1. Geometric Transformations
EE 7700
Geometric Transformations

2. Geometric Transformation
translation
Rotation matrix
scale
Scale matrix
rotation & scale
Rigid flow
Bahadir K. Gunturk

3. Affine Flow
Bahadir K. Gunturk

4. Perspective Flow
Bahadir K. Gunturk

5. Bahadir K. Gunturk

6. Bahadir K. Gunturk

7. Bahadir K. Gunturk

8. RANSAC: RANdom SAmple Consensus
EE 7730
RANSAC: RANdom SAmple Consensus

9. Outliers Consider the least squares fit for optical flow:
If some of the values are wrong, it will degrade the estimation.
Bahadir K. Gunturk

10. Outliers It is best not to include outliers in the estimation.
Line Fitting Problem: Given (x1,y1), …, (xN,yN); find the line y=ax+b
Outliers
Best fit is degraded due to the outliers
Bahadir K. Gunturk

11. Robust Estimation Two step process:
Classify data points as outliers or inliers
Use inliers only to fit a model
Bahadir K. Gunturk

12. RANSAC Repeat for k times:
Randomly choose n points (the smallest number of points required) from the data.
Estimate the parameters using these points.
For each data point other than these n points:
Check if the data point is within a threshold, t, distance of current model; if it is, the data point is consistent with current model.
The total number of data points that are consistent is model’s support.
If the support is larger than a predetermined number, d, then there is a good fit. Re-estimate the parameters using these inliers.
Choose the best fit with the smallest fitting error.
Bahadir K. Gunturk

13. RANSAC Two samples and their supports for line-fitting
Bahadir K. Gunturk

14. Example Find the perspective parameters Bahadir K. Gunturk
from Hartley & Zisserman
Bahadir K. Gunturk

15. Example Apply corner detectors to both images Bahadir K. Gunturk
from Hartley & Zisserman
Bahadir K. Gunturk

16. Example Find the best match within a search window. Bahadir K. Gunturk
from Hartley & Zisserman
Bahadir K. Gunturk

17. Example Initial match results
from Hartley & Zisserman
188 matched features in left image pointing
to locations of corresponding right image features
Bahadir K. Gunturk

18. Example Inliers and outliers after RANSAC 89 outliers 99 inliers
from Hartley & Zisserman
89 outliers
99 inliers
Bahadir K. Gunturk

19. Panoramic Image Reconstruction
Find features
Match features
Fit parametric model
Application: Mosaic construction
Bahadir K. Gunturk

20. EE7730
Stereo Vision

21. Pinhole Camera
Bahadir K. Gunturk

22. Review: Perspective Projection
Bahadir K. Gunturk

23. Stereo p p’ p p’ scene point image plane optical center
Bahadir K. Gunturk

24. Stereo Constraints M Image plane Epipolar Line Y1 p p’ Y2 X2 O1 Z1 X1
Epipole
Focal plane
Bahadir K. Gunturk

25. From Geometry to AlgebraP p p’
All vectors shown lie on the same plane.
Bahadir K. Gunturk

26. From Geometry to AlgebraP p p’
Bahadir K. Gunturk

27. Matrix form of cross product
a=axi+ayj+azk
a×b=|a||b|sin(η)u
b=bxi+byj+bzk
Bahadir K. Gunturk

28. The Essential Matrix
Essential matrix
Bahadir K. Gunturk

29. A Simple Stereo System Right image: Left image: target reference
LEFT CAMERA
RIGHT CAMERA
baseline
Elevation Zw
disparity
Depth Z
Right image:
target
Left image:
reference
Zw=0
Bahadir K. Gunturk

30. Parallel Cameras P Z xl xr f pl pr Ol Or Disparity: T
T is the stereo baseline
Bahadir K. Gunturk

31. Stereo View
Left View
Right View
Bahadir K. Gunturk
Disparity

32. Correlation Approach
LEFT IMAGE
(xl, yl)
(0). Essential Equation represents actually the epipolar plane in either the left or the right image
(1). Epipolar line in the right image given pl
(Epl)Tpr=0
zr = fr
extension of the equations in pr = (xr,yr,fr)
(2). Epipolar line in the left image given pr
(prTE) pl=0
zl = fl
For Each point (xl, yl) in the left image, define a window centered at the point
Bahadir K. Gunturk

33. Correlation Approach
RIGHT IMAGE
(xl, yl)
(0). Essential Equation represents actually the epipolar plane in either the left or the right image
(1). Epipolar line in the right image given pl
(Epl)Tpr=0
zr = fr
extension of the equations in pr = (xr,yr,fr)
(2). Epipolar line in the left image given pr
(prTE) pl=0
zl = fl
… search its corresponding point within a search region in the right image
Bahadir K. Gunturk

34. Correlation Approach
RIGHT IMAGE
(xr, yr) dx
(xl, yl)
(0). Essential Equation represents actually the epipolar plane in either the left or the right image
(1). Epipolar line in the right image given pl
(Epl)Tpr=0
zr = fr
extension of the equations in pr = (xr,yr,fr)
(2). Epipolar line in the left image given pr
(prTE) pl=0
zl = fl
… the disparity (dx, dy) is the displacement when the correlation is maximum
Bahadir K. Gunturk

35. Stereo results Data from University of Tsukuba Scene Ground truth
(Seitz)
Bahadir K. Gunturk

36. Results with window correlation
Estimated depth of field
Ground truth
(Seitz)
Bahadir K. Gunturk

37. Results with better method
A state of the art method
Boykov et al., Fast Approximate Energy Minimization via Graph Cuts,
International Conference on Computer Vision, September 1999.
Ground truth
Bahadir K. Gunturk
(Seitz)

38. Applications courtesy of Sportvision First-down line
Bahadir K. Gunturk

39. Applications courtesy of Princeton Video Image Virtual advertising
Bahadir K. Gunturk