3D reconstruction

1. 3D reconstruction

2. Three questions: • Correspondence geometry: Given an image point x in the first image, how does this constrain the position of the corresponding point x’ in the second image? • (ii) Camera geometry (motion): Given a set of corresponding image points {xi ↔x’i}, i=1,…,n, what are the cameras P and P’ for the two views? • (iii) Scene geometry (structure): Given corresponding image points xi ↔x’i and cameras P, P’, what is the position of (their pre-image) X in space?

3. P P L2 L2 m1 m1 m1 C1 C1 C1 M M L1 L1 l1 l1 e1 e1 lT1 l2 e2 e2 Canonical representation: l2 m2 m2 m2 l2 l2 Fundamental matrix (3x3 rank 2 matrix) C2 C2 C2 Epipolar geometry Underlying structure in set of matches for rigid scenes • Computable from corresponding points • Simplifies matching • Allows to detect wrong matches • Related to calibration

4. 3D reconstruction of cameras and structure reconstruction problem: given xi↔x‘i , compute P,P‘ and Xi for all i without additional informastion possible up to projective ambiguity

5. outline of reconstruction • Compute F from correspondences • Compute camera matrices from F • Compute 3D point for each pair of corresponding points computation of F use x‘iFxi=0 equations, linear in coeff. F 8 points (linear), 7 points (non-linear), 8+ (least-squares) (more on this next class) computation of camera matrices use triangulation compute intersection of two backprojected rays

6. Reconstruction Ambiguity • Without knowledge of a scene placement w.r.t. 3D coordinate frame, impossible to reconstruct absolute position or orientation of a scene from a pair of views: • Scene is determined at best up to a Euclidean transformation , i.e. rotation and translation w.r.t. world frame • Overall scale of the scene can also not be determined: • Example: Coin next to the integrated circuit to give an idea of size • Scene is determined by the image only up to a similarity transformation, i.e. rotation, translation and scaling

7. Reconstruction ambiguity: similarity Let Hs be a similarity transform Replace: Xi  Hs Xi P  P Hs -1 P’  P’ Hs -1 Then xi unchanged; Let P = K [ Rp | tp ] for some t’  Multiplying by Hs -1 does not change the intrinsic calibration matrix K Upshot: For calibrated cameras, reconstruction possible up to a similarity This is the only ambiguity of reconstruction ( Longuet and Higgins 1981) (a)

8. Reconstruction ambiguity: projective • Reconstruction from uncalibrated cameras possible up to a projective transformation. • Consider H any 4x4 invertible matrix • representing projective transformation • in P3 • Replace Xi  H Xi • P  P H-1 • P’  P’ H-1 • Image point xi does not change

9. Terminology xi↔x‘i Original scene Xi Projective, affine, similarity reconstruction = reconstruction that is identical to original up to projective, affine, similarity transformation Literature: Metric and Euclidean reconstruction = similarity reconstruction

10. The projective reconstruction theorem If a set of point correspondences in two views determine thefundamental matrix uniquely, then the scene and cameras may be reconstructed from these correspondences alone, and any two such reconstructions from these correspondences are projectively equivalent • key result: • allows reconstruction from pair of uncalibrated images • points lying on the line joining two camera centers are excluded, since they cannot be reconstructed uniquely even if camera matrices are determined • Projective reconstruction answers questions like: At what point does a line intersect a plane? What is the mapping between two views induced by planes?

