A method of triangular surface mesh smoothing is presented to improve angle quality by extending the original optimal Delaunay triangulation (ODT) to surface meshes. The mesh quality is improved by solving a quadratic optimization problem that minimizes the approximated interpolation error between a parabolic function and its piecewise linear interpolation defined on the mesh. A suboptimal problem is derived to guarantee a unique, analytic solution that is significantly faster with little loss in accuracy as compared to the optimal one. In addition to the quality-improving capability, the proposed method has been adapted to remove noise while faithfully preserving sharp features such as edges and corners of a mesh. Numerous experiments are included to demonstrate the performance of the method.

Isogeometric analysis (IGA) is a numerical simulation method which is directly based on the NURBS-based representation of CAD models. It exploits the tensor-product structure of 2- or 3-dimensional NURBS objects to parameterize the physical domain. Hence the physical domain is parameterized with respect to a rectangle or to a cube. Consequently, singularly parameterized NURBS surfaces and NURBS volumes are needed in order to represent non-quadrangular or non-hexahedral domains without splitting, thereby producing a very compact and convenient representation. The Galerkin projection introduces finite-dimensional spaces of test functions in the weak formulation of partial differential equations. In particular, the test functions used in isogeometric analysis are obtained by composing the inverse of the domain parameterization with the NURBS basis functions. In the case of singular parameterizations, however, some of the resulting test functions do not necessarily fulfill the required regularity properties. Consequently, numerical methods for the solution of partial differential equations cannot be applied properly. We discuss the regularity properties of the test functions. For one- and two-dimensional domains we consider several important classes of singularities of NURBS parameterizations. For specific cases we derive additional conditions which guarantee the regularity of the test functions. In addition we present a modification scheme for the discretized function space in case of insufficient regularity. It is also shown how these results can be applied for computational domains in higher dimensions that can be parameterized via sweeping.

Functional magnetic resonance imaging (fMRI) has become a popular technique for studies of human brain activity. Typically, fMRI is performed with >3-mm sampling, so that the imaging data can be regarded as two-dimensional samples that average through the 1.5-4-mm thickness of cerebral cortex. The increasing use of higher spatial resolutions, <1.5-mm sampling, complicates the analysis of fMRI, as one must now consider activity variations within the depth of the brain tissue. We present a set of surface-based methods to exploit the use of high-resolution fMRI for depth analysis. These methods utilize white-matter segmentations coupled with deformable-surface algorithms to create a smooth surface representation at the gray-white interface and pial membrane. These surfaces provide vertex positions and normals for depth calculations, enabling averaging schemes that can increase contrast-to-noise ratio, as well as permitting the direct analysis of depth profiles of functional activity in the human brain.

The smoothness and angle quality of a surface mesh are two important indicators of the "goodness" of the mesh for downstream applications such as visualization and numerical simulation. We present in this paper a novel surface mesh processing method not only to reduce mesh noise but to improve angle quality as well. Our approach is based on the local surface fitting around each vertex using the least square minimization technique. The new position of the vertex is obtained by finding the maximum inscribed circle (MIC) of the surrounding polygon and projecting the circle's center onto the analytically fitted surface. The procedure above repeats until the maximal vertex displacement is less than a pre-defined threshold. The mesh smoothness is improved by a combined idea of surface fitting and projection, while the angle quality is achieved by utilizing the MIC-based projection scheme. Results on a variety of geometric mesh models have demonstrated the effectiveness of our method.

This paper presents a technique for creating a smooth, closed surface from a set of 2D contours, which have been extracted from a 3D scan. The technique interprets the pixels that make up the contours as points in ℝ(3) and employs Multi-level Partition of Unity (MPU) implicit models to create a surface that approximately fits to the 3D points. Since MPU implicit models additionally require surface normal information at each point, an algorithm that estimates normals from the contour data is also described. Contour data frequently contains noise from the scanning and delineation process. MPU implicit models provide a superior approach to the problem of contour-based surface reconstruction, especially in the presence of noise, because they are based on adaptive implicit functions that locally approximate the points within a controllable error bound. We demonstrate the effectiveness of our technique with a number of example datasets, providing images and error statistics generated from our results.

In this paper we develop a set of inverse kinematics algorithms suitable for an anthropomorphic arm or leg. We use a combination of analytical and numerical methods to solve generalized inverse kinematics problems including position, orientation, and aiming constraints. Our combination of analytical and numerical methods results in faster and more reliable algorithms than conventional inverse Jacobian and optimization-based techniques. Additionally, unlike conventional numerical algorithms, our methods allow the user to interactively explore all possible solutions using an intuitive set of parameters that define the redundancy of the system.