1. Two-view geometry diagram Camera from F Structure computation Alg 12.1, HZ318 7-point algorithm normalized 8-pt alg for F HZ 11.2 Fundamental matrix HZ 9.2 Epipolar geometry HZ 9.1

2. Overview of 2-view geometry entering Part II of Hartley-Zisserman epipolar geometry –formalization of the structure between 2 views –used to extract depth information inherent in this stereo view how fundamental matrix F encodes epipolar geometry –the central structure in 2-view geometry is the fundamental matrix F: all of the camera and structure information is extracted directly or indirectly from F solving for F using point correspondences –8 point algorithm singularity constraint normalization –7 point algorithm how the fundamental matrix encodes the camera center and other camera information

3. Two-view geometry diagram Camera from F Structure computation Alg 12.1, HZ318 7-point algorithm normalized 8-pt alg for F HZ 11.2 Fundamental matrix HZ 9.2 Epipolar geometry HZ 9.1

4. Finding structure imagine two cameras (perhaps virtually by moving a single camera in time) camera centers c and c’ valid point correspondence (x,x’) we want to discover the 3D point X associated with this point correspondence finding X and finding c/c’ are the main goals of structure from motion we shall explore the geometric relationship between c,c’,x,x’,X

5. Epipolar geometry baseline = line cc’ between camera centers; demo: cardboard, 2 frames epipolar plane = any plane through the baseline –a pencil of planes epipole = intersection of baseline with image plane –equivalently, image of other camera center –this connection to camera center will be leveraged –crucial to computing structure and camera –note: may lie outside image epipolar line = intersection of an epipolar plane with an image plane –note: x has an associated epipolar plane (3 points define a plane), so an associated epipolar line –note: x’ lies on the epipolar line associated with x HZ239-241

6. Epipolar lines we have seen that x’ lies on the x epipolar line, and x lies on the x’ epipolar line offers a mechanism to tie the two points together: if we can compute the epipolar line l’ associated with a point x, then we have a constraint on the position of x’ –l’. x’ = 0 (x’ lies on l’) –this tells us something about the companion x’ the fundamental matrix will build epipolar lines from points so it is valuable in determining structure epipolar line = image of cx line

7. Epipoles from epipolar lines epipolar lines are tied up with epipoles: epipole lies on every epipolar line notice how the epipole can be found from these epipolar lines this gives information about the other camera center

8. Fundamental matrix the fundamental matrix relates two images the fundamental matrix F of two images is a 3x3 matrix such that: –F: points  epipolar lines –Fx is the epipolar line (in image 2) associated with the point x (in image 1) –also vice versa using F t : x’ in image 2 => F t x’ in image 1 how do we interpret this? x is in a 2-space, Fx is in another 2-space corollary: x t Fx’ = 0 if (x,x’) are a corresponding pair (i.e., image points of the same 3D point X) –proof: x lies on Fx’ F as epipolar line generator F as correspondence checker HZ242

9. Computing F: basics we will use x t Fx’ = 0 constraints from a few point correspondences to solve for F each point pair (x,x’) defines a linear equation in F 8 pairs should be enough to solve for the 8 degrees of freedom in F (projective 3x3) SIFT will be used to gather point correspondences detailed algorithms below

10. Epipoles from F F yields information about the epipoles (on top of information about epipolar lines and point correspondences) let e = epipole of image 1 e’ = epipole of image 2 both are found as null spaces Fe = 0 F t e’ = 0 proof below

11. Interpretation of F consider the epipolar line l’ associated with x l’ = e’ x x’ –e’ and x’ lie on l’ so l’ = [e’] x x’ –skew-symmetric rep but x’ = Hx for some homography H –can be understood as point transfer off a plane, but not necessary so l’ = [e’] x Hx we know l’ = Fx, so: F = [e’] x H HZ243

12. Resulting properties of F F is rank 2 –[e’] x is rank 2, H is rank 3 –that is, F is singular and has 1d null space –adds a constraint that is very helpful in guiding the computation of F (see below) proof that Fe = 0: –Fe = ([e’] x H) e = [e’] x (H e) = e’ x (He) = e’ x e’ = 0 image of e is e’ –thus, the epipole e may be found as the null space of F

13. CLAPACK OpenCV has SVD too www.netlib.org/clapack/ –download CLAPACK from www.netlib.org/clapack/clapack.tgz and www.netlib.org/clapack/clapack.hwww.netlib.org/clapack/clapack.tgz www.netlib.org/clapack/clapack.h install CLAPACK following www.netlib.org/clapack/readme.installwww.netlib.org/clapack/readme.install –generates lapack_LINUX.a and blas_LINUX.a optimize the BLAS for your machine (optional) see my Makefile see www.netlib.org/lapack for documentationwww.netlib.org/lapack download the manual pages for ready access: e.g., once you discover through the search engine that ‘sgesv’ solves Ax=b for you, ‘man sgesv’ gives its parameters. read CLAPACK/readme for caveats of style differences in calling LAPACK from C, such as column-major simulation and call by reference parameters. sgesvd for SVD