11. A B

12. Relation between Euclidean and Projective Reconstruction

13. Stratified reconstruction • Projective reconstruction • Affine reconstruction • Metric reconstruction

14. Projective to affine remember 2-D case

15. Projective to affine • The essence of affine reconstruction is to locate the plane at infinity; • Suppose we have somehow identified the plane at infinity. Projective reconstruction of the scene; (0,0,0,1) is the true coordinates of plane at infinity find transformation H that maps Π∞ to (0,0,01) (if D≠0) • Now apply H to all points and to two cameras. Plane at infinity has been correctly placed. • This reconstruction differs from true reconstruction up to projective transformation that fixes the plane at infinity. • But projective with fixed p∞ is affine transformation • Reconstruction is true up to an affine transformation • Affine reconstruction can be sufficient depending on application, • e.g. mid-point, centroid can be computed • Parallellism: lines constructed parallel to other lines and to planes • Question: how to identify plane at infinity need extra information

16. Translational motion to find plane at infinity • points at infinity are fixed for a pure translation • reconstruction of xi↔ xi is on p∞ • Get three points on plane at infinity to reconstruct it. Example: far away objects such as moon as we translate

17. Scene constraints to find plane at infinity • Parallel lines • parallel lines intersect at infinity • reconstruction of corresponding vanishing point yields point on plane at infinity • 3 sets of parallel lines (with different directions) allow to uniquely determine p∞ remark: in presence of noise determining the intersection of parallel lines is a delicate problem

18. Scene constraints

19. Scene constraints Distance ratios on a line to find plane at infinity known distance ratio along a line allow to determine point at infinity (same as 2D case) Given two intervals on a line with a known length ratio, the point at infinity on the line can be found  from an image of a line on which a world distance ratio is known, for example that three points are equally spaced, the vanishing point may be determined.

20. The infinite homography ∞ ∞ • A mapping that transfers points from the P image to the P’ image via • the plane at infinity • Ray corresponding to a point x meets Π∞ at X which can be • projected to point x’ in the other image • Infinite homography can be computed from an affine reconstruction and Vice versa. Infinite homography can be computed directly from corresponding image entries e.g. three vanishing points and F, or vanishing line, vanishing point and F

21. Affine to metric * * projection constraints • Identify absolute conic; it lies on the plane at infinity • Transform so that the absolute conic is mapped to the absolute conic in • Euclidean frame • Projective transformation relating original and reconstruction is a similarity transformation • in practice, find image of W∞ • image w∞back-projects to cone that intersects p∞ in W∞

22. Affine to metric Image of the absolute conic as seen by a camera with matrix: possible transformation from affine to metric is (cholesky factorisation) How to find the image of an absolute conic? constraints

23. Constraints arising from scene Ortogonality to find image of absolute conic vanishing points corresponding to orthogonal directions vanishing line and vanishing point corresponding to plane and normal direction

24. rectangular pixels square pixels Constraints arising from known internal parameters to find image of absolute conic

25. Constraints arising from Same camera for all images to find image of absolute conic e.g. moving cameras • same intrinsics  same image of the absolute conic • Property of absolute conic: its projection onto an image depends only on the calibration matrix of the camera, not on the position or orientation of camera • If P and P‘ have the same calibration matrix, i.e. both images taken with the same camera at different poses, then ω = ω’ • Since the absolute conic is on the plane at infinity, its image may be transfered from one view to the other via infinite homogrpahy: transfer of IAC: • Combine ω = ω’ with the above •  four contraints on ω; • ω has five dof; use some of the other contraints e.g. orthogonality or • known internal parameters.

26. a b

27. Direct metric reconstruction using w if ω is known, can directly do metric reconstruction, without stratified recon: approach : • Assume ω is known in both images, then can find K and K’ fortwo poses; • then apply essential matrix to metrically reconstruct: • Four cases; two are just miroor images; two are twisted pair; • All solutions but one can be ruled out: points only lie in front of camera approach 2 • compute projective reconstruction • back-project w from both images • Intersection of the two cones defines W∞ and its support plane p∞ • in general two cones intersect in two different plane • conics, each lying in a different support plane two solutions •  Twisted pair ambiguity

28. Direct reconstruction using ground truth Alternative: Relate ground control points to image measurements (2 lin. eq. in H-1per view,3 for two views;) • use control points XEi with known coordinates to go from projective to metric Since projective reconstruction Is related to to true reconstruction by a Homography: Each point correspondence provides 3 linearly indep. Eqns on elements of H; Since H has 15 elements 5 or more points a c b