Post on 20-Dec-2015
Camera models and single view geometry
Camera model
Camera: optical system
d2 1
thin lens
small angles:
Y
Z
11
Y
21
curvature radius
22
Y
Y
Z
incident light beam
deviated beam
deviation angle ? ’’
lens refraction index: n
)11
()1(21
Yn
n
)'sin(
)sin(
' 1
1
1
1
n
)'sin(
)''sin(
'
''
2
2
2
2
Thin lens rulesa) Y=0 = 0
)11
)(1(
1
21
n
f
f
Y
parallel rays converge onto a focal plane
b) f = Y
beams through lens center: undeviated
independent of y
r
f
Y
h
Where do all rays starting from a scene point P converge ?
Z
r
Y
h
Z
f
fr
Y
h
frZ
111Fresnel law
P
Obs. For Z ∞, r f
O
p?
d
f
a
Z
if d ≠ r …
focussed image:blurring circle) <image resolution
depth of field: range [Z1, Z2] where image is focussed
image plane P
p
O
r (blurring circle)=a (d-r)/r
image of a point = blurring circle
the image of a point P belongs to the line (P,O)
p
P
O
p = image of P = image plane ∩ line(O,P)
interpretation line of p: line(O,p) =locus of the scene points projecting onto image point p
image plane
r fHp: Z >> a
the image of a point P belongs to the line (P,O)
p
P
O
p = image of P = image plane ∩ line(O,P)
interpretation line of p: line(O,p) =locus of the scene points projecting onto image point p
image plane
r fHp: Z >> a
p
P
O Z
Y
X
c
y
x
Z
Xfx
Z
Yfy
perspective projection
f
-nonlinear-not shape-preserving-not length-ratio preserving
•Point [x,y]T expanded to [u,v,w]T
•Any two sets of points [u1,v1,w1]T and [u2,v2,w2]T
represent the same point if one is multiple of the other
•[u,v,w]T [x,y] with x=u/w, and y=v/w
•[u,v,0]T is the point at the infinite along direction (u,v)
• In 2D: add a third coordinate, w
Homogeneous coordinates
Transformations
translation by vector [dx,dy]T
scaling (by different factors in x and y)
rotation by angle
Homogeneous coordinates
In 3D: add a fourth coordinate, t
•Point [X,Y,Z]T expanded to [x,y,z,t]T
•Any two sets of points [x1,y1,z1,t1]T and [x2,y2,z2,t2]T
represent the same point if one is multiple of the other
•[x,y,z,t]T [X,Y,Z] with X=x/t, Y=y/t, and Z=z/t
•[x,y,z,0]T is the point at the infinite along direction (x,y,z)
Transformations
scaling
translation
rotation
Obs: rotation matrix is an orthogonal matrix
i.e.: R-1 = RT
Pinhole camera model
TT ZfYZfXZYX )/,/(),,(
101
0
0
1
Z
Y
X
f
f
Z
fY
fX
Z
Y
X
10100
000
000
Z
Y
X
f
f
w
v
u
with
w
ux
w
vy
Scene->Image mapping: perspective transformation
With “ad hoc” reference frames, for both image and scene
Let us recall them
O Z
Y
X
c
y
x
f
scene reference- centered on lens center- Z-axis orthogonal to image plane- X- and Y-axes opposite to image x- and y-axes
image reference- centered on principal point- x- and y-axes parallel to the sensor rows and columns- Euclidean reference
O
Z
YX
c
yx
f
scene reference- not attached to the camera
image reference- centered on upper left corner- nonsquare pixels (aspect ratio) noneuclidean reference
Actual references are generic
principal axis
principal point
Principal point offset
Too
T vZfYuZfXZYX )/,/(),,(
principal pointT
oo vu ),(
101
0
0
1
Z
Y
X
vf
uf
Z
ZvfY
ZufX
Z
Y
X
o
o
o
o
CCD camera
1oy
ox
vf
uf
K
11
1
o
o
vf
uf
aK
Scene-image relationship wrt actual reference frames
w
v
u
va
uas
w
v
u
o
o
100
0
)(1
Auimage
scene XtR
10
1
Z
Y
X
normally, s=0
XtR
XPu 10
