CS484: Introduction to Computer Visionvclab.kaist.ac.kr/cs484/slide07-EpipolarGeometry_v3.pdf · 2019-10-17 · Note that from this slide, the z-axis looks at the object to simplify

Lecturer: Min H. Kim (KAIST) CS484: Introduction to Computer Vision

CS484: Introduction to Computer Vision

Min H. KimKAIST School of Computing

Epipolar Geometry(Szeliski: 11)


SINGLE-VIEW GEOMETRY

2Acknowledgment: Many slides from James Hays, Derek Hoiem and Grauman & Leibe 2008 AAAI Tutorial


Single-view geometry

3Odilon Redon, Cyclops, 1914


Our goal: Recovery of 3D structure• Recovery of structure from one image is

inherently ambiguous

4

x

X?X?

X?




5




6


Ames Room

7

http://en.wikipedia.org/wiki/Ames_room

http://en.wikipedia.org/wiki/Ames_room


Our goal: Recovery of 3D structure• We will need multi-view geometry

8


3D-2D Projective mapping

9


Recall: Pinhole camera model

10

• Principal axis: line from the camera center perpendicular to the image plane

• Normalized (camera) coordinate system:camera center is at the origin and the principal axis is the z-axis

Note that from this slide, the z-axis looks at the object to simplify equations by removing the sign!


(X ,Y ,Z )! ( f X / Z , f Y / Z )XYZ1

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

!

f Xf YZ

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

f 0f 01 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

XYZ1

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Recall: Pinhole camera model

11x = PX


Principal point

12

• Principal point (p): point where principal axis intersects the image plane (origin of normalized coordinate system).

• Normalized coordinate system: origin is at the principal point.• Image coordinate system: origin is in the corner.• How to go from normalized coordinate system to image coordinate

system?

px

py


(X ,Y ,Z )! ( f X / Z + px , f Y / Z + py )

XYZ1

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

!

f X + Z pxf Y + Z pyZ

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

f px 0

f py 0

1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

XYZ1

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Principal point offset

13

principal point: ( px , py )

px

py


Principal point offset

14

P =K I | 0⎡⎣ ⎤⎦K =

f pxf py

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Camera matrix

principal point: ( px , py )

px

py

f X + Zpxf Y + ZpyZ

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

f pxf py

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1 01 01 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

XYZ1

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Projection(extrinsics)

Camera(intrinsics)


K =

mxmy

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

f pxf py

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

α x βxα y β y

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Pixel coordinates

15

• mx pixels per meter in horizontal direction, my pixels per meter in vertical direction

1mx

× 1my

Pixel size:

pixels/m m pixelsUnits:


!Xcam = R !X- !C( )

Camera rotation and translation

16

• In general, the camera coordinate frame will be related to the world coordinate frame by a rotation and a translation

3D coords. of point in camera frame

3D coords. of camera center in world frame

3D coords. of a pointin world frame (nonhomogeneous)


!Xcam = R !X- !C( )

Xcam = R −R !C0 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

!X1

⎛

⎝⎜⎞

⎠⎟= R −R !C

0 1

⎡

⎣⎢⎢

⎤

⎦⎥⎥X

x =K I | 0⎡⎣ ⎤⎦Xcam =K R |−R !C⎡⎣ ⎤⎦X P =K R | t⎡⎣ ⎤⎦ , t = −R !C

Camera rotation and translation

17

In non-homogeneous (~ symbol)coordinates:

Note: C is the null space of the camera projection matrix (PC=0)

Xcam and X are homogeneous coordinates


Camera parameters

18

• Intrinsic parameters– Principal point coordinates– Focal length– Pixel magnification factors– Skew (non-rectangular pixels)– Radial distortion

K =

mxmy

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

f pxf py

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

α x βxα y β y

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


Radial distortion

• To model lens distortion– Use above projection operation

instead of standard projection matrix multiplication19

2. Apply radial distortion

3. Apply focal length translate image center

Slide by Steve Seitz

1. Project to “normalized” image coordinates(where the center is the origin)

x, y, z( ) ′xn = x / z′yn = y / z

r 2 = ′xn2 + ′y 2

n

′xd = ′xn 1+ k1r2 + k2r

4( )′yd = ′yn 1+ k1r

2 + k2r4( )

′x = f ′xd + xc′y = f ′yd + yc

K =

f pxf py

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


Camera parameters• Intrinsic parameters

– Principal point coordinates– Focal length– Pixel magnification factors– Skew (non-rectangular pixels)– Radial distortion

• Extrinsic parameters– Rotation and translation relative to world coordinate

system

20


Camera Calibration (Affine Projection)

Source: D. Hoiem

λ xλ yλ

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=* * * ** * * ** * * *

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

XYZ1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

x =K R t⎡⎣

⎤⎦ X

PAffine projection

Also called “para-perspective”

How to find out?


