Loop Subdivision Research Articles

Construction of Almost Everywhere C2 Continuous Shape Interpolating Spaces Via Progressive Interpolation of Loop Subdivision

Construction of Almost Everywhere C2 Continuous Shape Interpolating Spaces Via Progressive Interpolation of Loop Subdivision

Neural subdivision

This paper introduces Neural Subdivision , a novel framework for data-driven coarse-to-fine geometry modeling. During inference, our method takes a coarse triangle mesh as input and recursively subdivides it to a finer geometry by applying the fixed topological updates of Loop Subdivision, but predicting vertex positions using a neural network conditioned on the local geometry of a patch. This approach enables us to learn complex non-linear subdivision schemes, beyond simple linear averaging used in classical techniques. One of our key contributions is a novel self-supervised training setup that only requires a set of high-resolution meshes for learning network weights. For any training shape, we stochastically generate diverse low-resolution discretizations of coarse counterparts, while maintaining a bijective mapping that prescribes the exact target position of every new vertex during the subdivision process. This leads to a very efficient and accurate loss function for conditional mesh generation, and enables us to train a method that generalizes across discretizations and favors preserving the manifold structure of the output. During training we optimize for the same set of network weights across all local mesh patches, thus providing an architecture that is not constrained to a specific input mesh, fixed genus, or category. Our network encodes patch geometry in a local frame in a rotation- and translation-invariant manner. Jointly, these design choices enable our method to generalize well, and we demonstrate that even when trained on a single high-resolution mesh our method generates reasonable subdivisions for novel shapes.

Research on Improved Triangular Mesh Surface Subdivision Algorithm

Subdivision surfaces have been widely used in computer graphics. They are also commonly used to subdivide surfaces in 3D geological modeling. The classic triangular mesh subdivision algorithms are Loop subdivision algorithm, Sqrt3 subdivision algorithm and Butterfly subdivision algorithm. Based on the realization of the three algorithms, the paper displays that the improved Butterfly subdivision algorithm, compared with the Sqrt3 subdivision algorithm and the Loop subdivision algorithm, can alleviate the contradiction between data volume and smoothness. It can also achieve a better fittness, and can deal with the smooth subdivision of the model well in 3D geological modeling and other applications.

Subdivision Surfaces — Boundary Element Accelerated by Fast Multipole for the Structural Acoustic Problem

A novel boundary element method based on subdivision surfaces is applied to simulate wave scattering problems governed by the Helmholtz equation. The Loop subdivision scheme widely used in the CAD (computer aided design) field is applied to represent the geometric model and approximate physical field variables. The novelty of this work is that it is the first time to apply the fast multipole method for subdivision surface boundary element method to accelerate the solution of the system of equations. It is demonstrated that the subdivision surfaces boundary element method based on the Loop subdivision scheme performs well in terms of accuracy, and automatically provides multiresolution subdivision hierarchy models to meet the requirements of different precision. This approach is then applied to several real-world applications to illustrate the potential for integrated engineering analysis.

Subdivision‐Specialized Linear Algebra Kernels for Static and Dynamic Mesh Connectivity on the GPU

AbstractSubdivision surfaces have become an invaluable asset in production environments. While progress over the last years has allowed the use of graphics hardware to meet performance demands during animation and rendering, high‐performance is limited to immutable mesh connectivity scenarios. Motivated by recent progress in mesh data structures, we show how the complete Catmull‐Clark subdivision scheme can be abstracted in the language of linear algebra. While this high‐level formulation allows for a fully parallel implementation with significant performance gains, the underlying algebraic operations require further specialization for modern parallel hardware. Integrating domain knowledge about the mesh matrix data structure, we replace costly general linear algebra operations like matrix‐matrix multiplication by specialized kernels. By further considering innate properties of Catmull‐Clark subdivision, like the quad‐only structure after refinement, we achieve an additional order of magnitude in performance and significantly reduce memory footprints. Our approach can be adapted seamlessly for different use cases, such as regular subdivision of dynamic meshes, fast evaluation for immutable topology and feature‐adaptive subdivision for efficient rendering of animated models. In this way, patchwork solutions are avoided in favor of a streamlined solution with consistent performance gains throughout the production pipeline. The versatility of the sparse matrix linear algebra abstraction underlying our work is further demonstrated by extension to other schemes such as and Loop subdivision.

