Variational Shape Approximation Research Articles

Converting pixel‐type topology optimization results to MMC‐representation based on sparse optimization and its applications

AbstractHow to realize the switching between various topology optimization approaches such as SIMP and moving morphable component (MMC) method, is a crucial challenge in the field of structural design. In this article, a robust conversion framework is proposed to convert a pixel‐type topology optimization result to MMC representation. Based on the sparse optimization approach, the framework enables the determination of the minimum number of components with a specified shape error. This method provides an efficient bridge for these two types of geometric descriptions, and promotes the free switching between the topology optimization frameworks with pixel‐based and MMC‐based design domains. The proposed procedure contains a pre‐processing of resolution improvement, symmetry axis extraction with sparse optimization, and variational shape approximation. Two practical applications are demonstrated using the proposed framework. First, it can be applied to the intermediate results of SIMP, to achieve faster optimization convergence. Furthermore, a stress‐based shape optimization approach can be applied to the obtained MMCs, and novel progressive continuity constraints are also introduced to maintain boundary continuity. Several examples demonstrate advantages of the proposed framework.

Part-to-Surface Mesh Segmentation for Mechanical Models Based on Multi-Stage Clustering

In this paper, we propose a part-to-surface mesh segmentation method for engineering models characterized by a multi-stage clustering of surface patches generated by the Variational Shape Approximation (VSA) method. Initially, border types between topologically adjacent surface patches are estimated and surface patches bonded by certain border types are grouped into clusters. Secondly, clusters of surface patches are iteratively aggregated into volumetric part-level sub-meshes according to the convexity variations of intermediate sub-meshes. At last, surface patches within their respective part-level sub-meshes are merged into surface-level sub-meshes that are well-fitted by basic primitive shapes based on an improved surface-aware similarity metric. Experimental results demonstrate that the proposed method can achieve better results on mechanical models both quantitatively and perceptually, especially for objects with non-zero genus topology and rich in tiny features.

A Novel OpenMVS-Based Texture Reconstruction Method Based on the Fully Automatic Plane Segmentation for 3D Mesh Models

The Markov Random Field (MRF) energy function, constructed by existing OpenMVS-based 3D texture reconstruction algorithms, considers only the image label of the adjacent triangle face for the smoothness term and ignores the planar-structure information of the model. As a result, the generated texture charts results have too many fragments, leading to a serious local miscut and color discontinuity between texture charts. This paper fully utilizes the planar structure information of the mesh model and the visual information of the 3D triangle face on the image and proposes an improved, faster, and high-quality texture chart generation method based on the texture chart generation algorithm of the OpenMVS. This methodology of the proposed approach is as follows: (1) The visual quality on different visual images of each triangle face is scored using the visual information of the triangle face on each image in the mesh model. (2) A fully automatic Variational Shape Approximation (VSA) plane segmentation algorithm is used to segment the blocked 3D mesh models. The proposed fully automatic VSA-based plane segmentation algorithm is suitable for multi-threaded parallel processing, which solves the VSA framework needed to manually set the number of planes and the low computational efficiency in a large scene model. (3) The visual quality of the triangle face on different visual images is used as the data term, and the image label of adjective triangle and result of plane segmentation are utilized as the smoothness term to construct the MRF energy function. (4) An image label is assigned to each triangle by the minimizing energy function. A texture chart is generated by clustering the topologically-adjacent triangle faces with the same image label, and the jagged boundaries of the texture chart are smoothed. Three sets of data of different types were used for quantitative and qualitative evaluation. Compared with the original OpenMVS texture chart generation method, the experiments show that the proposed approach significantly reduces the number of texture charts, significantly improves miscuts and color differences between texture charts, and highly boosts the efficiency of VSA plane segmentation algorithm and OpenMVS texture reconstruction.

Variational shape approximation of point set surfaces

In this work, we present a translation of the complete pipeline for variational shape approximation (VSA) to the setting of point sets. First, we describe an explicit example for the theoretically known non-convergence of the currently available VSA approaches. The example motivates us to introduce an alternate version of VSA based on a switch operation for which we prove convergence. Second, we discuss how two operations—split and merge—can be included in a fully automatic pipeline that is in turn independent of the placement and number of initial seeds. Third and finally, we present two approaches how to obtain a simplified mesh from the output of the VSA procedure. This simplification is either based on simple plane intersection or based on a variational optimization problem. Several qualitative and quantitative results prove the relevance of our approach.