Camera calibration• Given n points with known 3D coordinates Xi

and known image projections xi, estimate the camera parameters

22

? P

Xi

xi


Camera calibration: Linear method

23

λx i = PX i x i × PX i = 0 xiyi1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

×

PxTX i

PyTX i

PzTX i

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= 0

0 −X iT yiX i

T

X iT 0 −xiX i

T

− yiX iT xiX i

T 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

PxPyPz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

= 0

Two linearly independent equations

∵"a ×"b = "a

"b sinθ

? P

Xi

xi

P∈!3×4

Px ,y ,z ∈!4×1

X i ∈!4×1


Camera calibration: Linear method

24

• p has 11 degrees of freedom (3x4=12 parameters but the last element is 1, but scale is arbitrary)

• One 2D/3D correspondence gives us two linearly independent equations• Homogeneous least squares• 6 correspondences (6x2=12) needed for a minimal solution

∴Ap = 0

0T X1T − y1X1

T

X1T 0T −x1X1

T

! ! !0T Xn

T − ynXnT

XnT 0T −xnXn

T

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

PxPyPz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

= 0

Suppose we have n points’ coordinates

A p


Camera calibration: Linear method• Advantages: easy to formulate and solve• Disadvantages

– Doesn’t directly tell you camera parameters– Doesn’t model radial distortion– Can’t impose constraints, such as known focal length and

orthogonality

• Non-linear methods are preferred– Define error as difference between projected points and

measured points– Minimize error using Newton’s method or other non-line

ar optimization

Source: D. Hoiem


EPIPOLAR GEOMETRY

27Acknowledgment: Many slides from James Hays, Derek Hoiem and Grauman & Leibe 2008 AAAI Tutorial


Triangulation• Given projections of a 3D point in two or more

images (with known camera matrices), find the coordinates of the point

28

O1 O2

x1x2

X?


Triangulation• We want to intersect the two visual rays

corresponding to x1 and x2, but because of noise and numerical errors, they don’t meet exactly

29

O1 O2

x1x2

X?

R1R2


Triangulation: Geometric approach• Find shortest segment connecting the two viewing

rays and let X be the midpoint of that segment

30

O1 O2

x1x2

X


Recap: Triangulation: Linear approach

31

x × PX =xy1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥×

PxTX

PyTX

PzTX

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=

0 −XT y XT

XT 0 −x XT

− y XT x XT 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

PxPyPz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

= [x× ]PX

λ1x1 = P1Xλ2x2 = P2X

x1 × P1X = 0x2 × P2X = 0

[x1× ]P1X = 0[x2× ]P2X = 0

Cross product as matrix multiplication:

Pi ∈!3×4

X∈!4×1

x i ∈!3×1

λi ∈!1×1


Recap: Triangulation: Linear approach

32

Two independent equations each in terms of three unknown entries of X

λ1x1 = P1Xλ2x2 = P2X

x1 × P1X = 0x2 × P2X = 0

[x1× ]P1X = 0[x2× ]P2X = 0

Pi ∈!3×4

X∈!4×1

x i ∈!3×1

λi ∈!1×1


Triangulation: Nonlinear approach

minimize d 2(x1 ,P1X)+ d 2(x2 ,P2X)

33

• Find X that minimizes distances in the screen spaces

O1 O2

x1x2

X?

x’1

x’2

= ′x1 = ′x2


Two-view geometry

34


• Epipolar Plane – plane containing baseline (1D family)• Epipoles = intersections of baseline with image planes = projections of the other camera center

• Baseline – line connecting the two camera centers

Epipolar geometry

35

X

x x’


• Epipolar Plane – plane containing baseline (1D family)• Epipoles= intersections of baseline with image planes = projections of the other camera center

• Epipolar Lines - intersections of epipolar plane with imageplanes (always come in corresponding pairs)

• Baseline – line connecting the two camera centers

Epipolar geometry

36

X

x x’


Example: Converging cameras

37


Example: Motion parallel to image plane

38

Motion parallel


Example: Motion perpendicular to image plane

39



40


e

e’


41

Epipole has same coordinates in both images.Points move along lines radiating from e: “Focus of expansion”


Epipolar constraint

42

• If we observe a point x in one image, where can the corresponding point x’ be in the other image?

x x’

X


• Potential matches for x have to lie on the corresponding epipolar line l’.

• Potential matches for x’ have to lie on the corresponding epipolar line l.

Epipolar constraint

43

x x’

X

x’

X

x’

X


Epipolar constraint example

44


X

Epipolar constraint: Calibrated case

45

• Assume that the intrinsic and extrinsic parameters of the cameras are known

• We can also set the global coordinate system to the coordinate system of the first camera. Then the projection matrix of the first camera is [I | 0].

x’x



x x’

X

ˆ ′x = ′K -1 ′x = ′Xx =K-1x = XHomogeneous 2D point (3D ray towards X)in normalized screen coordinates

2D pixel coordinate (homogeneous)

3D scene point

3D scene point in 2nd

camera’s 3D coordinates

• We can multiply the projection matrix of each camera (and the image points) by the inverse of the calibration matrix to get normalized image coordinates

x x '


Camera1: projectionmatrix is [ I | 0 ], same as theworld coordinate. Point X has coordinate in the camera1’s normalized image coordinates


47

R, tCamera2: projection matrix is [ R | t ].Point X has coordinate in the camera2’snormalized image coordinates

NOTE: R and t are the coordinate transformation. (t = -RO’ )e.g., X’ can be written RX + t in the camera2’s normalized image coordinate.

