Geodesic Distance Function Research Articles

A Triangulation-Invariant Method for Anisotropic Geodesic Map Computation on Surface Meshes

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.

Spatial and Temporal Variations of Cortical Growth during Gyrogenesis in the Developing Ferret Brain

Spatial and temporal variations in cortical growth were studied in the neonatal ferret to illuminate the mechanisms of folding of the cerebral cortex. Cortical surface representations were created from magnetic resonance images acquired between postnatal day 4 and 35. Global measures of shape (e.g., surface area, normalized curvature, and sulcal depth) were calculated. In 2 ferrets, relative cortical growth was calculated between surfaces created from in vivo images acquired at P14, P21, and P28. The isocortical surface area transitions from a slower (12.7 mm(2)/day per hemisphere) to a higher rate of growth (36.7 mm(2)/day per hemisphere) approximately 13 days after birth, which coincides with the time of transition from neuronal proliferation to cellular morphological differentiation. Relative cortical growth increases as a function of relative geodesic distance from the origin of the transverse neurogenetic gradient and is related to the change in fractional diffusion anisotropy over the same time period. The methods presented here can be applied to study cortical growth during development in other animal models or human infants. Our results provide a quantitative spatial and temporal description of folding in cerebral cortex of the developing ferret brain, which will be important to understand the underlying mechanisms that drive folding.

Automated Segmentation Refinement of Small Lung Nodules in CT Scans by Local Shape Analysis

One of the most important problems in the segmentation of lung nodules in CT imaging arises from possible attachments occurring between nodules and other lung structures, such as vessels or pleura. In this report, we address the problem of vessels attachments by proposing an automated correction method applied to an initial rough segmentation of the lung nodule. The method is based on a local shape analysis of the initial segmentation making use of 3-D geodesic distance map representations. The correction method has the advantage that it locally refines the nodule segmentation along recognized vessel attachments only, without modifying the nodule boundary elsewhere. The method was tested using a simple initial rough segmentation, obtained by a fixed image thresholding. The validation of the complete segmentation algorithm was carried out on small lung nodules, identified in the ITALUNG screening trial and on small nodules of the lung image database consortium (LIDC) dataset. In fully automated mode, 217/256 (84.8%) lung nodules of ITALUNG and 139/157 (88.5%) individual marks of lung nodules of LIDC were correctly outlined and an excellent reproducibility was also observed. By using an additional interactive mode, based on a controlled manual interaction, 233/256 (91.0%) lung nodules of ITALUNG and 144/157 (91.7%) individual marks of lung nodules of LIDC were overall correctly segmented. The proposed correction method could also be usefully applied to any existent nodule segmentation algorithm for improving the segmentation quality of juxta-vascular nodules.

3D MORPHOLOGICAL ANALYSIS OF COMPOSITE MATERIALS WITH AGGREGATES OF SPHERICAL INCLUSIONS

Composite materials containing aggregates of spherical inclusions are studied from 3D images obtained by X-ray microtomography. Using two point statistics in different directions, and the empirical distribution of orientations of pairs of inclusions, interesting details concerning the anisotropy of the distribution of inclusions are obtained and are related to the method of construction for these materials. Some 3D morphological properties, available on the 3D images, give new information on the shape and the distribution of aggregates: tortuosity of shortest paths in the matrix, local volume fraction, geodesic distance function, local histograms of numbers of objects.

