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Comparing Two Motions

Comparing Two Motions. Jehee Lee Seoul National University. Distance Metric for Motion Sequences. How can we measure the difference between two stylistic walking motions? How do we represent motion sequences? Rotation/orientation makes problems We do we compare two time-series data?

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Comparing Two Motions

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  1. Comparing Two Motions Jehee Lee Seoul National University

  2. Distance Metric for Motion Sequences • How can we measure the difference between two stylistic walking motions? • How do we represent motion sequences? • Rotation/orientation makes problems • We do we compare two time-series data? • What if the duration of one series differ from the other?

  3. Affine Geometry Jehee Lee Seoul National University

  4. Geometric Programming • A way of handling geometric entities such as vectors, points, and transforms. • Traditionally, computer graphics packages are implemented using homogeneous coordinates. • We will review affine geometry and coordinate-invariant geometric programming.

  5. Example of coordinate-dependence • What is the “sum” of these two positions ? Point p Point q

  6. If you assume coordinates, … • The sum is (x1+x2, y1+y2) • Is it correct ? • Is it geometrically meaningful ? p = (x1, y1) q = (x2, y2)

  7. If you assume coordinates, … • Vector sum • (x1, y1) and (x2, y2) are considered as vectors from the origin to p and q, respectively. p = (x1, y1) (x1+x2, y1+y2) q = (x2, y2) Origin

  8. If you select a different origin, … • If you choose a different coordinate frame, you will get a different result p = (x1, y1) (x1+x2, y1+y2) q = (x2, y2) Origin

  9. Vector and Affine Spaces • Vector space • Includes vectors and related operations • No points • Affine space • Superset of vector space • Includes vectors, points, and related operations

  10. Points and Vectors • A point is a position specified with coordinate values. • A vector is specified as the difference between two points. • If an origin is specified, then a point can be represented by a vector from the origin. • But, a point is still not a vector in a coordinate-free point of view. Point q vector (p-q) Point p

  11. Vector spaces • A vector space consists of • Set of vectors, together with • Two operations: addition of vectors and multiplication of vectors by scalar numbers • A linear combination of vectors is also a vector

  12. Affine Spaces • An affine space consists of • Set of points, an associated vector space, and • Two operations: the difference between two points and the addition of a vector to a point

  13. Coordinate-Invariant Geometric Operations • Addition • Subtraction • Scalar multiplication • Linear combination • Affine combination

  14. Addition p + w u + v w v u p u + v is a vector p + w is a point u,v, w : vectors p, q : points

  15. Subtraction p p - w p - q -w u - v u v q p u - v is a vector p - q is a vector p - w is a point u,v, w : vectors p, q : points

  16. Scalar Multiplication scalar • vector = vector 1 • point = point 0 • point = vector c • point = (undefined) if (c≠0,1)

  17. Linear Combination • A linear space is spanned by a set of bases • Any vector in the space can be represented as a linear combination of bases

  18. Affine Combination

  19. Example • (p + q) / 2 : midpoint of line pq • (p + q) / 3 : Can you find a geometric meaning ? • (p + q + r) / 3 : center of gravity of ∆pqr • (p/2 + q/2 – r) : a vector from r to the midpoint of q and p

  20. Summary

  21. Rotation and Orientation:Fundamentals Jehee Lee Seoul National University

  22. What is Rotation ? • Not intuitive • Formal definitions are also confusing • Many different ways to describe • Rotation (direction cosine) matrix • Euler angles • Axis-angle • Rotation vector • Helical angles • Unit quaternions

  23. Orientation vs. Rotation • Rotation • Circular movement • Orientation • The state of being oriented • Given a coordinate system, the orientation of an object can be represented as a rotation from a reference pose

  24. Analogy (point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement) Reference coordinate system

  25. Analogy (point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement) point: the 3d location of the bunny vector: translational movement Reference coordinate system

  26. Analogy (point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement) point: the 3d location of the bunny vector: translational movement orientation: the 3d orientation of the bunny rotation : circular movement Reference coordinate system

  27. 2D Orientation Polar Coordinates

  28. 2D Orientation Although the motion is continuous, its representation could be discontinuous

  29. 2D Orientation Many-to-one correspondences between 2D orientations and their representations

  30. Extra Parameter

  31. Extra Parameter 2x2 Rotation matrix is yet another method of using extra parameters

  32. Complex Number Imaginary Real

  33. Complex Exponentiation Imaginary Real

  34. 2D Rotation • Complex numbers are good for representing 2D orientations, but inadequate for 2D rotations • A complex number cannot distinguish different rotational movements that result in the same final orientation • Turn 120 degree counter-clockwise • Turn -240 degree clockwise • Turn 480 degree counter-clockwise Imaginary Real

  35. 2D Rotation and Orientation • 2D Rotation • The consequence of any 2D rotational movement can be uniquely represented by a turning angle • 2D Orientation • The non-singular parameterization of 2D orientations requires extra parameters • Eg) Complex numbers, 2x2 rotation matrices

  36. Operations in 2D • (orientation) : complex number • (rotation) : scalar value • exp(rotation) : complex number

  37. 2D Rotation and Displacement Imaginary Real

  38. 2D Rotation and Displacement Imaginary Real

  39. 2D Orientation Composition Imaginary Real

  40. 2D Rotation Composition Imaginary Real

  41. Analogy

  42. 3D Rotation • Given two arbitrary orientations of a rigid object,

  43. 3D Rotation • We can always find a fixed axis of rotation and an angle about the axis

  44. Euler’s Rotation Theorem In other words, • Arbitrary 3D rotation equals to one rotation around an axis • Any 3D rotation leaves one vector unchanged The general displacement of a rigid body with one point fixed is a rotation about some axis Leonhard Euler (1707-1783)

  45. Rotation Vector • Rotation vector(3 parameters) • Axis-Angle(2+1 parameters)

  46. 3D Orientation • Unhappy with three parameters • Euler angles • Discontinuity (or many-to-one correspondences) • Gimble lock • Rotation vector (a.k.a Axis/Angle) • Discontinuity (or many-to-one correspondences)

  47. Using an Extra Parameter • Euler parameters

  48. Quaternions • William Rowan Hamilton (1805-1865) • Algebraic couples (complex number) 1833 where

  49. Quaternions • William Rowan Hamilton (1805-1865) • Algebraic couples (complex number) 1833 • Quaternions 1843 where where

  50. Unit Quaternions • Unit quaternions represent 3D rotations

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