Probabilistic solid form determined from 2D shape measurement

• Transformation of convex, symmetric solids into AR/HF space was studied.
• For each solid all possible silhouettes form a characteristic probability surface.
• Also, all possible 2D planar sections form a characteristic probability surface.
• Contrary to accepted stereology, forms of 3D particles can be deduced from 2D data.

Particle shape, a property of a three-dimensional solid, is an important property. Usually, and problematically, shape measurements of the solid particle are obtained only from two dimensions. It is well known that, in general, a given 2D shape measurement cannot be uniquely related to a solid. However, it is shown herein that unique, and therefore discriminating and useful, probabilistic relationships do exist between a convex solid's shape and 2D measurements taken in either section or projection. The key is to recognize that all possible 2D measurements of a solid make a characteristic probability density surface for an appropriate domain. The measurement domain studied in this paper is often used to measure actual particles. In principle a measured probability density surface of physical samples may be quantitatively approximated by mixtures of model solids.

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