Mesh Snapping: Robust Interactive Mesh Cutting Using Fast Geodesic Curvature Flow

This paper considers the problem of interactively finding the cutting contour to extract components from a given mesh. Some existing methods support cuts of arbitrary shape but require careful and tedious input from the user. Others need little user input however they are sensitive to user input and need a postprocessing step to smooth the generated jaggy cutting contours. The popular geometric snake can be used to optimize the cutting contour, but it cannot deal with the topology change. In this paper, we propose a geodesic curvature flow based framework to overcome all these problems. Since in many cases the meaningful cutting contour on a 3D mesh is locally shortest in the sense of some weighted curve length, the geodesic curvature flow is an ideal tool for our problem. It evolves the cutting contour to the nearby local minimum. We should mention that the previous numerical scheme, discretized geodesic curvature flow (dGCF) is too slow and has not been applied to mesh segmentation. With a careful observation to dGCF, we devise here a fast computation scheme called fast geodesic curvature flow (FGCF), which only needs to solve a smaller and easier problem. The initial cutting contour is generated by a variant of random walks algorithm, which is very fast and gives reasonable cutting result with little user input. Experiment results on the benchmark mesh segmentation data set show that our proposed framework is robust to user input and capable of producing good results reflecting geometric features and human shape perception.     

Part‐type Segmentation of Articulated Voxel‐Shapes using the Junction Rule

We present a part‐type segmentation method for articulated voxel‐shapes based on curve skeletons. Shapes are considered to consist of several simpler, intersecting shapes. Our method is based on the junction rule: the observation that two intersecting shapes generate an additional junction in their joined curve‐skeleton near the place of intersection. For each curve‐skeleton point, we construct a piecewise‐geodesic loop on the shape surface. Starting from the junctions, we search along the curve skeleton for points whose associated loops make for suitable part cuts. The segmentations are robust to noise and discretization artifacts, because the curve skeletonization incorporates a single user‐parameter to filter spurious curve‐skeleton branches. Furthermore, segment borders are smooth and minimally twisting by construction. We demonstrate our method on several real‐world examples and compare it to existing part‐type segmentation methods.     

A Hierarchical Grid Based Framework for Fast Collision Detection

We present a novel hierarchical grid based method for fast collision detection (CD) for deformable models on GPU architecture. A two‐level grid is employed to accommodate the non‐uniform distribution of practical scene geometry. A bottom‐to‐top method is implemented to assign the triangles into the hierarchical grid without any iteration while a deferred scheme is introduced to efficiently update the data structure. To address the issue of load balancing, which greatly influences the performance in SIMD parallelism, a propagation scheme which utilizes a parallel scan and a segmented scan is presented, distributing workloads evenly across all concurrent threads. The proposed method supports both discrete collision detection (DCD) and continuous collision detection (CCD) with self‐collision. Some typical benchmarks are tested to verify the effectiveness of our method. The results highlight our speedups over prior algorithms on different commodity GPUs.     

Approximating Gradients for Meshes and Point Clouds via Diffusion Metric

The gradient of a function defined on a manifold is perhaps one of the most important differential objects in data analysis. Most often in practice, the input function is available only at discrete points sampled from the underlying manifold, and the manifold is approximated by either a mesh or simply a point cloud. While many methods exist for computing gradients of a function defined over a mesh, computing and simplifying gradients and related quantities such as critical points, of a function from a point cloud is non-trivial.
In this paper, we initiate the investigation of computing gradients under a different metric on the manifold from the original natural metric induced from the ambient space. Specifically, we map the input manifold to the eigenspace spanned by its Laplacian eigenfunctions, and consider the so-called diffusion distance metric associated with it. We show the relation of gradient under this metric with that under the original metric. It turns out that once the Laplace operator is constructed, it is easier to approximate gradients in the eigenspace for discrete inputs (especially point clouds) and it is robust to noises in the input function and in the underlying manifold. More importantly, we can easily smooth the gradient field at different scales within this eigenspace framework. We demonstrate the use of our new eigen-gradients with two applications: approximating / simplifying the critical points of a function, and the Jacobi sets of two input functions (which describe the correlation between these two functions), from point clouds data.     

As‐Rigid‐AsPossible Distance Field Metamorphosis

Widely used for morphing between objects with arbitrary topology, distance field interpolation (DFI) handles topological transition naturally without the need for correspondence or remeshing, unlike surface‐based interpolation approaches. However, lack of correspondence in DFI also leads to ineffective control over the morphing process. In particular, unless the user specifies a dense set of landmarks, it is not even possible to measure the distortion of intermediate shapes during interpolation, let alone control it. To remedy such issues, we introduce an approach for establishing correspondence between the interior of two arbitrary objects, formulated as an optimal mass transport problem with a sparse set of landmarks. This correspondence enables us to compute non‐rigid warping functions that better align the source and target objects as well as to incorporate local rigidity constraints to perform as‐rigid‐aspossible DFI. We demonstrate how our approach helps achieve flexible morphing results with a small number of landmarks.     

