Computation Of Geodesic Distances Research Articles

Facial Similarity Measure for recognizing monozygotic twins utilizing 3D facial landmarks, efficient geodesic distance computation, and machine learning algorithms

Facial Similarity Measure for recognizing monozygotic twins utilizing 3D facial landmarks, efficient geodesic distance computation, and machine learning algorithms

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.

Parallel and Scalable Heat Methods for Geodesic Distance Computation.

In this paper, we propose a parallel and scalable approach for geodesic distance computation on triangle meshes. Our key observation is that the recovery of geodesic distance with the heat method [1] can be reformulated as optimization of its gradients subject to integrability, which can be solved using an efficient first-order method that requires no linear system solving and converges quickly. Afterward, the geodesic distance is efficiently recovered by parallel integration of the optimized gradients in breadth-first order. Moreover, we employ a similar breadth-first strategy to derive a parallel Gauss-Seidel solver for the diffusion step in the heat method. To further lower the memory consumption from gradient optimization on faces, we also propose a formulation that optimizes the projected gradients on edges, which reduces the memory footprint by about 50 percent. Our approach is trivially parallelizable, with a low memory footprint that grows linearly with respect to the model size. This makes it particularly suitable for handling large models. Experimental results show that it can efficiently compute geodesic distance on meshes with more than 200 million vertices on a desktop PC with 128 GB RAM, outperforming the original heat method and other state-of-the-art geodesic distance solvers.

A Singular Value Decomposition based Maximal Poisson-disk Sampling for adaptive Digital Elevation Model simplification.

The proposed method is to do simplification for Digital Elevation Model (DEM), which uses a few of original nodes representing the terrain surface while maintaining the accuracy. The original DEM nodes are sampled using the Maximal Poisson-disk Sampling (MPS), in which, the disk’s size of each sample is computed on basis of the Singular Value Decomposition (SVD). MPS can generate the hyper-uniformly distributed samples and was taken to do DEM adaptive sampling by being combined with the geodesic metric. However, the geodesic distance computation is complex and the requirement for memory is high. As such, this paper proposes an extension of the classic MPS based method for selecting quasi-randomly distributed points from DEM nodes based on the distribution of eigenvalues, accounting for surface heterogeneity. To achieve this objective, uniform MPS is conducted to sample the DEM nodes by setting the related disk radius to be inversely proportional to the local terrain complexity, which is defined as an index expressing the local terrain variation. Then, the geodesic metric related parameters are implicitly contained in the defined index. As a result, more samples are concentrated in the rugged regions, and vice versa. The proposed method shows better perfermance, at least the results are comparable with the geodesic distance based Poisson disk sampling method. Meanwhile, it greatly accelerates the sampling process and reduces the memory cost.

A Laplacian for Nonmanifold Triangle Meshes

AbstractWe describe a discrete Laplacian suitable for any triangle mesh, including those that are nonmanifold or nonorientable (with or without boundary). Our Laplacian is a robust drop‐in replacement for the usual cotan matrix, and is guaranteed to have nonnegative edge weights on both interior and boundary edges, even for extremely poor‐quality meshes. The key idea is to build what we call a “tufted cover” over the input domain, which has nonmanifold vertices but manifold edges. Since all edges are manifold, we can flip to an intrinsic Delaunay triangulation; our Laplacian is then the cotan Laplacian of this new triangulation. This construction also provides a high‐quality point cloud Laplacian, via a nonmanifold triangulation of the point set. We validate our Laplacian on a variety of challenging examples (including all models from Thingi10k), and a variety of standard tasks including geodesic distance computation, surface deformation, parameterization, and computing minimal surfaces.

Anisotropic tubular minimal path model with fast marching front freezing scheme

In this work, we introduce an anisotropic minimal path model based on a new Riemannian tensor integrating the crossing-adaptive anisotropic radius-lifted tensor field and the front freezing indicator by appearance and path features. The non-local path feature only can be obtained during the geodesic distance computation process by the fast marching method. The predefined criterion derived from path feature is able to steer the front evolution by freezing the point causing high bending of the geodesic to solve the shortcut problem. We performed qualitative and quantitative experiments on synthetic and real images (including retinal vessels, rivers and roads) and compare with the minimal path models with classical anisotropic Riemannian metric and dynamic isotropic metric, which demonstrated the proposed method can detect desired targets from complex tubular tree structures.

