Camera calibration and epipolar geometry - Computer …cs.unc.edu/~lazebnik/research/spring08/lec11_epipolar.pdfEpipolar constraint: Calibrated case • Assume that the intrinsic and

Camera calibration and epipolar geometry

Odilon Redon, Cyclops, 1914

Review: Alignment• What is the geometric relationship between

pictures taken by cameras that share the same center?

• How many points do we need to estimate a homography?

• How do we estimate a homography?

Geometric vision• Goal: Recovery of 3D structure

• What cues in the image allow us to do this?

Shading

Visual cues

Merle Norman Cosmetics, Los Angeles

Slide credit: S. Seitz

Visual cuesShading

Texture

The Visual Cliff, by William Vandivert, 1960


Visual cues

From The Art of Photography, Canon

Shading

Texture

Focus


Visual cuesShading

Texture

Focus

Perspective


Visual cuesShading

Texture

Focus

Perspective

MotionSlide credit: S. Seitz

Our goal: Recovery of 3D structure• We will focus on perspective and motion• We need multi-view geometry because

recovery of structure from one image is inherently ambiguous

x

X?X?

X?





)/,/(),,( ZYfZXfZYX a

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

10100

1ZYX

ff

ZYfXf

ZYX

a

Recall: Pinhole camera model

PXx =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

1010101

1ZYX

ff

ZYfXf

PXx = [ ]0|I)1,,(diagP ff=

Pinhole camera model

Camera coordinate system

• 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

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

Principal point offset

• Camera coordinate system: origin is at the prinicipal point

• Image coordinate system: origin is in the corner

principal point: ),( yx pp

)/,/(),,( yx pZYfpZXfZYX ++a

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛++

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

10100

1ZYX

pfpf

ZpZYfpZXf

ZYX

y

x

x

x

a



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛++

1010101

1ZYX

pfpf

ZZpYfZpXf

y

x

x

x


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1y

x

pfpf

K calibration matrix [ ]0|IKP =


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

111yy

xx

y

x

y

x

pfpf

mm

K βαβα

Pixel coordinates

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

Pixel size: yx mm

11×

pixels/m m pixels

( )C~-X~RX~ cam =

Camera rotation and translation

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

coords. of point in camera frame

coords. of camera center in world frame

coords. of a pointin world frame (nonhomogeneous)

( )C~-X~RX~ cam =

X10

C~RR1X~

10C~RRXcam ⎥

⎦

⎤⎢⎣

⎡ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡ −=

[ ] [ ]XC~R|RKX0|IKx cam −== [ ],t|RKP = C~Rt −=

Camera rotation and translation

In non-homogeneouscoordinates:

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

Camera parameters• Intrinsic parameters

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

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

111yy

xx

y

x

y

x

pfpf

mm

K βαβα

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

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

and known image projections xi, estimate the camera parameters

? P

Xi

xi

ii PXx =λ

Camera calibration

0PXx =× iiiT

T

T

i

i

yx

XPPP

1 3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡λ

0PPP

0XXX0X

XX0

3

2

1

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−

Tii

Tii

Tii

Ti

Tii

Ti

xyx

y

Two linearly independent equations

Camera calibration

• P has 11 degrees of freedom (12 parameters, but scale is arbitrary)

• One 2D/3D correspondence gives us two linearly independent equations

• Homogeneous least squares• 6 correspondences needed for a minimal solution

0pA =0PPP

X0XXX0

X0XXX0

3

2

1111

111

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

Tnn

TTn

Tnn

Tn

T

TTT

TTT

xy

xy

LLL

Camera calibration

• Note: for coplanar points that satisfy ΠTX=0,we will get degenerate solutions (Π,0,0), (0,Π,0), or (0,0,Π)

0Ap =0PPP

X0XXX0

X0XXX0

3

2

1111

111

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

Tnn

TTn

Tnn

Tn

T

TTT

TTT

xy

xy

LLL

Camera calibration• Once we’ve recovered the numerical form of

the camera matrix, we still have to figure out the intrinsic and extrinsic parameters

• This is a matrix decomposition problem, not an estimation problem (see F&P sec. 3.2, 3.3)

Two-view geometry• Scene geometry (structure): Given

corresponding points in two or more images, where is the pre-image of these points in 3D?

• Correspondence (stereo matching): Given a point in just one image, how does it constrain the position of the corresponding point x’ in another image?

• Camera geometry (motion): Given a set of corresponding points in two images, what are the cameras for the two views?