Topology‐based Smoothing of 2D Scalar Fields with C1‐Continuity

Data sets coming from simulations or sampling of real‐world phenomena often contain noise that hinders their processing and analysis. Automatic filtering and denoising can be challenging: when the nature of the noise is unknown, it is difficult to distinguish between noise and actual data features; in addition, the filtering process itself may introduce “artificial” features into the data set that were not originally present. In this paper, we propose a smoothing method for 2D scalar fields that gives the user explicit control over the data features. We define features as critical points of the given scalar function, and the topological structure they induce (i.e., the Morse‐Smale complex). Feature significance is rated according to topological persistence. Our method allows filtering out spurious features that arise due to noise by means of topological simplification, providing the user with a simple interface that defines the significance threshold, coupled with immediate visual feedback of the remaining data features. In contrast to previous work, our smoothing method guarantees a C¹‐continuous output scalar field with the exact specified features and topological structures.     

Smooth Shape‐Aware Functions with Controlled Extrema

Functions that optimize Laplacian‐based energies have become popular in geometry processing, e.g. for shape deformation, smoothing, multiscale kernel construction and interpolation. Minimizers of Dirichlet energies, or solutions of Laplace equations, are harmonic functions that enjoy the maximum principle, ensuring no spurious local extrema in the interior of the solved domain occur. However, these functions are only C⁰ at the constrained points, which often causes smoothness problems. For this reason, many applications optimize higher‐order Laplacian energies such as biharmonic or triharmonic. Their minimizers exhibit increasing orders of continuity but lose the maximum principle and show oscillations. In this work, we identify characteristic artifacts caused by spurious local extrema, and provide a framework for minimizing quadratic energies on manifolds while constraining the solution to obey the maximum principle in the solved region. Our framework allows the user to specify locations and values of desired local maxima and minima, while preventing any other local extrema. We demonstrate our method on the smoothness energies corresponding to popular polyharmonic functions and show its usefulness for fast handle‐based shape deformation, controllable color diffusion, and topologically‐constrained data smoothing.     

Generalized Discrete Ricci Flow

Surface Ricci flow is a powerful tool to design Riemannian metrics by user defined curvatures. Discrete surface Ricci flow has been broadly applied for surface parameterization, shape analysis, and computational topology. Conventional discrete Ricci flow has limitations. For meshes with low quality triangulations, if high conformality is required, the flow may get stuck at the local optimum of the Ricci energy. If convergence to the global optimum is enforced, the conformality may be sacrificed. This work introduces a novel method to generalize the traditional discrete Ricci flow. The generalized Ricci flow is more flexible, more robust and conformal for meshes with low quality triangulations. Conventional method is based on circle packing, which requires two circles on an edge intersect each other at an acute angle. Generalized method allows the two circles either intersect or separate from each other. This greatly improves the flexibility and robustness of the method. Furthermore, the generalized Ricci flow preserves the convexity of the Ricci energy, this ensures the uniqueness of the global optimum. Therefore the algorithm won't get stuck at the local optimum. Generalized discrete Ricci flow algorithms are explained in details for triangle meshes with both Euclidean and hyperbolic background geometries. Its advantages are demonstrated by theoretic proofs and practical applications in graphics, especially surface parameterization.     

Heat Walk: Robust Salient Segmentation of Non‐rigid Shapes

Segmenting three dimensional objects using properties of heat diffusion on meshes aim to produce salient results. The few existing algorithms based on heat diffusion do not use the full knowledge that can be gained from heat diffusion and are sensitive to varying kinds of perturbations. Our simple algorithm, Heat Walk, converts the implicit information in the heat kernel to explicit knowledge about the pathways for maximum heat flow capacity. We develop a two stage strategy for segmentation. In the first stage we quickly identify regions which are dominated by heat accumulators by employing a greedy algorithm. The second stage partitions out dissipative regions from the previously discovered accumulative regions by using a KL‐divergence based criterion. The resulting algorithm is both independent of human intervention and fast because of the globally aware directed walk along the maximal heat flow capacity. Extensive experimental evidence shows the method is robust to a variety of noise factors including topological short circuits, surface holes, pose variations, variations in tessellation, missing features, scaling, as well as normal and shot noise. Comparison with the Princeton Segmentation Benchmark (PSB) shows that our method is comparable with state of the art segmentation methods and has additional advantages of being robust and self contained. Based upon theoretical insight the convergence and stability of the Heat Walk is shown.     

