This study investigates the impact of the invariance of feature vectors for partial-to-partial point set registration under translation and rotation of input point sets, particularly in the realm of techniques based on deep learning and Gaussian mixture models (GMMs). We reveal both theoretical and practical problems associated with such deep-learning-based registration methods using GMMs, with a particular focus on the limitations of DeepGMR, a pioneering study in this line, to the partial-to-partial point set registration. Our primary goal is to uncover the causes behind such methods and propose a comprehensible solution for that. To address this, we introduce an attention-based reference point shifting (ARPS) layer, which robustly identifies a common reference point of two partial point sets, thereby acquiring transformation-invariant features. The ARPS layer employs a well-studied attention module to find a common reference point rather than the overlap region. Owing to this, it significantly enhances the performance of DeepGMR and its recent variant, UGMMReg. Furthermore, these extension models outperform even prior deep learning methods using attention blocks and Transformer to extract the overlap region or common reference points. We believe these findings provide deeper insights into registration methods using deep learning and GMMs. Our source code and datasets are available at https://github.com/tatsy/DGRM-ARPS.git.

Representing high-precision industrial assemblies characterized by complex structural features remains challenging. In this paper, we propose Hessian-Regularized Implicit Displacement Fields (HR-IDF), a framework that integrates a two-scale neural implicit representation with Hessian-based regularization. In a coarse-to-fine manner, our method generates a smooth base surface from mesh-sampled points and then refines it with a high-frequency displacement field to capture fine geometric details. Moreover, we introduce a relaxed off-surface loss that helps preserve a more consistent gradient in the generated SDF field, while suppressing ghost geometry and improving representation stability and fidelity. Extensive experiments on complex industrial assemblies and 3D models demonstrate that HR-IDF achieves a reliable solution for high-precision industrial applications.

In this article, we present a self-prior-based mesh inpainting framework that requires only an incomplete mesh as input, without the need for any training datasets. Additionally, our method maintains the polygonal mesh format throughout the inpainting process without converting the shape format to an intermediate one, such as a voxel grid, a point cloud, or an implicit function, which are typically considered easier for deep neural networks to process. To achieve this goal, we introduce two graph convolutional networks (GCNs): single-resolution GCN (SGCN) and multi-resolution GCN (MGCN), both trained in a self-supervised manner. Our approach refines a watertight mesh obtained from the initial hole filling to generate a complete output mesh. Specifically, we train the GCNs to deform an oversmoothed version of the input mesh into the expected complete shape. The deformation is described by vertex displacements, and the GCNs are supervised to obtain accurate displacements at vertices in real holes. To this end, we specify several connected regions of the mesh as fake holes, thereby generating meshes with various sets of fake holes. The correct displacements of vertices are known in these fake holes, thus enabling training GCNs with loss functions that assess the accuracy of vertex displacements. We demonstrate that our method outperforms traditional dataset-independent approaches and exhibits greater robustness compared with other deep-learning-based methods for shapes that infrequently appear in shape datasets.

X-ray CT scanners, due to the transmissive nature of X-rays, have enabled the non-destructive evaluation of industrial products, even inside their bodies. In light of its effectiveness, this study introduces a new approach to accelerate the inspection of many mechanical parts with the same shape in a bin. The input to this problem is a volumetric image (i.e., CT volume) of many parts obtained by a single CT scan. We need to segment the parts in the volume to inspect each of them; however, random postures and dense contacts of the parts prohibit part segmentation using traditional template matching. To address this problem, we convert both the scanned volumetric images of the template and the binned parts to simpler graph structures and solve a subgraph matching problem to segment the parts. We perform a distance transform to convert the CT volume into a distance field. Then, we construct a graph based on Morse theory, in which graph nodes are located at the extremum points of the distance field. The experimental evaluation demonstrates that our fully automatic approach can detect target parts appropriately, even for a heap of 50 parts. Moreover, the overall computation can be performed in approximately 30 min for a large CT volume of approximately 2000×2000×1000 voxels.

