On the use of moment invariants for the automated analysis of 3D particle shapes

A mathematical method is introduced to describe quantitatively the shape and shape evolution of precipitates in a two-phase microstructure. The method relies on the concept of moment invariants, i.e. combinations of second order moments that are invariant with respect to affine and/or similarity transformations. We introduce three invariants, one a shape discriminator, the other two aspect ratio discriminators. A normalized form of the invariants is defined and it is shown explicitly that any 3D shape must belong to a finite region of the normalized moment invariant space. The shape and bounds of this region are defined in terms of isoperimetric inequalities. Two examples of moment invariant applications are given: fitting the shape of second phase precipitates and the time evolution of a bimodal phase field simulation.