Geodesic distance numerical computation on compliant mechanical parts in the aircraft industry

The problem of geodesic distance computation in the Aircraft industry is considered in the paper. While simulating the assembly process, in some computational problems the calculation of the shortest distances between different objects is required. The aim of the work is to compare several approaches to geodesic distance numerical computation on test geometry and some of the most common joints of aircraft. The first considered approach represents graph-based algorithms. In the paper, geodesic distances are calculated in the constructed graph using Dijkstra’s algorithm or its modification with binary heap. The second approach is the fast marching method, which is based on the solution of the Eikonal equation and provides efficient geodesic distance computation on triangulated surfaces.

A minimalistic approach for fast computation of geodesic distances on triangular meshes

The computation of geodesic distances is an important research topic in Geometry Processing and 3D Shape Analysis as it is a basic component of many methods used in these areas. In this work, we present a minimalistic parallel algorithm based on front propagation to compute approximate geodesic distances on meshes. Our method is practical and simple to implement, and does not require any heavy pre-processing. The convergence of our algorithm depends on the number of discrete level sets around the source points from which distance information propagates. To appropriately implement our method on GPUs taking into account memory coalescence problems, we take advantage of a graph representation based on a breadth-first search traversal that works harmoniously with our parallel front propagation approach. We report experiments that show how our method scales with the size of the problem. We compare the mean error and processing time obtained by our method with such measures computed using other methods. Our method produces results in competitive times with almost the same accuracy, especially for large meshes. We also demonstrate its use for solving two classical geometry processing problems: the regular sampling problem and the Voronoi tessellation on meshes.

Navigating intrinsic triangulations

We present a data structure that makes it easy to run a large class of algorithms from computational geometry and scientific computing on extremely poor-quality surface meshes. Rather than changing the geometry, as in traditional remeshing, we consider intrinsic triangulations which connect vertices by straight paths along the exact geometry of the input mesh. Our key insight is that such a triangulation can be encoded implicitly by storing the direction and distance to neighboring vertices. The resulting signpost data structure then allows geometric and topological queries to be made on-demand by tracing paths across the surface. Existing algorithms can be easily translated into the intrinsic setting, since this data structure supports the same basic operations as an ordinary triangle mesh (vertex insertions, edge splits, etc. ). The output of intrinsic algorithms can then be stored on an ordinary mesh for subsequent use; unlike previous data structures, we use a constant amount of memory and do not need to explicitly construct an overlay mesh unless it is specifically requested. Working in the intrinsic setting incurs little computational overhead, yet we can run algorithms on extremely degenerate inputs, including all manifold meshes from the Thingi10k data set. To evaluate our data structure we implement several fundamental geometric algorithms including intrinsic versions of Delaunay refinement and optimal Delaunay triangulation, approximation of Steiner trees, adaptive mesh refinement for PDEs, and computation of Poisson equations, geodesic distance, and flip-free tangent vector fields.

Efficient Computation of Smoothed Exponential Maps

AbstractMany applications in geometry processing require the computation of local parameterizations on a surface mesh at interactive rates. A popular approach is to compute local exponential maps, i.e. parameterizations that preserve distance and angle to the origin of the map. We extend the computation of geodesic distance by heat diffusion to also determine angular information for the geodesic curves. This approach has two important benefits compared to fast approximate as well as exact forward tracing of the distance function: First, it allows generating smoother maps, avoiding discontinuities. Second, exploiting the factorization of the global Laplace–Beltrami operator of the mesh and using recent localized solution techniques, the computation is more efficient even compared to fast approximate solutions based on Dijkstra's algorithm.

Efficient Inter-Geodesic Distance Computation and Fast Classical Scaling.