Developable surface reconstruction from noisy data with $\boldsymbol L_{\rm 0}$-norm minimization

A novel method is proposed for developable surface reconstruction from noisy model. An optimization approach is employed to smooth normal vector field of given model via $L_0$-norm minimization. This optimization method has the capability of preserving geometric information concerning developability. Based on the optimized normal vector field, a variational shape approximation procedure is utilized to divide the target model into a sequence of planar segments. After each segment is assigned with an initial B-spline surface, a patch merging procedure is used to obtain a B-spline ruled surface which is nearly developable and covers the whole target model. Finally, a developable surface fitting process is used to improve the quality of fitting surface both in approximation error and developability. The presented method shows pleasing performance in recovering developable information in noisy model and consequently is capable of reconstructing high quality developable surfaces. Some experiments are included to demonstrate the performance of the presented method.

Feature curve extraction from point clouds via developable strip intersection

Abstract In this paper, we study the problem of computing smooth feature curves from CAD type point clouds models. The proposed method reconstructs feature curves from the intersections of developable strip pairs which approximate the regions along both sides of the features. The generation of developable surfaces is based on a linear approximation of the given point cloud through a variational shape approximation approach. A line segment sequencing algorithm is proposed for collecting feature line segments into different feature sequences as well as sequential groups of data points. A developable surface approximation procedure is employed to refine incident approximation planes of data points into developable strips. Some experimental results are included to demonstrate the performance of the proposed method.

Shape segmentation by hierarchical splat clustering

This paper presents a novel hierarchical shape segmentation method based on splats for 3D shapes. The major contribution is to propose a new similarity metric based on splats, which combines patch-aware similarity and part-aware similarity adaptively. An optimized L2,1 metric for VSA (variational shape approximation) method is used to get splats first, and such adaptive similarity metric is used to generate a hierarchy of components automatically through adaptive cluster. As a result, a binary tree is used to represent the hierarchy, in which low level segments are patch-aware regions while high level segments are part-aware components. Therefore, the combination and decomposition relations are clear between segments. Our method is designed to handle arbitrary models, such as CAD model, scanned object, organic shape, large-scale mesh and noisy model. A large number of experiments demonstrate the efficiency of our algorithm.

Mahalanobis centroidal Voronoi tessellations

Anisotropic centroidal Voronoi tessellations (CVT) are a useful tool for segmenting surfaces in geometric modeling. We present a new approach to anisotropic CVT, where the local distance metric is learned from the embedding of the shape. Concretely, we define the distance metric implicitly as the minimizer of the CVT energy. Constraining the metric tensors to have unit determinant leads to the optimal distance metric being the inverse covariance matrix of the data (i.e. Mahalanobis distances). We explicitly cover the case of degenerate covariance and provide an algorithm to minimize the CVT energy. The resulting technique has applications in shape approximation, particularly in the case of noisy data, where normals are unreliable. We also put our approach in the context of other techniques. Among others, we show that Variational Shape Approximation can be interpreted in the same framework by constraining the metric tensor based on another norm.

Geometric Features Sensitive Mesh Segmentation Orient to Patch-Based Fitting

Segmenting meshes into natural regions is useful for patch-based mesh fitting. In this paper, we present a novel algorithm for segmenting meshes into characteristic patches and provide a corresponding geometric proxy for each patch. We extend the powerful optimization technique of variational shape approximation by allowing for several different primitives to represent the geometric proxy of a surface region. Our method has the particular advantage of robustness. As the principal curvatures of the surfaces become more equal, the returned results are become closer to the surfaces of geometry primitives, i.e. planes, cylinders, or cones, or rotating surface which are the most common patch types in the reverse engineering. The expected result that we recover surface structures more robustly and thus get better approximations, is validated and demonstrated on a number of examples.

Exoskeleton: Curve network abstraction for 3D shapes

In this paper, we introduce the concept of an exoskeleton as a new abstraction of arbitrary shapes that succinctly conveys both the perceptual and the geometric structure of a 3D model. We extract exoskeletons via a principled framework that combines segmentation and shape approximation. Our method starts from a segmentation of the shape into perceptually relevant parts and then constructs the exoskeleton using a novel extension of the Variational Shape Approximation method. Benefits of the exoskeleton abstraction to graphics applications such as simplification and chartification are presented.

