Principal Curvature Directions Research Articles

CurveNet: Curvature-Based Multitask Learning Deep Networks for 3D Object Recognition

In computer vision fields, 3D object recognition is one of the most important tasks for many real-world applications. Three-dimensional convolutional neural networks (CNNs) have demonstrated their advantages in 3D object recognition. In this paper, we propose to use the principal curvature directions of 3D objects (using a CAD model) to represent the geometric features as inputs for the 3D CNN. Our framework, namely CurveNet, learns perceptually relevant salient features and predicts object class labels. Curvature directions incorporate complex surface information of a 3D object, which helps our framework to produce more precise and discriminative features for object recognition. Multitask learning is inspired by sharing features between two related tasks, where we consider pose classification as an auxiliary task to enable our CurveNet to better generalize object label classification. Experimental results show that our proposed framework using curvature vectors performs better than voxels as an input for 3D object classification. We further improved the performance of CurveNet by combining two networks with both curvature direction and voxels of a 3D object as the inputs. A Cross-Stitch module was adopted to learn effective shared features across multiple representations. We evaluated our methods using three publicly available datasets and achieved competitive performance in the 3D object recognition task.

3D volumetric isotopological meshing for finite element and isogeometric based reduced order modeling

This paper presents a generic framework to construct 3D structured volumetric meshes of complicated geometry and arbitrary topology. Structured meshes are well-suited for reduced order model applications with geometric parameters. For that purpose, we use the triangulated solid 3D model’s boundary provided from B-Rep CAD (Boundary-Representation in Computer Aided Design) models. The input triangulated mesh is decomposed into a set of cuboids in two steps: pants decomposition and cuboid decomposition. Both segmentations understand the geometry and features of meshes. Cuboid decomposition splits a surface into a set of quadrilateral patches which can define a volumetric layout of the associated boundary surface. Using aligned global parameterization, patches of the cuboid decomposition are re-positioned on the surface in a way to achieve low overall distortion, and alignment to principal curvature directions and sharp features. The optimization process is thought to design cross fields with topological and geometrical constraints. Using the optimized cuboid decomposition, a volumetric layout is extracted. Based on the global parameterization and the structured volumetric layout previously computed, a 3D volumetric parameterization is deducted. For different geometrical instances with the same topology but different geometries, the proposed method allows to have the same representation: 3D volumetric isotopological meshes holding the same connectivity. MEG-IsoHex method is introduced to compare fields on 3D hexahedral meshes. The efficiency and the robustness of the proposed approach are illustrated through a remeshing case for large deformations and reduced order models using isogeometric analysis.

A blend of stretching and bending in nematic polymer networks.

Nematic polymer networks are (heat and light) activable materials, which combine the features of rubber and nematic liquid crystals. When only the stretching energy of a thin sheet of nematic polymer network is minimized, the intrinsic (Gaussian) curvature of the shape it takes upon (thermal or optical) actuation is determined. This, unfortunately, produces a multitude of possible shapes, for which we need a selection criterion, which may only be provided by a correcting bending energy depending on the extrinsic curvatures of the deformed shape. The literature has so far offered approximate corrections depending on the mean curvature. In this paper, we derive the appropriate bending energy for a sheet of nematic polymer network from the celebrated neo-classical energy of nematic elastomers in three space dimensions. This task is performed via a dimension reduction based on a modified Kirchhoff-Love hypothesis, which withstands the criticism of more sophisticated analytical tools. The result is a surface elastic free-energy density where stretching and bending are blended together; they may or may not be length-separated, and should be minimized together. The extrinsic curvatures of the deformed shape not only feature in the bending energy through the mean curvature, but also through the relative orientation of the nematic director in the frame of the directions of principal curvatures.

