ISSN: 2180-1053 Vol. 5 No. 1 January-June 2013 Journal of Mechanical Engineering and Technology
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2.0 THE CLOSED-FORM SIMILARITY TRANSFORMATION
SOLUTION
We seek the values of c, θ in the rotation matrix
4
k the values of c,
in the rotation matrix 3
4
6
8
9
10
and the translation vector d given by
4
4
6
8
9
10
in Equation 1 such that
4
5
6
8
9
10
is minimized. As all summations in the paper are from i=1 to n, the summation bounds are omited from all further equations.
The c, θ and d values may be obtained through solving the simultaneous set
of equations
4
and d values may be obtained through solving the simultaneous set of equations 6
8
9
10
Substituing Equation 1 into Equation 5 and expanding gives
4
7
8
9
10
The irst of the simultaneous equations in 6 yields
4
8
9
10
Noting the deinitions
ISSN: 2180-1053 Vol. 5 No. 1 January-June 2013 A Closed-Form Solution for The Similarity Transformation Parameters of Two Planar Point Sets
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4
9
10
and
4
10
Eq. 8 may be solved for d as
5
11
12
13
14
15
16
17
18
The derivative of A is
5
12
13
14
15
16
17
18
and the second of the simultaneous equations in 6 yields
5
13
14
15
16
17
18
Substituting Equation 11 into Equation 13 eliminates d, and recognizing that as a rotation matrix, A
T
= A
-1
,
5
14 Observe that
15
16
17
18
Observe that
5
15
16
17
18
due to the skew-symmetry of Î. Equation 14 becomes
5
16
17
18
We can expand Equation 16 as
5
17
18
ISSN: 2180-1053 Vol. 5 No. 1 January-June 2013 Journal of Mechanical Engineering and Technology
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By expanding Equation 17 and solving for θ,
5
18
where
5
19
and ATAN2 is a four-quadrant inverse tangent function. This determines the proper value of
θ, so via Equation 3, A is now known. Since A is deined as a rotation matrix, it cannot result in a relection in the presence of “corrupted
data,” a possibility when an SVD is used as noted and corrected for by Umeyama Umeyama, 1991.
From the third of the simultaneous equations in 6,
6
“ ”
20
21
22
23
from which d may be eliminated by substituting Equation 11,
6
“ ”
21
22
23
Expanding,
6
“ ”
22
23
Recalling that A is known, c may be found as
6
“ ”
23
where
ISSN: 2180-1053 Vol. 5 No. 1 January-June 2013 A Closed-Form Solution for The Similarity Transformation Parameters of Two Planar Point Sets
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6
24
Substituting c and A into Equation 11 produces the value of d, and the similarity transformation is now known.
3.0 EXAMPLE
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