1. Homogeneous Coordinates and Tensor Notation - PRIMA

Coordonnées Homogènes Séance 1
Image Analysis and Formation
(Formation et Analyse d'Images)
James L. Crowley
ENSIMAG 3 - MMIS Option MIRV First Semester 2010/2011
Lesson 1 27 sept 2010
Lesson Outline:
Homogeneous Coordinates
1. Homogeneous Coordinates and Tensor Notation..........2
Tensor Notation: ......................................................................... 3
2. Cross Products ..............................................................5
Two point make a line................................................................. 5
Two lines make a point ............................................................... 6
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Coordonnées Homogènes Séance 1
1. Homogeneous Coordinates and Tensor Notation
Homogeneous coordinates allow us to express translation, rotation, scaling, and
projection all as matrix operations. The principle is to add an extra dimension to each
vector.
For example, points on a plane are expressed as:
r
P =
" x%
$ '
y
$
'
# 1&
Similarly, points in 3D space become
!
r
Q =
" x%
$ '
$
y
'
$ z'
$ '
# 1&
! The line equation, ax+by+c=0 can be expressed as a simple product:
r
L r T P = a b c
( )
" x%
$ '
y
$ ' = 0
# 1&
where
r
L T = a b c ( )
!
Similarly, for a plane equation: ax+by+cz+d=1: !
r
S r T P = a b c d
( )
" x%
$ '
$
y
' = 0 where
$ z'
$ '
# 1&
r
S = a b c d
( )
!
!
!
Note that in Homogeneous coordinates, all scalar multiplications are equivalent.
" x%
" x%
$ '
a
y
$ ' = b $ '
y
$ '
# 1&
# 1&
Any vector can be expressed in "canonical" form by normalizing the last coefficient
to 1.
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Coordonnées Homogènes Séance 1
" ax%
" ax /a%
" x%
$ '
ay
$ ' = $ '
ay /a
$ ' = $ '
y
$ '
# a & # a/a & # 1&
! Our camera model will have the form of a 3 x 4 matrix
M s
i
=
" m 11
m 12
m 13
m 14
%
$
'
m 21
m 22
m 23
m 24
$
'
# m 31
m 32
m 33
m 34 &
!
such that
"
$
$
$
$
$
#
%
' "
' $
' =
1 ' $
'
#
&
q 1
q3
q 2
q3
q 1
q 2
q 3
%
'
' = r i
Q = M s
P r
=
&
" x
" m 11
m 12
m 13
m 14
% s
%
$ '
$
'
m 21
m 22
m 23
m $
y s '
24
$
' $ z s
'
# m 31
m 32
m 33
m 34 & $ '
# 1 &
! We can express this in "canonical form" by dividing out the last coefficient:
M s
i
=
" m 11
m 12
m 13
m 14
%
$
'
m 21
m 22
m 23
m 24
$
'
# m 31
m 32
m 33
1 &
!
!
Notice that this gives 11 coefficients. This corresponds to our 11 parameters. The
Rank of M is 11.
Tensor Notation:
In tensor notation, the sign " → " is remplaced by subscripts and superscripts.
A super-script signifies a column vector.
r
For example the point P is P i
The line
"
P i $
=
$
#
p 1
p 2
p 3
%
'
!
'
&
r
L T is L i = (l 1 , l 2 , l 3 )
!
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Coordonnées Homogènes Séance 1
A matrix is a line matrix of column matrices (or a column of line matrices)
"
$
M j i
= $
$
#
m
1
1
m
1
2
m
1
3
m 2
1
m
2
2
m
2
3
m 3
1
m
3
2
m
3
3
%
'
'
'
&
! When homogeneous coordinates are used to represent transforms, these indices can
be used to indicate the reference frame.
For example: A transformation from the scene "s" to the image "i"
is a 3 x 4 matrix M
"
$
M i s
= $
$
#
m 1
1
m 1
2
m 1
3
m 2
1
m 2
2
m 2
3
m 3
1
m 3
2
m 3
3
%
'
'
1
'
&
m 4
1
m 4
2
The sub/super scripts indicate the source and destination reference frames.
i
M s
is ! a transformation from "S" (Scene) to "i" (image).
