Optimal designs for regression with spherical data

In this paper optimal designs for regression problems with spherical predictors of arbitrary dimension are considered. Our work is motivated by applications in material sciences, where crystallographic textures such as the missorientation distribution or the grain boundary distribution (depending on a four dimensional spherical predictor) are represented by series of hyperspherical harmonics, which are estimated from experimental or simulated data. For this type of estimation problems we explicitly determine optimal designs with respect to Kiefers op-criteria and a class of orthogonally invariant information criteria recently introduced in the literature. In particular, we show that the uniform distribution on the m-dimensional sphere is optimal and construct discrete and implementable designs with the same information matrices as the continuous optimal designs. Finally, we illustrate the advantages of the new designs for series estimation by hyperspherical harmonics, which are symmetric with respect to the first and second crystallographic point group

4D Hyperspherical Harmonic (HyperSPHARM) Representation of Multiple Disconnected Brain Subcortical Structures

Abstract. We present a novel surface parameterization technique using hyperspherical harmonics (HSH) in representing compact, multiple, disconnected brain subcortical structures as a single analytic function. The proposed hyperspherical harmonic representation (HyperSPHARM) has many advantages over the widely used spherical harmonic (SPHARM) parameterization technique. SPHARM requires flattening 3D surfaces to 3D sphere which can be time consuming for large surface meshes, and can’t represent multiple disconnected objects with single parameterization. On the other hand, HyperSPHARM treats 3D object, via simple stereographic projection, as a surface of 4D hypersphere with extremely large radius, hence avoiding the computationally demanding flattening process. HyperSPHARM is shown to achieve a better reconstruction with only 5 basis compared to SPHARM that requires more than 441.

Shape analysis of the human brain.

Autism is a complex developmental disability that has dramatically increased in prevalence, having a decisive impact on the health and behavior of children. Methods used to detect and recommend therapies have been much debated in the medical community because of the subjective nature of diagnosing autism. In order to provide an alternative method for understanding autism, the current work has developed a 3-dimensional state-of-the-art shape based analysis of the human brain to aid in creating more accurate diagnostic assessments and guided risk analyses for individuals with neurological conditions, such as autism. Methods: The aim of this work was to assess whether the shape of the human brain can be used as a reliable source of information for determining whether an individual will be diagnosed with autism. The study was conducted using multi-center databases of magnetic resonance images of the human brain. The subjects in the databases were analyzed using a series of algorithms consisting of bias correction, skull stripping, multi-label brain segmentation, 3-dimensional mesh construction, spherical harmonic decomposition, registration, and classification. The software algorithms were developed as an original contribution of this dissertation in collaboration with the BioImaging Laboratory at the University of Louisville Speed School of Engineering. The classification of each subject was used to construct diagnoses and therapeutic risk assessments for each patient. Results: A reliable metric for making neurological diagnoses and constructing therapeutic risk assessment for individuals has been identified. The metric was explored in populations of individuals having autism spectrum disorders, dyslexia, Alzheimers disease, and lung cancer. Conclusion: Currently, the clinical applicability and benefits of the proposed software approach are being discussed by the broader community of doctors, therapists, and parents for use in improving current methods by which autism spectrum disorders are diagnosed and understood