1 digital images worldcameradigitizer digital image (i) what determines where the image of a 3d...
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Digital Images
World Camera Digitizer DigitalImage
(i) What determines where the image of a 3D point appears on the 2D image?
(ii) What determines how bright that image point is?
Reflectance, radiometry
geometry
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1100
110
201
11
2
1
2y
x
y
x
y
x
y
x
yy
xx
21),( eejixx yxyxyx T
• change of coordinate system: the same pt in two different systems oxy and ox’y’• point transfomation: a point (x,y) is transformed (translated) into (x’,y’) within the same coordinate frame
Two different interpretations:
Review of some basic geometry(compulsary for vision, graphics and robotics)
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y
x
tyxy
tyxx
cossin
sincos
y
x
t
t
y
x
y
x22R
110
1
1222 y
x
y
xtR
R is a rotation matrix=orthonormal = orthogonal and unit vectors, 2*2 matrix (only 1 d.o.f.) such that
222222 IRR T
2D general Euclidean transformation:
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1
10
1
1333
z
y
x
z
y
x
tR333333 IRR T
cossin0
sincos0
001
cos0sin
010
sin0cos
100
0cossin
0sincos
)()()(33 xyz RRRR
One example of R might be:
3D Euclidean transformation:
Different sign
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prod.dot with nn AE nA nR
Naturally everything starts from the known vector space
• add two vectors• multiply any vector by any scalar• zero vector – origin • finite basis
One step further …vector, affine, and Euclidean spaces
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21),( eexx yxyx T
Affine geometry and affine coordinates:
E1 and e2 are any noncolinear vectors, not necessarily orthogonal unit ones
No more rotation as no perpendicularity (as no dot prod.)
fydxcy
eybxax
f
e
y
x
dc
ba
y
x
110
1
1222 y
x
y
xtA
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• Distances -- eucl. Coord
• Angles, ortho
• Ratios – affine coord.• parallelism
Dot product Linear dependency
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Given 3 points (2 vectors) on the plane, we can define an affine coordinate frame (affine basis),Any 4th point can be expressed in terms of affine coordinates …
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Geometric modeling of a camera
u
v
X
u
O
X’
u’
P3
P2
How to relate a 3D point X (in oxyz) to a 2D point in pixels (u,v)?
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Pinhole cameras
• Abstract camera model - box with a small hole in it
• Pinhole cameras work in practice
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Distant objects are smaller:
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Parallel lines meet:
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• each set of parallel lines (=direction) meets at a different point– The vanishing point for
this direction
• Sets of parallel lines on the same plane lead to collinear vanishing points. – The line is called the
horizon for that plane
• Good ways to spot faked images– scale and perspective
don’t work
– vanishing points behave badly
– supermarket tabloids are a great source.
Vanishing points:
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• Vector space to affine: isomorph, one-to-one (pt=vector)
• vector to Euclidean as an enrichment: scalar prod.
Pts, lines, parallelism
Angle, distances, circles
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Z
Y
f
y
Z
X
f
x ,
X
Y
Z
xy
u
v
X
x
O
f
Camera coordinate frame
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10100
0010
0001
Z
Y
X
f
y
x
Z
Y
X
f
y
x
In more familiar matrix form:
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xo
y
u
v
X
Y
Z
x y
u
v
X
xO
f
Image coordinate frame
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Image coordiante frame: intrinsic parameters
mm
pixels,
mm
pixels 00
y
vvk
x
uuk vu
0
0
vykv
uxku
v
u
1100
v0
u0
10v
0u
y
x
k
k
v
u
If u not perpendicular to v, but an angle alpha:
1100
v0
u)s(
10v
0u
y
x
k
k
v
u
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Camera calibration matrix
100
v0
u)s(
100
v0
u)s(
0v
0u
0v
0u
kf
kf
K
f
y
x
f
y
x
f
f
k
k
v
u
K
100
00
00
100
v0
u)s(
10v
0u
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• Focal length in horizontal/vertical pixels (2) (or focal length in pixels + aspect ratio)• the principal point (2)• the skew (1)
5 intrinsic parameters
one rough example: 135 film
In practice, for most of CCD cameras:
• alpha u = alpha v i.e. aspect ratio=1• alpha = 90 i.e. skew s=0• (u0,v0) the middle of the image• only focal length in pixels?
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Xw Yw
Zw
Xw
X
Y
Z
xy
u
v
X
x
O
f
World (object) coordinate frame
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World coordinate frame: extrinsic parameters
1
1
1w
w
w
c
c
c
Z
Y
X
Z
Y
X
0
tR
Relation between the abstract algebraic and geometric models is in the intrinsic/extrinsic parameters!
6 extrinsic parameters
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Finally, we should count properly ...
11
11
4333 Z
Y
X
Z
Y
X
v
u
pixel
C0
tR0IK
Finally, we have a map from a space pt (X,Y,Z) to a pixel (u,v) by
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Summary of camera modelling
• 3 coordinate frame• projection matrix• decomposition• intrinsic/extrinsic param