As‐Killing‐As‐Possible Vector Fields for Planar Deformation

Cartoon animation, image warping, and several other tasks in two‐dimensional computer graphics reduce to the formulation of a reasonable model for planar deformation. A deformation is a map from a given shape to a new one, and its quality is determined by the type of distortion it introduces. In many applications, a desirable map is as isometric as possible. Finding such deformations, however, is a nonlinear problem, and most of the existing solutions approach it by minimizing a nonlinear energy. Such methods are not guaranteed to converge to a global optimum and often suffer from robustness issues. We propose a new approach based on approximate Killing vector fields (AKVFs), first introduced in shape processing. AKVFs generate near‐isometric deformations, which can be motivated as direction fields minimizing an “as‐rigid‐as‐possible” (ARAP) energy to first order. We first solve for an AKVF on the domain given user constraints via a linear optimization problem and then use this AKVF as the initial velocity field of the deformation. In this way, we transfer the inherent nonlinearity of the deformation problem to finding trajectories for each point of the domain having the given initial velocities. We show that a specific class of trajectories — the set of logarithmic spirals — is especially suited for this task both in practice and through its relationship to linear holomorphic vector fields. We demonstrate the effectiveness of our method for planar deformation by comparing it with existing state‐of‐the‐art deformation methods.     

Anatomy‐Guided Multi‐Level Exploration of Blood Flow in Cerebral Aneurysms

For cerebral aneurysms, the ostium, the area of inflow, is an important anatomic landmark, since it separates the pathological vessel deformation from the healthy parent vessel. A better understanding of the inflow characteristics, the flow inside the aneurysm and the overall change of pre‐ and post‐aneurysm flow in the parent vessel provide insights for medical research and the development of new risk‐reduced treatment options. We present an approach for a qualitative, visual flow exploration that incorporates the ostium and derived anatomical landmarks. It is divided into three scopes: a global scope for exploration of the in‐ and outflow, an ostium scope that provides characteristics of the flow profile close to the ostium and a local scope for a detailed exploration of the flow in the parent vessel and the aneurysm. The approach was applied to five representative datasets, including measured and simulated blood flow. Informal interviews with two board‐certified radiologists confirmed the usefulness of the provided exploration tools and delivered input for the integration of the ostium‐based flow analysis into the overall exploration workflow.     

A Condition Number for Non‐Rigid Shape Matching

Despite the large amount of work devoted in recent years to the problem of non‐rigid shape matching, practical methods that can successfully be used for arbitrary pairs of shapes remain elusive. In this paper, we study the hardness of the problem of shape matching, and introduce the notion of the shape condition number, which captures the intuition that some shapes are inherently more difficult to match against than others. In particular, we make a connection between the symmetry of a given shape and the stability of any method used to match it while optimizing a given distortion measure. We analyze two commonly used classes of methods in deformable shape matching, and show that the stability of both types of techniques can be captured by the appropriate notion of a condition number. We also provide a practical way to estimate the shape condition number and show how it can be used to guide the selection of landmark correspondences between shapes. Thus we shed some light on the reasons why general shape matching remains difficult and provide a way to detect and mitigate such difficulties in practice.     

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the Gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive μ-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample.     

Shape Analysis Using the Auto Diffusion Function

Scalar functions defined on manifold triangle meshes is a starting point for many geometry processing algorithms such as mesh parametrization, skeletonization, and segmentation. In this paper, we propose the Auto Diffusion Function (ADF) which is a linear combination of the eigenfunctions of the Laplace-Beltrami operator in a way that has a simple physical interpretation. The ADF of a given 3D object has a number of further desirable properties: Its extrema are generally at the tips of features of a given object, its gradients and level sets follow or encircle features, respectively, it is controlled by a single parameter which can be interpreted as feature scale, and, finally, the ADF is invariant to rigid and isometric deformations.
We describe the ADF and its properties in detail and compare it to other choices of scalar functions on manifolds. As an example of an application, we present a pose invariant, hierarchical skeletonization and segmentation algorithm which makes direct use of the ADF.     

Efficient Image‐Based Proximity Queries with Object‐Space Precision

We present an efficient algorithm for object‐space proximity queries between multiple deformable triangular meshes. Our approach uses the rasterization capabilities of the GPU to produce an image‐space representation of the vertices. Using this image‐space representation, inter‐object vertex‐triangle distances and closest points lying under a user‐defined threshold are computed in parallel by conservative rasterization of bounding primitives and sorted using atomic operations. We additionally introduce a similar technique to detect penetrating vertices. We show how mechanisms of modern GPUs such as mipmapping, Early‐Z and Early‐Stencil culling can optimize the performance of our method. Our algorithm is able to compute dense proximity information for complex scenes made of more than a hundred thousand triangles in real time, outperforming a CPU implementation based on bounding volume hierarchies by more than an order of magnitude.     