Computing Intersection for Loop Subdivision Surface

Computing Intersection for Loop Subdivision Surface

Acoustic topology optimization of sound absorbing materials directly from subdivision surfaces with isogeometric boundary element methods

This paper presents an acoustic topology optimization approach using isogeometric boundary element methods based on subdivision surfaces to optimize the distribution of sound adsorption materials adhering to structural surfaces. The geometries are constructed from triangular control meshes through Loop subdivision scheme, and the associated Box-spline functions that generate limit smooth subdivision surfaces are employed to discretize the acoustic boundary integral equations. The effect of sound-absorbing materials on the acoustic response is characterized by acoustic impedance boundary conditions. The optimization problem is formulated in the framework of Solid Isotropic Material with Penalization methods and the sound absorption coefficients on elements are selected as design variables. The potential of the proposed topology optimization approach for engineering prototyping is illustrated by numerical examples.

3D iris model and reader for iris identification

SummaryIn the presence of eyelids and eyelashes movement, pupil dilation, poor lighting, blur due to movement during iris image acquisition, and factors that collectively cause distortion in the iris image, 2D image‐based iris identification techniques become limited and might lead to iris misclassification due to the dependency on the image appearance (texture) on the possibly distorted image. To alleviate this problem, we introduce a new 3D iris model and reader based upon which iris identification is performed. Using a set of at least two 2D images taken from different views, a small set of reliable and corresponding salient fiducial points (corner points taken from crypts, corona, and serpentine rings of iris pattern) in the two images are extracted, from which a set of 3D iris salient points are obtained using triangulation. Corresponding salient points in the 2D images are found using the Random Sampling Consensus (RANSAC) algorithm, which is robust in identifying the inlier points that correspond to each other in different views of the iris. From this small and reliable salient point set, a denser (high‐resolution) set of extra salient feature points is constructed at minimum cost using a loop subdivision method that yields corresponding extra salient points in the two images from which a high‐resolution 3D iris model is obtained. The points associated with the 3D high‐resolution model are tessellated to form a high‐resolution triangular mesh with the appearance of a triangular patch in the image imported onto the iris personalized 3D model rendering it a 3D iris surface with appearance. This 3D model construction method allows for a 3D classification of a test iris to one of many possible irises stored in the database. The classification is based on a geometric 3D point cloud error. For cross‐validation, the geometric‐based classification can be augmented by considering a mean squared error (MSE) based on the appearance of corresponding points on the test and base models to further disambiguate between irises with close geometric error values.

Isogeometric analysis for surface PDEs with extended Loop subdivision

We investigate the isogeometric analysis for surface PDEs based on the extended Loop subdivision approach. The basis functions consisting of quartic box-splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly, and construct the solution space for dependent variables as well, which is consistent with the concept of isogeometric analysis. The subdivision process is equivalent to the h-refinement of NURBS-based isogeometric analysis. The performance of the proposed method is evaluated by solving various surface PDEs, such as surface Laplace-Beltrami harmonic/biharmonic/triharmonic equations, which are defined on the limit surfaces of extended Loop subdivision for different initial control meshes. Numerical experiments show that the proposed method has desirable performance in terms of the accuracy, convergence and computational cost for solving the above surface PDEs defined on both open and closed surfaces. The proposed approach is proved to be second-order accuracy in the sense of L2-norm with theoretical and/or numerical results, which is also outperformed over the standard linear finite element by several numerical comparisons.

A merged tuning of binary and ternary Loop's subdivision

In the vicinity of extraordinary vertices, the action of a primal, symmetric subdivision scheme for the construction of arbitrary topology surfaces can be represented by structured matrices that form a hybrid matrix algebra related to the block-circulant algebra diagonalized by a block-FFT. Exploiting the block-diagonalization of such matrices and having in mind the target of a specific class of schemes, we can easily take into consideration the constraints to be satisfied by their eigenvalues and provide an efficient computational approach for determining the ranges of variability of the weights defining the extraordinary rules. Application examples of this computational strategy are shown to find the exact bounds of extraordinary rule weights for improved variants of two existing subdivision schemes.

