Smooth Spline Surfaces over Irregular Topology
Hui-xia XuWednesday, Apr. 4, 2007
Background
limitation
an inability of coping with surfaces of irregular topology, i.e., requiring the control meshes to form a regular quadrilateral structure
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
Subdivision Surfaces
Subdivision Surfaces---main idea
polygon mesh
iteratively applying
resultant meshconverging to
smooth surfacerefinement procedure
Subdivision Surfaces---magnum opus
Catmull-Clark surfaces E Catmull and J Clark. Recursively generated B-spline surface
s on arbitrary topological meshes, Computer Aided Design 10(1978) 350-355.
Doo-Sabin surfaces D Doo and M Sabin. Behaviour of recursive division surfaces n
ear extraordinary points, Computer Aided Design 10 (1978) 356-360.
About Subdivision Surfaces
advantagesimplicity and intuitive corner cutting
interpretation
shortageThe subdivision surfaces do not admit a
closed analytical expression
Spline Surfaces
Method 1
the technology of manifolds C Grimm and J Huges. Modeling surfaces of arbitrary topolog
y 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
Method 2
isolate irregular points C Loop and T DeRose. Generalised B-spline surfaces of arbitrary topo
logy, Proceedings of SIGGRAPH (1990) 347-356 J Peters. Biquartic C1-surface splines over irregular meshes, Comput
er-Aided Design 12(1995) 895-903 J J Zheng et al. Smooth spline surface generation over meshes of irr
egular topology, Visual Computer(2005) 858-864 J J Zheng et al. C2 continuous spline surfaces over Catmull-Clark me
shes, 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.
Smooth Spline Surface Generation over Meshes of Irregular Topo
logyJ J Zheng , J J Zhang, H J Zhou and L G She
nVisual Computer 21(2005), 858-864
What to Do
In this paper, an efficient method generates a generalized bi-quadratic B-spline surface and achieves C1 smoothness.
Zheng-Ball Patch A Zheng-Ball patch is a generation of a Sabin p
atch 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.
Zheng-Ball Patch An n-sided Zheng-Ball patch of degree m is def
ined by the following :
This patch model is able to smoothly
blend the surrounding regular patches
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
Fig 1. Control points for a six-sided quadratic Zheng-Ball patch
Spline Surface Generation---irregular closed mesh
Generate a new refined meshcarry out a single Catmull-Clark subdivision over th
e user-defined irregular mesh
Construct a C1 smooth spline surfaceregular vertex---a bi-quadratic Bézier patchOtherwise---a quadratic Zheng-Ball patch
Related Terms
ValenceThe valence of a point is the number of its
incident edges.
Regular vertexIf its valence is 4, the vertex is said to be
regular.
Regular faceA face is said to be regular if none of its
vertices are irregular vertices.
Catmull-Clark Surfaces---subdivision rules
Generation of geometric points
Construction of topology
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
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
Mesh Subdivision
Fig 2. Applying Catmull-Clark subdivision once to vertex V with valence n
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
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
Overall C1 Continuity
Fig 3. Two adjacent patches joined with C1 continuity
Geometric Model
Fig 4. Closed irregular mesh and the resulting geometric model
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
Subdivision Rules for Mesh Boundaries
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
Illustration
Fig 5. Subdivision around a boundary vertex v (n=3)
Boundary mesh subdivision for valence>3
For each vertex V of valence>3, n new vertices Wi (i=1,2, …,n) are created by
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
Concave Boundary Vertex
Fig 7. Left: Concave inner boundary vertex V of valence 4. Right: New boundary vertices W1 and W4 of valence 3
Boundary Patches
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
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 + di+1 ) is a corner control point if the valences of
di and di+1 are 3. ½*(d2i-1 + d) and ½*(d2i+1 + d) are the two mid-edge con
trol points if fi is not a boundary face. The centroid of face fi is a corner point if fi is not a bou
ndary face.
Generation Rules--- intermediate vertex
Fig 8. Intermediate vertex d (valence 5). Control points (○) for the patch corresponding to it
Geometric Model
Fig 9. Two models generated from open meshes by proposed method
Conclusions
Fig 10. Sphere produced with Loop’s method (left ) and with the proposed method (right )
Interactive Deformation of Irregular Surface Models
J J Zheng and J J ZhangLNCS 2330(2002), 239-248
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.
Background
Interactive deformation involves possibly the following user-controlled deformation operationsmoving control points of a patchspecifying geometric constraints for a patchdeforming a patch by exerting virtual forces
By far the most difficult task is to all these operations without violating their connection smoothness
Outline of the Proposed Research
This paper will concentrate on two issuesmodeling of irregular surface patches
Zheng-Ball model
the connection between different patchesformulate an explicit formula to degree elevatio
n and to insert a necessary number of extra control points
Zheng-Ball Patch This patch model can have any number of side
s 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
Zheng-Ball Patch
Fig 11. 3-sided cubic Zheng-Ball Patch with its control points
(m=3)
Explicit Formula of Degree Elevation
explicit
formula
Explicit Formula of Degree Elevation
The functions are defined by
The functions are defined by
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
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
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.
Global Energy Function
The new global energy functional is given by
where
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 !
Remarks
Typical G1 continuity constraints for the two patches
and can be expressed by the following:
Remarks
Fig 13. Two cubic patches share a common boundary
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.
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
Algorithm for Interactive Deforming
Fig 16. Algorithm if interactive deformation
Algorithm for Interactive Deforming
l>k. There are free variable left in linear constraints. So linear system can be solved.
l<=k. 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.
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.
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
Thank you!Thank you!
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