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Statistical Calculations of 3D-Orientation Parameters of Flat Symmetrical Polyhedrons

A. Manninen
Journal / Anthology

Pattern Recognition Letters
Year: 1993
Volume: 14
Page range: 207-211

The determination of orientational statistics of polyhedrons is encountered in radar imagery research of ice ridges. Especially first-year ice ridge sails are heaps of more or less flat polyhedrons of ice, which can be approximated with rectangles of flat discs when modelling the ridge structure. The radar response is sensitive to the distribution of the surface orientations of the ice blocks. Therefore radar satellites are promising instruments for locating ice ridges, which constitute a severe problem for winter navigation. The orientation of 3-dimensional objects can be described with the three Euler angles phi (horizontal rotation), Theta (vertical rotation) and psi (rotation in the plane of the object that originally was horizontal). For general 3D objects the three angles are independent of each other and have independent periodicity of 2pi. Then the statistical parameters for the three angles can be calculated separately. For a single periodical data set the best statistical description is obtained when the phase values are chosen so that the dispersion of the data is minimized. The calculation of the orientational statistics of general 3D-objects can be done using the methods developed for statistical calculations of 3-dimensional directions in general. Problems arise when the 3D-objects are symmetric, such as rectangular polyhedrons, so that their orientation is no more described with a single-valued angle triplet having a period of 2pi. Figure 1 shows the problem of deciding which end of a flat rectangular polyhedron has been originally towards the x-axis. For simplicity only one fact has been drawn, but this facet can be either the top or bottom facet, which means that the vertical rotation has period of pi. Moreover horizontal rotations phi plus or minus pi produce the same situation. An even more complex example of problems caused by symmetry is the case of a cube. To choose which facet has before the Euler rotations been horizontal one has six alternatives and to choose which side of the facet has originally been parallel to the s-axis one has still four alternatives. In this paper an algorithm for calculating the mean and standard deviation of the horizontal and vertical Euler angle of a set of flat polyhderons, whose main facet is symmetric with respect to a line, is presented in detail. Calculation of the statistics for the third Euler angle can in this case be carried out separately and is straightforward as mentioned before.

*Mathematics > Geometry > Solid Geometry
*Mathematics > Probability and Statistics