Point-sampled Surfaces Research Articles

Exploring differential geometry in neural implicits

We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions.To train a neural implicit function, we propose a loss functional that approximates a signed distance function, and allows terms with high-order derivatives, such as the alignment between the principal directions of curvature, to learn more geometric details. During training, we consider a non-uniform sampling strategy based on the curvatures of the point-sampled surface to prioritize points with more geometric details. This sampling implies faster learning while preserving geometric accuracy when compared with previous approaches.We also use the analytical derivatives of a neural implicit function to estimate the differential measures of the underlying point-sampled surface.

Editorial Issue 26.1

Reference EPFL-ARTICLE-207397doi:10.1002/cav.1634View record in Web of Science Record created on 2015-04-13, modified on 2016-08-09

Detail-generating geometry completion for point-sampled geometry

In this paper, we present a novel method for detail-generating geometry completion over point-sampled geometry. The main idea consists of converting the context-based geometry completion into the detail-based texture completion on the surface. According to the influence region of boundary points surrounding a hole, a smooth patch covering the hole is first constructed using radial base functions. By applying region-growing clustering to the patch, the patching units for further completion with geometry details is then produced, and using the trilateral filtering operator formulated by us, the geometry-detail texture of each sample point on the input geometry is determined. The geometry details on the smooth completed patch are finally generated by optimizing a constrained global texture energy function on the point-sampled surfaces. Experimental results demonstrate that the method can achieve efficient completed patches that not only conform with their boundaries, but also contain the plausible 3D surface details.

Point-Based Manifold Harmonics

This paper proposes an algorithm to build a set of orthogonal Point-Based Manifold Harmonic Bases (PB-MHB) for spectral analysis over point-sampled manifold surfaces. To ensure that PB-MHB are orthogonal to each other, it is necessary to have symmetrizable discrete Laplace-Beltrami Operator (LBO) over the surfaces. Existing converging discrete LBO for point clouds, as proposed by Belkin et al., is not guaranteed to be symmetrizable. We build a new point-wisely discrete LBO over the point-sampled surface that is guaranteed to be symmetrizable, and prove its convergence. By solving the eigen problem related to the new operator, we define a set of orthogonal bases over the point cloud. Experiments show that the new operator is converging better than other symmetrizable discrete Laplacian operators (such as graph Laplacian) defined on point-sampled surfaces, and can provide orthogonal bases for further spectral geometric analysis and processing tasks.

Graph-based representations of point clouds

This paper introduces a skeletal representation, called Point Cloud Graph, that generalizes the definition of the Reeb graph to arbitrary point clouds sampled from m-dimensional manifolds embedded in the d-dimensional space. The proposed algorithm is easy to implement and the graph representation yields to an effective abstraction of the data. Finally, we present experimental results on point-sampled surfaces and volumetric data that show the robustness of the Point Cloud Graph to non-uniform point distributions and its usefulness for shape comparison.

Defining, contouring, and visualizing scalar functions on point-sampled surfaces

This paper addresses the definition, contouring, and visualization of scalar functions on unorganized point sets, which are sampled from a surface in 3D space; the proposed framework builds on moving least-squares techniques and implicit modeling. Given a scalar function f : P → R , defined on a point set P , the idea behind our approach is to exploit the local connectivity structure of the k -nearest neighbor graph of P and mimic the contouring of scalar functions defined on triangle meshes. Moving least-squares and implicit modeling techniques are used to extend f from P to the surface M underlying P . To this end, we compute an analytical approximation f ̃ of f that allows us to provide an exact differential analysis of f ̃ , draw its iso-contours, visualize its behavior on and around M , and approximate its critical points. We also compare moving least-squares and implicit techniques for the definition of the scalar function underlying f and discuss their numerical stability and approximation accuracy. Finally, the proposed framework is a starting point to extend those processing techniques that build on the analysis of scalar functions on 2-manifold surfaces to point sets.

Segmenting point-sampled surfaces

Extracting features from point-based representations of geometric surface models is becoming increasingly important for purposes such as model classification, matching, and exploration. In an earlier paper, we proposed a multiphase segmentation process to identify elongated features in point-sampled surface models without the explicit construction of a mesh or other surface representation. The preliminary results demonstrated the strength and potential of the segmentation process, but the resulting segmentations were still of low quality, and the segmentation process could be slow. In this paper, we describe several algorithmic improvements to overcome the shortcomings of the segmentation process. To demonstrate the improved quality of the segmentation and the superior time efficiency of the new segmentation process, we present segmentation results obtained for various point-sampled surface models. We also discuss an application of our segmentation process to extract ridge-separated features in point-sampled surfaces of CAD models.

