16_Multiple_Cameras
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1.Computer vision: models, learning and inference Chapter 16 Multiple Cameras
2.Structure from motion 2 2 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Given an object that can be characterized by I 3D points projections into J images Find Intrinsic matrix Extrinsic matrix for each of J images 3D points
3.Structure from motion 3 3 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince For simplicity, we’ll start with simpler problem Just J=2 images Known intrinsic matrix
4.Structure 4 4 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Two view geometry The essential and fundamental matrices Reconstruction pipeline Rectification Multiview reconstruction Applications
5.Epipolar lines 5 5 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
6.Epipole 6 6 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
7.Special configurations 7 7 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
8.Structure 8 8 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Two view geometry The essential and fundamental matrices Reconstruction pipeline Rectification Multiview reconstruction Applications
9.The geometric relationship between the two cameras is captured by the essential matrix. Assume normalized cameras, first camera at origin. First camera: Second camera: The essential matrix 9 9 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
10.The essential matrix 10 10 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince First camera: Second camera: Substituting: This is a mathematical relationship between the points in the two images, but it’s not in the most convenient form.
11.The essential matrix 11 11 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Take cross product with t (last term disappears) Take inner product of both sides with x 2 .
12.The cross product term can be expressed as a matrix Defining: We now have the essential matrix relation The essential matrix 12 12 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
13.Properties of the essential matrix 13 13 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Rank 2: 5 degrees of freedom Nonlinear constraints between elements
14.Recovering epipolar lines 14 14 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Equation of a line: or or
15.Recovering epipolar lines 15 15 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Equation of a line: Now consider This has the form where So the epipolar lines are
16.Recovering epipoles 16 16 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Every epipolar line in image 1 passes through the epipole e 1 . In other words for ALL This can only be true if e 1 is in the nullspace of E . Similarly: We find the null spaces by computing , and taking the last column of and the last row of .
17.Decomposition of E 17 17 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Essential matrix: To recover translation and rotation use the matrix: We take the SVD and then we set
18.Four interpretations 18 18 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince To get the different solutions, we mutliply t by 1 and substitute
19.The fundamental matrix 19 19 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Now consider two cameras that are not normalised By a similar procedure to before, we get the relation or where Relation between essential and fundamental
20.Fundamental matrix criterion 20 20 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
21.When the fundamental matrix is correct, the epipolar line induced by a point in the first image should pass through the matching point in the second image and viceversa. This suggests the criterion If and then Unfortunately, there is no closed form solution for this quantity. Estimation of fundamental matrix 21 21 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
22.The 8 point algorithm 22 22 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Approach: solve for fundamental matrix using homogeneous coordinates closed form solution (but to wrong problem!) Known as the 8 point algorithm Start with fundamental matrix relation Writing out in full: or
23.The 8 point algorithm 23 23 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Can be written as: where Stacking together constraints from at least 8 pairs of points, we get the system of equations
24.The 8 point algorithm 24 24 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Minimum direction problem of the form , Find minimum of subject to . To solve, compute the SVD and then set to the last column of .
25.Fitting concerns 25 25 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince This procedure does not ensure that solution is rank 2. Solution: set last singular value to zero. Can be unreliable because of numerical problems to do with the data scaling – better to rescale the data first Needs 8 points in general positions (cannot all be planar). Fails if there is not sufficient translation between the views Use this solution to start nonlinear optimisation of true criterion (must ensure nonlinear constraints obeyed). There is also a 7 point algorithm (useful if fitting repeatedly in RANSAC)
26.Structure 26 26 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Two view geometry The essential and fundamental matrices Reconstruction pipeline Rectification Multiview reconstruction Applications
27.Two view reconstruction pipeline 27 27 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Start with pair of images taken from slightly different viewpoints
28.Two view reconstruction pipeline 28 28 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Find features using a corner detection algorithm
29.Two view reconstruction pipeline 29 29 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Match features using a greedy algorithm

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