Subdivision Surface Fitting Research Articles

Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting

In this paper, we propose a stochastic geometric iterative method (S-GIM) to approximate the high-resolution 3D models by finite loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite loop subdivision surface approximate the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of S-GIM and verify it by demonstrating error curves in the experiment. In addition, compared with existing geometric iterative methods, S-GIM has a shorter running time under the same number of iteration steps.

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.

Learning-based automated segmentation of the carotid artery vessel wall in dual-sequence MRI using subdivision surface fitting.

The quantification of vessel wall morphology and plaque burden requires vessel segmentation, which is generally performed by manual delineations. The purpose of our work is to develop and evaluate a new 3D model-based approach for carotid artery wall segmentation from dual-sequence MRI. The proposed method segments the lumen and outer wall surfaces including the bifurcation region by fitting a subdivision surface constructed hierarchical-tree model to the image data. In particular, a hybrid segmentation which combines deformable model fitting with boundary classification was applied to extract the lumen surface. The 3D model ensures the correct shape and topology of the carotid artery, while the boundary classification uses combined image information of 3D TOF-MRA and 3D BB-MRI to promote accurate delineation of the lumen boundaries. The proposed algorithm was validated on 25 subjects (48 arteries) including both healthy volunteers and atherosclerotic patients with 30% to 70% carotid stenosis. For both lumen and outer wall border detection, our result shows good agreement between manually and automatically determined contours, with contour-to-contour distance less than 1 pixel as well as Dice overlap greater than 0.87 at all different carotid artery sections. The presented 3D segmentation technique has demonstrated the capability of providing vessel wall delineation for 3D carotid MRI data with high accuracy and limited user interaction. This brings benefits to large-scale patient studies for assessing the effect of pharmacological treatment of atherosclerosis by reducing image analysis time and bias between human observers.

Segmentation-based semi-regular remeshing of 3D models using curvature-adapted subdivision surface fitting

This paper proposes a novel method of semi-regular remeshing for triangulated surfaces to achieve superior triangles lead to advanced visualization of 3D model. It is based on mesh segmentation and subdivision surface fitting which uses curvature-adapted polygon patches. Our contribution lies in building a sophisticated system with three stages, i.e., curvature-aware mesh segmentation, submesh surface fitting to generate a high-quality semi-regular mesh and finally, stitching the segments using an efficient algorithm. Our method uses centroidal Voronoi tessellation and Lloyd’s relaxation to generate curvature-adapted site centers. Geodesic distances from site centers are used for labeling segments and indexing corner vertices for each segment boundary. Using information of site centers and corner vertices, feature-adapted polygonal patches are generated for each segment. These patches are then subdivided and optimized using squared distance metric to adjust position of the subdivision sampling with segment details and prevent oversampling. At last, an efficient stitching algorithm is introduced to connect regular submeshes together and build the final semi-regular mesh. We have demonstrated the results of our semi-regular remeshing algorithm on meshes with different topology and complexity and compared them with known methods. Superior triangle quality with higher aspect ratio together with acceptable distortion error is achieved according to the experimental results.

Subdivision surface fitting to a dense mesh using ridges and umbilics

Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach.

Human eyeball model reconstruction and quantitative analysis.

Determining shape of the eyeball is important to diagnose eyeball disease like myopia. In this paper, we present an automatic approach to precisely reconstruct three dimensional geometric shape of eyeball from MR Images. The model development pipeline involved image segmentation, registration, B-Spline surface fitting and subdivision surface fitting, neither of which required manual interaction. From the high resolution resultant models, geometric characteristics of the eyeball can be accurately quantified and analyzed. In addition to the eight metrics commonly used by existing studies, we proposed two novel metrics, Gaussian Curvature Analysis and Sphere Distance Deviation, to quantify the cornea shape and the whole eyeball surface respectively. The experiment results showed that the reconstructed eyeball models accurately represent the complex morphology of the eye. The ten metrics parameterize the eyeball among different subjects, which can potentially be used for eye disease diagnosis.

