Cubics and conics geodesically associated to the points of a geometric surface

This work proposes a change-based segmentation method for applications to cultural heritage (CH) imaging to perform monitoring and assess changes at each surface point. It can be used as a support or component of the 3D sensors to analyze surface geometry changes. In this research, we proposed a new method to identify surface changes employing segmentation based on 3D geometrical data acquired at different time intervals. The geometrical comparison was performed by calculating point-to-point Euclidean distances for each pair of surface points between the target and source geometry models. Four other methods for local distance measurement were proposed and tested. In the segmentation method, we analyze the local histograms of the distances between the measuring points of the source and target models. Then the parameters of these histograms are determined, and predefined classes are assigned to target surface points. The proposed methodology was evaluated by considering two different case studies of restoration issues on CH surfaces and monitoring them over time. The results were presented with a colormap visualization for each category of the detected change in the analysis. The proposed segmentation method will help in the field of conservation and restoration for the documentation and quantification of geometrical surface change information. This analysis can help in decision-making for the assessment of damage and potential prevention of further damage, and the interpretation of measurement results.

We describe a method for creating semi-isometric flat maps of sections of the human cortical surface. These maps can be used for a wide range of applications including visualization of the distribution of functional activation over the unfolded brain surface and generating parametric models for the cortical surface that are appropriate for performing geometrical transformations and surface based inter-subject registration. In particular, their application in creating multi-resolution representations of arbitrarily shaped complex surfaces is presented in this paper. Using the property that every simultaneous conformal and equiareal mapping is isometric, we have formulated the calculation of an isometric mapping between surfaces as a constrained optimization problem. We have designed an energy function whose minima occur when the surface points are positioned in an unfolded configuration. Constraint functions imposing the requirements of preservation of angles and areas guarantee that the surface will deform to produce conformal and equiareal mappings. The constraints are imposed using penalty functions in an unconstrained conjugate gradient algorithm. The surface unfolds gradually, deforming to a near-flat form that corresponds to a minimum of the weighted sum of the energy and penalty function terms.

The main purpose of this initial paper is to demonstrate the application of order statistics in the estimation of form error from a CMM measurement. Nowadays, modern industry sets high standards for geometrical precision, surface texture and material properties. There are many parameters that can characterize mechanical part, out of which flatness error plays important in the assembly process and performance. Recently, due to the greater availability and price reduction, Coordinate Measurement Techniques have increased their popularity in the industry for on-line and off-line measurements as they allow automated measurements at relatively low uncertainty level. Data obtained from CMM measurements have to be processed and analyzed in order to evaluate component compliance with the required technical specification. The article presents an analysis of a minimal sample selection for the evaluation of flatness error by means of coordinate measurement. In the paper, a statistical approach was presented, assuming that, in the repetitive manufacturing process, the distribution of deviations between surface points and the reference plane is stable. Based on the known, statistical distribution, order statistics theorem was implemented to determine maximal and minimal point deviation statistics, as it played a dominant role in flatness error estimation. A brief analysis of normally distributed deviations was described in the paper. Moreover, the case study was presented for the set of the machined parts which were components of a machine tool mechanical structure. Empirical distributions were derived and minimal sample sizes were estimated for the given confidence levels using the proposed theorem. The estimation errors of flatness values for the derived sample sizes were analyzed and discussed in the paper.

Statistical Tolerancing based on Variation of Point-set

The objective of this work was to develop and validate a computational method for the registration (matching) of 3D cartilage plates from MR image data sets. The technique tracks local cartilage thickness changes over time. A 3D elastic registration technique was applied that identifies corresponding points of the bone-cartilage interface in MR data sets of 3D-reconstructed cartilage plates. In a first rigid preregistration step, the surfaces are aligned, using the principal axes decomposition to correct for different joint positions and orientations in the MR scanner. In a second step, the surfaces are deformed elastically, based on geometric surface features, until they are sufficiently similar to identify corresponding surface points. The method was validated against artificially corrupted cartilage surfaces and MR data obtained from in vivo and in vitro compression experiments. The in vivo reproducibility was tested on patellar data sets of volunteers, with repositioning of the joint in between replicate acquisitions.

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks.

In this paper, we present a method for accurately estimating the shape of an object by integrating the surface orientation measured by photometric stereo and the position measured by some range-measuring method. We first show that even if the knowledge of the reflectance/illumination is inaccurate, the first derivatives of the photometrically measured orientation can be accurately estimated at the surface points where they have small values. We propose a probabilistic framework to quantitate the (in)accuracy of the knowledge and connect it to the estimation accuracy of these derivatives. Based on this framework, we consider optimally integrating the surface orientation and position to obtain the object shape with higher accuracy. The integration reduces to an optimization problem, and it is efficiently solved by belief propagation. We present several experimental results showing the effectiveness of the proposed approach.

