Factorization.pptVIP

Previously Two view geometry: epipolar geometry Stereo vision: 3D reconstruction Today Orthographic projection Two views 3 or more views: factorization – simultaneous recovery of motion and shape Orthographic Projection (Reminder) Parallel projection rays, orthogonal to image plane Focal center at infinity Two Views Implies that Eliminating Z Since Therefore Epipolar lines This is a linear equation Further Simplification Select one point in first image and its corresponding point in the second image to be the origin of the two images In this coordinate frame translation is 0 Expression for epipolar lines: Epipolar Line Recovery We need 4 corresponding points: 1 to eliminate translation 3 to determine the 4 components of R up to scale The rest of the components cannot be determined In particular, cannot be determined from , because these components are known only up to scale Shape Recovery from Two Views Perspective: Translation recovered up to scale 3D shape recovered up to scale Recovery only if non-zero translation No calibration – recovery up to a projective transformation (“projective shape”) Orthographic: Rotation along epipolar line cannot be recovered 3D shape cannot be recovered Recovery is possible up to an affine transformation (“affine shape”) Recovery only if non-trivial rotation Translation along line at infinity = rotation Recovery from Three Views Under orthographic projection metric recovery is possible from three views Only rotation matters Rotation has three degrees of freedom Given an image, one rotation is in the image and two are out of plane rotations Ignoring the in-plane rotation we can associate the image with a point on the unit sphere Recovery from Three Views Recovery from Three Views Recovery from Three Views a, b, c are unknown a, b, g are known – angles between epipolar lines Can we determine lengths from angles? Recovery from Three Views In the plane the angles determine the sides of a t