Contributed Talk at the International Workshop on VISUALIZATION and MATHEMATICS 2002

Thomas Lewiner, Hélio Lopes, Geovan Tavares

Math&Media Laboratory,Department of Mathematics, PUC-Rio

Visualizing Forman's discrete vector field

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Outline of the Talk

• Differential Morse theory

• Forman’s discrete Morse theory

• Discrete gradient vector field

• Critical cells and topology

• Hypergraphs & hypertrees

• Algorithm description

• Applications

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Differential Morse Theory

The topology of a differentiable manifold is very closely related to the critical points of a real smooth map defined on it.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Gradient Vector Field

• f: X->R real differentiable map defined on a differential manifold X.

• Critical points: xX such that f(x) = 0.

• f Morse function: critical points are not degenerated.

• V Morse vector field iffthere exists a Morsefunction f such thatV= f.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

• Index of a critical point: number of negative eigenvalues of the Hessian matrix of f

• Critical points depends on the Morse vector field used

• Handlebody decomposition: each critical point is associated to a handle in the decomposition

Critical Points

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Forman’s Discrete Morse Theory(1995)

• General CW-complex

• Combinatorial structure

• Free of geometric embedding

• Tool for computational topology and geometry

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Cell Complexes (1)

• A cell of dimension k is a space homeomorphic to the ball of dimension k.

• A cell complex K is a collection of cells such that every intersection of the closures of two cells of K is also a cell of K.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Cell Complexes (2)

In Forman’s theory, cells are given by their incidences and not as a geometric realization.

Here we will consider only finite cell complexes

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Hasse Diagram

Simple oriented graph built out of K:

• Nodes represent the cells of K

• Links connect cells to their faces of codimension 1

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Discrete Gradient Vector Field (1)

A discrete gradient vector field V is an acyclic pairing in the Hasse diagram:

Acyclic : inverting the orientation of the links between paired cells do not create circuit.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

V discrete gradient vector field :• V: disjoint pairs of cells {,} with face of

V()= , V()=0• V is acyclic: there is no trivial closed V-path

V-path: 0, 0,..., r, r such that

V(p)= p face of p+1, p+1 p

Discrete Gradient Vector Field (2)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Critical Cells (1)

Critical cells of V: unpaired cells, i.e. unmatched nodes in the Hasse diagram.

Index of a critical cell is its dimension

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Critical Cells (2)

Depend on the gradientvector field used.

Empty discrete gradientvector field:every cell is critical.

Optimal discrete gradient vector field

Minimum number of critical cells.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Relation to Topology (1)

• Morse Inequalities

• The cell complex deformation retracts along the discrete gradient field

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

• We can build a cell complex L out of the only critical cells of K; and a homotopy from K to L.

• For polyhedra, the minimal number of critical cells is a topological invariant.

Relation to Topology (2)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Optimum Complexity

Optimum in each level of the Hasse diagram leads to the global optimum.

Reaching the optimum inside a level is aMAX SNP hard problem.

MAX SNP hard : polynomial approximations can be arbitrary far from the optimum.

(Eğecioğlu, 1995) (Hachimori, 2000)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Hypergraphs and the Hasse diagram

• A level of a Hasse diagram is a hypergraph.

• A level of the Hasse diagram of a discrete gradient vector field is a hypertree.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Hypergraphs• Regular links connect two nodes• Hyperlinks connect 1 or more than 3 nodes.

Orientation: each link has exactly one source node, and possibly many destination nodes.

Regular components: connected components of the graphs with the only regular links.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Hypertrees

Every node is the source of at most one hyperlink

Every regular component has the source node of at most one hyperlink

There is an orientation with no circuits

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Algorithm

Processing each level in the Hasse diagram separately, from the highest dimension to the lowest one:– first level is a normal graph– other levels are hypergraphs

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Example: EdgeBreaker on a Torus (1)

C

R

L

S

S*

E

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Example: EdgeBreaker on a Torus (2)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Example: EdgeBreaker on a Torus (3: dual tree)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Example: EdgeBreaker on a Torus (4: graph)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Algorithm Outline

• Successively for each level (hypergraph)(n,n-1), (n-1,n-2), ..., (3,2), (2,1):

• Orienting the remaining links of the 0-1 graph

1 - Selecting linksto build a hypertree

2 - Orienting links to define thediscrete gradient vector field

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Algorithm: Constructing Hypertrees

Kind of a greedy algorithm:

• spanning tree of the regular components

• adding one boundary link to each regularcomponent if any

• adding hyperlinks which do not create circuit with some priority.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Algorithm: Orienting links

Regular components are oriented tree.The root node is

critical or incident to the hyperlink.

Root nodes of the regular components are the source of the hyperlinks.

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Discrete Gradient Field Visualization (1)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Discrete Gradient Field Visualization (2)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Discrete Gradient Field Visualization (3)

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Visualizing Abstract Complex

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Topological Decomposition

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Next Steps

Tools to understand the topology of 3-manifolds:– auditory discrete Morse theory– visual investigation of 3-manifolds– graphical navigation guided by the topology

Topology consistent morphing

Conditions for reaching optimality in polynomial time

Applications to volumetriccompression

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VisMath 2002 Thomas Lewiner, Hélio Lopes, Geovan Tavares. PUC-Rio

Classical Morse Theory•Gradient Vector Field•Critical PointsForman’s Theory•Cell Complexes•Hasse DiagramDiscrete Gradient FieldCritical cells & topology•Critical Cells•Relation to Topology•Optimal ComplexityHypergraph, HypertreeAlgorithm•EdgeBreaker Example•OutlineApplications•Discrete Gradient Field Visualization•Visualizing Abstract Complex•Topological Decomposition

Morse Inequalities

m(k) : number of critical points of index k

(k) : k-th Betti number

• Strong Morse Inequalities(k) - (k-1) + … (0) m(k) - m(k-1) + …

m(0)

• Weak Morse Inequalities(k) m(k)

• Euler Characteristic(n) - (n-1) + … (0) = m(n) - m(n-1) + …

m(0)