Extracting Manifold and Feature-Enhanced Mesh Surfaces From Binary Volumes

Abstract This paper presents an approach to automatically recover mesh surfaces with sharp edges for solids from their binary volumetric discretizations (i.e., voxel models). Our method consists of three steps. The topology singularity is first eliminated on the binary grids so that a topology correct mesh M0 can be easily constructed. After that, the shape of M0 is refined, and its connectivity is iteratively optimized into Mn. The shape refinement is governed by the duplex distance fields derived from the input binary volume model. However, the refined mesh surface lacks sharp edges. Therefore, we employ an error-controlled variational shape approximation algorithm to segment Mn into nearly planar patches and then recover sharp edges by applying a novel segmentation-enhanced bilateral filter to the surface. Using the technique presented in this paper, smooth regions and sharp edges can be automatically recovered from raw binary volume models without scalar field or Hermite data Compared to other related surface recovering methods on binary volume, our algorithm needs less heuristic coefficients.

A Robust Two‐Step Procedure for Quad‐Dominant Remeshing

Abstract We propose a new technique for quad‐dominant remeshing which separates the local regularity requirements from the global alignment requirements by working in two steps. In the first step, we apply a slight variant of variational shape approximation in order to segment the input mesh into patches which capture the global structure of the processed object. Then we compute an optimized quad‐mesh for every patch by generating a finite set of candidate curves and applying a combinatorial optimization procedure. Since the optimization is performed independently for each patch, we can afford more complex operations while keeping the overall computation times at a reasonable level. Our quad‐meshing technique is robust even for noisy meshes and meshes with isotropic or flat regions since it does not rely on the generation of curves by integration along estimated principal curvature directions. Instead we compute a conformal parametrization for each patch and generate the quad‐mesh from curves with minimum bending energy in the 2D parameter domain. Mesh consistency between patches is guaranteed by simply using the same set of sample points along the common boundary curve. The resulting quad‐meshes are of high‐quality locally (shape of the quads) as well as globally (global alignment) which allows us to even generate fairly coarse quad‐meshes that can be used as Catmull‐Clark control meshes. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Geometric algorithms, languages, and systems

Structure Recovery via Hybrid Variational Surface Approximation

Aiming at robust surface structure recovery, we extend the powerful optimization technique of variational shape approximation by allowing for several different primitives to represent the geometric proxy of a surface region. While the original paper only considered planes, we also include spheres, cylinders, and more complex rollingball blend patches. The motivation for this choice is the fact that most technical CAD objects consist of patches from these four categories. The robust segmentation and global optimization properties which have been observed for the variational shape approximation carry over to our hybrid extension. Hence, we can use our algorithm to segment a given mesh model into characteristic patches and provide a corresponding geometric proxy for each patch. The expected result that we recover surface structures more robustly and thus obtain better approximations with a smaller number of primitives, is validated and demonstrated on a number of examples.

Truly selective polygonal mesh hierarchies with error control

In this work a new multi-resolution model is proposed for polygonal meshes. It is based on the dual edge collapse, which performs face clustering instead of vertex clustering. The new hierarchical mesh representation combines a truly selective refinement scheme with a strict control of the two-sided Hausdorff distance. The proposed approach allows to build hierarchical meshes directly over non-triangulated polygonal models. As most dependencies in the hierarchy are in the form of trees, the resulting representation is very compact and allows for a compressed in-core representation. Furthermore an optimization scheme is proposed for the hierarchy based on variational shape approximation.

Variational shape approximation

A method for concise, faithful approximation of complex 3D datasets is key to reducing the computational cost of graphics applications. Despite numerous applications ranging from geometry compression to reverse engineering, efficiently capturing the geometry of a surface remains a tedious task. In this paper, we present both theoretical and practical contributions that result in a novel and versatile framework for geometric approximation of surfaces. We depart from the usual strategy by casting shape approximation as a variational geometric partitioning problem. Using the concept of geometric proxies, we drive the distortion error down through repeated clustering of faces into best-fitting regions. Our approach is entirely discrete and error-driven, and does not require parameterization or local estimations of differential quantities. We also introduce a new metric based on normal deviation, and demonstrate its superior behavior at capturing anisotropy.