Motion Blending (Multidimensional Interpolation)

1. Motion Blending(Multidimensional Interpolation) Jehee Lee

2. Data-Driven Approach • It is difficult to understand basic principles of the real world • Instead, we would like to sample the real world ! • What kind of data is available ? • Pictures (camera), video (camcorder) • Motion, facial expression (motion capture) • Geometry (3D scanner) • Voice, sound (recorder) • Tactile, physical properties, … • Data-driven approaches try to reconstruct the real world in a computer from a rich set of samples

3. The real world is multi-dimensional • Simple (object space) classification • One dimension: Sketch, curve • Two dimension: Pictures, surface, tactile • Three dimension: 3D geometry, video • Four dimension: Particle motion(position + time) • High dimension: Articulated figure motion • Many interpretations (parameterizations) are possible independent of object space dimensions

4. Dimensionality of Motion • Motion can have many parameters (dimensionality) • Physical dimensions: Velocity, turning angle, reach position • Ambiguous dimensions: Style, emotion, mood [Rose et al. 98]

5. The Space of Motion • The space of every possible human walk • M-dimensional object space • N-dimensional parameter space Object space Parameter space

6. Multi-Dimensional Interpolation • 3-parameters (velocity, turning angle, happiness) • n-samples (motion clips) • Interpolation gives a motion for any given parameters • We want to compute a continuous function

7. How many samples are needed ? • One dimension: 1000 (points)

8. How many samples are needed ? • Two dimension: 1000x1000 = 1000000 (pixels)

9. How many samples are needed ? • High-dimension • We cannot acquire enough samples • The interpolating function should be reconstructed from scattered (possibly sparse) samples We have to solve a multi-dimensional interpolation problem with scattered samples

10. Multi-Dimensional Functions • One parameter (Curve) • Two parameters (Surface) • Three parameters (Volume) C(t) t v S(u,v) u v V(u,v,w) u w

11. Tensor-Product Surfaces • Cubic polynomial curve • Tensor-product surface • A curve is “multiplied” by a curve

12. Tensor-Product Functions • Tensor-product is a standard technique for increasing the dimension of parametric functions • Bezier surfaces, B-spline surfaces, NURBS surfaces, … • It works fine for low-dimensional parametric spaces • Great for surfaces • Maybe good enough for volumes • It can be problematic for higher-dimensions • Too many control points • High degree of basis functions

13. Radial Basis Functions • Radial refers to the pattern that you get when straight lines are drawn from the center of a circle to a number of points round the edge • Radial basis function • A real valued function • having a center in a parameter space • The function value is determined by a distance from the center

14. Scattered Data Interpolation using RBF • Find a function that interpolate given points such that • RBF interpolation • We use M basis functions (same as # of given data points) • We don’t have a grid structure (compare to tensor-product surfaces) • Radial basis function is easily defined in any dimensional spaces

15. Popular Choices for RBF • Thin-Plate • Multiquadric • Gaussian • Biharmonic • Triharmonic • See demo: RBFwithoutLinearApproximation.exe for some constant c for some constant c

16. Augmented Polynomial Function • A linear function or a low-order polynomial is augmented for better extrapolation • Under-specified linear system is obtained • Orthogonality conditions give a unique solution

17. Motion Blending • S. I. Park, et al., On-line Locomotion Generation Based on Motion Blending, Symposium on Computer Animation, 2002 • Movie

18. Application to Motion Blending • The length of time • The first motion is 3 seconds long, the second motion is 5 seconds long, and so on. • Those motions are normalized in time and blended • What is the length of the blend ? • The length of the blend can be negative • RBF interpolation doesn’t have convex hull property • Is it make sense to create a “backward walk” by blending “forward walks” ?

19. Application to Motion Blending • Handling unit quaternions (or rotation matrices) • What if we apply RBF interpolation component-wisely ? • Represent the surface as an affine combination • Construct cardinal basis functions using RBF

20. Application to Motion Blending • Inverse problem • Precise control usually requires numerical iterations • May not have a unique solution • Eg) Find a set of weights (coordinates in the parameter space) to create a walking motion of 8 m/s Actual walking speed Parameter 5.5 m/s 3 m/s 7.5 m/s 12 m/s

21. Other applications • Surface reconstruction [Carr et al. 2002]

22. Other applications • Shape morphing [Turk and O’Brien 99]

23. Summary • Multidimensional interpolation • Sparse samples in high-dimensional spaces • Time-series data • Scatter data interpolation using radial basis functions • The basis function is determined by applications • All basis function explained are not locally supported • For some applications, locally supported basis functions should be chosen