Local contact characteristics of threaded surfaces in a planetary roller screw mechanism

This study investigates the local contact characteristics of the threaded surface meshing of a planetary roller screw mechanism (PRSM). First, according to the threaded surface structure and threaded surface meshing characteristics, expressions for the principal curvature and principal direction of the contact ellipse at a contact point are derived based on the differential geometry theory. Next, based on a force analysis and threaded surface equations, an analytical model is established to calculate the dimensions and principal vector direction of contact ellipses on threaded surfaces. The elastic deformation and maximum contact stress are determined using Hertz elastic contact theory. Then, finite element (FE) numerical models for a single pair of threads at the screw–roller interface and the roller–nut interface are developed to calculate the contact area and maximum contact stress. The results are compared with those of the analytical model to demonstrate the validity of the analytical model. Finally, based on the analytical model proposed in this article, the local contact characteristics of threaded surfaces with various thread pitches, flank angles, and profile radii of roller threads are analyzed in detail.

Novel method of boundary-free mesh parameterization.

Unless the targeted mesh is developable, metric distortion is inevitable during the process of surface mesh parameterization, thus one important objective of all involved parametric studies is to reduce the metric distortion. In order to further reduce area and angle distortion, a novel method of boundary-free mesh parameterization is presented in the paper. Firstly, the initial boundary-fixed conformal parameterization from 3D surface mesh patch to a plane is performed in the method. Then, based on the initial parameterization, the iterations of boundary-free quasi-harmonic parameterization are developed, where the tensor field is updated in each iterative step and the principal curvature direction is utilized to terminate the iteration. The solution of the novel method is convenient to calculate since it involves a series of linear systems. In our novel parameterization method, lower metric distortion and considerable efficiency have been obtained in experiments.

Curvature radius measurement by optical profiler and determination of the residual stress in thin films

The Stoney formula, based on the measurement of the substrate curvature, is often used for the determination of the thin films' residual stress. In this study, titanium nitride coatings were deposited by DC reactive magnetron sputtering on silicon substrates. An optical profiler was used to determine the curvature of the surface before and after coating. Two radii were then obtained, along the principal perpendicular directions of the surface curvature. A simple and efficient method to determine the experimental error on the stress calculation was developed taking into account the film thickness dispersion and the radii dispersion. Using constant deposition parameters, some samples' characteristics were tested: film and substrate thickness, size, shape and crystallographic orientations of the substrates. With the help of the developed error method, we analyzed what can be conclude about the influence of these characteristics on the calculated stress values, obtained from the experimental measurements.

A robust and efficient framework for fast cylinder detection

In this work, a complete solution is provided for detecting and identifying cylindrical shapes, which are commonly found in household and industrial environments, using consumer-grade RGB-D cameras. Most standard approaches to detect and identify cylinders are not robust to outliers (e.g. points on other objects in the scene), which limits their applicability in realistic scenes. In addition, these methods fail to benefit from environmental constraints, e.g. the fact that cylinders often lie or stand on flat surfaces. To tackle the aforementioned limitations, we introduce three main novelties: (i) a point cloud soft voting scheme with curvature information that reduces the influence of outliers and noise, (ii) a selective sampling of the orientation space that favorsorientations known a priori, and (iii) a deep-learning based classifier to filter out objects with non-cylindrical appearance in the 2D images, thus further improving robustness to outliers.A set of experiments with synthetically generated data are used to assess the robustness of our fitting method to different levels of outliers and noise. The results demonstrate that incorporating the principal curvature direction within the orientation voting process allows for large improvements on cylinders parameters estimation. Furthermore, we demonstrate that combining the 2D deep-learning cylinder classifier with the 3D orientation voting scheme allows for large speed-up and accuracy improvements on cylinder identification. The qualitative and quantitative results with real data acquired from a consumer RGB-D camera, confirm the advantages of the proposed framework.

