lecture 6.1 basic epipolar geometry - universitetet i two-view geometry involve several new...
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Lecture 6.1 Basic epipolar geometry
Thomas Opsahl
Weekly overview β Stereo imaging
2
Introduction
β’ Single-view geometry β Camera model ππΏοΏ½ = ποΏ½ β Finite projective camera π = πΎ π π β Undistortion β Estimating π from 3D-2D correspondences β Calibration β PnP
3
π₯πΆ π¦πΆ
π§πΆ πΆ
π§πΆ = 1
π πΏ
π’
π£
π π
Introduction
β’ Single-view geometry β Camera model ππΏοΏ½ = ποΏ½ β Finite projective camera π = πΎ π π β Undistortion β Estimating π from 3D-2D correspondences β Calibration β PnP
β’ Two-view geometry
β Epipolar geometry is the geometric relationship between two perspective cameras
β Two camera models π1πΏοΏ½ = ποΏ½1, π2πΏοΏ½ = ποΏ½2 β Next week - General two-view
4
π₯πΆ1
π¦πΆ1
π§πΆ1
π₯πΆ2 π¦πΆ2
π§πΆ2
π₯πΆ π¦πΆ
π§πΆ πΆ
π§πΆ = 1
π πΏ
π’
π£
π π
Introduction
β’ Single-view geometry β Camera model ππΏοΏ½ = ποΏ½ β Finite projective camera π = πΎ π π β Undistortion β Estimating π from 3D-2D correspondences β Calibration β PnP
β’ Two-view geometry
β Epipolar geometry is the geometric relationship between two perspective cameras
β Two camera models π1πΏοΏ½ = ποΏ½1, π2πΏοΏ½ = ποΏ½2 β Next week - General two-view β This week - Stereo view
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π₯πΆ π¦πΆ
π§πΆ πΆ
π§πΆ = 1
π πΏ
π’
π£
π π
π₯πΆ1 π¦πΆ1
π§πΆ1
π₯πΆ2 π¦πΆ2
π§πΆ2
β’ Two-view geometry involve several new geometrical entities compared to single-view geometry
Epipolar geometry
πͺ1
π1 π2
πͺ2
πΏ
β’ Two-view geometry involve several new geometrical entities compared to single-view geometry β’ The epipolar plane is the plane containing πΏ and the two camera centers πͺ1 and πͺ2
Epipolar geometry
Epipolar plane
πͺ1
π1 π2
πͺ2
πΏ
β’ Two-view geometry involve several new geometrical entities compared to single-view geometry β’ The epipolar plane is the plane containing πΏ and the two camera centers πͺ1 and πͺ2 β’ The baseline is the line joining the two camera centers
Epipolar geometry
Epipolar plane
πͺ1
π1 π2
πͺ2
πΏ
baseline
β’ Two-view geometry involve several new geometrical entities compared to single-view geometry β’ The epipolar plane is the plane containing πΏ and the two camera centers πͺ1 and πͺ2 β’ The baseline is the line joining the two camera centers β’ The epipolar lines are where the epipolar plane intersect the image planes
Epipolar geometry
Epipolar plane
πͺ1
π1 π2
πͺ2
πΏ
baseline
Epipolar line Epipolar line
β’ Two-view geometry involve several new geometrical entities compared to single-view geometry β’ The epipolar plane is the plane containing πΏ and the two camera centers πͺ1 and πͺ2 β’ The baseline is the line joining the two camera centers β’ The epipolar lines are where the epipolar plane intersect the image planes β’ The epipoles are where the baseline intersects the two image planes
Epipolar geometry
Epipolar plane
πͺ1
π1 π2
πͺ2
πΏ
baseline
Epipolar line Epipolar line
Epipole Epipole
π1 π2
β’ Two-view geometry involve several new geometrical entities compared to single-view geometry β’ The epipolar plane is the plane containing πΏ and the two camera centers πͺ1 and πͺ2 β’ The baseline is the line joining the two camera centers β’ The epipolar lines are where the epipolar plane intersect the image planes β’ The epipoles are where the baseline intersects the two image planes β’ The baseline and epipoles are uniquely defined by the two camera matrices π1 and π2 β’ The epipolar plane and epipolar lines depends on the observed point πΏ
Epipolar geometry
πͺ1
π1 π2
πͺ2
πΏ
baseline
Epipolar line Epipolar line
Epipole Epipole
Epipolar plane
π1 π2
Example
12
Example
13
β’ Corresponding points lie on corresponding epipolar lines β’ Both epipoles are outside of the visible part of the image planes
Example
14
Example
15
β’ Corresponding points lie on corresponding epipolar lines β’ Both epipoles are visible as the intersection of epipolar lines
Summary
β’ Epipolar geometry β Epipolar planes β Epipolar lines β Epipoles
β’ Topics ahead
β Stereo imaging β Representing epipolar geometry β Estimating epipolar geometry β 3D from epipolar geometry β Relative pose from epipolar geometry β More viewsβ¦
β’ Additional reading: β Szeliski: 11 introduction & 11.1
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πͺ1
π1 π2
πͺ2
πΏ
baseline
Epipolar line Epipolar line
Epipole Epipole
Epipolar plane