Adaptive fitting algorithm of progressive interpolation for Loop subdivision surface

Subdivision surface and data fitting have been applied in data compression and data fusion a lot recently. Moreover, subdivision schemes have been successfully combined into multi-resolution analysis and wavelet analysis. This makes subdivision surfaces attract more and more attentions in the field of geometry compression. Progressive interpolation subdivision surfaces generated by approximating schemes were presented recently. When the number of original vertices becomes huge, the convergence speed becomes slow and computation complexity becomes huge. In order to solve these problems, an adaptive progressive interpolation subdivision scheme is presented in this article. The vertices of control mesh are classified into two classes: active vertices and fixed ones. When precision is given, the two classes of vertices are changed dynamically according to the result of each iteration. Only the active vertices are adjusted, thus the class of active vertices keeps running down while the fixed ones keep rising, which saves computation greatly. Furthermore, weights are assigned to these vertices to accelerate convergence speed. Theoretical analysis and numerical examples are also given to illustrate the correctness and effectiveness of the method.

SiSSR: Simultaneous subdivision surface registration for the quantification of cardiac function from computed tomography in canines.

Recent improvements in cardiac computed tomography (CCT) allow for whole-heart functional studies to be acquired at low radiation dose (<2mSv) and high-temporal resolution (<100ms) in a single heart beat. Although the extraction of regional functional information from these images is of great clinical interest, there is a paucity of research into the quantification of regional function from CCT, contrasting with the large body of work in echocardiography and cardiac MR. Here we present the Simultaneous Subdivision Surface Registration (SiSSR) method: a fast, semi-automated image analysis pipeline for quantifying regional function from contrast-enhanced CCT. For each of thirteen adult male canines, we construct an anatomical reference mesh representing the left ventricular (LV) endocardium, obviating the need for a template mesh to be manually sculpted and initialized. We treat this generated mesh as a Loop subdivision surface, and adapt a technique previously described in the context of 3-D echocardiography to register these surfaces to the endocardium efficiently across all cardiac frames simultaneously. Although previous work performs the registration at a single resolution, we observe that subdivision surfaces naturally suggest a multiresolution approach, leading to faster convergence and avoiding local minima. We additionally make two notable changes to the cost function of the optimization, explicitly encouraging plausible biological motion and high mesh quality. Finally, we calculate an accepted functional metric for CCT from the registered surfaces, and compare our results to an alternate state-of-the-art CCT method.

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

SummaryWe introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C1 continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.

Introduction and Applications of 3D Printing in Radiological Technology

A 3D printing emerges as a common procedure in clinical radiology practice after installation of a module that converts the digital imaging and communications in medicine (DICOM) dataset into stereolithography (STL) data on medical workstations. However, they did not conventionally provide the appropriate filtering, sculpting, hollowing out, and Boolean (subtraction) operations on STL data. These functions are indispensable to handle the STL data to fabricate the smooth, low-cost, and sophisticated models. Here are some tips for handling the 3D data with three software packages through making a sample lumbar spine model. Because they are all free- and open-source software with the exception of Boolean operations, they could make it easy for anyone to fabricate their 3D model imaged by CT or MRI. We tested the loop subdivision surface algorithms for the smoothing, the sculpting function for removing a sharp prick, and the hollowing function to save the cost. Computer-aided design (CAD) is also used to fabricate the devices in medical research. We designed and developed a cap attached to a glass dosimeter to show the effectiveness of CAD in radiological research. Lastly, we discuss the important matters for 3D printing and examples of the clinical applications.

Variational reconstruction using subdivision surfaces with continuous sharpness control

We present a variational method for subdivision surface reconstruction from a noisy dense mesh. A new set of subdivision rules with continuous sharpness control is introduced into Loop subdivision for better modeling subdivision surface features such as semi-sharp creases, creases, and corners. The key idea is to assign a sharpness value to each edge of the control mesh to continuously control the surface features. Based on the new subdivision rules, a variational model with L1 norm is formulated to find the control mesh and the corresponding sharpness values of the subdivision surface that best fits the input mesh. An iterative solver based on the augmented Lagrangian method and particle swarm optimization is used to solve the resulting non-linear, non-differentiable optimization problem. Our experimental results show that our method can handle meshes well with sharp/semi-sharp features and noise.