A robust and efficient method for direct projection on point-sampled surfaces

A novel and robust approach for computing the direct projection onto a point cloud for a given 3D point with associated projection vector is proposed. An energy function is presented to model the active contour, and the curve evolving method is applied to track the projection in certain iterations. In each step of the iteration, the energies of the related points are used to estimate the step length and orientation of the evolution. Then the point’s position is continuously updated and approximates the solution until the given accuracy is reached. The proposed approach directly operates on the point cloud, which is followed by a refining process to correct the results for under-sampled regions. Since the algorithm is carried out on a small point set (i.e., the neighborhood of the test point, generally composed of 30∼60 points) it is not expensive computationally. By numerical experiments, the method has been demonstrated to be more robust and efficient than the existing direct projection approaches on processing discrete point clouds.

Simplification of Point Set Surfaces using Bilateral Filter and Multi-Sized Splats

We have developed a new algorithm that addresses the two major issues that are critical for the use of point-based rendering in real-world applications: rendering performance and rendering quality. The proposed algorithm improves rendering performance by reducing the number of points to be rendered. We generated points with different sizes in order to improve the quality of the point-splatting-type rendering. In this paper, we propose a feature-preserved simplification algorithm for point-sampled surfaces that generates models with different levels of details. The proposed method automatically balances the sampling density and point sizes. Our algorithm iteratively reduces the number of points using a bilateral filtering algorithm. We validate our method on the basis of the rendering results of different models with different resolutions.

Similarity-based denoising of point-sampled surfaces

A non-local denoising (NLD) algorithm for point-sampled surfaces (PSSs) is presented based on similarities, including geometry intensity and features of sample points. By using the trilateral filtering operator, the differential signal of each sample point is determined and called “geometry intensity”. Based on covariance analysis, a regular grid of geometry intensity of a sample point is constructed, and the geometry-intensity similarity of two points is measured according to their grids. Based on mean shift clustering, the PSSs are clustered in terms of the local geometry-features similarity. The smoothed geometry intensity, i.e., offset distance, of the sample point is estimated according to the two similarities. Using the resulting intensity, the noise component from PSSs is finally removed by adjusting the position of each sample point along its own normal direction. Experimental results demonstrate that the algorithm is robust and can produce a more accurate denoising result while having better feature preservation.

GPU-based rendering of point-sampled water surfaces

Particle-based simulations are widely used to simulate fluids. We present a real-time rendering method for the results of particle-based simulations of water. Traditional approaches to visualize the results of particle-based simulations construct water surfaces that are usually represented by polygons. To construct water surfaces from the results of particle-based simulations, a density function is assigned to each particle and a density field is computed by accumulating the values of the density functions of all particles. However, the computation of the density field is time consuming. To address this problem, we propose an efficient calculation of density field using a graphics processing unit (GPU). We present a rendering method for water surfaces sampled by points. The use of the GPU permits efficient simulation of optical effects, such as refraction, reflection, and caustics.

High frequency geometric detail manipulation and editing for point-sampled surfaces

In this paper, based on the new definition of high frequency geometric detail for point-sampled surfaces, a new approach for detail manipulation and a detail-preserving editing framework are proposed. Geometric detail scaling and enhancement can always produce fantastic effects by directly manipulating the geometric details of the underlying geometry. Detail-preserving editing is capable of preserving geometric details during the shape deformation of point-sampled model. For efficient editing, the point set of the model is first clustered by a mean shift scheme, according to its anisotropic geometric features and each cluster is abstracted as a simplification sample point (SSP). Our editing operation is implemented by manipulating the SSP first and then diffusing the deformation to all sample points on the underlying geometry. As a postprocessing step, a new up-sampling and relaxation procedure is proposed to refine the deformed model. The effectiveness of the proposed method is demonstrated by several examples.

Differentials-Based Segmentation and Parameterization for Point-Sampled Surfaces

Efficient parameterization of point-sampled surfaces is a fundamental problem in the field of digital geometry processing. In order to parameterize a given point-sampled surface for minimal distance distortion, a differentials-based segmentation and parameterization approach is proposed in this paper. Our approach partitions the point-sampled geometry based on two criteria: variation of Euclidean distance between sample points, and angular difference between surface differential directions. According to the analysis of normal curvatures for some specified directions, a new projection approach is adopted to estimate the local surface differentials. Then a k-means clustering (k-MC) algorithm is used for partitioning the model into a set of charts based on the estimated local surface attributes. Finally, each chart is parameterized with a statistical method -- multidimensional scaling (MDS) approach, and the parameterization results of all charts form an atlas for compact storage.