0100
000
000
100
0
01
f
f
va
u
o
o
tKRKtR
IP 10
0
100
0
0
o
o
vaf
uf
K upper triangular: intrinsic camera parameters
scene-camera tranformation
extrinsic camera parameters
orthogonal (3D rotation) matrix
P: 10-11 degrees of freedom (10 if s=0)
1
z
y
x
oIMXoIMu
)(1
oxMx
oIMu
i.e., defining x = [x, y, z]T
oIMoMMmMtKRKP
with RKM and mMo 1
The locus of the points x whose image is u is a straight line through o having direction
mMo 1 is independent of u
o is the camera viewpoint (perspective projection center)
uMd 1
line(o, d) = Interpretation line of image point u
Interpretation of o:
u is image of x if uMox 1)(
i.e., if uMox 1
Intrinsic and extrinsic parameters from P
RKM
M K and R
1111 KRKRM T
RQ-decomposition of a matrix: as the product betweenan orthogonal matrix and an upper triangular matrix
M and m t
tRKtKRKtKR)mMo 1T1 T (1
Rot
Camera anatomy
Camera centerColumn pointsPrincipal planeAxis planePrincipal pointPrincipal ray
Camera center
0P O
null-space camera projection matrix
Oλ)(1λAX
Oλ)P(1λPAPXx
For all A all points on AO project on image of A,
therefore O is camera center
Image of camera center is (0,0,0)T, i.e. undefined
Finite cameras:
1
M
1
1moO
Infinite cameras: 0Md,0
d
o
Column vectors
0
0
1
0
mpppp 3212
Image points corresponding to X,Y,Z directions and origin
Row vectors
1p
p
p
0 3
2
1
Z
Y
X
y
x
T
T
T
Image of a point on the principal plane (the plane of the thin lens) is at the infinity
w = 0
is the principal plane
0
1
p3
Z
Y
X
T
T3p
1p
p
p0
3
2
1
Z
Y
X
w
yT
T
T
note: p1,p2 dependent on image reparametrization
0
1
p2
Z
Y
X
T T2p is the plane through the u-axis
T1p is the plane through the v-axis similarly,
The principal point
principal point
0,,,p̂ 3332313 ppp
33o Mmp̂Pc
The principal axis vector
3m
vector defining front side of camera
the direction of the normal to the principal plane
3m
Action of projective camera on point
PXx
MdDm|MPDx
Forward projection
Back-projection
xPX 1PPPP
TT IPP
(pseudo-inverse)
0P O
OλxPλX
1
m-μxM
1
mM-
0
xMμλX
-1-1-1
xMd -1
OD
Depth of points
oO X~
mXPXPT3T3T3 w
(dot product)(PO=0)
1m;0det 3 MIf , then m3 unit vector in positive direction
3m
)sign(detMPX;depth
T
w TX X,Y,Z,T
Camera matrix decomposition
Finding the camera center
0P O (use SVD to find null-space)
m,p,pdet 32X m,p,pdet 31Y
m,p,pdet 21Z m,p,pdet 21T
Finding the camera orientation and internal parameters
KRM (use RQ decomposition ~QR)
Q R=( )-1= -1 -1QR
(if only QR, invert)
When is skew non-zero?
1yx
xx
p
ps
K
1
arctan(1/s)
for CCD/CMOS, always s=0
Image from image, s≠0 possible(non coinciding principal axes)
HPresulting camera:
Euclidean vs. projective
homography 44
0100
0010
0001
homography 33P
general projective interpretation
Meaningfull decomposition in K,R,t requires Euclidean image and space
Camera center is still valid in projective space
Principal plane requires affine image and space
Principal ray requires affine image and Euclidean space
Camera calibration
11
v
u
Z
Y
X
ii uX
?
3
2
1
P
P
P
P
from scene-point to image point correspondence…
…to projection matrix
Basic equations
ii PXu 0u ii PX
0
0
01
10
3
2
1
i
ii
i
i
P
P
P
uv
u
v
X
0
0
01
10
0
01
10
3
2
1
3
2
1
TT
TT
TT
P
P
P
uv
u
v
P
P
P
uv
u
v
i
i
i
ii
i
i
i
i
i
ii
i
i
X
X
X
X
X
X
Basic equations ctd.
