Interpolation of Transformation

Transcription

1 Spatial Rotation

2 Interpolation of Transformation I 3,0 U 12 K v V Kv B 2 U 11 B 1

3 Interpolation of Transformation I 3,0 R,t U 12 K v V Kv B 2 U 11 B 1

4 Interpolation of Transformation I 3,0 R,t U 12 K v V Kv B 2 U 11 B 1

5 Recall: Rotate and then, Translate Camera C R W, t C 3D world rx1 rx2 rx3 t x C C = = C RW t ry1 ry2 ry3 t y 1 rz1 rz2 rz3 t z Ground plane C = ( x C, y C, z C ) = ( x, y, z ) Origin at world coordinate where camera. C t is translation from world to camera seen from Rotate and then, translate. C t Camera World

6 Recall: Translate and the, Rotate Camera C R W, t C 3D world rx1 rx2 rx3 t x C C = = C RW t ry1 ry2 ry3 t y 1 rz1 rz2 rz3 t z Ground plane C = ( x C, y C, z C ) = ( x, y, z ) Origin at world coordinate where camera. C t is translation from world to camera seen from Rotate and then, translate. cf) Translate and then, rotate. C Camera World rx1 rx2 r x3 1 -C x C C = RW C = ry1 ry2 ry3 1 -C y where C world. 1 r r r 1 -C z1 z2 z3 z is translation from world to camera seen from

7 Interpolation of Translation I 3,0 R,t 3 - u K R t KR I C 1 1 U 12 K v V Kv B 2 U 11 B 1

8 Interpolation of Translation I 3,0 R,t 3 - u K R t KR I C 1 1 Rot. Trans. Trans. Rot. U 12 Translation is independent on rotation. K v V Kv B 2 U 11 B 1 How to interpolate translation? C C C x x 1 2 y y 1 C1 C2 C2 z z C1 C2

9 Interpolation of Translation I 3,0 R,t 3 - u K R t KR I C 1 1 Rot. Trans. Trans. Rot. U 12 Translation is independent on rotation. K v V Kv B 2 U 11 B 1 How to interpolate translation? C C C x x 1 2 y y 1 C1 C2 C2 z z C1 C2 Interpolated camera center: C wc + ( w ) C w [0,1] i 1 1 2

10 Interpolation of Rotation 2D coordinate transform: cos x = y 1 sin1 cos 1 y x sin = ( xy, ) = ( x, y )

11 Interpolation of Rotation 2D coordinate transform: cos x = y 1 sin1 cos 1 y x sin = ( xy, ) = ( x, y ) cos1 sin1 2 2 det = cos 1sin 11 sin1 cos 1 1

12 Interpolation of Rotation 2D coordinate transform: cos x = y 1 sin1 cos 1 y x sin = ( xy, ) = ( x, y ) cos x = y 2 sin2 cos 2 y x sin

13 Interpolation of Rotation 2D coordinate transform: cos x = y 1 sin1 cos 1 y x sin = ( xy, ) = ( x, y ) w (1 w) 1 2 w [0,1] 2 cos x = y 2 sin2 cos 2 y x sin

14 Interpolation of Rotation in 3D = ( x, y, z ) = ( x 2, y 2, z 2 ) 2 rx1 rx2 r x3 = r r r = R rz1 rz2 rz3 y1 y2 y3 1 1

15 Interpolation of Rotation in 3D = ( x, y, z ) = ( x 2, y 2, z 2 ) 2 rx1 rx2 r x3 = r r r = R rz1 rz2 rz3 y1 y2 y3 1 1 How to interpolate between two coordinates? R R 1 2 # dof: 3 # of parameters: 9

16 Axis Angle Representation

17 Axis Angle Representation

18 Axis Angle Representation e e = e e x y z

19 Axis Angle Representation x y z e = e e e Any displacement of a rigid body such that a point on the rigid body remains fixed, is euivalent to a single rotation about some axis that runs through the fixed point. Leonhard Euler

20 Axis Angle Representation Leonhard Euler e e = e e x y z Rotation: e where e =1 Any displacement of a rigid body such that a point on the rigid body remains fixed, is euivalent to a single rotation about some axis that runs through the fixed point.

21 Im x = rexp( i ) r cos isin x 1 r 1 Re

22 Im x = rexp( i ) r cos isin x 1 Ref) exp( x ) x k k! k 0 r 1 Re

23 Im x = rexp( i ) r cos isin x 2 x 1 x2 = exp( i ) x1 r cos i sin cos1 i sin1 1 r Re r cos cos sin sin i cos sin sin cos r cos( ) isin( ) r cos isin

24 2D Exponential Im x = rexp( i ) r cos isin x 2 x 1 x2 = exp( i ) x1 r cos i sin cos1 i sin1 1 r Re r cos cos sin sin i cos sin sin cos r cos( ) isin( ) r cos isin