In industrial manufacturing, reverse engineering, which can be applied to analyze or re-design, is an essential process, whereby scanned data of real objects are used to generate a CAD model. CAD data have various features such as fillets; therefore, robust and versatile reverse engineering is still difficult. Particularly, an adequate segmentation from real scanned data is challenging, and it is a major obstacle for reverse engineering. In this study, we propose an innovative segmentation from CT scanned data, wherein user-friendly segmentation is achieved by directly computing the curvature gradient from a CT sinogram. We used the proposed method to patch a B-spline surface onto each segmented region. The experimental results show that the proposed method offers robust, versatile, and user-friendly reverse engineering that is faster than conventional manual modeling. In addition, as it is supported by X-ray CT simulation, the proposed method can be applied to various surface mesh data without requiring any actual X-ray CT equipment.

This paper proposes a novel approach, the sinogram polygonizer, for directly reconstructing 3D shapes from sinograms (i.e., the primary output from X-ray computed tomography (CT) scanners consisting of projection image sequences of an object shown from different viewing angles). To obtain a polygon mesh approximating the surface of a scanned object, a grid-based isosurface polygonizer, such as Marching Cubes, has been conventionally applied to the CT volume reconstructed from a sinogram. In contrast, the proposed method treats CT values as a continuous function and directly extracts a triangle mesh based on tetrahedral mesh deformation. This deformation involves quadratic error metric minimization and optimal Delaunay triangulation for the generation of accurate, high-quality meshes. Thanks to the analytical gradient estimation of CT values, sharp features are well approximated, even though the generated mesh is very coarse. Moreover, this approach eliminates aliasing artifacts on triangle meshes.

In segmentation for reverse engineering, an input triangular mesh is usually split into segments for each of which a parametric surface is generated. The segment boundaries must be smooth for fine surfaces to be generated, but today's segmentation methods cannot necessarily generate a sufficient level of smoothness. The authors of this paper were inspired by the work reported in, and here try to extend it to efficiently segment a mesh obtained by scanning an object. Fuzzy boundary curves between two neighboring regions are globally rectified using a graph cut approach in which new energy functions are constructed. However, some curves are still not smooth since there are no straight lines around them in the original scanned mesh models, although the corresponding physical curves are smooth. To achieve further smoothing, we split some triangles by adding points and edges based on virtual triangle subdivision and the graph cut approach. As a result, significant further straightening of boundaries is achieved with simple straightest geodesics.

Abstract We propose a novel method for smoothing partition of unity (PU) implicit surfaces consisting of sets of non‐conforming linear functions with spherical supports. We derive new discrete differential operators and Laplacian smoothing using a spherical covering of PU as a grid‐like data structure. These new differential operators are applied to the smoothing of PU implicit surfaces. First, Laplacian smoothing is performed for the vector field defined by the gradient of the PU implicit surface, which is then updated to reflect the smoothing of the gradient field. This process achieves a method for noise robust surface reconstruction from scattered points.

We introduce a method for extracting boundary surfaces from volumetric models of mechanical parts by X-ray CT scanning. When the volumetric model is composed of two materials, one for the object and the other for the background (Air), these boundary surfaces can be extracted as isosurfaces using a contouring method such as Marching Cubes [Lorensen and Cline 1987]. For a volumetric model composed of more than two materials, we need to classify the voxel types into segments by material and use a generalized Marching Cubes algorithm that can deal with both CT values and material types. Here we propose a method for precisely classifying the volumetric model into its component materials using a modified and combined method of two well-known algorithms in image segmentation, region growing and Graph-cut. We then apply the generalized Marching Cubes algorithm to generate triangulated mesh surfaces. In addition, we demonstrate the effectiveness of our method by constructing high-quality triangular mesh models of the segmented parts.

We introduce an algorithm for computing discrete shortest paths on smooth surface representations. For given source and target points on a surface, an initial shortest path is computed by using the Dijkstra’s algorithm. Based on such an initial curve, locally-refined discrete paths are repeatedly calculated so as to get more accurate shortest path. Our algorithm is simple, easy to implement and much fast because it requires only the Dijkstra’s algorithm to compute the path. Our algorithm is a versatile approach for a wide variety of smooth surface representations.

The skeletal structures of solid objects play an important role in medical and industrial applications. Given a volumetrically sampled solid object, our method extracts a nice-looking skeletal structure represented as a polygon mesh. The purpose is to achieve a noise-robust extraction of the skeletal mesh from a real-world object obtained using a scanning technology such as the CT scan method. We first approximate the input through a set of spherically supported polynomials that provide an adaptively smoothed intensity field, and then perform a polygonization process to find the extremal sheet of the field, which is regarded as a skeletal sheet in this research. In our polygonization, a subset of the weighted Delaunay tetrahedrization defined by a set of spherical supports is used as an adaptively sampled grid. The derivatives for detecting extremality are analytically evaluated at the tetrahedron vertices. We also demonstrate the effectiveness of our method by extracting skeletal meshes from noisy CT images.

