x - Universität Freiburg

Transcription

The Group of Affine Maps A
The group of affine maps results from choosing an arbitrary regular
matrix A (det(A) ≠ 0) and a translation a:
t' = At + a
This map has 6 degrees of freedom (4 for the general matrix A, and two
for the translation vector a).
The affine map describes the general steric motion of a planar master
illustration followed by parallel projection on the camera plane, which is
to be shown now.
H. Burkhardt, Institut für Informatik, Universität Freiburg
ME-I, Kap. 2b
1
Motion of an arbitrary curve in space with
subsequent parallel projection on the camera plane


x1(t)
x(t) =  x2(t) 
x3(t)
x3
x2
x1
special case:
plane steric curve:
x3(t) 0
ME-I, Kap. 2b
2
The arc length calculates as follows:
s=
t
t0
t 2
x 1(t) + x 22(t) + x 23(t) dt
x (t) dt =
t0
with:




x 1(t)
dx1/dt
x (t) =  x 2(t)  =  dx2/dt 
dx3/dt
x 3(t)
ME-I, Kap. 2b
3
Euclidian Space Motion (Rotation
and Translation):
x ( t) = A x ( t) + b
With the orthogonal rotation matrix:
A=
A11 = c2c3 c1s 2s 3 A12 = c2s 3 c1s 2c3
A21 = s 2c3 + c1c2s 3 A22 = s 2s 3 + c1c2c3
A31 = s 1s 3
A32 = s1c3
A13 = s1s 2
A23 = s 1c2
A33 = c1
whereas:
c1 = cos ϑ, c2 = cos ψ, c3 = cos ϕ
s 1 = sin ϑ, s2 = sin ψ, s 3 = sin ϕ
and the
translation:
b=
b1
b2
b3
ME-I, Kap. 2b
4
zI
Euler Angles:
1. nutation ϑ (Theta)
2. precession ψ (Psi)
y
3. mere rotation ϕ (Phi)
z
ϑ
O
ϕ
yI
ψ
N
x
xI
To exchange from the inertial system into the body coordinate system the following three rotations are needed:
1.
rotation ϕ about the zI - axis: the x-axis results/merges in the so-called line of nodes O-N.
2.
rotation ϑ about the line of nodes O-N: the inertial zI - axis merges in the body z-axis.
3.
rotation ψ about the z - axis: The body (x,y,z)-coordinate system results.
ME-I, Kap. 2b
5
Factorization of the Rotation Matrix:
c2 − s2 0 
A =  s2 c2 0 
 0
0 1 
1 0 0 
⋅ 0 c1 − s1 
0 s1 c1 
c3 − s3 0 
⋅  s3 c3 0 
 0
0 1 
rotation about z-axis (ψ ) rotation about x-axis (ϑ ) rotation about z-axis (ϕ )
ME-I, Kap. 2b
6
Properties of the Rotation Matrix A:
• The row and column vectors of A are orthogonal and
normalized and thus
• A rotation matrix A is orthogonal, i.e.:
A-1=AT and thus also ATA=AAT =I
• Furthermore: det(A)=1 (mere rotation, without
reflection)
• Thus, for rotation matrices applies:
( ABC) −1 = ( ABC)T = CT BT AT
ME-I, Kap. 2b
7
And thus:
 c3
A −1 = AT =  − s3
 0
s3
c3
0
0  1 0 0   c2
0  ⋅ 0 c1 s1  ⋅  − s2
1  0 − s1 c1   0
s2
c2
0
0
0 
1 
Rotation backwards can be achieved by inversion of the order!
ME-I, Kap. 2b
8
A more compact notation:
R 2