25 Im x = rexp( i ) r cos isin x 2 x 1 x2 = exp( i ) x1 r cos i sin cos1 i sin1 2 r r cos cos 1 sin sin 1 icos sin 1 sin cos 1 1 Re r cos( ) isin( ) r 2 1 cos isin

26 3D Exponential Map: Quaternion e e = e e x y z exp e cos sin e e e i j k i j k ijk -1 x y z i Rotation: e where e =1 k j ij k jk i ki j

27 Exercise e e = e e x y z exp e cos sin e e e i j k x y z Find a uaternion such that it describes a rotation of 60 degrees about the axis a=[3, 4, 0]. Rotation: e where e =1

28 Exercise e e = e e x y z exp e cos sin e e e i j k x y z Find a uaternion such that it describes a rotation of 60 degrees about the axis a=[3, 4, 0] e a / a i j k Unit vector cos sin i j k cos sin i j k Rotation: e where e = i j k

29 3D Exponential Map: Quaternion e e = e e x y z exp e cos sin e e e i j k i j k ijk -1 Broom Bridge, Dublin, Ireland x y z i Rotation: e where e =1 k j ij k jk i ki j

30 Quaternion Product W i j k Rotate and then, p: p W i j k pw ip jp kp W pw p p p i W p pw p p j p p p p k p p p p W W W W ˆ pˆ pˆ ˆ ˆ pˆ W pw W pw where ˆ i j + k p pw ip jp kp

31 Quaternion Product Example Rotating 90 degrees about axis. Rotating 90 degrees about axis. / 2 / 2 1 cos jsin 2 2 / 2 / 2 2 cos ksin ?

32 Quaternion Product Example ˆ ˆ ˆ ˆ ˆ ˆ W pw W pw p p p p / 2 / 2 1 cos jsin 2 2 / 2 / 2 2 cos ksin 2 2

33 Quaternion Product Example ˆ ˆ ˆ ˆ ˆ ˆ W pw W pw p p p p / 2 / 2 1 cos jsin 2 2 / 2 / 2 2 cos ksin 2 2 θ = e= i j k i j i j k i j k i j k

34 Quaternion in 4D Sphere W i j k W W 1 W

35 Quaternion in 4D Sphere Interpolation in the sphere W i j k W W 2 1 W

36 Quaternion in 4D Sphere Interpolation in the sphere 2 1 W i j k W W cos 1 2 Plane 1 W

37 Quaternion Interpolation Interpolation in the sphere 2 1 W i j k W W cos p 1 2 Plane w 1 W

38 Quaternion Interpolation Interpolation in the sphere 2 1 W i j k W W cos p 1 2 W Plane p w 2 1 = 1 cosw sin 1 cos w sin sincosw cossin w + sinw 1 2 sin sin(1- w) + sinw sin 1 2

39 Quaternion Interpolation Interpolation in the sphere 2 1 W i j k W W cos p 1 2 W Plane p w 2 1 = 1 cosw sin 1 cos w sin sincosw cossin w + sinw 1 2 sin sin(1- w) + sinw sin 1 2

40 Quaternion Interpolation Interpolation in the sphere 2 1 W i j k W W cos p 1 2 W Plane p w 2 1 = 1 cosw sin 1 cos w sin sincosw cossin w + sinw 1 2 sin sin(1- w) + sinw sin 1 2

41 Quaternion Interpolation Interpolation in the sphere 2 1 W i j k W W cos p 1 2 W Plane p w 2 1 = 1 cosw sin 1 cos w sin sincosw cossin w + sinw 1 2 sin sin(1- w) + sinw sin 1 2

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43 function CameraInterpolation R1 = eye(3); C1 = [0;0;0]; 2 = 2*(rand(4,1)-0.5); R2 = Quaternion2Rotation(2); C2 = 2*rand(3,1); [Rset, Cset] = InterpolateCoordinate(R1, C1, R2, C2, 20); CameraInterpolation.m

44 function CameraInterpolation R1 = eye(3); C1 = [0;0;0]; 2 = 2*(rand(4,1)-0.5); R2 = Quaternion2Rotation(2); C2 = 2*rand(3,1); [Rset, Cset] = InterpolateCoordinate(R1, C1, R2, C2, 20); CameraInterpolation.m function [Rset, Cset] = InterpolateCoordinate(R1, C1, R2, C2, n) Cx = linspace(c1(1), C2(1), n+1); Cy = linspace(c1(2), C2(2), n+1); Cz = linspace(c1(3), C2(3), n+1); Cset = [Cx; Cy; Cz]; w = 0 : 1/n : 1; 1 = Rotation2Quaternion(R1); 2 = Rotation2Quaternion(R2); omega = acos(1'*2); for i = 1 : length(w) = sin(omega*(1-w(i)))/sin(omega) * 1 + sin(omega*w(i))/sin(omega) * 2; Rset{i} = Quaternion2Rotation(); end InterpolateCoordinate.m

45 View Interpolation Looking left Looking right

46 View Interpolation (HW #3)

47 Single View Review (Where am I?)