Parameter optimization criteria guided 3D point cloud classification

3D point cloud classification is one of the basic topics in multimedia analysis and understanding. By the construction of the discriminant model and efficient parameter optimization, point cloud classification can be achieved after the training. However, most parameter optimization methods do not guarantee the highest global classification accuracy with a high classification accuracy on smaller classes. In addition, geometric features of the point cloud are not sufficiently utilized. In this paper, we use local geometric shape features including the nearest neighbor tetrahedral volume, Gaussian curvature, the neighbourhood normal vector consistency and the neighbourhood minimum principal curvature direction consistency. We propose three discrete criteria for parameter optimization to design explicit functions, and we present concrete algorithms, in which Monte Carlo method and Probabilistic Neural Network method are employed to estimate these parameters respectively. Experimental results show that our criteria can be applied to the classification of the 3D point cloud of the scene, and can be used to improve the classification accuracy of small-scale point sets when different classes have great disparities in the number.

Analytical prediction of the geometry of contact ellipses and kinematics in a roller screw versus experimental results

This paper brings together the geometry of threaded contact areas and roller screw kinematics in order to shine some light on the local phenomena which dissipate power in the mechanism. The subject has received little attention in current literature, often because only the global kinematics is studied. Moreover, principal directions of curvature are usually assumed from the start. In this paper, differential geometry is used to calculate the result, which proves to be different from published research. Next, the classic Hertzian theory is adapted for slightly conforming contacts, which can be encountered in common roller screw designs. The generalized equations are used to deduce the shape, size and orientation of the contact ellipses for both the roller-screw and the roller-nut contacts. The two appear to be very different in terms of local kinematics. A new stationary model is introduced, which can calculate the sliding velocity field at any point within the contact area. The model has only one degree of freedom in the form of a slip ratio, which depends on lubricant properties and dynamics. An experimental setup was designed to measure this ratio and thus allow comparison with analytical models available in the literature.

Structural Intensity Analysis on Irregular Shells

A method is presented to assess the transmission path of vibration energy and to localize sources or sinks on shells with arbitrary shape, constant thickness, and isotropic material properties. The derived equations of the structural intensity (SI) are based on the Kirchhoff–Love postulates and are formulated in terms of displacements, Lamé parameters, principal curvatures, and their partial derivatives with respect to the principal curvilinear coordinates (PCC). To test the accuracy of the method, two numerical models of thin shells with nonuniform curvatures were developed. The coordinates, displacements, and principal curvature directions (PCDs) at the shell's outer surface were used to estimate the SI vector fields and the energy density at the shell's middle surface. The power estimated from the surface integral of the divergence of the SI over the source was compared to the actual power injected in the shell. The absolute error in both models did not exceed 1%, showing that, in theory, the method is able to handle the high-order spatial derivatives of the displacement and geometry data. The qualitative effect of varying the internal damping in the models on the energy transmission was also investigated.

FlowRep

We present FlowRep , an algorithm for extracting descriptive compact 3D curve networks from meshes of free-form man-made shapes. We infer the desired compact curve network from complex 3D geometries by using a series of insights derived from perception, computer graphics, and design literature. These sources suggest that visually descriptive networks are cycle-descriptive , i.e their cycles unambiguously describe the geometry of the surface patches they surround. They also indicate that such networks are designed to be projectable , or easy to envision when observed from a static general viewpoint; in other words, 2D projections of the network should be strongly indicative of its 3D geometry. Research suggests that both properties are best achieved by using networks dominated by flowlines , surface curves aligned with principal curvature directions across anisotropic regions and strategically extended across sharp-features and isotropic areas. Our algorithm leverages these observation in the construction of a compact descriptive curve network. Starting with a curvature aligned quad dominant mesh we first extract sequences of mesh edges that form long, well-shaped and reliable flowlines by leveraging directional similarity between nearby meaningful flowline directions We then use a compact subset of the extracted flowlines and the model's sharp-feature, or trim, curves to form a sparse, projectable network which describes the underlying surface. We validate our method by demonstrating a range of networks computed from diverse inputs, using them for surface reconstruction, and showing extensive comparisons with prior work and artist generated networks.

