Two-view geometry diagram

1
Two-view geometry diagram
Fundamental matrix HZ 9.2
Epipolar geometry HZ 9.1
Camera from F
7-point algorithm
normalized 8-pt alg for F HZ 11.2
Structure computation Alg 12.1, HZ318
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
Fundamental matrix HZ 9.2
Epipolar geometry HZ 9.1
Camera from F
7-point algorithm
normalized 8-pt alg for F HZ 11.2
Structure computation Alg 12.1, HZ318
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 Ft x in image 2 gt Ft x
  in image 1
• how do we interpret this? x is in a 2-space, Fx
  is in another 2-space
• corollary xtFx 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 xtFx 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
• Ft 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 ex 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 ex Hx
• we know l Fx, so
• F ex H
• HZ243

12
Resulting properties of F

• F is rank 2
• ex 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 (ex H) e ex (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/clapa
  ck.tgz and www.netlib.org/clapack/clapack.h
• install CLAPACK following www.netlib.org/clapack/r
  eadme.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 documentation
• download the manual pages for ready access e.g.,
  once you discover through the search engine that
  sgesv solves Axb 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