1. T ENSOR P RODUCT V OLUMES AND M ULTIVARIATE M ETHODS CAGD Presentation by Eric Yudin June 27, 2012

2. M ULTIVARIATE M ETHODS : O UTLINE Introduction and Motivation Theory Practical Aspects Application: Free-form Deformation (FFD)

3. I NTRODUCTION AND M OTIVATION Until now we have discussed curves ( ) and surfaces ( ) in space. Now we consider higher dimensional – so-called “multivariate” objects in

4. I NTRODUCTION AND M OTIVATION Examples Scalar or vector-valued physical fields (temperature, pressure, etc. on a volume or some other higher-dimensional object)

5. I NTRODUCTION AND M OTIVATION Examples Spatial or temporal variation of a surface (or higher dimensions)

6. I NTRODUCTION AND M OTIVATION Examples Freeform Deformation

7. M ULTIVARIATE M ETHODS : O UTLINE Introduction and Motivation Theory Practical Aspects Application: Free-form Deformation (FFD)

8. T HEORY – G ENERAL F ORM Definition (21.1) : The tensor product B-spline function in three variables is called a trivariate B- spline function and has the form It has variable u i, degree k i, and knot vector  i in the i th dimension

9. T HEORY – G ENERAL F ORM Definition (21.1) (cont.): Generalization to arbitrary dimension q : Determining the vector of polynomial degree in each of the q dimensions, n, (?), forming q knot vectors  i, i=1, …, q Let  =(u 1, u 2, …, u q ) Let i = (i 1, i 2, …, i q ), where each i j, j=1, …, q q -variate tensor product function: Of degrees k 1, k 2, …, k q in each variable

10. T HEORY – G ENERAL F ORM is a multivariate function from to given that If d > 1, then T is a vector (parametric function)

11. T HEORY – G ENERAL F ORM

12. T HEORY From here on, unless otherwise specified, we will concern ourselves only with Bézier trivariates and multivariates.

13. T HEORY – O PERATIONS & P ROPERTIES

14. T HEORY – O PERATIONS AND P ROPERTIES Boundary surfaces : The boundary surfaces of a TPB volume are TPB surfaces. Their Bézier nets are the boundary nets of the Bézier grid. Boundary curves : The boundary curves of a TPB volume are Bézier curve segments. Their Bézier polygons are given by the edge polygons of the Bézier grid. Vertices : The vertices of a TPB volume coincide with the vertices of its Bézier grid.

15. T HEORY – O PERATIONS AND P ROPERTIES

18. T HEORY – T ERMINOLOGY

19. T HEORY – CONSTRUCTORS Extruded Volume & Ruled Volume

20. T HEORY – CONSTRUCTORS

22. M ULTIVARIATE M ETHODS : O UTLINE Introduction and Motivation Theory Practical Considerations Application: Free-form Deformation (FFD)

23. A PPLICATION : F REE - FORM D EFORMATION Introduction & Motivation: Embed curves, surfaces and volumes in the parameter domain of a free-form volume Then modify that volume to warp the inner objects on a ‘global’ scale [DEMO]

24. A PPLICATION : F REE - FORM D EFORMATION

25. Process (Bézier construction): 1. Obtain/construct control point structure (the FFD) 2. Transform coordinates to FFD domain 3. Embed object into the FFD equation From the paper: Sederberg, Parry: “Free-form Deormation of Solid Geometric Models.” ACM 20 (1986) 151-160.

26. A PPLICATION : F REE - FORM D EFORMATION Obtain/construct control point structure (the FFD): A common example is a lattice of points P such that: Where is the origin of the FFD space S, T, U are the axes of the FFD space l, m, n are the degrees of each Bézier dimension i, j, k are the indices of points in each dimension Edges mapped into Bézier curves

27. A PPLICATION : F REE - FORM D EFORMATION

32. Examples Surfaces (solid modeling) Text (one dimension lower): Text Sculpt [DEMO]

33. M ULTIVARIATE M ETHODS : O UTLINE Introduction and Motivation Theory Practical Considerations Application: Free-form Deformation (FFD)

34. P RACTICAL A SPECTS – E VALUATION Tensor Product Volumes are composed of Tensor Product Surfaces, which in turn are composed of Bezier curves. Everything is separable, so each component can be handled independently

35. P RACTICAL A SPECTS – V ISUALIZATION Marching Cubes Algorithm Split space up into cubes For each cube, figure out which points are inside the iso-surface 2 8 =256 combinations, which map to 16 unique cases via rotations and symmetries Each case has a configuration of triangles (for the linear case) to draw within the current cube

36. P RACTICAL A SPECTS – V ISUALIZATION Marching Cubes Algorithm : 2D case With ambiguity in cases 5 and 10

37. P RACTICAL A SPECTS – V ISUALIZATION Marching Cubes Algorithm : 3D case. Generalizable by 15 families via rotations and symmetries.