A ternary three-point scheme for curve designing

In this paper, a stationary ternary three-point approximating subdivision scheme is presented, which generates C 2 limiting curve and its limiting function has a support on [−3, 2]. The analysis of the scheme is shown using the Laurent polynomial method. Two examples are illustrated for comparison with the ternary three-point approximating subdivision scheme developed by Hassan and Dodgson [Ternary and three point univariate subdivision schemes, in Curve and Surface Fitting: Sant-Malo 2002, A. Cohen, J.-L. Merrien, and L.L. Schumaker, eds., Nashboro Press, Brentwood, 2003, pp. 199–208.], which show that the scheme presented in the paper is better.

Semi-sharp subdivision surface fitting based on feature lines approximation

This paper presents an algorithm for approximating arbitrary polygonal meshes with subdivision surfaces, with the objective of preserving the relevant features of the object while searching the coarsest possible control mesh. The main idea is to firstly extract the feature lines of the object, and secondly construct the subdivision surface over this network. Control points are created by approximating these lines while the connectivity is built with respect to the anisotropy of the object. Our algorithm reinforces the similarity between the subdivision surface and the original shape by affecting an integer sharpness degree to each control edge in order to accurately reproduce the different curvature radii of corresponding fillets and blends.

Design and Analysis of Optimization Methods for Subdivision Surface Fitting

We present a complete framework for computing a subdivision surface to approximate unorganized point sample data, which is a separable nonlinear least squares problem. We study the convergence and stability of three geometrically-motivated optimization schemes and reveal their intrinsic relations with standard methods for constrained nonlinear optimization. A commonly-used method in graphics, called point distance minimization, is shown to use a variant of the gradient descent step and thus has only linear convergence. The second method, called tangent distance minimization, which is well-known in computer vision, is shown to use the Gauss-Newton step, and thus demonstrates near quadratic convergence for zero residual problems but may not converge otherwise. Finally, we show that an optimization scheme called squared distance minimization, recently proposed by Pottmann et al., can be derived from the Newton method. Hence, with proper regularization, tangent distance minimization and squared distance minimization are more efficient than point distance minimization. We also investigate the effects of two step size control methods -- Levenberg-Marquardt regularization and the Armijo rule -- on the convergence stability and efficiency of the above optimization schemes.

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three-dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle-triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.

Complete 3D surface reconstruction from unstructured point cloud

In this study, a complete 3D surface reconstruction method is proposed based on the concept that the vertices of surface model can be completely matched to the unstructured point cloud. In order to generate the initial mesh model from the point cloud, the mesh subdivision of bounding box and shrink-wrapping algorithm are introduced. The control mesh model for well representing the topology of point cloud is derived from the initial mesh model by using the mesh simplification technique based on the original QEM algorithm, and the parametric surface model for approximately representing the geometry of point cloud is derived by applying the local subdivision surface fitting scheme on the control mesh model. And, to reconstruct the complete matching surface model, the insertion of isolated points on the parametric surface model and the mesh optimization are carried out. Especially, the fast 3D surface reconstruction is realized by introducing the voxel-based nearest-point search algorithm, and the simulation results reveal the availability of the proposed surface reconstruction method.

조직화되지 않은 점군으로부터의 3차원 완전 형상 복원

In this study. a complete 3D surface reconstruction method is proposed based on the concept that the vertices of surface model can be completely matched to the unstructured point cloud. In order to generate the initial mesh model from the point cloud, the mesh subdivision of bounding box and shrink-wrapping algorithm are introduced, The control mesh model for well representing the topology of point cloud is derived from the initial mesh model by using the mesh simplification technique based on the original QEM algorithm, and the parametric surface model for approximately representing the geometry of point cloud is derived by applying the local subdivision surface fitting scheme on the control mesh model. And, to reconstruct the complete matching surface model, the insertion of isolated points on the parametric surface model and the mesh optimization are carried out. Especially, the fast 3D surface reconstruction is realized by introducing the voxel-based nearest-point search algorithm. and the simulation results reveal the availability of the proposed surface reconstruction method.