X (=X’ in the camera2 frame)

x x '

x x '

ˆ ′x = Rx + t


X’=RX + t


48

Those three vectors are coplanar.Writing in the camera2’s normalized image coordinates,the vectors , , and t are coplanar.(we only care about direction!)

Camera1: extrinsic matrix is [ I | 0 ]Point X has coordinate in the camera1’s normalized image coordinate

Camera2: extrinsic matrix is [ R | t ]Point X has coordinate in the camera2’snormalized image coordinate

t = -RO’

(x in the camera2 frame) Rx

x '

ˆ ′x = Rx + t ˆ ′x ⋅ t × Rx⎡⎣ ⎤⎦ = 0

Rx x '

x x '


Essential Matrix(Longuet-Higgins, 1981)


49

′x ⋅[t × (Rx)]= 0 ′x TEx = 0 with E = [t× ]R

X

The vectors , and t are coplanar.

essential matrix gives us the relative rotation and translation

t

Rx x '

(x in the camera2 frame) Rx

x '


X


50

• forms the epipolar line l’ associated with ( )• forms the epipolar line l associated with ( )• Epipoles and satisfy that and • E is singular (rank two)• E has five degrees of freedom (rotation + translation)

′x ⋅[t × (Rx)]= 0 ′x TEx = 0 with E = [t× ]R

Ex E

T x ' x

x '∀p '∈ l ', p 'Ex = 0∀p∈ l, pET x ' = 0

x x '

e e ' Ee = 0 ET e ' = 0


Epipolar constraint: Uncalibrated case

51

• The camera matrices K and K’ of the two cameras are unknown

• We can write the epipolar constraint in terms of unknown normalized coordinates :

• Here are screen coordinates

X

ˆ ˆT¢ =x Ex 0 x =Kx, ′x = ′K ˆ ′x

K =

f pxf py

1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

x and ˆ ′x

x’x

x and x '

x x '



52

X

Fundamental Matrix(Faugeras and Luong, 1992)

x =K −1xˆ ′x = ′K −1 ′x

weak calibration

ˆ ˆT¢ =x Ex 0 1withT T- -¢ ¢= =x Fx K EK0 F

x’x

x x '



53

• F x forms the epipolar line l’ associated with x (∀p’∈l’, p’Fx=0)• FTx’ forms the epipolar line l associated with x’ (∀p∈l, pFTx’=0)• Epipoles satisfy that F e = 0 and FTe’ = 0• F is singular (rank two)• F has seven degrees of freedom

X

ˆ ˆT¢ =x Ex 0 1withT T- -¢ ¢= =x Fx K EK0 F

x x x '

x’


The eight-point algorithm

54

• Meaning of error

sum of Euclidean distances between points x’iand epipolar lines Fxi (or points xi and epipolarlines FTx’i) multiplied by a scale factor

• Nonlinear approach:

minimize 2 2

1d ( , ) d ( , )

N

i i i ii=

¢ ¢é ù+ë ûå Tx Fx x F x

2

1( ) :

NTi i

i=

¢å x Fx


The eight-point algorithm

55

x = (u, v, 1)T, x’ = (u’, v’, 1)T

Minimize:

under the constraintF33 = 1

2

1( )

NTi i

i=

¢å x Fx

, where both x and x’ are screen coordinates.


Example with eight-point algorithm

56


Example with eight-point algorithm

57

• Poor numerical conditioning• Can be fixed by rescaling the data


The normalized eight-point algorithm

• Center the image data at the origin, and scale it so the mean squared distance between the origin and the data points is 2 pixels

• Use the eight-point algorithm to compute F from the normalized points

• Enforce the rank-2 constraint (for example, take SVD of F and throw out the smallest singular value)

• Transform fundamental matrix back to original units: if T and T’ are the normalizing transformations in the two images, than the fundamental matrix in original coordinates is T’T F T

58(Hartley, 1995)


Comparison of estimation algorithms

59

8-point Normalized 8-point Nonlinear least squares

Av. Dist. 1 2.33 pixels 0.92 pixel 0.86 pixel

Av. Dist. 2 2.18 pixels 0.85 pixel 0.80 pixel


From epipolar geometry to camera calibration

• Estimating the fundamental matrix is known as “weak calibration”

• If we know the calibration matrices of the two cameras, we can estimate the essential matrix: E = K’TFK

• The essential matrix gives us the relative rotation and translation between the cameras, or their extrinsic parameters

60

CS484: Introduction to Computer Visionvclab.kaist.ac.kr/cs484/slide07-EpipolarGeometry_v3.pdf · 2019-10-17 · Note that from this slide, the z-axis looks at the object to simplify

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