Amyloid positivity is an early indicator of Alzheimer's disease and is necessary to determine the disease. In this study, a deep generative model is utilized to predict the amyloid positivity of cognitively normal individuals using proxy measures, such as structural MRI scans, demographic variables, and cognitive scores, instead of invasive direct measurements. Through its remarkable efficacy in handling imperfect datasets caused by missing data or labels, and imbalanced classes, the model outperforms previous studies and widely used machine learning approaches with an AUROC of 0.8609. Furthermore, this study illuminates the model's adaptability to diverse clinical scenarios, even when feature sets or diagnostic criteria differ from the training data. We identify the brain regions and variables that contribute most to classification, including the lateral occipital lobes, posterior temporal lobe, and APOE ϵ4 allele. Taking advantage of deep generative models, our approach can not only provide inexpensive, non-invasive, and accurate diagnostics for preclinical Alzheimer's disease, but also meet real-world requirements for clinical translation of a deep learning model, including transferability and interpretability.

This paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner.

Ridges are characteristic curves of a surface that mark salient intrinsic features of its shape and are therefore valuable for shape matching, surface quality control, visualization and various other applications. Ridges are loci of points on a surface where either of the principal curvatures attain a critical value in its respective principal direction. These curves have complex behavior near umbilics on a surface, and may also pass through certain turning points causing added complexity for ridge computation. We present a new algorithm for numerically tracing ridges on B-Spline surfaces that also accurately captures ridge behavior at umbilics and ridge turning points. The algorithm traverses ridge segments by detecting ridge points while advancing and sliding in principal directions on a surface in a novel manner, thereby computing connected curves of ridge points. The output of the algorithm is a set of curve segments, some or all of which, may be selected for other applications such as those mentioned above. The results of our technique are validated by comparison with results from previous research and with a brute-force domain sampling technique.

We present robust and efficient algorithms for computing Voronoi diagrams of planar freeform curves. Boundaries of the Voronoi diagram consist of portions of the bisector curves between pairs of planar curves. Our scheme is based on computing critical structures of the Voronoi diagrams, such as self-intersections and junction points of bisector curves. Since the geometric objects we consider in this paper are represented as freeform NURBS curves, we were able to reformulate the solution to the problem of computing those critical structures into the zero-set solutions of a system of nonlinear piecewise rational equations in parameter space. We present a new algorithm for computing error-bounded bisector curves using a distance surface constructed from error-bounded offset approximations of planar curves. This error-bounded algorithm is fast and produces bisector curves that are correct both in topology and geometry. Once bisectors are computed, both local and global self-intersections of the bisector curves are located and trimmed away by solving a system of three piecewise rational equations in three variables. Further, our method computes junction points at which three or more trimmed bisector curves intersect by transforming them into the solutions to a system of piecewise rational equations in the merged parameter space of the planar curves. The bisectors are trimmed at those self-intersection and global junction points. The Voronoi diagram is then computed from the trimmed bisectors using a pruning algorithm. We demonstrate the effectiveness of our approach with several experimental results.

The distribution of geometric features is anisotropic by its nature. Intrinsic properties of surfaces such as normal curvatures, for example, varies with direction. In this paper this characteristic of a shape is used to create a new anisotropic geodesic (AG) distance map on parametric surfaces. We first define local distance (LD) from a point as a function of both the surface point and a unit direction in its tangent plane and then define a total distance as an integral of that local distance. The AG distance between points on the surface is then defined as their minimum total distance. The path between the points that attains the minimum is called the anisotropic geodesic path. This differs from the usual geodesic in ways that enable it to better reveal geometric features. Minimizing total distances to attain AG distance is performed by associating the LD function with the tensor speed function that controls wave propagation of the convex Hamilton-Jacobi (H-J) equation solver. We present two different, but related metrics for the local distance function, a curvature tensor and a difference curvature tensor. Each creates a different AG distance. Some properties of both new AG distance maps are presented, including parametrization invariance. We then demonstrate the effectiveness of the proposed geodesic map as a shape discriminator in several applications, including surface segmentation and partial shape matching.

We present an efficient and robust algorithm for computing the minimum distance between a point and freeform curve or surface by lifting the problem into a higher dimension. This higher dimensional formulation solves for all query points in the domain simultaneously, therefore providing opportunities to speed computation by applying coherency techniques. In this framework, minimum distance between a point and planar curve is solved using a single polynomial equation in three variables (two variables for a position of the point and one for the curve). This formulation yields two-manifold surfaces as a zero-set in a 3D parameter space. Given a particular query point, the solution space's remaining degrees-of-freedom are fixed and we can numerically compute the minimum distance in a very efficient way. We further recast the problem of analyzing the topological structure of the solution space to that of solving two polynomial equations in three variables. This topological information provides an elegant way to efficiently find a global minimum distance solution for spatially coherent queries. Additionally, we extend this approach to a 3D case. We formulate the problem for the surface case using two polynomial equations in five variables. The effectiveness of our approach is demonstrated with several experimental results.

We propose an algorithm for contouring k-manifolds (k = 1,2) embedded in an arbitrary n-dimensional space. We assume (n -k) geometric constraints are represented as polynomial equations in n variables. The common zero-set of these (n-k) equations is computed as an 1-or 2-manifold, respectively, for k = 1 or k = 2. In the case of 1-manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitly-defined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2-manifolds, the algorithm is similar to techniques for contouring iso-surfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 -2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zero-set as a piecewise linear 1or 2-manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.

We present efficient and robust algorithms for intersecting a freeform surface with a ringed surface or a ruled surface. A ringed surface is given as a one-parameter family of circles. By computing the intersection between a freeform surface and each circle in the family, we can solve the intersection problem. We propose two approaches which are closely related to each other. The first approach detects certain critical points; and the intersection curve is constructed by connecting them in a correct topology. The second approach converts the intersection problem to that of finding the zero set of two polynomial equations in the parameter space. The intersection between a freeform surface and a ruled surface can be computed in a similar way.

We present an algorithm for computing Minkowski sums among surfaces of revolution and surfaces of linear extrusion, generated by slope-monotone closed curves. The special structure of these simple surfaces allows the process of normal matching between two surfaces to be expressed as an explicit equation. Based on this insight, we also present an efficient algorithm for computing the distance between two simple surfaces, even though they may in general be non-convex. Using an experimental implementation, the distance between two surfaces of revolution was computed in less than 0.5 msec on average.

We present a simple method for constructing a physical 3D trackball that can input 3D rotation about an arbitrary axis. This input device is based on multiple sensors that can detect the tangential velocities at certain points on the boundary sphere of the trackball. A mathematical analysis is given for the optimal locations of multiple sensors. A prototype hardware device has been built for the 3D trackball. We demonstrate the effectiveness of this input device in 3D rotation by comparing its performance with the Magellan/SPACE MOUSE.