1、Epipolar Geometry class 11,Multiple View Geometry Comp 290-089 Marc Pollefeys,Content,Background: Projective geometry (2D, 3D), Parameter estimation, Algorithm evaluation. Single View: Camera model, Calibration, Single View Geometry. Two Views: Epipolar Geometry, 3D reconstruction, Computing F, Comp
2、uting structure, Plane and homographies. Three Views: Trifocal Tensor, Computing T. More Views: N-Linearities, Multiple view reconstruction, Bundle adjustment, auto-calibration, Dynamic SfM, Cheirality, Duality,Multiple View Geometry course schedule (subject to change),More Single-View Geometry,Proj
3、ective cameras and planes, lines, conics and quadrics.Camera calibration and vanishing points, calibrating conic and the IAC,Two-view geometry,Epipolar geometry3D reconstructionF-matrix comp.Structure comp.,Correspondence geometry: Given an image point x in the first view, how does this constrain th
4、e position of the corresponding point x in the second image?,(ii) Camera geometry (motion): Given a set of corresponding image points xi xi, i=1,n, what are the cameras P and P for the two views?,(iii) Scene geometry (structure): Given corresponding image points xi xi and cameras P, P, what is the p
5、osition of (their pre-image) X in space?,Three questions:,The epipolar geometry,C,C,x,x and X are coplanar,The epipolar geometry,What if only C,C,x are known?,The epipolar geometry,All points on p project on l and l,The epipolar geometry,Family of planes p and lines l and l Intersection in e and e,T
6、he epipolar geometry,epipoles e,e = intersection of baseline with image plane = projection of projection center in other image = vanishing point of camera motion direction,an epipolar plane = plane containing baseline (1-D family),an epipolar line = intersection of epipolar plane with image(always c
7、ome in corresponding pairs),Example: converging cameras,Example: motion parallel with image plane,Example: forward motion,e,e,The fundamental matrix F,algebraic representation of epipolar geometry,we will see that mapping is (singular) correlation (i.e. projective mapping from points to lines) repre
8、sented by the fundamental matrix F,The fundamental matrix F,geometric derivation,mapping from 2-D to 1-D family (rank 2),The fundamental matrix F,algebraic derivation,(note: doesnt work for C=C F=0),The fundamental matrix F,correspondence condition,The fundamental matrix satisfies the condition that
9、 for any pair of corresponding points xx in the two images,The fundamental matrix F,F is the unique 3x3 rank 2 matrix that satisfies xTFx=0 for all xx,Transpose: if F is fundamental matrix for (P,P), then FT is fundamental matrix for (P,P) Epipolar lines: l=Fx & l=FTx Epipoles: on all epipolar lines
10、, thus eTFx=0, x eTF=0, similarly Fe=0 F has 7 d.o.f. , i.e. 3x3-1(homogeneous)-1(rank2) F is a correlation, projective mapping from a point x to a line l=Fx (not a proper correlation, i.e. not invertible),The epipolar line geometry,l,l epipolar lines, k line not through e l=Fkxl and symmetrically l
11、=FTkxl,(pick k=e, since eTe0),Fundamental matrix for pure translation,Fundamental matrix for pure translation,Fundamental matrix for pure translation,example:,motion starts at x and moves towards e, faster depending on Z,pure translation: F only 2 d.o.f., xTexx=0 auto-epipolar,General motion,Geometr
12、ic representation of F,Fs: Steiner conic, 5 d.o.f. Fa=xax: pole of line ee w.r.t. Fs, 2 d.o.f.,Geometric representation of F,Pure planar motion,Steiner conic Fs is degenerate (two lines),Projective transformation and invariance,Derivation based purely on projective concepts,F invariant to transforma
13、tions of projective 3-space,unique,not unique,canonical form,Projective ambiguity of cameras given F,previous slide: at least projective ambiguity this slide: not more!,lemma:,(22-15=7, ok),Canonical cameras given F,F matrix corresponds to P,P iff PTFP is skew-symmetric,Possible choice:,Canonical re
14、presentation:,The essential matrix,fundamental matrix for calibrated cameras (remove K),5 d.o.f. (3 for R; 2 for t up to scale),E is essential matrix if and only if two singularvalues are equal (and third=0),Four possible reconstructions from E,(only one solution where points is in front of both cameras),Next class: 3D reconstruction,
