Theoretical foundations of hypertrace-transform: scanning techniques, mathematical apparatus and experimental verification

We consistently describe the theoretical basis of a new geometric method of analysis and recognition of three-dimensional (3D) images. The description of a scanning technique for forming a hypertrace transform and its mathematical model are given. This method, unlike the existing ones, enables 3D images to be analyzed directly from their 3D shape, without first simplifying them or constructing plane projections. We substantiate the selection of a particular scanning tool and the need to construct a reference spherical grid to address the problem of the rotational invariance of the 3D image recognition. A mathematical apparatus of the stochastic realization of the scanning technique based on stochastic geometry and functional analysis is developed. We introduce a new mathematical tool for 3D image analysis - a hypertrex matrix that allows spatial objects of complex shape and structure to be recognized by constructing a single mathematical model of the 3D image. We describe a new type of 3D image features that have an analytic structure - hypertryplet features, whose analytical structure makes possible an automatic generation of a large number of features with predetermined properties. Results of the experimental verification are presented, demonstrating the accurate calculation of features for 3D image recognition and proving the adequacy of the developed mathematical apparatus.