A=
 0
and:
0  1 0 0   R 3




0  ⋅ 0
⋅
R1   0
0 1  0
0

0
0 1 
R i−1 = R Ti
ME-I, Kap. 2b
9
Now: orthogonal projection of the sterical
curve x(t) into the (e1,e2)-plane with the
projection operator P12:
 x1 (t ) 
x12 =  x2 (t )  = P12 x
 0 
with:
1 0 0 
P12 = 0 1 0 
0 0 0 
The projection operator P12 is idempotent, i.e.:
P122 = P12
ME-I, Kap. 2b
10
From the general motion in space with
subsequent projection follows:
x '' = P12 x ' = P12 ( Ax + b )
The projection has
the following effect on A:
 A11
P12 A =  A21
 0
A12
A22
0
A13 
A23 
0 
Since the curve is planar (x3≡0).
the 3rd column can be omitted:
 A11
A
 21
 0
A12
A22
0
0
0 
0 
ME-I, Kap. 2b
11
And so the description can be reduced to 2 dimensions:
 A11
0
x' = A 'x' = 
 A21
A12   x10 
 
A22   x20 
with:
or expanded:
 c2 c3 − c1s2 s3
A' = 
 s2 c3 + c1c2 s3
−c2 s3 − c1s2 c3 
− s2 s3 + c1c2 c3 
ME-I, Kap. 2b
12
Rotating a plane curve (x3≡0) in a 3D space and
projecting it afterwards in a (e1,e2)-plane is
equivalent to describing it with a
corresponding two-dimensional affine map!
ME-I, Kap. 2b
13
Discussing the Degrees of Freedom:
• We see only three degrees of freedom c1,c2,c3, but
the general affine map has 4!
• As described here, the objects are only compressed
and not enlarged. Adding a general dilation
(enlarging/compressing) with a factor k, 4 degrees
of freedom can be obtained!
0 
δ 1 0 
1
 0 δ  = δ1  0 δ / δ 

2

2
1
dilation general
compression
compressing along y
ME-I, Kap. 2b
14
Invariants of geometric maps:
Congruent maps (AAT=I) are length preserving:
!
x ' =< x ', x ' >=< Ax, Ax >=< x, A Ax > = < x, x >= x
2
*
2
only applies for: A* A = I ⇒ A −1 = A* (orthogonal rotation matrix)
ME-I, Kap. 2b
15
The group of similarities guarantees angle
preserving (conform) maps:
< x ', y ' >
< Ax, Ay >
cos ϕ =
=
x' y'
< Ax, Ax >1/ 2 < Ay , Ay >1/ 2
with: A* A = µ I follows:
< x, A* Ay >
cos ϕ =
=
*
1/ 2
*
1/ 2
< x, A Ax > < y , A Ay >
µ < x, y >
< x, y >
=
1/ 2
1/ 2
1/ 2
1/ 2
µ < x, x > µ < y , y >
x y
ME-I, Kap. 2b
16
The group of affine maps preserves
parallelisms:
g1=r1+αl
linear equations
r1
l
g2=r2+αl
for all pairs of points on straight
lines parallel to l applies: x2 -x1= αl
r2
affine transformed: x´=Ax+t and thus:
x′2 − x1′ = Ax 2 + t − Ax1 − t = A(x 2 − x1 ) = α Al = α l′
that are points on the parallel straight line with gradient l´
ME-I, Kap. 2b
17
Interpolation
For rotation and translation about fractions of the
sampling interval a grayscale interpolation of the
images is required (non gridconforming maps).
In the easiest way this can be realized by
a) the Nearest-Neighbour-Rule:
ME-I, Kap. 2b
18
Or better: with bilinear interpolation:
xc
x1 = xa +
x2 = xc +
ε
∆
ε
∆
x
( xb − xa ) = x ( xa , xb , ε )
( xd − xc ) = x ( xc , xd , ε )
xa
and so:
x = x1 +
η
∆
= xa +
ε
∆
( xb − xa ) +
η
∆
( xc − xa ) +
εη
∆2
( xa − xb − xc + xd )
xd
c
η
a
( x2 − x1 ) = x ( x1 , x2 ,η )
x2
x1
xb
b
ε
d
∆
∆
By exchanging xa for xc and ε for η, one gets the same result, i.e. one could
interpolate first with (xa, xa, η) and (xb, xd, η) and then interpolate the result
linearly.
ME-I, Kap. 2b
19

x - Universität Freiburg

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