448
S. Schiele et al.
where the vector v j represents vertices in the crystal’s V-representation as defined in
Eq. (9).
Finally, symmetry conditions were applied to ensure that all faces belonging to
the same facet group had the same distance to the crystal center. The procedure is
described by Kovaˇ cevi´ c et al. [1].
As measures how well investigated crystals were fitted, the mean quadratic deviation, d quad , a volume ratio, r vol , and the volume deviation, d vol were introduced. d quad
is a measure to determine how much measured points differ from their fit. Therein
long distances between model and measurement are weighted stronger. The volume
ratio is a measure how well the volume is conserved and the volume deviation is a
scaled measure of how much the fitted shape mismatches the measured shape. Details
on how these measures were calculated can be retrieved from Kovaˇ cevi´ c et al. [1].
The method described in this section enables to fit crystal models to 3D μCT
images obtained from regular potash alum crystals. Results of the fits are shown in
Fig. 11. Irregular particles that resulted from e.g. breakage or agglomeration were
identified by high volume deviations and high quadratic deviations.
In conclusion, the methods that were developed enable face identification in 3D
images. It was described how these faces can be matched to predefined particle
models. The method was applied to the model substance potash alum and it was
shown that single particles can be described with only few parameters by appropriate
crystal models.
2.3 Shape Identification of Crystal Agglomerates
Even more challenging than identifying single crystals is the identification of several
primary crystals constituting a crystal agglomerate. Kovaˇ cevi´ c et al. [2] presents an
approach how concepts of identification of single crystals can be transferred to the
identification of agglomerates.
Generally, one needs to find all primary particles in an agglomerate and then use
the methods described by Kovaˇ cevi´ c et al. [1] and in the previous section to model
their structure with individual H-representations. Thus, each agglomerate would be
described by a hand full of H–representations of its primary particles.
To separate agglomerates into their primary particles an algorithm based on
the seeded watershed segmentation and region recombination was developed (see
Fig. 12). The steps defined in Fig. 12 are illustrated with an example agglomerate in
Fig. 13. The first step of the agglomerate segmentation algorithm is to find concavity
points that are an indicator for contact points of primary particles. The 2D approach
of Fernandez et al. [29] and Indhumathi et al. [30] was used and transferred to the
presented 3D problem: A cubic mask was centered at each surface point of the measurement and a concavity value c p was calculated. It was defined as the quotient of
the number of voxels that contain material N foreground mask and the number of voxels
inside the mask N mask .
S. Schiele et al.
where the vector v j represents vertices in the crystal’s V-representation as defined in
Eq. (9).
Finally, symmetry conditions were applied to ensure that all faces belonging to
the same facet group had the same distance to the crystal center. The procedure is
described by Kovaˇ cevi´ c et al. [1].
As measures how well investigated crystals were fitted, the mean quadratic deviation, d quad , a volume ratio, r vol , and the volume deviation, d vol were introduced. d quad
is a measure to determine how much measured points differ from their fit. Therein
long distances between model and measurement are weighted stronger. The volume
ratio is a measure how well the volume is conserved and the volume deviation is a
scaled measure of how much the fitted shape mismatches the measured shape. Details
on how these measures were calculated can be retrieved from Kovaˇ cevi´ c et al. [1].
The method described in this section enables to fit crystal models to 3D μCT
images obtained from regular potash alum crystals. Results of the fits are shown in
Fig. 11. Irregular particles that resulted from e.g. breakage or agglomeration were
identified by high volume deviations and high quadratic deviations.
In conclusion, the methods that were developed enable face identification in 3D
images. It was described how these faces can be matched to predefined particle
models. The method was applied to the model substance potash alum and it was
shown that single particles can be described with only few parameters by appropriate
crystal models.
2.3 Shape Identification of Crystal Agglomerates
Even more challenging than identifying single crystals is the identification of several
primary crystals constituting a crystal agglomerate. Kovaˇ cevi´ c et al. [2] presents an
approach how concepts of identification of single crystals can be transferred to the
identification of agglomerates.
Generally, one needs to find all primary particles in an agglomerate and then use
the methods described by Kovaˇ cevi´ c et al. [1] and in the previous section to model
their structure with individual H-representations. Thus, each agglomerate would be
described by a hand full of H–representations of its primary particles.
To separate agglomerates into their primary particles an algorithm based on
the seeded watershed segmentation and region recombination was developed (see
Fig. 12). The steps defined in Fig. 12 are illustrated with an example agglomerate in
Fig. 13. The first step of the agglomerate segmentation algorithm is to find concavity
points that are an indicator for contact points of primary particles. The 2D approach
of Fernandez et al. [29] and Indhumathi et al. [30] was used and transferred to the
presented 3D problem: A cubic mask was centered at each surface point of the measurement and a concavity value c p was calculated. It was defined as the quotient of
the number of voxels that contain material N foreground mask and the number of voxels
inside the mask N mask .