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!