Sulcal fundi are 3-D curves that lie in the depths of the cerebral cortex and, in addition to their intrinsic value in brain research, are often used as landmarks for downstream computations in brain imaging. In this paper, we present a geometric algorithm that automatically extracts the sulcal fundi from magnetic resonance images and represents them as spline curves lying on the extracted triangular mesh representing the cortical surface. The input to our algorithm is a triangular mesh representation of an extracted cortical surface as computed by one of several available software packages for performing automated and semi-automated cortical surface extraction. Given this input we first compute a geometric depth measure for each triangle on the cortical surface mesh, and based on this information we extract sulcal regions by checking for connected regions exceeding a depth threshold. We then identify endpoints of each region and delineate the fundus by thinning the connected region while keeping the endpoints fixed. The curves, thus, defined are regularized using weighted splines on the surface mesh to yield high-quality representations of the sulcal fundi. We present the geometric framework and validate it with real data from human brains. Comparisons with expert-labeled sulcal fundi are part of this validation process.

In this paper, a framework for smoothing gridlike digital terrain elevation data, which achieves a fair shape by means of minimizing an energy functional, is presented. The minimization is performed under the side condition of hard constraints, which comes from available horizontal and vertical accuracy bounds in the standard elevation specification. In this paper, the framework is introduced, and the suitability of this method for the tasks of accuracy-constrained smoothing, feature-preserving smoothing, and filling of data voids is demonstrated

We present a system for automatic reassembly of broken 3D solids. Given as input 3D digital models of the broken fragments, we analyze the geometry of the fracture surfaces to find a globally consistent reconstruction of the original object. Our reconstruction pipeline consists of a graph-cuts based segmentation algorithm for identifying potential fracture surfaces, feature-based robust global registration for pairwise matching of fragments, and simultaneous constrained local registration of multiple fragments. We develop several new techniques in the area of geometry processing, including the novel integral invariants for computing multi-scale surface characteristics, registration based on forward search techniques and surface consistency, and a non-penetrating iterated closest point algorithm. We illustrate the performance of our algorithms on a number of real-world examples.

This paper presents a new method for the recognition and reconstruction of surfaces from 3D data. Line element geometry, which generalizes both line geometry and the Laguerre geometry of oriented planes, enables us to recognize a wide class of surfaces (spiral surfaces, cones, helical surfaces, rotational surfaces, cylinders, etc.), by fitting linear subspaces in an appropriate seven-dimensional image space. In combination with standard techniques such as PCA and RANSAC, line element geometry is employed to effectively perform the segmentation of complex objects according to surface type. Examples show applications in reverse engineering of CAD models and testing mathematical hypotheses concerning the exponential growth of sea shells

Variational interpolation in curved geometries has many applications, so there has always been demand for geometrically meaningful and efficiently computable splines in manifolds. We extend the definition of the familiar cubic spline curves and splines in tension, and we show how to compute these on parametric surfaces, level sets, triangle meshes, and point samples of surfaces. This list is more comprehensive than it looks, because it includes variational motion design for animation, and allows the treatment of obstacles via barrier surfaces. All these instances of the general concept are handled by the same geometric optimization algorithm, which minimizes an energy of curves on surfaces of arbitrary dimension and codimension.

An active contour model for parametric curve and surface approximation is presented. The active curve or surface adapts to the model shape to be approximated in an optimization algorithm. The quasi-Newton optimization procedure in each iteration step minimizes a quadratic function which is built up with the help of local quadratic approximants of the squared distance function of the model shape and an internal energy which has a smoothing and regularization effect. The approach completely avoids the parametrization problem. We also show how to use a similar strategy for the solution of variational problems for curves on surfaces. Examples are the geodesic path connecting two points on a surface and interpolating or approximating spline curves on surfaces. Finally we indicate how the latter topic leads to the variational design of smooth motions which interpolate or approximate given positions.