Change coordinate system so that center of the coordinate system is at pinhole and Z axis is along viewing direction Perspective projection The projection equation Is this equation linear Can this equation be represented by a matrix multiplication ID: 1021181
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1. Homogenous coordinates
2. Putting everything togetherChange coordinate system so that center of the coordinate system is at pinhole and Z axis is along viewing directionPerspective projection
3. The projection equationIs this equation linear?Can this equation be represented by a matrix multiplication?
4. Is projection linear?
5. Can projection be represented as a matrix multiplication?Matrix multiplicationPerspective projection
6. The space of raysEvery point on a ray maps it to a point on image planePerspective projection maps rays to pointsAll points ) map to the same image point (x,y,1) O(x,y,1)()
7. Projective spaceStandard 2D space (plane) : Each point represented by 2 coordinates (x,y)Projective 2D space (plane) : Each “point” represented by 3 coordinates (x,y,z), BUT:Mapping to (points to rays):Mapping to (rays to points):
8. Projective space and homogenous coordinatesMapping to (points to rays):Mapping to (rays to points):A change of coordinatesAlso called homogenous coordinates
9. Homogenous coordinatesIn standard Euclidean coordinates2D points : (x,y)3D points : (x,y,z)In homogenous coordinates2D points : (x,y,1)3D points : (x,y,z,1)
10. Why homogenous coordinates?Homogenous coordinates of world pointHomogenous coordinates of image point
11. Why homogenous coordinates?Perspective projection is matrix multiplication in homogenous coordinates!
12. Why homogenous coordinates?Translation is matrix multiplication in homogenous coordinates!
13. Homogenous coordinates
14. Homogenous coordinates
15. Perspective projection in homogenous coordinates
16. More about matrix transformations3 x 4 : Perspective projection4 x 4 : Translation 4 x 4 : Affine transformation (linear transformation + translation)
17. More about matrix transformationsEuclidean
18. More about matrix transformationsSimilarity transformation
19. More about matrix transformationsAnisotropic scaling and translation
20. More about matrix transformationsGeneral affine transformation
21. Matrix transformations in 2D
22. Perspective projection in homogenous coordinates
23. 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
24. 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
25. Final perspective projectionCamera parameters
26. 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
27. Camera calibrationYXZOX’Y’Z’O’
28. Camera calibrationNeed to estimate PHow many parameters does P have?Size of P : 3 x 4But: P can only be known upto a scale3*4 - 1 = 11 parameters
29. 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.
30. 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
31. Camera calibrationSuppose we know that (X,Y,Z) in the world projects to (x,y) in the image.How many equations does this provide?
32. Camera calibrationSuppose we know that (X,Y,Z) in the world projects to (x,y) in the image.How many equations does this provide?
33. 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?
34. Camera calibrationIn matrix vector form: Ap = 06 points give 12 equations, 12 variables to solve forBut can only solve upto scale
35. 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
36. 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? Eigenvector with 0 eigenvalue! s.t
37. 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!
38. Camera calibrationYXZOX’Y’Z’O’