Structure from Motion

Transcription

1 Structure from Motion Lecture-13 Moving Light Display 1

2 Shape from Motion Problem Given optical flow or point correspondences, compute 3-D motion (translation and rotation) and shape (depth). 2

3 S. Ullman Hanson & Riseman Webb & Aggarwal T. Huang Heeger and Jepson Chellappa Faugeras Zisserman Kanade Structure from Motion Pentland Van Gool Pollefeys Seitz & Szeliski Shahsua Irani Vidal & Yi Ma Medioni Fleet Tian & Shah - Photosynth 3

4 Tomasi and Kanade Factorization Orthographic Projection Assumptions The camera model is orthographic. The positions of p points in F frames (F>=3), which are not all coplanar, and have been tracked. The entire sequence has been acquired before starting (batch mode). Camera calibration not needed, if we accept 3D points up to a scale factor. 4

5 Feature Points Image points (This is not optical flow Mean Normalize Feature Points (A) 5

6 Orthographic Projection Orthographic Projection 3D world point (C) Orthographic projection i, j, k are unit vectors along X, Y, Z 6

7 If Origin of world is at the centroid of object points 7

8 (B) 3XP 2FX3 Rank of S is 3, because points in 3D space are not Co-planar Rank Theorem Without noise, the registered measurement matrix is at most of rank three. Because W is a product of two matrices. The maximum rank of S is 3. 8

9 Translation From (A) From (B) From (C) (D) is projection of camera translation along x-axis Translation 2FXP 2FX3 3XP 2FX1 1XP 9

10 Translation Projected camera translation can be computed: From (D) Noisy Measurements Without noise, the matrix must be at most of rank 3. When noise corrupts the images, however, will not be rank 3. Rank theorem can be extended to the case of noisy measurements. 10

11 Approximate Rank SVD 2FXP 2FXP PXP PXP Singular Value Decomposition (SVD) Theorem: Any m by n matrix A, for which,can be written as is diagonal mxn mxn nxn nxn are orthogonal 11

12 Singular Value Decomposition (SVD) If A is square, then are all square. nxn nxn nxn nxn Approximate Rank 3 P-3 2F 3 P-3 3 P-3 3 P-3 P 12

13 Approximate Rank The best rank 3 approximation to the ideal registered measurement matrix. Rank Theorem for noisy measurement The best possible shape and rotation estimate is obtained by considering only 3 greatest singular values of together with the corresponding left, right eigenvectors. 13

14 Approximate Rank Approximate Rotation matrix Approximate Shape matrix This decomposition is not unique Q is any 3X3 invertable matrix Approximate Rank How to determine Q? R and S are linear transformation of approximate Rotation and shape matrices Rows of rotation matrix are unit vectors and orthogonal 14

15 How to determine Q: Newton s Method is error Compute SVD of define Compute Algorithm Compute 15

16 Results..\..\CAP6411\Fall2002\tomasiTr92Figures.pdf Hotel Sequence 16

17 Results (rotations) Selected Features 17

18 Reconstructed Shape Comparison 18

19 House Sequence Reconstructed Walls 19

20 Further Reading C. Tomasi and T. Kanade. Shape and motion from image streams under orthography---a factorization method. International Journal on Computer Vision, 9(2): , November Computer Vision: Algorithms and Applications, Richard Szeliski, Section