0Ap
0
0
01
10
0
01
10
3
2
1
3
2
1
TT
TT
TT
P
P
P
uv
u
v
P
P
P
uv
u
v
i
i
i
ii
i
i
i
i
i
ii
i
i
X
X
X
X
X
X
00
0
3
2
1
T
T
T
TT
TT
P
P
P
u
v
iii
iii
XX
XX
withT
321 PPP p (12x1)
singular matrix
minimal solution
over-determined solution
5½ correspondences needed (say 6)
P has 11 dof, 2 independent eq./points
n 6 points
Apminimize subject to constraint 1p
0Ap
p: eigenvector of ATA associated to its smallest eigenvalue
Degenerate configurations
More complicate than 2D case (see Ch.21)
(i) Camera and points on a twisted cubic
(ii) Points lie on plane or single line passing through projection center
Data normalization
32ii UXX
~
ii Tuu~
(i) translate origin to gravity center(ii) (an)isotropic scaling
from line correspondences
Extend DLT to lines
ilPT
0PXl 1T ii
(back-project image line)
0PXl 2T ii
(2 independent eq.)
Geometric error
Gold Standard algorithmObjective
Given n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P
Algorithm
(i) Linear solution:
(a) Normalization:
(b) DLT:
(ii) Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error:
(iii) Denormalization:
ii UXX~ ii Txx~
UP~
TP -1
~ ~~
Calibration example
(i) Canny edge detection(ii) Straight line fitting to the detected edges(iii) Intersecting the lines to obtain the images corners
typically precision <1/10
(HZ rule of thumb: 5n constraints for n unknowns
Exterior orientation
Calibrated camera, position and orientation unkown
Pose estimation
6 dof 3 points minimal (4 solutions in general)
short and long focal length
Radial distortion
Radial distortion
Correction of distortion
Choice of the distortion function and center
Computing the parameters of the distortion function(i) Minimize with additional unknowns(ii) Straighten lines(iii) …
Properties of perspective transformations
1) vanishing points
V image of the unproper point along direction d
dMd
mMuV 0
VuMd 1
the interpretation line of V is parallel to d
O
V
Pd
The images of parallel lines are concurrent lines
2) cross ratio invariance
Given four colinear points 4321 ,,, pppp
42
32
41
31
4321 ,,,
xx
xxxx
xx
CR
pppp
let 4321 ,,, xxxx be their abscissae
Properties of perspective transformations ctd.
Cross ratio invariance under perspective transformation
TT ],0,0,[],,,[ txtzyx Xa point on the line y=0=z
xPu T 144
1],[t
xwu
p
p
its image belongs to a line
its coordinate u
XPu T],,[ wvu
4231
43214321 ,det,det
,det,det,,,
uuuu
uuuuuuuu CR
42143114
43142114
,detdet,detdet
,detdet,detdet
xxPxxP
xxPxxP
4231
4321
,det,det
,det,det
xxxx
xxxx 4321 ,,, xxxxCR
Object localization 1: three colinear points
geometric model of an objecta perspective image of the object
position and orientation
of the object ?