Multidimensional scaling (MDS) is a dimensionality reduction tool used for information analysis, data visualization and manifold learning. Most MDS procedures embed data points in low-dimensional euclidean (flat) domains, such that distances between the points are as close as possible to given inter-point dissimilarities. We present an efficient solver for classical scaling, a specific MDS model, by extrapolating the information provided by distances measured from a subset of the points to the remainder. The computational and space complexities of the new MDS methods are thereby reduced from quadratic to quasi-linear in the number of data points. Incorporating both local and global information about the data allows us to construct a low-rank approximation of the inter-geodesic distances between the data points. As a by-product, the proposed method allows for efficient computation of geodesic distances.

Minimal Paths for Tubular Structure Segmentation With Coherence Penalty and Adaptive Anisotropy.

The minimal path method has proven to be particularly useful and efficient in tubular structure segmentation applications. In this paper, we propose a new minimal path model associated with a dynamic Riemannian metric embedded with an appearance feature coherence penalty and an adaptive anisotropy enhancement term. The features that characterize the appearance and anisotropy properties of a tubular structure are extracted through the associated orientation score. The proposed the dynamic Riemannian metric is updated in the course of the geodesic distance computation carried out by the efficient single-pass fast marching method. Compared to the state-of-the-art minimal path models, the proposed minimal path model is able to extract the desired tubular structures from a complicated vessel tree structure. In addition, we propose an efficient prior path-based method to search for vessel radius value at each centerline position of the target. Finally, we perform the numerical experiments on both synthetic and real images. The quantitive validation is carried out on retinal vessel images. The results indicate that the proposed model indeed achieves a promising performance.

Immersion of self-intersecting solids and surfaces

Self-intersecting, or nearly self-intersecting, meshes are commonly found in 2D and 3D computer graphics practice. Self-intersections occur, for example, in the process of artist manual work, as a by-product of procedural methods for mesh generation, or due to modeling errors introduced by scanning equipment. If the space bounded by such inputs is meshed naively, the resulting mesh joins ("glues") self-overlapping parts, precluding efficient further modeling and animation of the underlying geometry. Similarly, near self-intersections force the simulation algorithm to employ an unnecessarily detailed mesh to separate the nearly self-intersecting regions. Our work addresses both of these challenges, by giving an algorithm to generate an "un-glued" simulation mesh, of arbitrary user-chosen resolution, that properly accounts for self-intersections and near self-intersections. In order to achieve this result, we study the mathematical concept of immersion, and give a deterministic and constructive algorithm to determine if the input self-intersecting triangle mesh is the boundary of an immersion. For near self-intersections, we give a robust algorithm to properly duplicate mesh elements and correctly embed the underlying geometry into the mesh element copies. Both the self-intersections and near self-intersections are combined into one algorithm that permits successful meshing at arbitrary resolution. Applications of our work include volumetric shape editing, physically based simulation and animation, and volumetric weight and geodesic distance computation on self-intersecting inputs.

Gradient-domain processing within a texture atlas

Processing signals on surfaces often involves resampling the signal over the vertices of a dense mesh and applying mesh-based filtering operators. We present a framework to process a signal directly in a texture atlas domain. The benefits are twofold: avoiding resampling degradation and exploiting the regularity of the texture image grid. The main challenges are to preserve continuity across atlas chart boundaries and to adapt differential operators to the non-uniform parameterization. We introduce a novel function space and multigrid solver that jointly enable robust, interactive, and geometry-aware signal processing. We demonstrate our approach using several applications including smoothing and sharpening, multiview stitching, geodesic distance computation, and line integral convolution.

Localized constraint based deformation framework for triangle meshes

Traditional Verlet integration frameworks have been successful with their robustness and efficiency to simulate deformable bodies ranging from simple cloth to geometrically complex solids. However, the existing frameworks deform the models as a whole. We present a Verlet integration framework which provides local surface deformation on the arbitrary selected area of the mesh without giving any global deformation impact to the whole model. The framework is designed to take the advantage of physics to deform simple triangle meshes with the position-based approach. Our framework provides an interactive selection of the deformation influence area by using geodesic distance computation based on heat kernel. Additionally, the framework computes several geometric constraints that provides an interconnected particle system simulation with environmental interactions such as collision. The proposed framework is robust and easy to implement since it is based on highly accurate geodesic distance computation and solving the projected geometric constraints. We demonstrate the benefits of our framework with the results obtained from various triangle mesh models to present its potential in terms of practicability and effectiveness.

