morse theory on triangulated meshes - mustafa hajij · morse theory on meshes suppose that we have...
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Morse Theory on Triangulated Meshes
MUSTAFA HAJIJ
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What is Morse theory ?
Morse Theory is a tool in differential topology that is concerned with the relations between the geometric and topological aspects of manifolds and the real-valued functions defined on them.
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What is Morse theory ?
Morse Theory is a tool in differential topology that is concerned with the relations between the geometric and topological aspects of manifolds and the real-valued functions defined on them.
One of the primary interests in this theory is the relationship between the topology of a smooth manifold M and the critical points of a real-valued smooth function f defined on M.
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What is Morse theory ?
Consider the following example. Let M be a 2-dimensional torus.
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What is Morse theory ?
Consider the following example. Let M be a 2-dimensional torus.
Define a function f on M to be the function that sends every point (x,y,z) on the torus to z coordinate.
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What is Morse theory ?
Consider the inverse image of the value z1 under the function f.
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What is Morse theory ?
Consider the inverse image of the value z1 under the function f.
As we increase z1 to z2, the topology of the level set changes from one circle to 2 circles
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Consider the inverse image of the value z1 under the function f.
As we increase z1 to z2, the topology of the level set changes from one circle to 2 circles
We call the point on M at which this topological change occurs a critical point
What is Morse theory ?
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Morse theory studies carefully the topological changes that happen to the manifold M as we pass from a critical point.
What is Morse theory ?
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Morse theory on smooth surfaces
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Morse Lemma
minimum saddle maximum
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Handle decomposition
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Handle decomposition
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Morse Theory on Meshes In practice we approximate surfaces by triangulated meshes.
Examples of triangulated meshes
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Morse Theory on Meshes In practice we approximate surfaces by triangulated meshes.
Examples of triangulated meshes
Mesh components
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Morse Theory on Meshes Suppose that we have a mesh M and suppose that f is a scalar function defined on the set of vertices of M.
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Morse Theory on Meshes Suppose that we have a mesh M and suppose that f is a scalar function defined on the set of vertices of M.
Consider a local ring neighborhood around a vertex and consider the values of v on the ring
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Morse theory on Meshes Suppose that we have a mesh M and suppose that f is a scalar function defined on the set of vertices of M.
Consider a local ring neighborhood around a vertex and consider the values of v on the ring
Locally, the Morse scalar function around a vertex has one of the following possibilities
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Morse Scalar Functions on Meshes
minimum maximum regular saddle
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Morse Scalar Functions on Meshes
minimum maximum saddle
minimum maximum regular saddle
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Types of vertices Given a Morse function f on a triangulated mesh M. Then we can classify the vertices of M as follows
Definition: A scalar function f on a triangulated mesh M is PL Morse function if each vertex is either regular or simple critical (minimum, maximum or saddle with m=1) and the function values of the vertices are distinct.
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Morse Scalar Functions on Meshes We will represent the isolines as follows:
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Morse Scalar Functions on Meshes In the following example we represent :
minimum point by a blue spheremaximum point by a red spheresaddle point by a green sphere
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Morse Scalar Functions on Meshes
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Applications of Morse TheoryMorse theory has found many applications recently in geometric processing
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Applications of Morse TheoryMorse theory has found many applications recently in geometric processing
Extracting Surface feature lines (Sahner et. el.)
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Applications of Morse TheoryMorse theory has found many applications recently in geometric processing
Extracting Surface feature lines (Sahner et. el.)
Surface quadrangulation (Dong et. el.)
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Applications of Morse TheoryMorse theory has found many applications recently in geometric processing
Extracting Surface feature lines (Sahner et. el.)
Surface quadrangulation (Dong et. el.)
Cutting a surface into a disk (Ni et. el)
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Reeb Graph
Given a surface M and a scalar function defined on it, we can define a combinatorial structure called the Reeb graphof M and f by collapsing the level sets of f as illustrated in the figure
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Thank You