Geometric Spanners for Weighted Point Sets

Let (S,d) be a finite metric space, where each element p∈S has a non-negative weight w (p). We study spanners for the set S with respect to the following weighted distance function: $$\mathbf{d}_{\omega}(p,q)=\left\{\begin{array}{ll}0&\mbox{ if $p=q$,}\\ \operatorname {w}(p)+\mathbf{d}(p,q)+ \operatorname {w}(q)&\mbox{ if $p\neq q$.}\end{array}\right.$$ We present a general method for turning spanners with respect to the d-metric into spanners with respect to the d ω -metric. For any given ε>0, we can apply our method to obtain (5+ε)-spanners with a linear number of edges for three cases: points in Euclidean space ℝ d , points in spaces of bounded doubling dimension, and points on the boundary of a convex body in ℝ d where d is the geodesic distance function. We also describe an alternative method that leads to (2+ε)-spanners for weighted point points in ℝ d and for points on the boundary of a convex body in ℝ d . The number of edges in these spanners is O(nlog n). This bound on the stretch factor is nearly optimal: in any finite metric space and for any ε>0, it is possible to assign weights to the elements such that any non-complete graph has stretch factor larger than 2−ε.

Möbius Transformations For Global Intrinsic Symmetry Analysis

AbstractThe goal of our work is to develop an algorithm for automatic and robust detection of global intrinsic symmetries in 3D surface meshes. Our approach is based on two core observations. First, symmetry invariant point sets can be detected robustly using critical points of the Average Geodesic Distance (AGD) function. Second, intrinsic symmetries are self‐isometries of surfaces and as such are contained in the low dimensional group of Möbius transformations. Based on these observations, we propose an algorithm that: 1) generates a set of symmetric points by detecting critical points of the AGD function, 2) enumerates small subsets of those feature points to generate candidate Möbius transformations, and 3) selects among those candidate Möbius transformations the one(s) that best map the surface onto itself. The main advantages of this algorithm stem from the stability of the AGD in predicting potential symmetric point features and the low dimensionality of the Möbius group for enumerating potential self‐mappings. During experiments with a benchmark set of meshes augmented with human‐specified symmetric correspondences, we find that the algorithm is able to find intrinsic symmetries for a wide variety of object types with moderate deviations from perfect symmetry.

Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs

We introduce an original approach for the cerebral white matter connectivity mapping from diffusion tensor imaging (DTI). Our method relies on a global modeling of the acquired magnetic resonance imaging volume as a Riemannian manifold whose metric directly derives from the diffusion tensor. These tensors will be used to measure physical three-dimensional distances between different locations of a brain diffusion tensor image. The key concept is the notion of geodesic distance that will allow us to find optimal paths in the white matter. We claim that such optimal paths are reasonable approximations of neural fiber bundles. The geodesic distance function can be seen as the solution of two theoretically equivalent but, in practice, significantly different problems in the partial differential equation framework: an initial value problem which is intrinsically dynamic, and a boundary value problem which is, on the contrary, intrinsically stationary. The two approaches have very different properties which make them more or less adequate for our problem and more or less computationally efficient. The dynamic formulation is quite easy to implement but has several practical drawbacks. On the contrary, the stationary formulation is much more tedious to implement; we will show, however, that it has many virtues which make it more suitable for our connectivity mapping problem. Finally, we will present different possible measures of connectivity, reflecting the degree of connectivity between different regions of the brain. We will illustrate these notions on synthetic and real DTI datasets.

Reduce the stretch in surface flattening by finding cutting paths to the surface boundary

This paper presents a method for finding cutting paths on a 3D triangular mesh surface to reduce the stretch in the flattened surface. The cutting paths link the surface boundary and the nodes where the Gaussian curvature is high, and their total length is minimized. First, a linear algorithm for computing an approximate boundary geodesic distance map is introduced; the map encapsulates the undirected geodesic distance from every triangular node to the surface boundary approximately. This is followed by determining the undirected shortest paths passing through all the nodes where the Gaussian curvature is larger than a threshold. The cutting paths walk along the triangular edges of the given surface. Compared with other similar approaches, our method reaches a faster speed, and can deal with surfaces with widely distributed curvatures.

Path planning by fractional differentiation

In path planning design, potential fields can introduce force constraints to ensure curvature continuity of trajectories and thus facilitate path-tracking design. The parametric thrift of fractional potentials permits smooth variations of the potential in function of the distance to obstacles without requiring design of geometric charge distribution. In the approach we use, the fractional order of differentiation is the risk coefficient associated to obstacles. A convex danger map towards a target and a convex geodesic distance map are defined. Real-time computation can also lead to the shortest minimum danger trajectory, or to the least dangerous of minimum length trajectories.

