Polynomial Splines Over Tmeshes

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Polynomial Splines Over T-meshes

• Falai Chen
• Department of Mathematics
• University of Science and Technology
• of China

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Outlines

• Motivation
• T-meshes
• Spline spaces over T-meshes
• Applications
• Future work

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• 1. Motivations

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Problems with NURBS

• No local refinement properties two global knot
  vectors.

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Problems with NURBS

• Superfluous control points in modeling local
  details to maintain a tensor product structure

Sederberg, et al., 2003
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Previous Solutions

• D. Forsey R.H. Bartels Hierarchical B-spline,
  1988
• F. Weller H. Hagen Tensor-product spline with
  knot segments, 1990
• T. W. Sederberg et. al T-spline, 20032004

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Hierarchical B-splines
A hierarchical B-spline surface is the
overlap of a series of layers, with each later
layer B-spline has a resolution twice that of the
previous B-spline. The basis functions
should be refinable.
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Hierarchical B-splines

• Hierarchical B-spline is a special type of
  B-spline over T-meshes which allows local
  refinement.

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TP splines with knot segments

• Defined over a more general T-mesh, where
  crossing, T-junctional and L-junctional vertices
  are allowed.
• Dimension are estimated.
• Semi-regular bases.

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TP splines with knot segments
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T-Splines

• Sederberg et al., SIGGRAPH03 04
• T-splines are point-based splines
• They are rational, and are piecewise in each
  cell.
• Admit local refinement, but the complexity is
  uncertain.

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T-Splines
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T-splines comparison with NURBS
pwfwWcNL
Sederberg et al. 2003
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Our Solution

• Polynomial Splines
• over T-meshes

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• 2.T-meshes

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T-meshes

• T-meshes are generalization of TP meshes, they
  are formed by rectangular grids which allows
  T-junctions.
• T-meshes have appeared in
• scientific computing, geometric
• modeling. But the theory of
• splines over T-meshes has not
• been fully build up yet.

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T-meshes extension
Adaptability
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T-meshes simplification
TP mesh to T-mesh by simplification
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T-meshes refinement
TP mesh to T-mesh by refinement
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Quantities

• VT number of T-vertices
• V number of crossing vertices
• Vb number of boundary vertices
• Total number of vertices V VT V Vb
• F number of cells
• E number of edges
• Eb number of boundary
• edges
• EL number of large edges

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Topological Identities

• Eulers formula F E V 1
• Regular T-mesh
• 3F E V 1
• F V EL 1

F 10, E 28, V19, V 3, EL 6
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• 3.Splines over T-meshes

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Definition

• T T-mesh, F cells in T, ? region occupied by
  T.
• S(m,n,?,?, T )s(x,y)?C?,?(?) s? is a
  bivariate polynomial of bidegree (m,n) , for any
  ? ? F

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Dimension formula

• For m?2?1, n?2?1,
• dim S F(m1)(n1) Eh(m1)(?1)
  Ev(?1)(n1) V(?1)(?1)
• where F number of faces Ehnumber of
  horizontal edges Ev number of vertical edges
  V number of inner vertices

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dim S(1,1,00) and dim S(3,3,1,1)

• dim S(1,1,0,0) VbV
• dim S(3,3,1,1) 4(Vb V)
• Here Vb is the number of boundary vertices,
  V is the number of crossing vertices.
• For the T-mesh in the right figure, dim
  S(1,1,0,0) 14
• dim S(3,3,1,1) 4(113) 56

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Dimension Calculation

• Three methods
• Smoothing cofactor method
• B-net method
• Syzygy method
• A general dimension formula for general T-meshes
  is unknown.
• Is the dimension independent on geometry?

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Basis functions

• A set of basis functions bi(u,v) is constructed
  with the following properties
• Non-negativity
• Local support
• Partition of Unity
• For S(3,3,1,1),each crossing vertex and each
  boundary vertex (which is called basis vertex)
  corresponds to four basis functions.

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Hierarchical T-mesh
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Basis construction over hierarchical T-meshes

• At each new level, basis functions in the
  previous level are modified, and add new basis
  functions corresponding to each new basis vertex.

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Illustrations of basis functions
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General T-meshes

• For general T-meshes,basis functions for
  S(3,3,1,1) can be constructed analogously.
• The construction process can be generalized to
  the cases m?2?1, n?2?1.

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Polynomial Spline surfaces over T-meshes

• Let bi(u,v) be a set of basis functions over
  a T-mesh, and let Ci be the corresponding control
  points, then the polynomial spline surface over
  the T-mesh (PT-Splines for short) is defined by
• PT-Splines have similar properties like
  B-splines, such as convex property, affine
  invariant property, etc.

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• 4. Applications

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Adaptive Surface Fitting

• Given a mesh model, we fit the mesh model using
  PT-splines by successive refinement.

19231 points, 38,388 triangles. The result is
obtained in less than one second
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Adaptive Surface Fittingmethod

• Keep unchanged the control points associating
  with the old basis functions, and compute the
  control points for the new basis functions
• Determine the cells whose fitting errors are
  greater than some threshold e, and then subdivide
  these cells into sub-cells to form a new mesh,
  and construct basis functions for the new mesh.
• Only linear system of equations with small size
  are involved.

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Adaptive Surface Fitting--Examples
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Adaptive Surface Fitting--Examples
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Adaptive Surface Fitting--Examples
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Adaptive Surface Fitting--Examples
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Adaptive Surface Fitting--Examples
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Adaptive Surface Fitting--Examples
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Adaptive Surface Fitting--Timing
Time(Sec.)
Mesh
level
Facets
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Stitching PT-Splines
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Stitching PT-Splines
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NURBS Conversion

• PT-splines converted to NURBS

One patch 17 patches
47 patches
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NURBS Conversion

• NURBS converted to PT-Splines
• Remove superfluous control points

8835 CPs 8280 Patches 2592
CPs, 555 Patches, errorlt1.2
1260 CPs,
312 Patches, errorlt1.8
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Simplification

• Given a PH-Spline, Simplify it with a PH-Spline
  with less number of patches

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Future work

• Splines over general T-meshes
• Dimension calculation
• Basis construction
• Applications of splines over T-meshes
• Adaptive surface modeling
• Adaptive finite elements for solving PDEs
• Relation with wavelet and subdivision

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• Thank you for
• your attention

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• Domain

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TP Splines based on knot segments

• TP splines are defined on general T-meshes which
  allow T-junctions and L-junctions.
• Basis functions are semi-regular

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Local refinement
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Edges and Faces

• Edges E
• Horizontal edge Eh
• Vertical edge Ev
• Composite edge
• Faces F