Smooth Spline Surfaces over Irregular Topology

1
Smooth Spline Surfaces over Irregular Topology

• Hui-xia Xu
• Wednesday, Apr. 4, 2007

2
Background

• limitation
• an inability of coping with surfaces of irregular
  topology, i.e., requiring the control meshes to
  form a regular quadrilateral structure

3
Improved Methods

• To overcome this limitation, a number of methods
  have been proposed. Roughly speaking, these
  methods are categorized into two groups
• Subdivision surfaces
• Spline surfaces

4
Subdivision Surfaces
5
Subdivision Surfaces---main idea
iteratively applying
resultant mesh converging to smooth surface
polygon mesh
refinement procedure
6
Subdivision Surfaces---magnum opus

• Catmull-Clark surfaces
• E Catmull and J Clark. Recursively generated
  B-spline surfaces on arbitrary topological
  meshes, Computer Aided Design 10(1978) 350-355.
• Doo-Sabin surfaces
• D Doo and M Sabin. Behaviour of recursive
  division surfaces near extraordinary points,
  Computer Aided Design 10 (1978) 356-360.

7
About Subdivision Surfaces

• advantage
• simplicity and intuitive corner cutting
  interpretation
• shortage
• The subdivision surfaces do not admit a closed
  analytical expression

8
Spline Surfaces
9
Method 1

• the technology of manifolds
• C Grimm and J Huges. Modeling surfaces of
  arbitrary topology using manifolds, Proceedings
  of SIGGRAPH (1995) 359-368
• J Cotrina Navau and N Pla Garcia. Modeling
  surfaces from meshes of arbitrary topology,
  Computer Aided Geometric Design 17(2000) 643-671

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Method 2

• isolate irregular points
• C Loop and T DeRose. Generalised B-spline
  surfaces of arbitrary topology, Proceedings of
  SIGGRAPH (1990) 347-356
• J Peters. Biquartic C1-surface splines over
  irregular meshes, Computer-Aided Design 12(1995)
  895-903
• J J Zheng et al. Smooth spline surface generation
  over meshes of irregular topology, Visual
  Computer(2005) 858-864
• J J Zheng et al. C2 continuous spline surfaces
  over Catmull-Clark meshes, Lecture Notes in
  Computer Science 3482(2005) 1003-1012
• J J Zheng and J J Zhang. Interactive deformation
  of irregular surface models, Lecture Notes in
  Computer Science 2330(2002) 239-248
• etc.

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Smooth Spline Surface Generation over Meshes of
Irregular Topology

• J J Zheng , J J Zhang, H J Zhou and L G Shen
• Visual Computer 21(2005), 858-864

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What to Do

• In this paper, an efficient method generates a
  generalized bi-quadratic B-spline surface and
  achieves C1 smoothness.

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Zheng-Ball Patch

• A Zheng-Ball patch is a generation of a Sabin
  patch that is valid for 3- or 5-sided areas. For
  more details, the following can be referred
• J J Zheng and A A Ball. Control point surface
  over non-four sided areas, Computer Aided
  Geometric Design 14(1997)807-820.
• M A Sabin. Non-rectangular surfaces suitable for
  inclusion in a B-spline surface, Hagen, T. (ed.)
  Eurographics (1983) 57-69.

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Zheng-Ball Patch

• An n-sided Zheng-Ball patch of degree m is
  defined by the following

This patch model is able to smoothly blend the
surrounding regular patches
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Zheng-Ball Patch

• the n-ple subscripts,
• n parameters of which only
  two are independent
• denotes the control points in ,as
  shown in Fig 1.
• the associated basis functions

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Fig 1. Control points for a six-sided quadratic
Zheng-Ball patch
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Spline Surface Generation---irregular closed
mesh

• Generate a new refined mesh
• carry out a single Catmull-Clark subdivision over
  the user-defined irregular mesh
• Construct a C1 smooth spline surface
• regular vertex---a bi-quadratic Bézier patch
• Otherwise---a quadratic Zheng-Ball patch

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Related Terms

• Valence
• The valence of a point is the number of its
  incident edges.
• Regular vertex
• If its valence is 4, the vertex is said to be
  regular.
• Regular face
• A face is said to be regular if none of its
  vertices are irregular vertices.

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Catmull-Clark Surfaces---subdivision rules

• Generation of geometric points
• Construction of topology

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Geometric Points

• new face points
• averaging of the surrounding vertices of the
  corresponding surface
• new edge points
• averaging of the two vertices on the
  corresponding edge and the new face points on the
  two faces adjacent to the edge
• new vertex points
• averaging of the corresponding vertices and
  surrounding vertices

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Topology

• connect each new face point to the new edge
  points surrounding it
• Connect each new vertex point to the new edge
  points surrounding it

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Mesh Subdivision
Fig 2. Applying Catmull-Clark subdivision once to
vertex V with valence n
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Mesh Subdivision

• new faces four-sided
• The valence of a new edge point is 4
• The valence of the new vertex point v remains n
• The valence of a new face point is the number of
  edges of the corresponding face of the initial
  mesh

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Patch Generation

• For a regular vertex, a bi-quadratic Bézier patch
  is used
• For an extraordinary vertex, an n-sided quadratic
  Zheng-Ball patch will be generated

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Overall C1 Continuity
Fig 3. Two adjacent patches joined with C1
continuity
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Geometric Model
Fig 4. Closed irregular mesh and the resulting
geometric model
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Spline Surface Generation---irregular open mesh