Texture mapping using surface flattening via multidimensional scaling

Presents a novel technique for texture mapping on arbitrary surfaces with minimal distortion by preserving the local and global structure of the texture. The recent introduction of the fast marching method on triangulated surfaces has made it possible to compute a geodesic distance map from a given surface point in O(n lg n) operations, where n is the number of triangles that represent the surface. We use this method to design a surface flattening approach based on multi-dimensional scaling (MDS). MDS is a family of methods that map a set of points into a finite-dimensional flat (Euclidean) domain, where the only data given is the corresponding distance between every pair of points. The MDS mapping yields minimal changes of the distances between the corresponding points. We then solve an "inverse" problem and map a flat texture patch onto a curved surface while preserving the structure of the texture.

Good metric spaces without good parameterizations

A classical problem in geometric topology is to recognize when a topological space is a topological manifold. This paper addresses the question of when a rnetric space admits a quasisymmetric parameterization by providing counterexamples to the obvious optimistic conjectures, or, in other words, by providing examples of spaces with many Euclidean-like properties which are nonetheless substantially different from Euclidean geometry. These examples are geometrically self-similar versions of classical topologically self-similar examples from geometric topology, and they can be realized as codimension 1 subsets of Euclidean spaces. Unlike earlier examples going back to Rickman, these sets enjoy good bounds on their geodesic distance functions and good mass bounds (AhIfors regularity). They are also smooth except for reasonably tame degenerations near small sets, they are uniform1y rectifiable, and they have good properties in terms of analysis (like Sobolev and Poincaré inequalities). The construction also produces uniform domains which have many nice properties but which are not quasiconformally equivalent to balls.

Invariant correlations in simplicial gravity.

Some first results are presented regarding the behavior of invariant correlations in simplicial gravity, with an action containing both a bare cosmological term and a lattice higher derivative term. The determination of invariant correlations as a function of geodesic distance by numerical methods is a difficult task, since the geodesic distance between any two points is a function of the fluctuating background geometry, and correlation effects become rather small for large distances. Still, a strikingly different behavior is found for the volume and curvature correlation functions. While the first one is found to be negative definite at large geodesic distances, the second one is always positive for large distances. For both correlations the results are consistent in the smooth phase with an exponential decay, turning into a power law close to the critical point at $G_c$. Such a behavior is not completely unexpected, if the model is to reproduce the classical Einstein theory at distances much larger than the ultraviolet cutoff scale.

Topographic distance and watershed lines

The watershed line is the basic tool for segmenting images in mathematical morphology. A rigorous definition is given in terms of a distance function called topographic distance. If the topographical function is itself a distance function, then the topographical distance becomes identical with the geodesic distance function and the watershed becomes identical with the skeleton by zone of influence. The classical shortest paths algorithms of the graph theory are then revisited in order to derive new watershed algorithms, which are either new or more easy to implement in hardware.

Fractal structure of two-dimensional gravity coupled to c=-2 matter.

We present the results of numerical calculations of internal fractal dimensions of a dynamically triangulated two-dimensional gravity coupled to c=-2 matter. Internal fractal dimensions as a function of internal geodesic distance are measured from the number of boundaries and the total length of boundaries, and show clear scaling behavior, while the number of triangles starts to scale only around the maximum size. We apply a recursive sampling algorithm for c=-2, which makes it possible to generate the most probable triangulations of spherical topology, consisting of 5\ifmmode\times\else\texttimes\fi{}${10}^{6}$ triangles.

On subvarieties of a Hermitian manifold

We make use of the variational properties of the geodesic distance function of a Riemannian manifold and the technique of the “blowing-up” ( σ \sigma -process) on a complex manifold to derive the nonexistence of compact complex analytic subvarieties in a simply connected, complete Hermitian manifold with nonpositive sectional curvature.