!
Einstein summation convention:
The summation symbol is implicit when a superscript adn subscript have the same
letter.
L i P i = l 1 p 1 + l 2 p 2 + l 3 p 3
for a matrix and vector, this gives a new vector:
p j = T i j p i
! This example transforms the point p in reference i to a point p in reference j.
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!
Coordonnées Homogènes Séance 1
2. Cross Products
Two point make a line
A line is a set of points that satisfies a constraint. The constraint is the line equation.
r
L r T P = ax + by + c = 0
The constraint can be determined from any two points on the line P 1 = (x 1 ,y 1 ) and
P 2 = (x 2 ,y 2 ).
Classically, we subsitute P 1 and P 2 in the line equation to obtain two equations
ax 1 + by 1 + c = 0
ax 2 + by 2 + c = 0
We then solve the two equations for a, b and c. This should give :
a = (y 1 –y 2 ) b = (x 2 – x 1 )
c = –(a x 1 – b y 1 ) = – x 1 (y 1 –y 2 )– y 1 (x 2 – x 1 ) = x 1 y 2 +x 2 y 1
if we normalize such that || (a, b) || = 1 then
d = ax + by + c
where d is a signed perpendicular distance from the line, positive to the left and
negative to the right.
In fact the constraint imposed by the line can be discovered from the cross product of
the two points:
L T = ⎜ ⎜⎛ a
b ⎝ c⎠ ⎟⎟⎞
= P → 1 x P → 2 = ⎜ ⎜⎛ 0 –1 y 1 x 2
1 0 –x 1
⎝ –y ⎠ ⎟⎟⎞
1 x 1 0 ⎝ ⎜⎜⎛ y 2
⎠ ⎟⎟⎞ = ⎜ ⎜⎛ y 1 –y 2
x 2 –x 1
1 ⎝ x ⎠ ⎟⎟⎞
1 y 2 –x 2 y 1
This is equivalent to a setting the determinant to zero with free variables in one of the
rows.
x y 1
x 1
y 1
1 = x(y 1
– y 2
)+ y(x 2
– x 1
)+ x 1
y 2
– x 2
y 1
= 0
x 2
y 2
1
!
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Coordonnées Homogènes Séance 1
This is a simple example of a general principle that can be useful in higher
dimensions.
Two lines make a point
There is a perfect duality between points and lines. So naturally, two lines make a
point.
Given non-parallel lines L and M.
L: a 1
x + b 1
y + c 1
= 0
and M : a 2
x + b 2
y + c 2
= 0.
In classic notation L T = (a 1
, b 1
, c 1 ) et M T =(a 2
, b 2
, c 2 )
The unique point is
y = c 1a 2
" c 2
a 1
c 1
a 2
" c 2
a 1
The cross product is
!
P → = L → x M → =
x = b 1c 2
" b 2
c 1
a 1
b 2
" a 2
b 1
!
v
P = L v
" M v # 0 –c 1
b 1
&#
%
(%
=
c 1
0 –a 1
%
(%
$ –b 1
a 1
0 ' $
a 2
b 2
c 2
& #–c 1
b 2
+ b 1
c 2
& b 1 c 2 –c 1 b 2
(
( = % (
c 1
a 2
– a 1
c 2
= % ( ⎝ ⎜⎜⎛ c 1 a 2 –a 1 c 2
a ⎠ ⎟⎟⎞
' $ –a 2
b 1
+ a 1
b 1 b 2 –a 2 b 1
2 '
as before the cross ! product is equivalent to setting a determinant to zero. In this case:
a b c
a 1
b 1
c 1
= a(b 1
c 2
– c 1
b 2
)+ b(c 1
a 2
– a 1
c 2
)+ c(a 1
b 2
– b 1
a 2
) = 0
a 2
b 2
c 2
!
The unique point is
x = b 1 c 2 " b 2 c 1
a 1
b 2
" a 2
b 1
!
y = c 1 a 2 " c 2 a 1
c 1
a 2
" c 2
a 1
!
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