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

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 t : 1-t P(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

Quaternion Bezier Spline Find a smooth quaternion Bezier spline that interpolates given unit quaternions –Catmull-Rom’s derivative estimation –Bezier control points (q i, a i, b i, q i+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

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