• Step 1 subdividing the mesh to make all faces
  four-sided
• Step 2 constructing a surface patch
  corresponding to each vertex
• The main task is to deal with the mesh boundaries

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Subdivision Rules for Mesh Boundaries
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Boundary mesh subdivision for 2- and 3-valent
vertices

• face point Centroid of the i-th face incident to
  V
• edge point averaging of the two endpoints in the
  associated edge
• vertex point equivalent to n-valent vertex V of
  the initial mesh

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Illustration
Fig 5. Subdivision around a boundary vertex v
(n3)
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Boundary mesh subdivision for valencegt3

• For each vertex V of valencegt3, n new vertices Wi
  (i1,2, ,n) are created by

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Convex Boundary Vertex
Fig 6. Left Convex boundary vertex V0 of valence
4. Right New boundary vertices V0 , W1 and W4
of valence 2 or 3
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Concave Boundary Vertex
Fig 7. Left Concave inner boundary vertex V of
valence 4. Right New boundary vertices W1 and
W4 of valence 3
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Boundary Patches
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Some Definitions

• Boundary vertex vertex on the boundary of the
  new mesh
• Boundary face at lease one of its vertices is a
  boundary vertex
• Intermediate vertex not a boundary vertex, but
  at least one of its surrounding faces is a
  boundary face
• Inner vertex none of the faces surrounding is a
  boundary face

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Generation Rules--- intermediate vertex

• d is a central control point
• d2i is a corner point if its valence is 2
• d2i-1 is a mid-edge control point if its valence
  is 3
• ½(di di1 ) is a corner control point if the
  valences of di and di1 are 3.
• ½(d2i-1 d) and ½(d2i1 d) are the two
  mid-edge control points if fi is not a boundary
  face.
• The centroid of face fi is a corner point if fi
  is not a boundary face.

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Generation Rules--- intermediate vertex
Fig 8. Intermediate vertex d (valence 5). Control
points (?) for the patch corresponding to it
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Geometric Model
Fig 9. Two models generated from open meshes by
proposed method
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Conclusions
Fig 10. Sphere produced with Loops method (left
) and with the proposed method (right )
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Interactive Deformation of Irregular Surface
Models

• J J Zheng and J J Zhang
• LNCS 2330(2002), 239-248

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Background

• Interactive deformation of surface models is an
  important research topic in surface modeling.
• However, the presence of irregular surface
  patches has posed a difficulty in surface
  deformation.

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Background

• Interactive deformation involves possibly the
  following user-controlled deformation operations
• moving control points of a patch
• specifying geometric constraints for a patch
• deforming a patch by exerting virtual forces
• By far the most difficult task is to all these
  operations without violating their connection
  smoothness

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Outline of the Proposed Research

• This paper will concentrate on two issues
• modeling of irregular surface patches
• Zheng-Ball model
• the connection between different patches
• formulate an explicit formula to degree elevation
  and to insert a necessary number of extra control
  points

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Zheng-Ball Patch

• This patch model can have any number of sides and
  is able to smoothly blend the surrounding regular
  patches
• This surface model is control-point based and to
  a large extent similar to Bézier surfaces

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Zheng-Ball Patch
Fig 11. 3-sided cubic Zheng-Ball Patch with its
control points
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Explicit Formula of Degree Elevation
(m3)
explicit
formula
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Explicit Formula of Degree Elevation

• The functions are defined by
• The functions are defined by

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After Degree Elevation
Fig 12. Quartic patches with control points after
degree elevation. The circles represent the
control points contributing to the C0 condition,
the black dots represent the control points
contributing G1 condition, and the square in the
middle represents the free central control point
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Central Control Point

• The central control point has provided an extra
  degree of freedom.
• Moving this control point will deform the shape
  of the blending patch intuitively, without
  violating the continuity conditions

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Energy function

• For an arbitrary patch , an energy function
  is defined by
• where Vi, Ki and Fi are the control point
  vector,
• stiffness matrix and force vector,
  respectively.

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Global Energy Function

• The new global energy functional is given by
• where

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Deformation Function

• The continuity constraints are defined by the
  following linear matrix equation

Minimising the global energy function subject to
the continuity constraints leads to the
production of a deformed model consisting of both
regular and irregular patches !
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Remarks

• Typical G1 continuity constraints for the two
  patches
• and can be expressed by the
  following

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Remarks
Fig 13. Two cubic patches share a common boundary
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Illustration
Fig 14. Model with 3- and 5-sided patches (green
patches). (Middle and Right) Deformed models.
There are eight triangular patches on the outer
corners of the model, and eight pentagonal
patches on the inner corners of the model.
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Algorithm for Interactive Deforming

• If physical forces are applied to the surface,
  the following linear system is generated by
  minimising the quadratic form
• Subjuect to linear constraints

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Algorithm for Interactive Deforming
Fig 16. Algorithm if interactive deformation
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Algorithm for Interactive Deforming

• lgtk. There are free variable left in linear
  constraints. So linear system can be solved.
• lltk. There is no free variable left in linear
  constraints. So linear system is not solvable.
• In the latter case, extra degrees of freedom are
  needed to solve linear system.

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Smooth Models
Fig 17. A smooth model with 3- and 5-sided cubic
surface patches (left). Deformed model after
twice degree elevation (right). Arrows indicate
the forces applied on the surface points.
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Conclusions

• Proposed a surface deformation technique
• no assumption is made for the degrees of freedom
• all surface patches can be deformed in the
  unified form
• during deformation process, the smoothness
  conditions between patches will be maintained
• Derived an explicit formula for degree elevation
  of irregular patches

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Thank you!