Discovery of Intrinsic Primitives on Triangle Meshes

The discovery of meaningful parts of a shape is required for many geometry processing applications, such as parameterization, shape correspondence, and animation. It is natural to consider primitives such as spheres, cylinders and cones as the building blocks of shapes, and thus to discover parts by fitting such primitives to a given surface. This approach, however, will break down if primitive parts have undergone almost‐isometric deformations, as is the case, for example, for articulated human models. We suggest that parts can be discovered instead by finding intrinsic primitives, which we define as parts that posses an approximate intrinsic symmetry. We employ the recently‐developed method of computing discrete approximate Killing vector fields (AKVFs) to discover intrinsic primitives by investigating the relationship between the AKVFs of a composite object and the AKVFs of its parts. We show how to leverage this relationship with a standard clustering method to extract k intrinsic primitives and remaining asymmetric parts of a shape for a given k. We demonstrate the value of this approach for identifying the prominent symmetry generators of the parts of a given shape. Additionally, we show how our method can be modified slightly to segment an entire surface without marking asymmetric connecting regions and compare this approach to state‐of‐the‐art methods using the Princeton Segmentation Benchmark.     

Filtering Relocations on a Delaunay Triangulation

Updating a Delaunay triangulation when its vertices move is a bottleneck in several domains of application. Rebuilding the whole triangulation from scratch is surprisingly a very viable option compared to relocating the vertices. This can be explained by several recent advances in efficient construction of Delaunay triangulations. However, when all points move with a small magnitude, or when only a fraction of the vertices move, rebuilding is no longer the best option. This paper considers the problem of efficiently updating a Delaunay triangulation when its vertices are moving under small perturbations. The main contribution is a set of filters based upon the concept of vertex tolerances. Experiments show that filtering relocations is faster than rebuilding the whole triangulation from scratch under certain conditions.     

Soft Articulated Characters with Fast Contact Handling

Fast contact handling of soft articulated characters is a computationally challenging problem, in part due to complex interplay between skeletal and surface deformation. We present a fast, novel algorithm based on a layered representation for articulated bodies that enables physically-plausible simulation of animated characters with a high-resolution deformable skin in real time. Our algorithm gracefully captures the dynamic skeleton-skin interplay through a novel formulation of elastic deformation in the pose space of the skinned surface. The algorithm also overcomes the computational challenges by robustly decoupling skeleton and skin computations using careful approximations of Schur complements, and efficiently performing collision queries by exploiting the layered representation. With this approach, we can simultaneously handle large contact areas, produce rich surface deformations, and capture the collision response of a character/s skeleton.     

Real‐Time Modelling of Fibrous Muscle

Relatively recently it has become apparent that providing human kind with a better healthcare requires personalized, predictive and integrative medicine, for which the building of virtual physiological human (VPH) framework accessible via virtual patient avatar is necessary. Real‐time modelling and visual exploration of such a complex avatar is a challenging task. In this paper, we propose a real‐time method for automatic modelling of an arbitrarily large number of muscle fibres in the volume of a muscle represented by its surface mesh. The method is based on an iterative morphing of predefined fibres template into the muscle volume exploiting harmonic scalar field computed on the surface of muscle. Experiments with muscles of thighs and pelvis show that the method produces realistic shapes of fibres. Our sequential VTK‐based C++ implementation is capable of producing 64 fine fibres within a muscle of 10K triangles in less than 170 ms on commodity hardware making the method suitable for VPH purposes as well as for interactive educational medical software.     

Positional, Metric, and Curvature Control for Constraint-Based Surface Deformation

We present a geometry processing framework that allows direct manipulation or preservation of positional, metric, and curvature constraints anywhere on the surface of a geometric model. Target values for these properties can be specified point-wise or as integrated quantities over curves and surface patches embedded in the shape. For example, the user can draw several curves on the surface and specify desired target lengths, manipulate the normal curvature along these curves, or modify the area or principal curvature distribution of arbitrary surface patches. This user input is converted into a set of non-linear constraints. A global optimization finds the new deformed surface that best satisfies the constraints, while minimizing adaptable measures for metric and curvature distortion that provide explicit control of the deformation semantics. We illustrate how this approach enables flexible surface processing and shape editing operations not available in current systems.