Similarity maps and field-guided T-splines

A variety of techniques were proposed to model smooth surfaces based on tensor product splines (e.g. subdivision surfaces, free-form splines, T-splines). Conversion of an input surface into such a representation is commonly achieved by constructing a global seamless parametrization, possibly aligned to a guiding cross-field (e.g. of principal curvature directions), and using this parametrization as domain to construct the spline-based surface. One major fundamental difficulty in designing robust algorithms for this task is the fact that for common types, e.g. subdivision surfaces (requiring a conforming domain mesh) or T-spline surfaces (requiring a globally consistent knot interval assignment) reliably obtaining a suitable parametrization that has the same topological structure as the guiding field poses a major challenge. Even worse, not all fields do admit suitable parametrizations, and no concise conditions are known as to which fields do. We present a class of surface constructions (T-splines with halfedge knots ) and a class of parametrizations ( seamless similarity maps ) that are, in a sense, a perfect match for the task: for any given guiding field structure, a compatible parametrization of this kind exists and a smooth piecewise rational surface with exactly the same structure as the input field can be constructed from it. As a byproduct, this enables full control over extraordinary points. The construction is backward compatible with classical NURBS. We present efficient algorithms for building discrete conformal similarity maps and associated T-meshes and T-spline surfaces.

On surfaces that are intrinsically surfaces of revolution

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.

The generation of axially aligned seams on triangulated pipe faces

This paper presents a method for creating seams on pipe faces which are effective for reducing the distortion of the face flattening and thus promote the generation of high quality meshes on the faces. The method is applicable to groups of connected B-rep triangulated faces whose underlying surfaces resemble Generalised Cylinders. Isoparametric curves of Generalised Cylinder parametrisations are approximated by establishing singularity-free cross-fields starting with estimated principal curvature directions from which the seams are derived.

Towards an automatic isogeometric analysis suitable trivariate models generation—Application to geometric parametric analysis

This paper presents an effective method to automatically construct trivariate tensor-product spline models of complicated geometry and arbitrary topology. Our method takes as input a solid model defined by its triangulated boundary surface. Using cuboid decomposition, an initial polycube approximating the input boundary mesh is built. This polycube serves as the parametric domain of the tensor-product spline representation required for isogeometric analysis. The polycube’s nodes and arcs decompose the input model’s boundary into quadrilateral patches, and these patches form hexahedral domains. Using aligned global parameterization, the nodes are re-positioned and the arcs are re-routed across the surface in a way to achieve low overall patch distortion, and alignment to principal curvature directions and sharp features. The optimization process is based on one of the main contributions of this paper: a novel way to design cross fields with topological (i.e., imposed singularities) and geometrical (i.e., imposed directions) constraints by solving only sparse linear systems. Based on the optimized polycube and parameterization, compatible B-spline boundary surfaces are reconstructed. Finally, the interior volumetric parameterization is computed using Coon’s interpolation. In the context of parametric studies based on geometrical parameters, this method can be used to compute the morphing required for reduced order modeling. For different parametric instances with the same topology but different geometries, this method allows to have the same representation: i.e., meshes (or parameterizations) with the same topology. The efficiency and the robustness of the proposed approach are illustrated by several examples.

Scale‐Invariant Directional Alignment of Surface Parametrizations

AbstractVarious applications of global surface parametrization benefit from the alignment of parametrization isolines with principal curvature directions. This is particularly true for recent parametrization‐based meshing approaches, where this directly translates into a shape‐aware edge flow, better approximation quality, and reduced meshing artifacts. Existing methods to influence a parametrization based on principal curvature directions suffer from scale‐dependence, which implies the necessity of parameter variation, or try to capture complex directional shape features using simple 1D curves. Especially for non‐sharp features, such as chamfers, fillets, blends, and even more for organic variants thereof, these abstractions can be unfit. We present a novel approach which respects and exploits the 2D nature of such directional feature regions, detects them based on coherence and homogeneity properties, and controls the parametrization process accordingly. This approach enables us to provide an intuitive, scale‐invariant control parameter to the user. It also allows us to consider non‐local aspects like the topology of a feature, enabling further improvements. We demonstrate that, compared to previous approaches, global parametrizations of higher quality can be generated without user intervention.

