![]() The explicit expressions for interfacial Minkowski tensors are confirmed in detailed simulations. Here, we analytically calculate the expected Minkowski tensors of arbitrary rank for the level sets of Gaussian random fields. ![]() Anisotropy in such systems can sensitively and comprehensively be characterized by the so-called Minkowski tensors from integral geometry. Gaussian random fields are among the most important models of amorphous spatial structures and appear across length scales in a variety of physical, biological, and geological applications, from composite materials to geospatial data. While a variety of methods exist for point cloud registration, the method proposed in this paper is notably different as registration is achieved by a closed form solution that employs the UME low dimensional representation of the shapes to be registered. In this paper, we extend the UME framework to the special case where it is a priori known that the geometric transformations are rigid. Therefore registration, matching and classification can be solved as linear problems in a low dimensional linear space. The UME nonlinearly maps functions related by certain types of geometric transformations of coordinates to the same linear subspace of some Euclidean space while retaining the information required to recover the transformation. The solution is obtained by adapting the general framework of the universal manifold embedding (UME) to the case where the transformations the object may undergo are rigid. We present a closed form solution to the problem of registration of fully overlapping 3D point clouds undergoing unknown rigid transformations, as well as for detection and registration of sub-parts undergoing unknown rigid transformations.
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