Perspective projection in homogenous coordinates Matrix transformations in 2D Translation Scaling of Image x and y conversion from meters to pixels Added skew if image x and y axes are ID: 1019100
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1. Camera calibrationTriangulation
2. Perspective projection in homogenous coordinates
3. Matrix transformations in 2DTranslationScaling of Image x and y (conversion from “meters” to “pixels”)Added skew if image x and y axes are not perpendicular
4. Final perspective projectionCamera intrinsics: how your camera handles pixel. Changes if you change your camera Camera extrinsics: where your camera is relative to the world. Changes if you move the camera
5. Final perspective projectionCamera parameters
6. Camera calibrationGoal: find the parameters of the cameraWhy?Tells you where the camera is relative to the world/particular objectsEquivalently, tells you where objects are relative to the cameraCan allow you to ”render” new objects into the scene
7. Camera calibrationYXZOX’Y’Z’O’
8. Camera calibrationSuppose we know that (X,Y,Z) in the world projects to (x,y) in the image.How many equations does this provide?Need to convert equivalence into equality.
9. Camera calibrationSuppose we know that (X,Y,Z) in the world projects to (x,y) in the image.How many equations does this provide?Note: is unknown
10. Camera calibrationSuppose we know that (X,Y,Z) in the world projects to (x,y) in the image.How many equations does this provide?
11. Camera calibrationSuppose we know that (X,Y,Z) in the world projects to (x,y) in the image.How many equations does this provide?
12. Camera calibrationSuppose we know that (X,Y,Z) in the world projects to (x,y) in the image.How many equations does this provide?2 equations!Are the equations linear in the parameters?How many equations do we need?
13. Camera calibrationIn matrix vector form: Ap = 06 points give 12 equations, 12 variables to solve forBut can only solve upto scale
14. Camera calibrationIn matrix vector form: Ap = 0We want non-trivial solutionsIf p is a solution, p is a solution tooLet’s just search for a solution with unit normHow do you solve this? s.t
15. Camera calibrationIn matrix vector form: Ap = 0We want non-trivial solutionsIf p is a solution, p is a solution tooLet’s just search for a solution with unit norm s.t
16. Camera calibrationWhat happens if there are more than 6 points?What if there is noise in the point locations?s.t
17. Camera calibrationWhat happens if there are more than 6 points?What if there is noise in the point locations?Look at eigenvector of ATA with the smallest eigenvalue!s.t
18. Camera calibration>=6 points with known 3D coordinates + known image coordinatesIn matrix vector form: want Ap = 0Resilience to noise: Look at eigenvector of ATA with the smallest eigenvalue!s.t
19. Camera calibrationWe need 6 world points for which we know image locationsWould any 6 points work?What if all 6 points are the same?Need at least 6 non-coplanar points!
20. Camera calibrationYXZOX’Y’Z’O’
21. Camera calibrationHow do we get K, R and t from P?Need to make some assumptions about KWhat if K is identity?
22. Camera calibrationHow do we get K, R and t from P?Need to make some assumptions about KWhat if K is upper triangular?Added skew if image x and y axes are not perpendicular
23. Camera calibrationHow do we get K, R and t from P?Need to make some assumptions about KWhat if K is upper triangular?P = K [ R t]First 3 x 3 matrix of P is KR“RQ” decomposition: decomposes an n x n matrix into product of upper triangular and rotation matrix
24. Camera calibrationHow do we get K, R and t from P?Need to make some assumptions about KWhat if K is upper triangular?P = K [ R t]First 3 x 3 matrix of P is KR“RQ” decomposition: decomposes an n x n matrix into product of upper triangular and rotation matrixt = K-1P[:,2] last column of P
25. Camera calibration and pose estimation
26. TriangulationSuppose we have two camerasCalibrated: parameters knownAnd a pair of corresponding pixelsFind 3D location of point!
27. TriangulationSuppose we have two camerasCalibrated: parameters knownAnd a pair of corresponding pixelsFind 3D location of point!(x1,y1)(x2,y2)
28. Triangulation
29. Triangulation
30. Triangulation1 image gives 2 equationsNeed 2 images!Solve linear equations to get 3D point location
31. Linear vs non-linear optimization
32. Linear vs non-linear optimizationReprojection error
33. Linear vs non-linear optimizationReprojection error is the squared error between the true image coordinates of a point and the projected coordinates of hypothesized 3D pointActual error we care aboutMinimize total sum of reprojection error across all imagesNon-linear optimization Reprojection error
34.