Rotation and Orientation Affine Combination Jehee Lee Seoul

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 4 – Project onto the sphere

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

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