We propose a new approach for surface reconstruction from any unorganized point set, with a fixed amount of parameters. First, we consider some local approximations by compactly supported radial basis functions (CSRBFs), then blend them by a partition of unity method (PU). In our scheme, we start by selecting CSRBFs centers, then construct partitions, finally parameters are calculated in order to minimize, on each sub-domain, the mean square error with a regularization constraint. We demonstrate the effectiveness of our approach on several models with different points distributions.

Sparse low-degree implicit (SLIM) surface is a recently developed non-conforming surface representation. In this paper, a method for free-form deformation of SLIM surfaces is presented. By employing a Laplacian based mesh deformation technique, a global deformation is realized as movements of the centers of the local function supports. A graph connectivity for defining the Laplacians is simply created using inclusion of other centers in the supports. By following the global deformation according to the movements of the centers, we update each local function and its support size via several times of local least square fittings by using the new positions of the neighboring centers. Since the additional computations for updating local functions are very computationally cheap, the proposed SLIM surface deformation achieves an interactive surface deformation similarly to Laplacian based mesh deformation techniques. Further, because a SLIM surface requires a smaller number of elements than a mesh for representing the same geometrical details, the computational effort for a global Laplacian based deformation is dramatically reduced

In this paper, we propose a new method for approximating an unorganized set of points scattered over a piecewise smooth surface by a triangle mesh. The method is based on the Garland-Heckbert local quadric error minimization strategy. First an adaptive spherical cover and auxiliary points corresponding to the cover elements are generated. Then the intersections between the spheres of the cover are analyzed and the auxiliary points are connected. Finally the resulting mesh is cleaned from non-manifold parts. The method allows us to control the approximation accuracy, process noisy data, and reconstruct sharp edges and corners. Further, the vast majority of the triangles of the generated mesh have their aspect ratios close to optimal. Thus our approach integrates the mesh reconstruction, smoothing, decimation, feature restoration, and remeshing stages together.

Feature sensitive mesh segmentation is important for many computer graphics and geometric modeling applications. In this paper, we develop a mesh segmentation method, which is capable of producing high-quality shape partitioning. It respects fine shape features and works well on various types of shapes, including natural shapes and mechanical parts. The method combines a procedure for clustering mesh normals with a modification of the mesh clarification technique. For clustering of mesh normals, we adopt Mean Shift, a powerful general purpose technique for clustering scattered data. We demonstrate advantages of our method by comparing it with two state-of-the-art mesh segmentation techniques.

We develop an adaptive RBF fitting procedure for a high quality approximation of a set of points scattered over a piecewise smooth surface. We use compactly supported RBFs whose centers are randomly chosen from the points. The randomness is controlled by the point density and surface geometry. For each RBF, its support size is chosen adoptively according to surface geometry at a vicinity of the RBF center. All these lead to a noise-robust high quality approximation of the set. We also adapt our basic technique for shape reconstruction from registered range scans by taking into account measurement confidences. Finally, an interesting link between our RBF fitting procedure and partition of unity approximations is established and discussed.

We propose a simple and effective method for detecting view-and scale-independent ridge-valley lines defined via first- and second-order curvature derivatives on shapes approximated by dense triangle meshes. A high-quality estimation of high-order surface derivatives is achieved by combining multi-level implicit surface fitting and finite difference approximations. We demonstrate that the ridges and valleys are geometrically and perceptually salient surface features, and, therefore, can be potentially used for shape recognition, coding, and quality evaluation purposes.

We propose a new interpolatory subdivision scheme for generating nice-looking curvature-continuous curves of round shapes. The scheme is based on a diffusion of normals. Given a subdivided polyline, the new polyline vertices inserted at the splitting step are updated in order to fit diffused (averaged with appropriate weights) normals. Although the resulting interpolatory subdivision scheme is nonstationary, nonlinear, and nonuniform from the traditional point of view, the scheme is easy to implement because the same simple geometric procedure for generating new vertices is used at each subdivision step. According to our experiments, the scheme is robust and demonstrates very good convergence properties.

