Rotation and Orientation: Affine Combination

Rotation and Orientation:Affine Combination Jehee Lee Seoul National University

Applications • What do we do with quaternions ? • Curve construction • Keyframe animation

Applications • What do we do with quaternions ? • Filtering • Convolution

Applications • What do we do with quaternions ? • Statistical analysis • Mean

Applications • What do we do with quaternions ? • Curve construction • Keyframe animation • Filtering • Convolution • Statistical analysis • Mean • It’s all about weighted sum !

Weighted Sum • How to generalize slerp for n-points • Affine combination of n-points • Methods • Re-normalization • Multi-linear • Global linearization • Functional Optimization

Inherent problem • Weighted sum may have multiple solutions • Spherical structure • Antipodal equivalence

Re-normalization • Expect result to be on the sphere • Weighed sum in R • Project onto the sphere 4

Re-normalization • Pros • Simple • Efficient • Cons • Linear precision • Singularity: The weighted sum may be zero

Multi-Linear Method • Evaluate n-point weighted sum as a series of slerps Slerp Slerp

De Casteljau Algorithm • A procedure for evaluating a point on a Bezier curve t : 1-t P(t) t : 1-t t : 1-t

Quaternion Bezier Curve • Multi-linear construction • Replace linear interpolation by slerp • Shoemake (1985)

Quaternion Bezier Spline • Find a smooth quaternion Bezier spline that interpolates given unit quaternions • Catmull-Rom’s derivative estimation

Quaternion Bezier Spline • Find a smooth quaternion Bezier spline that interpolates given unit quaternions • Catmull-Rom’s derivative estimation • Bezier control points (qi, ai, bi, qi+1) of i-th curve segment

Multi-Linear Method Slerp is not associative

Multi-Linear Method • Pros • Simple, intuitive • Inherit good properties of slerp • Cons • Need ordering • Eg) De Casteljau algorithm • Algebraically complicated

Global Linearization

Global Linearization • Pros • Easy to implement • Versatile • Cons • Depends on the choice of the reference frame • Singularity near the antipole

Functional Optimization • In vector spaces • We assume that this weighted sum was derived from a certain energy function

Functional Optimization • In vector spaces Functional Minimize Weighted sum

Functional Optimization • In orientation space • Buss and Fillmore (2001) • Spherical distance • Affine combination satisfies

Functional Optimization • Pros • Theoretically rigorous • Correct (?) • Cons • Need numerical iterations (Newton-Rapson) • Slow

Summary • Re-normalization • Practically useful for some applications • Multi-linear method • Slerp ordering • Global linearization • Well defined reference frame • Functional optimization • Rigorous, correct

Summary • We don’t have an ultimate solution • An appropriate solution may be determined by application • More specific problems may have better solutions • For convolution filters, points have an ordering

Rotation and Orientation: Affine Combination