Mysteries of Parameterizing Camera Motion - Part 2

Transcription

1 Mysteries of Parameterizing Camera Motion - Part 2 Instructor - Simon Lucey Advanced Computer Vision Apps

2 Today Gauss-Newton and SO(3) Alternative Representations of SO(3) Exponential Maps SL(3) for Homographies

3 Example - Extrinsic Estimation x 1 x 2 x 4 x 3 NX arg min n=1 x n (w n ; ) 2 2 =[, ] extrinsics ( )! R 3 : R 2 projection function

4 Example - Extrinsic Estimation arg min y x F(y) 2 2

5 Example - Structure from Motion arg min y x F(y) 2 2

6 Reminder: Non-Linear Least Squares Many problems in vision can be expressed as solving a nonlinear least-squares objective. x F(y) 2 2 y

7 Reminder: Gauss-Newton Attempts to solve a non-linear least-squares problem by approximating it at as a sequence of linear least-squares problems. x F(y) 2 2 y If initialized properly, Gauss-Newton has quadratic convergence properties unlike steepest descent which is at most linear. 7

8 Reminder: Gauss-Newton Algorithm Gauss-Newton algorithm common strategy for optimizing non-linear least-squares problems. arg min y x F(y) 2 2 s.t. F : R N! R M Step 1: Carl Friedrich Gauss arg min y T y 2 2 Step 2: y! y + y Problem keep applying steps until y converges. Isaac Newton 8

9 Reminder: Non-Convex Set What happens if y belongs to a non-convex set? 9

10 Reminder: SO(3) is not a Convex Domain As is a rotation matrix it is constrained by the following, T = I det( ) =1 We refer to these matrices as belonging to the Special Orthogonal Group - SO(3). 1, 2 2 SO(3) 1 +(1 ) 2 2/ SO(3), 8 s.t. 0 apple apple 1

11 Today Gauss-Newton and SO(3) Alternative Representations of SO(3) Exponential Maps SL(3) for Homographies

12 Non-Uniqueness of Euler Angles We can define a rotation matrix through Euler angles. ( u, v, w )= ( u ) ( v ) ( w ) Euler angles do form a convex set. apple apple However, the representation is not unique as it is possible, ( u, v, w )= ( 0 u, 0 v, 0 w) u 6= 0 u v 6= 0 v w 6= 0 w 12

13 Example - Gimbal Lock 13

14 Quaternions Extension of complex numbers they take the form, q 0 + q 1 i + q 2 j + q 3 k where i, j and k are fundamental quaternion units. Unlike Euler angles, unit quaternions give a global parameterization of SO(3), at the cost of using four numbers instead of three to represent a rotation. 2 3 q = q 0 6q 1 4 q 2 q s.t. q 2 2 =1 William Rowan Hamilton 14

15 Quaternions - Problem apple q0 q 1 q = q 1 q 0 q 2 2 =1 Does q form a convex set? 15

16 Today Gauss-Newton and SO(3) Alternative Representations of SO(3) Exponential Maps SL(3) for Homographies

17 Right-hand rule Given three orthonormal 3D vectors (x, y, z) Then, z = x y

18 Uniquely Defining Rotation n n 2 =1 apple apple 18

19 Exponential Map n w(0) w( ) w( ) = (n, )w(0) 19

20 Exponential Map =[n] w(0) 20

21 Exponential Map n w( ) w( ) (I + [n] )w(0) 21

22 Exponential Map n w(0) w( ) w( ) =(I + [n] + 2! [n]2 + 3! [n]3 +...)w(0) 22

23 Exponential Map n w( ) w( ) =exp( [n] )w(0) 23

24 Exponential Map To more compactly represent a 3D rotation matrix, =exp( [n] ) Would this work in MATLAB??? >> skew [0,-n(3),n(2); n(3),0,n(1); -n(2),n(1),0]; >> R = exp(theta*skew(n)); >> R = expm(theta*skew(n)); 24

25 Exponential Map = Convex Set (!) =exp([!] ) where,! = n n 2 =1 apple apple therefore,! 2 R 3 s.t.! 2 apple!

26 Rodrigues Formula Rodrigues formula parametrizes a 3D rotation matrix uniquely in terms of the axis n and angle. = I +sin( )[n] +[1 cos( )][n] 2 where, 2 0 a 3 a 2 3 a b =[a] b = 4 a 3 0 a 1 a 2 a [a] 2 = aa T a 2 2I [a] 3 = a 2 2[a] R.M. Murray, Z. Li, and S.S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, Benjamin Olinde Rodrigues (6 October December 1851) 26

27 Short Angle Approximation For many optimization problems in vision involving rotation matrices people try to take advantage of the short angle approximation. sin( ) cos( ) 1 2 In MATLAB type, >> x = linspace(-1,1); >> plot(x,sin(x),x,x,x,cos(x),x,1-x.*x); 27

28 Short Angle Approximation? f( ) f( ) =sin( ) f( ) = f( ) = cos( ) f( ) =1 2 How could this be used to simplify linearization of? 28

29 Linearizing Naively we would linearize, (! +!) x! x y! y z! z where! =[! x,! y,! z ] T Using the short angle approximation why is it better to? (!) (0 +!) x! x y! y z! z Hint: (!) I +[!] when! 2 2 is small!!!! 29

30 Reminder: Gauss-Newton Algorithm Gauss-Newton algorithm common strategy for optimizing non-linear least-squares problems. arg min y x F(y) 2 2 s.t. F : R N! R M Step 1: Step 2: arg min y x y! y + T y 2 2 Is the update additive? Carl Friedrich Gauss keep applying steps until y converges. Isaac Newton 30

31 Today Gauss-Newton and SO(3) Alternative Representations of SO(3) Exponential Maps SL(3) for Homographies

32 Reminder: Homographies = 2 1 1

33 Homographies are not Convex Scale Ambiguous Scale ambiguity is controlled by det( )=+1 Constraint is not convex!!!

34 SL(3) Matrices Turns out any 3x3 matrix that has the constraint, det( )=+1 called Special Linear Group of dimension 3 - SL(3). Similar trick can be employed such that, 8X ( )=exp k=1 ka k! 2 R 8

35 SL(3) Matrices ( )=exp 8X k=1 ka k! A 1 = , A 3 = , A 5 = , A 7 = A 2 = , A 4 = , A 6 = , A 8 = (4 C. Mei, S. Benhimane, E. Malis and P. Rives, Homography-based Tracking for Central Catadioptric Cameras

36 Lie Algebra Exponential maps on the SO(3) and SL(3) groups are related to the much broader topic of Lie Algebra. More details on this topic can be found at in Murray et al Another group of particular interest in this group will be the SE(3) group. T = apple 0 T 1 Sophus Lie

37 More to read A Mathematical Introduction to Robotic Manipulation Richard M. Murray California Institute of Technology Zexiang Li Hong Kong University of Science and Technology S. Shankar Sastry University of California, Berkeley R.M. Murray, Z. Li, and S.S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, (Excellent!!!) c 1994, CRC Press All rights reserved This electronic edition is available from murray/mlswiki. Hardcover editions may be purchased from CRC Press, This manuscript is for personal use only and may not be reproduced, in whole or in part, without written consent from the publisher.