1. CSCE 643 Computer Vision: Structure from Motion
Jinxiang Chai

2. Stereo reconstruction
Given two or more images of the same scene or object, compute a representation of its shape
known
camera
viewpoints

3. Stereo reconstruction
Given two or more images of the same scene or object, compute a representation of its shape
known
camera
viewpoints
How to estimate camera parameters?
- where is the camera?
- where is it pointing?
- what are internal parameters, e.g. focal length?

4. Calibration from 2D motion
Structure from motion (SFM)
- track points over a sequence of images
- estimate for 3D positions and camera positions
- calibrate intrinsic camera parameters before hand
Self-calibration:
- solve for both intrinsic and extrinsic camera parameters

5. SFM = Holy Grail of 3D Reconstruction
Take movie of object
Reconstruct 3D model
Would be
commercially
highly viable

6. How to Get Feature Correspondences
Feature-based approach
- good for images
- feature detection (corners or sift features)
- feature matching using RANSAC (epipolar line)
Pixel-based approach
- good for video sequences
- patch based registration with lucas-kanade algorithm
- register features across the entire sequence

7. A Brief Introduction on Feature-based Matching
Find a few important features (aka Interest Points)
Match them across two images
Compute image transformation function h

8. Feature Detection
Two images taken at the same place with different angles
Projective transformation H3X3

9. Feature Matching ?
Two images taken at the same place with different angles
Projective transformation H3X3

10. How do we match features across images? Any criterion?
Feature Matching ?
Two images taken at the same place with different angles
Projective transformation H3X3
How do we match features across images? Any criterion?

11. How do we match features across images? Any criterion?
Feature Matching ?
Two images taken at the same place with different angles
Projective transformation H3X3
How do we match features across images? Any criterion?

12. Feature Matching Intensity/Color similarity
The intensity of pixels around the corresponding features should have similar intensity

13. Feature Matching Feature similarity (Intensity or SIFT signature)
The intensity of pixels around the corresponding features should have similar intensity
Cross-correlation, SSD

14. Feature Matching Feature similarity (Intensity or SIFT signature)
The intensity of pixels around the corresponding features should have similar intensity
Cross-correlation, SSD
Distance constraint
The displacement of features should be smaller than a given threshold

15. Feature Matching Feature similarity (Intensity or SIFT signature)
The intensity of pixels around the corresponding features should have similar intensity
Cross-correlation, SSD
Distance constraint
The displacement of features should be smaller than a given threshold
Epipolar line constraint
The corresponding pixels satisfy epipolar line constraints.

16. Feature Matching Feature similarity (Intensity or SIFT signature)
The intensity of pixels around the corresponding features should have similar intensity
Cross-correlation, SSD
Distance constraint
The displacement of features should be smaller than a given threshold
Epipolar line constraint
The corresponding pixels satisfy epipolar line constraints.
Fundamental matrix H

17. Feature-space Outlier Rejection
bad
Good

18. Feature-space Outlier Rejection
Can we now compute H3X3 from the blue points?

19. Feature-space Outlier Rejection
Can we now compute H3X3 from the blue points?

20. Feature-space Outlier Rejection
Can we now compute H3X3 from the blue points?
No! Still too many outliers…

21. Feature-space Outlier Rejection
Can we now compute H3X3 from the blue points?
No! Still too many outliers…
What can we do?

22. Feature-space Outlier Rejection
Can we now compute H3X3 from the blue points?
No! Still too many outliers…
What can we do?
Robust estimation!

23. Robust Estimation: A Toy Example
How to fit a line based on a set of 2D points?

24. RANSAC for Estimating Projective Transformation
RANSAC loop:
Select four feature pairs (at random)
Compute the transformation matrix H (exact)
Compute inliers where SSD(pi’, H pi) < ε
Keep largest set of inliers
Re-compute least-squares H estimate on all of the inliers
For more detail, check -
- Philip H. S. Torr (1997). "The Development and Comparison of Robust Methods for Estimating the Fundamental Matrix". International Journal of Computer Vision 24 (3): 271–300

25. Structure from Motion Two Principal Solutions
Bundle adjustment (nonlinear optimization)
Factorization (SVD, through orthographic approximation, affine geometry)

26. Projection Matrix Perspective projection:
2D coordinates are just a nonlinear function of its 3D coordinates and camera parameters: K R T P 26

27. Nonlinear Approach for SFM
What’s the difference between camera calibration and SFM?

28. Nonlinear Approach for SFM
What’s the difference between camera calibration and SFM?
- camera calibration: known 3D and 2D

29. Nonlinear Approach for SFM
What’s the difference between camera calibration and SFM?
- camera calibration: known 3D and 2D
- SFM: unknown 3D and known 2D

30. Nonlinear Approach for SFM
What’s the difference between camera calibration and SFM?
- camera calibration: known 3D and 2D
- SFM: unknown 3D and known 2D
- what’s 3D-to-2D registration problem?