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

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

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

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

O1 O2

x1x2

X

Triangulation: Linear approach

baba ][0

00

×=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=×

z

y

x

xy

xz

yz

bbb

aaaa

aa

XPxXPx

222

111

==

λλ

0XPx0XPx

22

11

=×=×

0XP][x0XP][x

22

11

==

×

×

Cross product as matrix multiplication:

Triangulation: Linear approach

XPxXPx

222

111

==

λλ

0XPx0XPx

22

11

=×=×

0XP][x0XP][x

22

11

==

×

×

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

Triangulation: Nonlinear approachFind X that minimizes

O1 O2

x1x2

X?

x’1

x’2

),(),( 222

112 XPxdXPxd +

• Epipolar Plane – plane containing baseline (1D family)• Epipoles= intersections of baseline with image planes = projections of the other camera center= vanishing points of camera motion direction• Epipolar Lines - intersections of epipolar plane with imageplanes (always come in corresponding pairs)

• Baseline – line connecting the two camera centers

Epipolar geometryX

x x’

Example: Converging cameras

Example: Motion parallel to image plane

e

e’

Example: Forward motion

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

Epipolar constraint

• 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

x x’

X

x’

X

x’

X

Epipolar constraint example

Source: K. Grauman

X

x x’

Epipolar constraint: Calibrated case

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

• 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

• We can also set the global coordinate system to the coordinate system of the first camera

X

x x’


Camera matrix: [I|0]X = (u, v, w, 1)T

x = (u, v, w)T

Camera matrix: [RT | –RTt]Vector x’ in second coord. system has coordinates Rx’ in the first one

Rt

The vectors x, t, and Rx’ are coplanar

Essential Matrix(Longuet-Higgins, 1981)


0)]([ =′×⋅ xRtx RtExExT ][with0 ×==′

X

x x’

The vectors x, t, and Rx’ are coplanar

X

x x’


• E x’ is the epipolar line associated with x’ (l = E x’)• ETx is the epipolar line associated with x (l’ = ETx)• E e’ = 0 and ETe = 0• E is singular (rank two)• E has five degrees of freedom (up to scale)

0)]([ =′×⋅ xRtx RtExExT ][with0 ×==′

Epipolar constraint: Uncalibrated case

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

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

X

x x’

0ˆˆ =′xExT xKxxKx ′′=′= ˆ,ˆ

Epipolar constraint: Uncalibrated caseX

x x’

Fundamental Matrix(Faugeras and Luong, 1992)

0ˆˆ =′xExT

xKxxKx′′=′

=ˆˆ

1with0 −− ′==′ KEKFxFx TT

Epipolar constraint: Uncalibrated case

0ˆˆ =′xExT 1with0 −− ′==′ KEKFxFx TT

• F x’ is the epipolar line associated with x’ (l = F x’)• FTx is the epipolar line associated with x (l’ = FTx)• F e’ = 0 and FTe = 0• F is singular (rank two)• F has seven degrees of freedom

X

x x’

The eight-point algorithm

x = (u, v, 1)T, x’ = (u’, v’, 1)T

Minimize:

under the constraint|F|2 = 1

2

1)( i

N

i

Ti xFx ′∑

=

The eight-point algorithm

• Meaning of error

sum of Euclidean distances between points xi and epipolar lines Fx’i (or points x’i and epipolar lines FTxi) multiplied by a scale factor

• Nonlinear approach: minimize

:)( 2

1i

N

i

Ti xFx ′∑

=

[ ]∑=

′+′N

ii

Tiii xFxxFx

1

22 ),(d),(d

Problem with eight-point algorithm

Problem with eight-point algorithm

Poor numerical conditioningCan 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 TT F T’

(Hartley, 1995)

Comparison of estimation algorithms

0.80 pixel0.85 pixel2.18 pixelsAv. Dist. 2

0.86 pixel0.92 pixel2.33 pixelsAv. Dist. 1

Nonlinear least squaresNormalized 8-point8-point

Epipolar transfer• Assume the epipolar geometry is known• Given projections of the same point in two

images, how can we compute the projection of that point in a third image?

x1 x2? x3

Epipolar transfer• Assume the epipolar geometry is known• Given projections of the same point in two

images, how can we compute the projection of that point in a third image?

x1 x2 x3 l32l31

l31 = FT13 x1

l32 = FT23 x2

When does epipolar transfer fail?

Next time: Stereo

Documents

Camera calibration and epipolar geometry - Computer …cs.unc.edu/~lazebnik/research/spring08/lec11_epipolar.pdfEpipolar constraint: Calibrated case • Assume that the intrinsic and