Projective Geometry
CS 4495/7495
Computer Vision
Frank Dellaert, Fall 07
Photographs are Projections
Vertical vanishing
point
(at infinity)
Vanishing
line
Vanishing
point
Slide Courtesy Antonio Criminisi
Vanishing
point
2D Coordinate Frames &
Points
• coordinates x and y
j
x = op ! i
p =(x,y)T
y = op ! j
i
o
2D Lines
• Line l = ax+by=c
j
p=(x,y)T
c
(a,b)T
i
Homogeneous Coordinates
• Uniform treatment of points and
lines
j
• Line-point incidence: lTp=0
stay the same when scaled:
p=(x,y,1)T~(kx,kx,k)T
c
l=(a,b,c)T~(ka,kb,kc)T
(a,b)T
i
Join = cross product !
• Join of two lines is a point:
p=l1xl2
• Join of two points is a line:
l=p1xp2
Automatic estimation of
vanishing points and lines
Joining two parallel lines ?
(a,b,c)
i j k
&bd ' cb #
p = a b c = $$ca ' ad !!
a b d $% 0 !"
(a,b,c)
(a,b,d)
Points at Infinity !
(-b,a,0)T
Line at infinity
linf=(0,0,1)T
j
l=(a,b,c)T
i
(-b,a,0)T
Method 1: Determinants
• 2D Line between two 2D points
– Point x on l must be linear comb of p1
& p2:
xT
x
p1T = x1
p2T
x2
y
y1
y2
w
w1 = 0 " x = p1 ! p2
w2
cameras01.m
Method 2: Homogenous
Equations
• we want lTp=0 for all lines !:
&l1T #
&0 #
$ T ! &x# $ !
l 2 ! $ ! $0 !
$
Ap = T ' $ y ! =
$l3 !
$0 !
$ T ! $% w!" $ !
$%l4 !"
%0 "
Live Demo!
cameras02.m
cameras03.m
Solving Homogeneous
Equations
• Much nicer than in book:
2
Ap = USV p
&l1T #
&0 #
$ T ! &x# $ !
l 2 ! $ ! $0 !
$
Ap = T ' $ y ! =
$l3 !
$0 !
$ T ! $% w!" $ !
$%l4 !"
%0 "
2
= p T VS T SV T p
write
x =VT p
arg min xT S T Sx
2
arg min Ap s.t. p = 1
T
!
! x = (0,0,...,1)T
SVD Figure
A=U S VT
4x4
3
R
column space
4x3
3x3
2
3 =P
R
row space
nullspace(AT)
R
nullspace(A)=empty
3d eigenvector smallest contribution
Fundamental Theorem
(Strang)
Ax
x
x+y
y
Strang Course 18.06 Linear Algebra,
on MIT OpenCourseWare, ocw.mit.edu
Ax
Applied to line fitting
& x1T #
$ T!
$ x2 !
$ x3T !
$ T!
$ x4 ! a
$ xT ! & #
$ 5T ! ' $b ! = 0
$ x6 ! $ !
$ xT ! $% c !"
$ 7!
$ x8T !
$ T!
$ x9 !
$% x10T !"
cameras04.m
Caveat: Normalization matters !
Hierarchy of 2D
Transforms
• Subgroup Structure:
– Translation (2DOF)
– Rigid 2D (3DOF)
– Affine (6DOF)
– Projective (8DOF)
Rigid 2D Transform
• Take Notes
A special case, planes
Homography matrix (3x3)
Observer
2D
Image plane
(retina, film, canvas)
2D
World plane
A plane to plane projective transformation
Homography
• Take Notes
•u=
•v=
Analysing patterns and shapes
What is the shape of the b/w floor pattern?
y
h
p
a
r
og
m
o
H
The floor
(enlarged)
Automatical
ly
rectified
SVD For Homography
• Cross-product Trick
• Take Notes
In 3D: Same Story
•
•
•
•
•
3D points: (x,y,z,w)T
3D planes: (a,b,c,d)T
join of three points = plane
join of three planes = point
plane at infinity: (0,0,0,1)T
Join of 3D points
X Y
PT
X 1 Y1
P1T
=
T
X 2 Y2
P2
X 3 Y3
P3T
Z T
Z1 T1
=0
Z 2 T2
Z 3 T3
Determinant minors:
(analog of cross product)
Y1
X Y2
Y3
Z1 T1
X1
Z 2 T2 ! Y X 2
Z 3 T3
X3
Z1 T1
X1
Z 2 T2 + Z X 2
Z 3 T3
X3
Y1 T1
X1
Y2 T2 ! T X 2
Z 3 T3
X3
Y1
Y2
Z3
Z1
Z2 = 0
Z3
Plane fitting
&a #
&X # $ !
$ ! $b !
$X ! ' $c ! = 0
$X ! $ !
% " d
% "
T
1
T
2
T
3
cameras05.m
Hierarchy of 3D
Transforms
• Subgroup Structure:
– Translation (3DOF)
– Rigid 3D (6DOF)
– Affine (12DOF)
– Projective (15DOF)
Rigid Body
Transformations
• Need a way to specify the six degreesof-freedom of a rigid body.
• Why are their 6 DOF?
A rigid body is a
collection of points
whose positions
relative to each
other can’t change
Fix one point,
three DOF
Fix second point,
two more DOF
(must maintain
distance constraint)
Third point adds
one more DOF,
for rotation
around line
Notations
•
•
•
Superscript references coordinate frame
AP is coordinates of P in frame A
BP is coordinates of P in frame B
• Example :
" A x%
$A '
A
P = $ y ' ( OP =
$ A z'
# &
kA
!
iA
OA
!
!
!
!
jA
P
(
A
) (
x • iA +
A
) (
y • jA +
A
z • kA
)
Translation
B
A
B
P= P+ (OA )
kB
!
kA
!
iA
OA
!
jA
!
!
iB
!
P
!
!
!
OB
jB
Translation
• Using homogeneous coordinates,
translation can be expressed as a matrix
multiplication.
B
P = A P + BO A
! B P" ! I
# $=#
%1 & % 0
B
OA " ! A P "
$# $
1 & %1 &
• Translation is communicative
Rotation
uuuv
OP = (i A
jA
B
B
A
R
! Ax "
#A $
k A )# y $ = (iB
# Az $
% &
jB
! Bx "
#B $
k B )# y $
# Bz $
% &
P = AB R A P
means describing frame A in
The coordinate system of
frame B
Rotation
! i A .i B
B
# i .j
R
=
A
# A B
#% i A .k B
= !# B i A
jA .i B
j A . jB
jA .k B
B
jA
k A .i B "
k A . jB $
$
k A .k B $&
B
k A "$
! A iTB "
#A T$
= # jB $
# A k TB $
%
&
Orthogonal matrix!
Example: Rotation about z
axis
What is the rotation matrix?
Rotation in homogeneous
coordinates
• Using homogeneous coordinates, rotation can
be expressed as a matrix multiplication.
B
B
A
A
P= R P
! B P " ! AB R 0" ! A P "
# $=#
$# $
%1 & % 0 1& %1 &
• Rotation is not communicative
Rigid transformations
B
B
A
A
B
P = R P + OA
Rigid transformations
(con’t)
• Unified treatment using homogeneous
coordinates.
! B P " !1
# $=#
%1 & %0
! AB R
=# T
%0
B
O A " ! AB R 0" ! A P "
$#
$# $
1 & % 0 1& %1 &
B
OA " ! A P "
$# $
1 & %1 &
! B P " B ! AP "
# $ = AT # $
%1&
%1&