31. Nonlinear Approach for SFM
What’s the difference between camera calibration and SFM?
- camera calibration: known 3D and 2D
- SFM: unknown 3D and known 2D
- what’s 3D-to-2D registration problem?

32. SFM: Bundle Adjustment
SFM = Nonlinear Least Squares problem
Minimize through
Gradient Descent
Conjugate Gradient
Gauss-Newton
Levenberg Marquardt common method
Prone to local minima

33. Count # Constraints vs #Unknowns
M camera poses
N points
2MN point constraints
6M+3N + 4 (unknowns)
Suggests: need 2mn  6m + 3n+4
But: Can we really recover all parameters???

34. Count # Constraints vs #Unknowns
M camera poses
N points
2MN point constraints
6M+3N+4 unknowns (known intrinsic camera parameters)
Suggests: need 2mn  6m + 3n+4
But: Can we really recover all parameters???
Can’t recover origin, orientation (6 params)
Can’t recover scale (1 param)
Thus, we need 2mn  6m + 3n+4 - 7

35. Are We Done?
No, bundle adjustment has many local minima.

36. SFM Using Factorization
Assume an orthographic camera
Image
World

37. SFM Using Factorization
Assume orthographic camera
Image
World
Subtract the mean

38. SFM Using Factorization
Stack all the features from the same frame:

39. SFM Using Factorization
Stack all the features from the same frame:
Stack all the features from all the images: W

40. SFM Using Factorization
Stack all the features from the same frame:
Stack all the features from all the images: W

41. SFM Using Factorization
Stack all the features from all the images:
Factorize the matrix into two matrix using SVD: W

42. SFM Using Factorization
Stack all the features from all the images:
Factorize the matrix into two matrix using SVD:

43. SFM Using Factorization
Stack all the features from all the images:
Factorize the matrix into two matrix using SVD:
How to compute the matrix ? W

44. SFM Using Factorization
M is the stack of rotation matrix:

45. SFM Using Factorization
M is the stack of rotation matrix: 1
Orthogonal constraints from rotation matrix 1 1 1

46. SFM Using Factorization
M is the stack of rotation matrix: 1
Orthogonal constraints from rotation matrix 1 1 1

47. SFM Using Factorization
Orthogonal constraints from rotation matrices: 1 1 1 1

48. SFM Using Factorization
Orthogonal constraints from rotation matrices: 1 1 1
QQ: symmetric 3 by 3 matrix 1

49. SFM Using Factorization
Orthogonal constraints from rotation matrices: 1 1 1
QQ: symmetric 3 by 3 matrix 1
How to compute QQT?
least square solution
- 4F linear constraints, 9 unknowns (6 independent due to symmetric matrix)

50. SFM Using Factorization
Orthogonal constraints from rotation matrices: 1 1 1
QQ: symmetric 3 by 3 matrix 1
How to compute QQT?
least square solution
- 4F linear constraints, 9 unknowns (6 independent due to symmetric matrix)
How to compute Q from QQT:
SVD again:

51. SFM Using Factorization
M is the stack of rotation matrix: 1
Orthogonal constraints from rotation matrix 1 1 1
QQT: symmetric 3 by 3 matrix
Computing QQT is easy:
3F linear equations
6 independent unknowns

52. SFM Using Factorization
Form the measurement matrix
Decompose the matrix into two matrices and using SVD
Compute the matrix Q with least square and SVD
Compute the rotation matrix and shape matrix:
and

53. Weak-perspective Projection
Factorization also works for weak-perspective projection (scaled orthographic projection): z0 d

54. Factorization for Full-perspective Cameras
[Han and Kanade]

55. SFM for Deformable Objects
For detail, click here 55

56. SFM for Articulated Objects
For video, click here

57. SFM Using Factorization
Bundle adjustment (nonlinear optimization)
- work with perspective camera model
- work with incomplete data
- prone to local minima
Factorization:
- closed-form solution for weak perspective camera
- simple and efficient
- usually need complete data
- becomes complicated for full-perspective camera model
Phil Torr’s structure from motion toolkit in matlab (click here)
Voodoo camera tracker (click here) 57

58. All Together Video Click here - feature detection
- feature matching (epipolar geometry)
- structure from motion
- stereo reconstruction
- triangulation
- texture mapping 58