T-splines Speaker : 周 联

Mian works Sederberg,T.W., Zheng,J.M., Bakenov,A., Nasri,A., T-splines and T-NURCCS. SIGGRAPH Sederberg,T.W., David L. C., Zheng, J.M., Lyche,T., T-spline Simplication and Local Refinement. SIGGRAPH 2004

Authors Thomas W. Sederberg, Brigham Young University Jianmin Zheng, Nanyang Technological University Almaz Bakenov, Embassy of Kyrgyz Republic Washington, D.C. Ahmad Nasri, American University of Beirut Tom Lyche, University of Oslo David L. Cardon, Brigham Young University

Other works Song W.H., Yang X.N., Free-form deformation with weighted T-spline, The Visual Computer 2005 Xin Li, Jiansong Deng, Falai Chen, Dimensions of Spline Spaces Over 3D Hierarchical T-Meshes, Journal of Information and Computational Science, Vol.3, No.3, , (EI) Zhangjing Huang, Jiansong Deng, Yuyu Feng, and Falai Chen, New Proof of Dimension Formula of Spline Spaces over T-meshes via Smoothing Cofactors, Journal of Computational Mathematics, Vol.24, No.4, , 2006 Jiansong Deng, Falai Chen, Yuyu Feng, Dimensions of spline spaces over T- meshes, Journal of Computational and Applied Mathematics, Vol.194, No.2, , Xin Li, Jiansong Deng, Falai Chen, Surface Modeling with Polynomial Splines over Hierarchical T-meshes, accepted by CAD/CG'2007, and published on The Visual Computer, Jiansong Deng, Falai Chen, etal., Polynomial splines over hierarchical T-meshes, submitted to Graphical Models,

What are T-Splines? T-Splines: a generalization of non-uniform B-spline surfaces "T-Splines are the next thing...They have opened up possibilities to work with surfaces that were simply impossible before." -- Eric Allen, Production Manager, DAZ

Why use T-Splines? Add detail only where you need it Create even the most complex shapes as a single, editable surface Create natural edge flow and non- rectangular topology

Why use T-Splines? Fits into your Workflow

T-Splines vs. NURBS Reduce the number of superfluous control points.

T-Splines vs. NURBS Remove unwanted ripples.

T-Splines vs. NURBS Remove gap

Other methods hierarchical B splines  D. Forsey, R.H. Bartels, Hierarchical B-spline refinement, Comput. Graphics 22 (4) (1988) 205–212 a spline space over a more general T-mesh, where crossing, T-junctional, and L-junctional vertices are allowed.  F. Weller, H. Hagen, Tensor-product spline spaces with knot segments, in: M. Dalen, T. Lyche, L.L. Schumaker (Eds.), Mathematical Methods for Curves and Surfaces, Vanderbilt University Press, Nashville, TN, 1995, pp. 563–572.

Polar Form Definition: Examples:

Polar Form Definition: Ramshaw, L Blossoms are polar forms. Computer Aided Geometric Design 6,

Polar Form

Knot Intervals

PB-splines whose control points have no topological relationship with each other it is point based instead of grid based

PB-splines

T-mesh A T-spline is a PB-spline by means of a control grid called a T-mesh.

T-mesh Infer knot vectors from T-grid

Two rules for T-mesh

Control Point Insertion (2003)

Local knot insertion

Create features

Extract Bezier Patches

Merge B-splines into a T-spline Traditional way Use cubic NURSSes (SIGGRAPH 98)

Merge B-splines into a T-spline

Examples

A problem A local knot insertion sometimes requires that other local knot insertions must be performed. Are there cases in which these prerequisites cannot all be satisfied?

T-spline Local Refinement (2004)

Blending Function Refinement

T-spline Spaces T-spline Spaces : the set of all T-splines that have the same T-mesh topology, knot intervals, and knot coordinate system.

T-spline Spaces

Local Refinement Algorithm

= +

Compared with old one always work requires far fewer unrequested control point insertions

T-spline Simplication

Compared with B-spline wavelet decomposition

Create a T-spline model

Convert a T-spline into a B-spline surface

Standard T-splines

Semi-Standard T-splines

Some open questions What T-mesh configurations yield a standard T-spline? Are T-spline blending functions always linearly independent? What are the fairness properties of PB-splines?

Free-form deformation with weighted T-spline Wenhao Song, XunnianYang The Visual Computer (2005)

Weighted 3D PB-splines

T-lattice A T-lattice is a rectangular parallelepiped, that allows T-junctions.  Rule 1. In each minimal cell, the sum of knot intervals in the same direction must be equal.  Rule 2. Any edge must be a cell edge.  Rule 3. There are no zero edges in T lattices.

Automatic generation of T-lattice 1. Define the initial lattice to be deformed. 2. If the cell contains any vertex of the model, subdivide it by applying the octree subdivision. 3. Repeat steps 2 and 3 until a user-specified threshold is reached.

Automatic generation of T-lattice

Parametrization Nonlinear conjugate gradient method

Deformation algorithm Step 1. Define the initial region of the model to be deformed. Step 2. Generate the multiresolution lattices and set initial weights for T-spline volumes. Step 3. Calculate the parametric coordinates (u, v, w) for each point. Step 4. Modify the control points or weights and evaluate the new locations of the points.

Some open questions To generate control lattices with arbitrary topology that is better than octree subdivision lattices; To support hierarchical deformation; To import T-spline surfaces by creating a T-spline volume in which the surface is an isosurface; To implement w-TFFD by hardware acceleration

Dimensions of spline spaces over T-meshes Jiansong Deng, Falai Chen,Yuyu Feng Journal of Computational and Applied Mathematics 194 (2006) 267–283

The spline space over the given T-mesh

Dimension formula

Some open questions construction of a set of basis functions with good properties do not know whether the dimension relies on the geometry of the T-mesh or not