We present two simple yet effective mesh denoising methods. The methods combine a diffusion (repeated local averaging) of mesh normals with histogram-based filtering procedures. According to our experimental results, the methods outperform conventional smoothing techniques in denoising meshes with sharp features.

We propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. An original point set and its coarse-to-fine hierarchy of interpolated sets is presented. According to our numerical experiments, the method is essentially faster than the state-of-the-art scattered data approximation with globally supported RBFs (Carr et al., 2001) and much simpler to implement.

We present a new shape representation, the multi-level partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape.Our approach gives us considerable flexibility in the choice of local shape functions, and in particular we can accurately represent sharp features such as edges and corners by selecting appropriate shape functions. An error-controlled subdivision leads to an adaptive approximation whose time and memory consumption depends on the required accuracy. Due to the separation of local approximation and local blending, the representation is not global and can be created and evaluated rapidly. Because our surfaces are described using implicit functions, operations such as shape blending, offsets, deformations and CSG are simple to perform.

This paper presents frameworks to extend the mean and median filtering schemes in image processing to smoothing noisy 3D shapes given by triangle meshes. The frameworks consist of the application of the mean and median filters to face normals on triangle meshes and the editing of mesh vertex positions to make them fit the modified normals. We also give a quantitative evaluation of the proposed mesh filtering schemes and compare them with conventional mesh smoothing methods such as Laplacian smoothing flow and mean curvature flow. The quantitative evaluation is performed in error metrics on mesh vertices and normals. Experimental results demonstrate that our mesh mean and median filtering methods are more stable than conventional Laplacian and mean curvature flows. We propose thee new mesh smoothing methods as one possible solution of the oversmoothing problem.

The paper develops a simple and effective method for adaptive smoothing tangential direction fields defined on piecewise smooth surfaces approximated by triangle meshes. The method consists of simultaneous and iterative updating each vertex direction by a weighted sum of the directions at the neighboring vertices. The weights themselves depend on directions: if the direction at a given vertex differs strongly from the direction at a neighboring vertex, then the weight associated with that neighboring direction is chosen small. Choosing such nonlinear weights allows us to preserve important direction field discontinuities during the smoothing process. Applications of the developed smoothing technique to curvature extrema detection and non-photorealistic rendering with curvature lines are given.

The paper presents a novel approach for accurate polygonization of implicit surfaces with sharp features. The approach is based on mesh evolution towards a given implicit surface with simultaneous control of the mesh vertex positions and mesh normals.

In this paper, we propose a method for improvement of isosurface polygonizations. Given an initial polygonization of an isosurface, we introduce a mesh evolution process initialized by the polygonization. The evolving mesh converges quickly to its limit mesh which provides with a high quality approximation of the isosurface even if the isosurface has sharp features, boundary, complex topology. To analyze how close the evolving mesh approaches its destined isosurface, we introduce error estimators measuring the deviations of the mesh vertices from the isosurface and mesh normals from the isosurface normals. A new technique for mesh editing with isosurfaces is also proposed. In particular, it can be used for creating carving effects.

A computer graphics object reconstructed from real-world data often contains undesirable noise and small-scale oscillations. An important problem is how to remove the noise and oscillations while preserving desirable geometric features of the object. We develops methods for polyhedral surface smoothing and denoising with simultaneous increasing mesh regularity. We also propose an adaptive smoothing method allowing to reduce possible oversmoothing. Roughly speaking, our smoothing schemes consist of moving every vertex in the direction defined by the Laplacian flow with speed equal to a properly chosen function of the mean curvature at the vertex.

A new method for visual and interactive topology is presented. While many classical and modern textbooks on topology contain beautiful illustrations for learners, and much excellent video software has been produced for animating geometrical phenomena, they are all based on static scenarios. This means their geometrical presentations fail to meet the demand for interactivity. With the progress of multimedia technology, dynamic scenarios, which enable learners to make mathematical experiments as many times as they like, are now well within reach. However, an appropriate methodology for interactive mathematical visualization is still lacking. The authors present a dual visualizer method in which the operational visualizer and the geometrical visualizer play complementary roles. The process graph is a key technique in the operational visualizer The method illuminates crucial factors to be considered in the methodology.