446
S. Schiele et al.
into
B = UDW
T
(21)
and
M = diag(1, 1, det(U), det(W ))
(22)
Provided S M and S D were known, one could solve Wahba’s problem given in
Eq. (18). However, S M and S D were unknown and one could have found a solution
to the problem by testing all possible combinations of S M , S D and R:
L(R, S D , S M ) =
1
2
i
ω i
(S D · A F ) i −
(S M · A)R
T
i
(23)
This had involved the optimization over a huge parameter space, which is computationally inefficient. Kovaˇ cevi´ c et al. [1] developed an algorithm to efficiently solve
this problem and described it in detail. Within the scope of this contribution it is
sufficient to understand that a numerically efficient solution exists that yields R, S D
and S M .
To obtain a description of a measured crystal by a crystal model so far a rotation
matrix, a mapping, and a filtering matrix has been found. The face distances remained
to be determined. This could have been done quite easily by finding the ρ l that
maximize Eq. (15) for faces that have been measured. However, for faces that have
grown out of the crystal, and are hence not found in the measurement data, this
does not work. For these cases, the measured data was cut into planar slices that
are orthogonal on a face normal vector, and are therefore defined by ρn
θ i , φ j
. The
triangles in Fig. 9 show, that slices inside the measured data contain an outline of the
crystal at distance ρ from the center. The ratio between the number of pixels on the
outline and the area of the convex hull of the outline was then maximized by finding
an appropriate face distance ρ. This principle is illustrated in Fig. 9. Therein the arrow
indicates a normal vector of a face of a potash alum crystal that is not expressed in
the measured data. The triangles represent the outline for a face distance that is too
small, and squares indicate a slice where the mentioned ratio reaches its maximum.
In the last two steps towards a successful fit, it needed to be assured that all
identified faces lied within the crystal model and fulfilled the symmetry conditions
defined by the crystal model. The first issue is illustrated in Fig. 10: If a face distance
was identified to be too long, a face lied outside of the model.
To assure that all identified crystal faces lied within the fitted model, invalid entries
in the vector h were identified according to Reinhold and Briesen [11], and Borchert
and Sundmacher [12] and then modified according to
h i =
max
1≤i≤n H ,1≤j≤n V
a i , v j
(24)
S. Schiele et al.
into
B = UDW
T
(21)
and
M = diag(1, 1, det(U), det(W ))
(22)
Provided S M and S D were known, one could solve Wahba’s problem given in
Eq. (18). However, S M and S D were unknown and one could have found a solution
to the problem by testing all possible combinations of S M , S D and R:
L(R, S D , S M ) =
1
2
i
ω i
(S D · A F ) i −
(S M · A)R
T
i
(23)
This had involved the optimization over a huge parameter space, which is computationally inefficient. Kovaˇ cevi´ c et al. [1] developed an algorithm to efficiently solve
this problem and described it in detail. Within the scope of this contribution it is
sufficient to understand that a numerically efficient solution exists that yields R, S D
and S M .
To obtain a description of a measured crystal by a crystal model so far a rotation
matrix, a mapping, and a filtering matrix has been found. The face distances remained
to be determined. This could have been done quite easily by finding the ρ l that
maximize Eq. (15) for faces that have been measured. However, for faces that have
grown out of the crystal, and are hence not found in the measurement data, this
does not work. For these cases, the measured data was cut into planar slices that
are orthogonal on a face normal vector, and are therefore defined by ρn
θ i , φ j
. The
triangles in Fig. 9 show, that slices inside the measured data contain an outline of the
crystal at distance ρ from the center. The ratio between the number of pixels on the
outline and the area of the convex hull of the outline was then maximized by finding
an appropriate face distance ρ. This principle is illustrated in Fig. 9. Therein the arrow
indicates a normal vector of a face of a potash alum crystal that is not expressed in
the measured data. The triangles represent the outline for a face distance that is too
small, and squares indicate a slice where the mentioned ratio reaches its maximum.
In the last two steps towards a successful fit, it needed to be assured that all
identified faces lied within the crystal model and fulfilled the symmetry conditions
defined by the crystal model. The first issue is illustrated in Fig. 10: If a face distance
was identified to be too long, a face lied outside of the model.
To assure that all identified crystal faces lied within the fitted model, invalid entries
in the vector h were identified according to Reinhold and Briesen [11], and Borchert
and Sundmacher [12] and then modified according to
h i =
max
1≤i≤n H ,1≤j≤n V
a i , v j
(24)