This paper presents (on the base of the classical and some additional sandwich assumptions) a geometrically nonlinear theory for sandwich shells with seven kinematic degrees of freedom which is capable to describe the global as well as the local (load singularities, wrinkling) structural behaviour. For the kinematic specification a position vector is used to describe a point of the geometrical middle surface and a corresponding director (no unit vector) as well as a further scalar degree of freedom which could be explained as the linear part of the transversal strain. According to the definition of Green-Lagrangian strains and second Piola-Kirchhoff stress resultants, the nonlinear field equation of force equilibrium with pertinent boundary conditions will be gained from the principal of virtual work. Finally, the equation according to the first order theory for plane structures will be mentioned as a simple special case.

Geometric segmentation of 3D scanned surfaces

The geometric error modeling method for assembly[1] proposes a feasible basis for the assembly process considering manufacturing error. However, contacting two mounting surfaces with geometric error and finding their actual contact points are two main difficulties in the study of real three-dimensional parts assembly methods. A high-precision and fast solution method for linearized assembly contact points considering geometric errors is proposed by calculating the point cloud spatial pose of the actual contact surface. This method is suitable for spatial pose calculation in high precision plane assembly of rigid parts, which enables fast, accurate and quantitative solution of the positional relationship of components during assembly and measurement, and improves the accuracy and efficiency of spatial pose prediction and control.

Application d'un solveur à l'identification de surfaces réelles et au contrôle des spécifications ISO

In this paper, we propose a novel shape descriptor that is robust in differentiating intrinsic symmetric points on geometric surfaces. Our motivation is that even the state‐of‐theart shape descriptors and non‐rigid surface matching algorithms suffer from symmetry flips. They cannot differentiate surface points that are symmetric or near symmetric. Hence a left hand of one human model may be matched to a right hand of another. Our Symmetry Robust Descriptor (SRD) is based on a signed angle field, which can be calculated from the gradient fields of the harmonic fields of two point pairs. Experiments show that the proposed shape descriptor SRD results in much less symmetry flips compared to alternative methods. We further incorporate SRD into a stand‐alone algorithm to minimize symmetry flips in finding sparse shape correspondences. SRD can also be used to augment other modern non‐rigid shape matching algorithms with ease to alleviate symmetry confusions.

In stereo vision-based reconstruction of grinding surface topography with micron-level roughness, complex microstructures and measurement noise often lead to incomplete point cloud data, particularly in the underlying surface layers. To address this, we propose PointGAN, a generative adversarial network designed for high-fidelity completion of grinding surface point clouds. The network incorporates a Multi-Scale Feature Extraction (MLFE) module that captures both global and fine-grained surface characteristics across resolutions. Leveraging the self-affine nature of grinding surfaces, a fractal-inspired decoder and a channel attention discriminator are employed to refine microstructural features and suppress reconstruction noise. PointGAN directly predicts missing geometries from partial point clouds, yielding completed data that closely preserve the original texture, spatial continuity, and roughness metrics. Experimental results demonstrate that the proposed method outperforms existing approaches in both geometric accuracy and surface parameter restoration. This framework offers a robust tool for enhanced surface quality assessment, roughness parameter recovery, and intelligent monitoring in precision grinding processes.

This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality f(x,y,z) ≥ 0. The volume spline is based on use of the Green's function for interpolation of scalar function values of a chosen “carrier” solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross‐sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD for bodies with complex surfaces.

QuickTrace is a new, fast contact detection algorithm. It searches for contact typically between a tool, modelled by some kind of geometrical surface facets, and deforming material. Contact is searched for at material surface points called contact nodes. A contact node is in contact with a facet when the contact node is inside of the tool and the negative outer normal on the material surface at the contact node ‘leaves’ the tool surface through that facet. When multiple facets are intersected, only the closest facet to the contact node should be considered. From this definition of contact, the penetration depth results automatically as a by‐product.In QuickTrace, the m facets of the tool are packaged in boxes that are hierarchically ordered in a search tree with an average depth of log4 m. The computational complexity for one contact search is proportional to this average depth. So, for n contact nodes the computational complexity is O(n log4 m). This places QuickTrace amongst the best performing contact detection algorithms. At the same time there are no approximations or tricks involved and the contact detection is absolutely correct. The algorithm can easily be extended to material–material or tool–tool contact. Copyright © 2002 John Wiley & Sons, Ltd.