algorithms is the ordering of the available simplifications based on a priority schema.
Dimension-independent multi-resolution Morse complexes
2012, Computers and Graphics (Pergamon)Citation Excerpt :These complexes are widely used in shape analysis and modeling, and they have been applied in scientific visualization for understanding and analyzing the critical features of a scalar field. Simplification of Morse and Morse–Smale complexes has been an important research topic in these last years [1–3]. We have defined atomic simplification (and the inverse refinement) operators [3], which have the important property of forming a minimally complete basis for modifying Morse and Morse–Smale complexes in arbitrary dimension.
A Morse theoretic approach to the geometrical feature terms specified in ISO 25178-2 and ISO 16610-85
2021, Surface Topography: Metrology and PropertiesIf You Must Choose Among Your Children, Pick the Right One
2020, Proceedings of the 32nd Canadian Conference on Computational Geometry, CCCG 2020A local update and refinement method of the global terrain nature grid
2019, IOP Conference Series: Earth and Environmental Science