Multidirectional and Topography-based Dynamic-scale Varifold Representations with Application to Matching Developing Cortical Surfaces

The human cerebral cortex is marked by great complexity as well as substantial dynamic changes during early postnatal development. To obtain a fairly comprehensive picture of its age-induced and/or disorder-related cortical changes, one needs to match cortical surfaces to one another, while maximizing their anatomical alignment. Methods that geodesically shoot surfaces into one another as currents (a distribution of oriented normals) and varifolds (a distribution of non-oriented normals) provide an elegant Riemannian framework for generic surface matching and reliable statistical analysis. However, both conventional current and varifold matching methods have two key limitations. First, they only use the normals of the surface to measure its geometry and guide the warping process, which overlooks the importance of the orientations of the inherently convoluted cortical sulcal and gyral folds. Second, the ‘conversion’ of a surface into a current or a varifold operates at a fixed scale under which geometric surface details will be neglected, which ignores the dynamic scales of cortical foldings. To overcome these limitations and improve varifold-based cortical surface registration, we propose two different strategies. The first strategy decomposes each cortical surface into its normal and tangent varifold representations, by integrating principal curvature direction field into the varifold matching framework, thus providing rich information of the orientation of cortical folding and better characterization of the complex cortical geometry. The second strategy explores the informative cortical geometric features to perform a dynamic-scale measurement of the cortical surface that depends on the local surface topography (e.g., principal curvature), thereby we introduce the concept of a topography-based dynamic-scale varifold. We tested the proposed varifold variants for registering 12 pairs of dynamically developing cortical surfaces from 0 to 6 months of age. Both variants improved the matching accuracy in terms of closeness to the target surface and the goodness of alignment with regional anatomical boundaries, when compared with three state-of-the-art methods: (1) diffeomorphic spectral matching, (2) conventional current-based surface matching, and (3) conventional varifold-based surface matching.

Three-dimensional (3D) facial recognition and prediction

This paper provides solution to the problem in identifying humans from their 3D facial characteristics. For this reason, a standard 3D facial recognition system was built and used in this research work. This work proposes two novel fusion schemes where the first one employs a confidence-aided combination approach, and the second one implements a two-level serial integration method. Recognition simulations performed on the 3DRMA and the FRGC databases show that: (1) generic face template-based rigid registration of faces is better than the non-rigid variant, (2) principal curvature directions and surface normal have better discriminative power, (3) representing faces using local patch descriptors can both reduce the feature dimensionality and improve the identification rate, and (4) confidence-assisted fusion rules and serial two-stage fusion schemes have a potential to improve the accuracy when compared to other decision-level fusion rules.

Retiling scheme: a novel approach of direct anisotropic quad-dominant remeshing

Remeshing has been an active research topic in Digital Geometry Processing. In this paper, a novel approach of direct anisotropic quad-dominant remeshing is proposed. We apply the retiling method to the particular problem of quad-dominant remeshing. Compared with other methods, this method can simply partition the surface of an original triangular mesh into connected quads with the mesh edges aligning to the principal directions. The first step in this method is to estimate and smooth the curvature tensor field of the surface at the vertices, and then the quad-dominant mesh is obtained by retiling the quad surface so that quadrilateral edges are parallel to the local principal curvature directions. In addition, to preserve the sharp feature information during remeshing processes, the feature lines can be extracted using mesh segmentation method, and the intersections between the feature lines and the orthogonal planes can be found during the process of retiling. A feature fusion process is presented to join the feature edges and feature points into the quad-dominant mesh. The experiment results show that this new remeshing method is simple and easy to implement. The resolution of the quadrilateral mesh can be controlled during the remeshing. It is applicable to arbitrary genus meshes and can generate high-quality quad-dominant mesh.

Parallel and dual surfaces of cuspidal edges

We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify relations between singularities of parallel and dual surfaces and differential geometric properties of initial cuspidal edges.