13 Morphological Modelling and Simulation …
443
[21], abrasion occurs through impact of crystals on the stirrer. This impact causes
breakage predominantly at corners and edges of the particles [22, 23]. This explains
that roundness increased and the mean width of the particles decreased through
intensive agitation.
In conclusion, limitations of the measurement of roundness from 2D projections
of particles were revealed and quantified. Further, a new method was demonstrated
that is able—under the previously mentioned limitations—to quantify the particle
roundness and mean width during crystallization by using stereoscopic imaging in
combination with appropriate image analysis.
2.2 Shape Identification of 3D Single Crystals
Similar image analysis concepts as described in the previous subsection can be
applied to 3D images of crystals. 3D images can be obtained by micro-computed
tomography (μCT) as described by Kovaˇ cevi´ c et al. [1]. This imaging technique
allows capturing the full shape information of particles. However, images are much
harder to obtain.
The goal here was to find faces of measured crystals and fit them to a predefined crystal model. Here again, the image analysis involved face identification
using Hough transform. Subsequently face normals of the predefined crystal models—given by a crystal specific matrix A—were matched to the faces found in a
3D image. The main difference to the problem in Sect. 2.1—except for the higher
dimensionality—is that a crystal model was predefined and then fitted to the 3D
images. The goal was to find faces that appear in a measured crystal.
In a first step, it was assumed that a 3D image of a crystal is given as a set of regularly distributed surface points. This data set was extracted from μCT measurements.
A 3D polar coordinate system was chosen so that it originated from the arithmetic
mean of these surface points. In this coordinate system face normal vectors were
described by an azimuthal angle θ and a polar angle φ. Distances from the origin
were described by ρ. Both angles θ and φ were then discretized using the HEALPix
algorithm [24].
Figure 7 visualizes the HEALPix discretization of a unit sphere with 12 principal
elements (solid outline), with N
2
side = 16 subelements of equal area (dashed outline).
For calculations in this work N side was chosen as 20. Each of these 12 · N
2
side subelements were represented by a potential face normal vector and face distance. The face
distance was also discretized. To apply the Hough transform, each grid point that
was defined by θ i , φ j and ρ l was assigned a bin value b
θ i , φ j , ρ l
that represented
the number of surface points within it. To obtain the directions of the face normals
it was sufficient to define one bin value ˜
b
θ i , φ j
per subelement:
˜
b
θ i , φ j
= max
ρ
b
θ i , φ j , ρ l
(15)
443
[21], abrasion occurs through impact of crystals on the stirrer. This impact causes
breakage predominantly at corners and edges of the particles [22, 23]. This explains
that roundness increased and the mean width of the particles decreased through
intensive agitation.
In conclusion, limitations of the measurement of roundness from 2D projections
of particles were revealed and quantified. Further, a new method was demonstrated
that is able—under the previously mentioned limitations—to quantify the particle
roundness and mean width during crystallization by using stereoscopic imaging in
combination with appropriate image analysis.
2.2 Shape Identification of 3D Single Crystals
Similar image analysis concepts as described in the previous subsection can be
applied to 3D images of crystals. 3D images can be obtained by micro-computed
tomography (μCT) as described by Kovaˇ cevi´ c et al. [1]. This imaging technique
allows capturing the full shape information of particles. However, images are much
harder to obtain.
The goal here was to find faces of measured crystals and fit them to a predefined crystal model. Here again, the image analysis involved face identification
using Hough transform. Subsequently face normals of the predefined crystal models—given by a crystal specific matrix A—were matched to the faces found in a
3D image. The main difference to the problem in Sect. 2.1—except for the higher
dimensionality—is that a crystal model was predefined and then fitted to the 3D
images. The goal was to find faces that appear in a measured crystal.
In a first step, it was assumed that a 3D image of a crystal is given as a set of regularly distributed surface points. This data set was extracted from μCT measurements.
A 3D polar coordinate system was chosen so that it originated from the arithmetic
mean of these surface points. In this coordinate system face normal vectors were
described by an azimuthal angle θ and a polar angle φ. Distances from the origin
were described by ρ. Both angles θ and φ were then discretized using the HEALPix
algorithm [24].
Figure 7 visualizes the HEALPix discretization of a unit sphere with 12 principal
elements (solid outline), with N
2
side = 16 subelements of equal area (dashed outline).
For calculations in this work N side was chosen as 20. Each of these 12 · N
2
side subelements were represented by a potential face normal vector and face distance. The face
distance was also discretized. To apply the Hough transform, each grid point that
was defined by θ i , φ j and ρ l was assigned a bin value b
θ i , φ j , ρ l
that represented
the number of surface points within it. To obtain the directions of the face normals
it was sufficient to define one bin value ˜
b
θ i , φ j
per subelement:
˜
b
θ i , φ j
= max
ρ
b
θ i , φ j , ρ l
(15)