Real-time human segmentation from RGB-D video sequence based on adaptive geodesic distance computation

In this paper, we propose a method for extracting humans in the foreground of video frames using color and depth information. To ensure real-time performance and to increase accuracy, we classify a video frame into two parts by degree of noise: head region with high noise level, and non-head region with low noise level. Then, we apply a high-computational geodesic matting algorithm to the noisy head region that includes hair, and a low-computational hole filling with smoothing method to other regions. Additionally, we modify the traditional color-based geodesic segmentation algorithm to consider additional depth information. Then, we apply temporal/spatial smoothing to the blended foreground mask in order to enhance the coherence between video frames. Experimental results show that the proposed method outperforms a previous approach by accuracy and performance.

Subdivision exterior calculus for geometry processing

This paper introduces a new computational method to solve differential equations on subdivision surfaces. Our approach adapts the numerical framework of Discrete Exterior Calculus (DEC) from the polygonal to the subdivision setting by exploiting the refin-ability of subdivision basis functions. The resulting Subdivision Exterior Calculus (SEC) provides significant improvements in accuracy compared to existing polygonal techniques, while offering exact finite-dimensional analogs of continuum structural identities such as Stokes' theorem and Helmholtz-Hodge decomposition. We demonstrate the versatility and efficiency of SEC on common geometry processing tasks including parameterization, geodesic distance computation, and vector field design.

Discretized Riemannian Delaunay Triangulations

Anisotropic meshes are desirable for various applications, such as the numerical solving of partial differential equations and graph- ics. In this paper, we introduce an algorithm to compute discrete approximations of Riemannian Voronoi diagrams on 2-manifolds. This is not straightforward because geodesics, shortest paths between points, and therefore distances cannot in general be computed exactly.Our implementation employs recent developments in the numerical computation of geodesic distances and is accelerated through the use of an underlying anisotropic graph structure.We give conditions that guarantee that our discrete Riemannian Voronoi diagram is combinatorially equivalent to the Riemannian Voronoi diagram and that its dual is an embedded triangulation, using both approximate geodesics and straight edges. Both the theoretical guarantees on the approximation of the Voronoi diagram and the implementation are new and provide a step towards the practical application of Riemannian Delaunay triangulations.

Bayesian sensitivity analysis with the Fisher–Rao metric

We propose a geometric framework to assess sensitivity of Bayesian procedures to modelling assumptions based on the nonparametric Fisher–Rao metric. While the framework is general, the focus of this article is on assessing local and global robustness in Bayesian procedures with respect to perturbations of the likelihood and prior, and on the identification of influential observations. The approach is based on a square-root representation of densities, which enables analytical computation of geodesic paths and distances, facilitating the definition of naturally calibrated local and global discrepancy measures. An important feature of our approach is the definition of a geometric ϵ-contamination class of sampling distributions and priors via intrinsic analysis on the space of probability density functions. We demonstrate the applicability of our framework to generalized mixed-effects models and to directional and shape data.

Multiple Depth Maps Integration for 3D Reconstruction Using Geodesic Graph Cuts

Depth images, in particular depth maps estimated from stereo vision, may have a substantial amount of outliers and result in inaccurate 3D modelling and reconstruction. To address this challenging issue, in this paper, a graph-cut based multiple depth maps integration approach is proposed to obtain smooth and watertight surfaces. First, confidence maps for the depth images are estimated to suppress noise, based on which reliable patches covering the object surface are determined. These patches are then exploited to estimate the path weight for 3D geodesic distance computation, where an adaptive regional term is introduced to deal with the "shorter-cuts" problem caused by the effect of the minimal surface bias. Finally, the adaptive regional term and the boundary term constructed using patches are combined in the graph-cut framework for more accurate and smoother 3D modelling. We demonstrate the superior performance of our algorithm on the well-known Middlebury multi-view database and additionally on real-world multiple depth images captured by Kinect. The experimental results have shown that our method is able to preserve the object protrusions and details while maintaining surface smoothness.