A
B
C
C’
A’
B’O
calibrated camera: mMP A
B
C
known
known interpretation lines
A
B
C
C’
A’
B’O
∞
V
a) orientation
cb
caCBACRVCBACR
),,,(,',','
Cross ratio invariance:
solve for V (image of ∞)
V: vanishing point of the direction of (A,B,C)
interpretation line of V parallel to (A,B,C) VuM 1direction
b) position (e.g., distance(O,A))
A
B
C
C’
A’
B’O
V
VuM 1
CuM 1
AuM 1
interpretation lines
angles and
sin
sinACOA
Object localization 2: four coplanar points
O
(i) orientation of (A,E,C)(ii) orientation of (B,E,D)(iii) distance (O,A)
E
A
B
CD
a’
b’
b”
a”
Find vanishing point of the field-bottom direction
Off-side
images of symmetric segments
a and b: abscissae of the endpoints of a segment
c=(a+b)/2: abscissa of segment midpoint,
d=∞: point at the infinite along the segment direction
a c b d
1,,,
cb
ca
dbdacbca
dcbaCR
(a’,b’) and (a”, b”) are image of symmetric segments
same image of the midpoint c’, same vanishing point d’
Harmonic 4-tuple (a,b,c,d)
solve
1
''''
''
''
',',','
dbda
cb
ca
dcbaCR
1
''''''
''''''
',','',''
dbda
cbca
dcbaCR
{ for c’, d’
system of two linear equations in (c’d’) and (c’+d’)
two degree equation, whose solutions are c’ and d’
among the two solutions, the one for d’ is the value external to the range [a’,b’]
Action of projective camera on planes
1ppp
10ppppPXx 4214321 Y
XYX
The most general transformation that can occur between a scene plane and an image plane under perspective imaging is a plane projective transformation
Action of projective camera on lines
forward projection
μbaμPBPAμB)P(AμX
back-projection
PXx
lPT
0xlT
with
X0PXl TT
Interpretation plane of line l
PxxY
X
T
Y
X
4214214321 ppp
1
ppp0
ppppPXx
Image of a conic
xCx0C 1-TTT PPxx
therefore
1-TCC' PP
Action of projective camera on conics
back-projection of a conic C to cone
C
coQ
coQ
CPPQ Tco
XQX0CPXPXCxx coTTTT
000CKK0|KC
0KQ
TT
T
co
example:
0CxxT
PXx with
Interpretation cone of a conic C
back-projection of a conic C to cone coQ
Images of smooth surfaces
The contour generator is the set of points X on S at which rays are tangent to the surface. The corresponding apparent contour is the set of points x which are the image of X, i.e. is the image of
The contour generator depends only on position of projection center, depends also on rest of P
Action of projective camera on quadrics
apparent contour of a quadric Q
TPPQC **
0lPPQlQ T*T*T
dual quadric1* QQ is a plane quadric:
0Q*T the set of planes tangent to Q
Let us consider only those planes that are backprojection of image lines
lPT
with its dual is1*CC
The plane containing the apparent contour of a quadric Q from a camera center O follows from pole-polar relationship
The cone with vertex V and tangent to the quadric Q is
TTCO (QV)(QV)-QV)QV(Q 0VQCO
back-projection to cone
=QO
What does calibration give?
xKd 1
0d0]|K[Ix
21-T-T
211-T-T
1
2-1-TT
1
2T
21T
1
2T
1
)xK(Kx)xK(Kx
)xK(Kx
dddd
ddcos
An image line l defines a plane through the camera center
with normal n=KTl measured in the camera’s Euclidean
frame. In fact the backprojection of l is PTl=KTl
The image of the absolute conic
KRd0
dO]|KR[IPXx
mapping between ∞ to an image is given by the planar
homogaphy x=Hd, with H=KR
absolute conic (IAC), represented by I3 within ∞ (w=0)
1-T-1T KKKKω 1TCHHC
(i) IAC depends only on intrinsics(ii) angle between two rays(iii) DIAC=*=KKT
(iv) K (Cholesky factorization)(v) image of circular points belong
to (image of absolute conic)
2T
21T
1
2T
1
ωxxωxx
ωxxcos
its image (IAC)
A simple calibration device
(i) compute Hi for each square
(corners (0,0),(1,0),(0,1),(1,1))
(ii) compute the imaged circular points Hi [1,±i,0]T
(iii) fit a conic to 6 imaged circular points
(iv) compute K from K-T K-1 through Cholesky factorization (= Zhang’s calibration method)
Orthogonality = pole-polar w.r.t. IAC
The calibrating conic
1T K1
11
KC
Vanishing points
λKdaλPDPAλPXλx
KdλKda limλ xlimvλλ
KdPXv
Vanishing lines
Orthogonality relation
2T
21T
1
2T
1
ωvvωvv
ωvvcos
0ωvv 2T
1
0lωl 2*T
1
Calibration from vanishing points and lines
Calibration from vanishing points and lines