A level set based segmentation approach for point-sampled surfaces

Segmenting a complex 3D surface model into some visually meaningful sub-parts is one of the fundamental problems in digital geometry processing. In this paper, a novel segmentation approach of point-sampled surfaces is proposed, which is based on the level set evolution scheme. To segment the model so as to align the patch boundaries with high curvature zones, the driven speed function for the zero level set inside narrow band is defined by the extended curvature field, which approaches zero speed as the propagating front approaches high curvature zone. The effectiveness of the proposed approach is demonstrated by our experimental results. Furthermore, two applications of model segmentation are illustrated, such as piecewise parameterization and local editing for point-sampled geometry.

Mean shift denoising of point-sampled surfaces

This paper presents an anisotropic denoising/smoothing algorithm for point-sampled surfaces. Motivated by the impressive results of mean shift filtering on image denoising, we extend the concept to 3D surface smoothing by taking the vertex normal and the curvature as the range component and the vertex position as the spatial component. Then the local mode of each vertex on point-based surfaces is computed by a 3D mean shift procedure dependent on local neighborhoods that are adaptively obtained by a kdtree data structure. Clustering pieces of point-based surfaces of similar local mode provides a meaningful model segmentation. Based on the adaptively clustered neighbors, we finally apply a trilateral point filtering scheme that adjusts the position of sample points along their normal directions to successfully reduce noise from point-sampled surfaces while preserving geometric features.

A quasi-Monte Carlo method for computing areas of point-sampled surfaces

A novel and efficient quasi-Monte Carlo method for computing the area of a point-sampled surface with associated surface normal for each point is presented. Our method operates directly on the point cloud without any surface reconstruction procedure. Using the Cauchy–Crofton formula, the area of the point-sampled surface is calculated by counting the number of intersection points between the point cloud and a set of uniformly distributed lines generated with low-discrepancy sequences. Based on a clustering technique, we also propose an effective algorithm for computing the intersection points of a line with the point-sampled surface. By testing on a number of point-based models, experiments suggest that our method is more robust and more efficient than those conventional approaches based on surface reconstruction.

Efficient Raytracing of Deforming Point‐Sampled Surfaces

We present efficient data structures and caching schemes to accelerate ray-surface intersections for deforming point-sampled surfaces. By exploiting spatial and temporal coherence of the deformation during the animation, we are able to improve rendering performance by a factor of two to three compared to existing techniques. Starting from a tight bounding sphere hierarchy for the undeformed object, we use a lazy updating scheme to adapt the hierarchy to the deformed surface in each animation step. In addition, we achieve a significant speedup for ray-surface intersections by caching per-ray intersection points. We also present a technique for rendering sharp edges and corners in point-sampled models by introducing a novel surface clipping algorithm. © The Eurographics Association and Blackwell Publishing 2005.

Touch-based haptics for interactive editing on point set surfaces

A modeling paradigm for haptics-based editing on point set surfaces exploits implicit surfaces, physics-based modeling, point-sampled surfaces, and haptic. We propose a point-based geometry representation that we initially designed for dynamic physics-based sculpting, but can easily generalize to other relevant applications such as data modeling and human-computer interaction. By extending the idea of the local reference domain in the moving least square (MLS) surface model to the construction of a local and global surface distance field, we naturally incorporate Hua and Qin's dynamic implicit volumetric model into our deformation of the point-based geometry, which not only facilitates topology change but also affords dynamic sculpting and deformation.

Context-based surface completion

Sampling complex, real-world geometry with range scanning devices almost always yields imperfect surface samplings. These "holes" in the surface are commonly filled with a smooth patch that conforms with the boundary. We introduce a context-based method: the characteristics of the given surface are analyzed, and the hole is iteratively filled by copying patches from valid regions of the given surface. In particular, the method needs to determine best matching patches, and then, fit imported patches by aligning them with the surrounding surface. The completion process works top down, where details refine intermediate coarser approximations. To align an imported patch with the existing surface, we apply a rigid transformation followed by an iterative closest point procedure with non-rigid transformations. The surface is essentially treated as a point set, and local implicit approximations aid in measuring the similarity between two point set patches. We demonstrate the method at several point-sampled surfaces, where the holes either result from imperfect sampling during range scanning or manual removal.

Shape modeling with point-sampled geometry

We present a versatile and complete free-form shape modeling framework for point-sampled geometry. By combining unstructured point clouds with the implicit surface definition of the moving least squares approximation, we obtain a hybrid geometry representation that allows us to exploit the advantages of implicit and parametric surface models. Based on this representation we introduce a shape modeling system that enables the designer to perform large constrained deformations as well as boolean operations on arbitrarily shaped objects. Due to minimum consistency requirements, point-sampled surfaces can easily be re-structured on the fly to support extreme geometric deformations during interactive editing. In addition, we show that strict topology control is possible and sharp features can be generated and preserved on point-sampled objects. We demonstrate the effectiveness of our system on a large set of input models, including noisy range scans, irregular point clouds, and sparsely as well as densely sampled models.