eecs 274 computer vision geometric camera models
TRANSCRIPT
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EECS 274 Computer Vision
Geometric Camera Models
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Geometric Camera Models
• Elements of Euclidean geometry• Intrinsic camera parameters• Extrinsic camera parameters• General form of perspective projection
• Reading: Chapter 1 of FP, Chapter 2 of S
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Euclidean Geometry
Geometric camera calibration
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z
y
x
zyxOP
OPz
OPy
OPx
Pkji
k
j
i
.
.
.
Euclidean coordinate system
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1
and where
00
,],,[,],,[
00
z
y
x
d
c
b
a
dczbyax
dOAcbazyxP
OAOPAPTT
PΠ
PΠ
nn
nnn
homogenous coordinate
Planes
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OBP = OBOA + OAP , BP = BOA+ APAP: point P in frame A
Pure translation
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BABABA
BABABA
BABABABA R
kkkjki
jkjjji
ikijii
...
...
...
AB
AB
AB kji
TB
A
TB
A
TB
A
k
j
i
1st column:iA in the basis of (iB, jB, kB)
3rd row:kB in the basis of (iA, jA, kA)
Pure rotation
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100
0cossin
0sincos
RBA
Rotation about z axis
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Rotation matrix
R=R x R y R z , described by three angles
Elementary rotation
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• Its inverse is equal to its transpose, R-1=RT , and
• Its determinant is equal to 1.
Or equivalently:
• Its rows (or columns) form a right-handedorthonormal coordinate system.
Properties of rotation matrix
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Rotation group and SO(3)
• Rotation group: the set of rotation matrices, with matrix product– Closure, associativity, identity, invertibility
• SO(3): the rotation group in Euclidean space R3 whose determinant is 1– Preserve length of vectors– Preserve angles between two vectors– Preserve orientation of space
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PRP
z
y
x
z
y
x
OP
ABA
B
B
B
B
BBBA
A
A
AAA
kjikji
Pure rotations
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ABAB
AB OPRP
Rigid transformation
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2221
1211
2221
1211
BB
BBB
AA
AAA
What is AB ?
2222122121221121
2212121121121111
BABABABA
BABABABAAB
Homogeneous Representation of Rigid Transformations
11111
PT
OPRPORP ABA
ABAB
AA
TA
BBA
B
0
Block matrix manipulation
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Rigid transformations as mappings
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Rotation about the k Axis
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Affine transformation
• Images are subject to geometric distortion introduced by perspective projection
• Alter the apparent dimensions of the scene geometry
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Affine transformation
• In Euclidean space, preserve– Collinearity relation between points
• 3 points lie on a line continue to be collinear
– Ratio of distance along a line• |p2-p1|/|p3-p2| is preserved
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Shear matrix
Horizontal shear
Vertical shear
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2D planar transformations
See Szeliski Chapter 2
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2D planar transformations
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2D planar transformations
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3D transformation
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Pinhole Perspective Equation
z
yfy
z
xfx
''
''Idealized coordinate system
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Camera parameters
• Intrinsic: relate camera’s coordinate system to the idealized coordinated system
• Extrinsic: relate the camera’s coordinate system to a fix world coordinate system
• Ignore the lens and nonlinear aberrations for the moment
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Normalized ImageCoordinates
Physical Image Coordinates (f ≠1)
Units:
k,l : pixel/m
f : m(See EXIF tags)pixel
Intrinsic camera parameters
Scale parameters: k, l (image sensor may not be square)Offset: u0, v0
Manufacturing error: θ
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Calibration matrix κ
The perspectiveprojection Equation
TzyxP )1,,,(
Intrinsic camera parameters
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In reality
• Physical size of pixel and skew are always fixed for a given camera, and in principal known during manufacturing
• Some parameters often available in EXIF tag• Focal length may vary for zoom lenses when
optical axis is not perpendicular to image plane
• Change focus affects the magnification factor• From now on, assume camera is focused at
infinity
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Extrinsic camera parameters
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denotes the i-th row of R, tx, ty, tz, are the coordinates of t can be written in terms of the corresponding anglesR can be written as a product of three elementary rotations, and described by three angles
M is 3 × 4 matrix with 11 parameters5 intrinsic parameters: α, β, u0, v0, θ6 extrinsic parameters: 3 angles defining R and 3 for t
TirT
ir
Explicit form of projection Matrix
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Note:
M is only defined up to scale in this setting!!
Tir : i-th row of R
Explicit form of projection Matrix
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Theorem (Faugeras, 1993)
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Projection equation
• The projection matrix models the cumulative effect of all parameters• Useful to decompose into a series of operations
ΠXx
1****
****
****
Z
Y
X
s
sy
sx
110100
0010
0001
100
'0
'0
31
1333
31
1333
x
xx
x
xxcy
cx
yfs
xfs
00
0 TIRΠ
projectionintrinsics rotation translation
identity matrix
Camera parametersA camera is described by several parameters
• Translation T of the optical center from the origin of world coords• Rotation R of the image plane
• focal length f, principle point (x’c, y’c), pixel size (sx, sy)
• blue parameters are called “extrinsics,” red are “intrinsics”
• Definitions are not completely standardized– especially intrinsics—varies from one book to another
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Camera calibration toolbox
• Matlab toolbox by Jean-Yves Bouguethttp://www.vision.caltech.edu/bouguetj/calib_doc/
• Extract corner points from checkerboard