A new CAD mesh segmentation method, based on curvature tensor analysis

This paper presents a new and efficient algorithm for the decomposition of 3D arbitrary triangle meshes and particularly optimized triangulated CAD meshes. The algorithm is based on the curvature tensor field analysis and presents two distinct complementary steps: a region based segmentation, which is an improvement of that presented by Lavoue et al. [Lavoue G, Dupont F, Baskurt A. Constant curvature region decomposition of 3D-meshes by a mixed approach vertex-triangle, J WSCG 2004;12(2):245–52] and which decomposes the object into near constant curvature patches, and a boundary rectification based on curvature tensor directions, which corrects boundaries by suppressing their artefacts or discontinuities. Experiments conducted on various models including both CAD and natural objects, show satisfactory results. Resulting segmented patches, by virtue of their properties (homogeneous curvature, clean boundaries) are particularly adapted to computer graphics tasks like parametric or subdivision surface fitting in an adaptive compression objective.

Subdivision surfaces for CAD—an overview

Subdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surface modelling are addressed. Several key topics, such as scheme construction, property analysis, parametric evaluation and subdivision surface fitting, are discussed. Some other important topics are also summarized for potential future research and development. Several examples are provided to highlight the modelling capability of subdivision surfaces for CAD applications.

A direct approach for subdivision surface fitting from a dense triangle mesh

This article presents a new and direct approach for fitting a subdivision surface from an irregular and dense triangle mesh of arbitrary topological type. All feature edges and feature vertices of the original mesh model are first identified. A topology- and feature-preserving mesh simplification algorithm is developed to further simplify the dense triangle mesh into a coarse mesh. A subdivision surface with exactly the same topological structure and sharp features as that of the simplified mesh is finally fitted from a subset of vertices of the original dense mesh. During the fitting process, both the position masks and subdivision rules are used for setting up the fitting equation. Some examples are provided to demonstrate the proposed approach.

曲線集合からの細分割曲面生成手法

In CAID (Computer Aided Industrial Design) systems, a free form surface design generally starts with defining characteristic curves in 3D space. Based on these characteristic curves, basic curved surfaces are then generated so as to interpolate them. It is required that these curves be connected in order to compute interpolating surface patches. However, it is desirable to give the designers more freedom to allow them to define the characteristic curves without considering their connectivity. In this research, we propose a method called "Curve Wrapping" to generate the connectivity from a set of unconnected characteristic curves by using "a graph searching mesh" technique. And using this connectivity, the curves are adapted to form a graph so that subdivision surfaces are generated using a subdivision surface fitting method and the "combined subdivision" method. A prototype system was developed to evaluate the approach using some examples.

Direct Reconstruction of a Displaced Subdivision Surface from Unorganized Points

In this paper we describe the generation of a displaced subdivision surface directly from a set of unorganized points. The displaced subdivision surface is an efficient mesh representation that defines a detailed mesh with a displacement map over a smooth domain surface and has many benefits including compression, rendering, and animation, which overcome limitations of an irregular mesh produced by an ordinary mesh reconstruction scheme. Unlike previous displaced subdivision surface reconstruction methods, our method does not rely on a highly detailed reconstructed mesh. Instead, we efficiently create a coarse base mesh, which is used to sample displacements directly from unorganized points, and this results in a simple process and fast calculation. We suggest a shrink-wrapping-like shape approximation and a point-based mesh simplification method that uses the distance between a set of points and a mesh as an error metric to generate a domain surface that optimally approximates the given points. We avoid time-consuming energy minimization by employing a local subdivision surface fitting scheme. Finally, we show several reconstruction results that demonstrate the usability of our algorithm.