Isogeometric finite element approximation of minimal surfaces based on extended loop subdivision

In this paper, we investigate the formulation of isogeometric analysis for minimal surface models on planar bounded domains by extended Loop surface subdivision approach. The exactness of the physical domain of interest is fixed on the coarsest level of the triangular discretization with any topological structure, which is thought of as the initial control mesh of Loop subdivision. By performing extended Loop subdivision, the control mesh can be repeatedly refined, and the geometry is described as an infinite set of quartic box-spline while maintaining its original exactness. The limit function representation of extended Loop subdivision forms our finite element space, which possesses C1 smoothness and the flexibility of mesh topology. We establish its inverse inequalities which resemble the ones of general finite element spaces. We develop the approximation estimate with the aid of H1 convergence property of the corresponding linear models. It enables us to overcome the difficulty of proving the boundedness of the gradient of finite element solutions appearing in the coefficient of minimal surface models. Numerical examples are given with the comparison to the classical linear finite element method which is consistent with our theoretical results.

Smooth interpolation of key frames in a Riemannian shell space

Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation.

Truncated Hierarchical Loop Subdivision Surfaces and application in isogeometric analysis

Subdivision Surface provides an efficient way to represent free-form surfaces with arbitrary topology. Loop subdivision is a subdivision scheme for triangular meshes, which is C2 continuous except at a finite number of extraordinary vertices with G1 continuous. In this paper we propose the Truncated Hierarchical Loop Subdivision Surface (THLSS), which generalizes truncated hierarchical B-splines to arbitrary topological triangular meshes. THLSS basis functions are linearly independent, form a partition of unity, and are locally refinable. THLSS also preserves the geometry during adaptive h-refinement and thus inherits the surface continuity of Loop subdivision surface. Adaptive isogeometric analysis is performed with the THLSS basis functions on several complex models with extraordinary vertices to show the potential application of THLSS.

Influence of surface viscosity on droplets in shear flow

The behaviour of a single droplet in an immiscible external fluid, submitted to shear flow is investigated using numerical simulations. The surface of the droplet is modelled by a Boussinesq–Scriven constitutive law involving the interfacial viscosities and a constant surface tension. A numerical method using Loop subdivision surfaces to represent droplet interface is introduced. This method couples boundary element method for fluid flows and finite element method to take into account the stresses due to the surface dilational and shear viscosities and surface tension. Validation of the numerical scheme with respect to previous analytic and computational work is provided, with particular attention to the viscosity contrast and the shear and dilational viscosities characterized both by a Boussinesq number$B_{q}$. Then, influence of equal surface viscosities on steady-state characteristics of a droplet in shear flow are studied, considering both small and large deformations and for a large range of bulk viscosity contrast. We find that small deformation analysis is surprisingly predictive at moderate and high surface viscosities. Equal surface viscosities decrease the Taylor deformation parameter and tank-treading angle and also strongly modify the dynamics of the droplet: when the Boussinesq number (surface viscosity) is large relative to the capillary number (surface tension), the droplet displays damped oscillations prior to steady-state tank-treading, reminiscent from the behaviour at large viscosity contrast. In the limit of infinite capillary number$Ca$, such oscillations are permanent. The influence of surface viscosities on breakup is also investigated, and results show that the critical capillary number is increased. A diagram$(B_{q};Ca)$of breakup is established with the same inner and outer bulk viscosities. Additionally, the separate roles of shear and dilational surface viscosity are also elucidated, extending results from small deformation analysis. Indeed, shear (dilational) surface viscosity increases (decreases) the stability of drops to breakup under shear flow. The steady-state deformation (Taylor parameter) varies nonlinearly with each Boussinesq number or a linear combination of both Boussinesq numbers. Finally, the study shows that for certain combinations of shear and dilational viscosities, drop deformation for a given capillary number is the same as in the case of a clean surface while the inclination angle varies.

Feature-aligned Surface Parameterization Using Secondary Laplace Operator and Loop Subdivision

Surface parameterization is of great importance for many applications such as quadrangulation, texture mapping and surface fitting. An important issue for surface parameterization is how to align parametric lines with feature directions. To address this issue, in this paper we first utilize Loop subdivision basis functions and isogeometric analysis (IGA) to calculate eigenfunctions of the secondary Laplace operator (SLO) on triangle meshes. Eigenfunctions are then used for centroidal Voronoi tessellation (CVT) based surface segmentation, and boundaries of the segmented regions are extracted as feature lines which contain concave creases and convex ridges. Along each feature line, adjacent triangles are defined as guidance triangles to parameterize the surface using a constrained cross field method, where feature lines are preserved and aligned to parametric lines. Several examples are presented in the end to verify the robustness of our algorithm.