This paper introduces compressed eigenfunctions of the Laplace‐Beltrami operator on 3D manifold surfaces. They constitute a novel functional basis, called the compressed manifold basis, where each ...function has local support. We derive an algorithm, based on the alternating direction method of multipliers (ADMM), to compute this basis on a given triangulated mesh. We show that compressed manifold modes identify key shape features, yielding an intuitive understanding of the basis for a human observer, where a shape can be processed as a collection of parts. We evaluate compressed manifold modes for potential applications in shape matching and mesh ion. Our results show that this basis has distinct advantages over existing alternatives, indicating high potential for a wide range of use‐cases in mesh processing.
We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a ...function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, we propose a band‐by‐band spectrum computation algorithm and an out‐of‐core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. We also propose a limited‐memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care.
We introduce an algorithm for the automatic computation of global parameterizations on arbitrary simplicial 2‐manifolds, whose parameter lines are guided by a given frame field, for example, by ...principal curvature frames. The parameter lines are globally continuous and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2‐manifold and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.
Interpolating vertex positions among triangle meshes with identical vertex‐edge graphs is a fundamental part of many geometric modelling systems. Linear vertex interpolation is robust but fails to ...preserve local shape. Most recent approaches identify local affine transformations for parts of the mesh, model desired interpolations of the affine transformations, and then optimize vertex positions to conform with the desired transformations. However, the local interpolation of the rotational part is non‐trivial for more than two input configurations and ambiguous if the meshes are deformed significantly. We propose a solution to the vertex interpolation problem that starts from interpolating the local metric (edge lengths) and mean curvature (dihedral angles) and makes consistent choices of local affine transformations using shape matching applied to successively larger parts of the mesh. The local interpolation can be applied to any number of input vertex configurations and due to the hierarchical scheme for generating consolidated vertex positions, the approach is fast and can be applied to very large meshes.
Non‐rigid registration of 3D shapes is an essential task of increasing importance as commodity depth sensors become more widely available for scanning dynamic scenes. Non‐rigid registration is much ...more challenging than rigid registration as it estimates a set of local transformations instead of a single global transformation, and hence is prone to the overfitting issue due to underdetermination. The common wisdom in previous methods is to impose an ℓ2‐norm regularization on the local transformation differences. However, the ℓ2‐norm regularization tends to bias the solution towards outliers and noise with heavy‐tailed distribution, which is verified by the poor goodness‐of‐fit of the Gaussian distribution over transformation differences. On the contrary, Laplacian distribution fits well with the transformation differences, suggesting the use of a sparsity prior. We propose a sparse non‐rigid registration (SNR) method with an ℓ1‐norm regularized model for transformation estimation, which is effectively solved by an alternate direction method (ADM) under the augmented Lagrangian framework. We also devise a multi‐resolution scheme for robust and progressive registration. Results on both public datasets and our scanned datasets show the superiority of our method, particularly in handling large‐scale deformations as well as outliers and noise.
This article presents an analysis of the content of geometric transformations in three collections of mathematics textbooks for high school selected by the last Programa Nacional do Livro Didático ...(PNLD). In this analysis, we sought to identify how the geometric transformations of translation, reflection, rotation and homotetia are presented, what are the tasks proposed on the content and the assignments for teachers who use these books. The analyzes were constituted from literature review of task, including problem situation, and exercise. It was found that there was a lack of deepening in some units of the analyzed books, the absence of a significant number of problems instead of exercises and the scarcity of suggestions for a more significant teacher’s performance at High School.
With the increase in demand for identification of authenticity of the digital images, researchers are widely studying the image forgery detection techniques. Copy-move forgery is amongst the commonly ...used forgery, which is performed by copying a part of an image and then pasting it on the same or different image. This results in the concealing of image content. Most of the existing copy-move forgery detection techniques are subjected to degradation in results, under the effect of geometric transformations. In this paper, a Discrete Cosine Transformation (DCT) and Singular Value Decomposition (SVD) based technique is proposed to detect the copy-move image forgery. DCT is used to transform the image from the spatial domain to the frequency domain and SVD is used to reduce the feature vector dimension. Combination of DCT and SVD makes the proposed scheme robust against compression, geometric transformations, and noise. For classification of images as forged or authentic, Support Vector Machine (SVM) classifier is used on the feature set. Once the image is detected as forged, then for the localization of forged region, K-means clustering is used on the feature vector. According to the distance threshold, similar blocks are identified and marked. The application of SVD provides stability and invariance from geometric transformations. Evaluation of the proposed scheme is done with and without post-processing operations on the images, both at the pixel level and image level. The proposed scheme outperforms the various state-of-the-art techniques of Copy-Move Forgery Detection (CMFD) in terms of accuracy, precision, recall and F
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parameters. Moreover, the proposed scheme also provides better results against rotation, scaling, noise and JPEG compression.
The development of the future's smart cities will increasingly depend on autonomous vehicles. For example, object detection, tracking, path planning, and sentiment or intent recognition, among other ...things, have all been the subject of several suggestions in recent years in an effort to address specific components of the working pipeline and develop a useful end-to-end system. This article presents a simple benchmark to evaluate the performance of object identification algorithms under deteriorating image quality. In this article, the input image is taken from the car object detection dataset, then, it gets preprocessed. The preprocessing involved Histogram equalization-based contrast enhancement, geometric transformations, and image augmentation. After getting preprocessed the region of interest is segmented with the help of a mask convolutional recurrent neural network (MCRNN). Then, the segmented region is classified using the long short-term memory (LSTM) with the self-improved chimp optimization algorithm. Object detection is performed through the proposed model called MCRNN-SICLSTM. The implementation is performed using MATLAB software. The performance of the proposed model is compared with the existing techniques using the performance metrics accuracy, precision, recall, F-Measure, mean absolute error, mean square error, false positive rate, false negative rate. The accuracy achieved is 99.23%.
A set D of disks in the plane is said to be pierced by a point set P if each disk in D contains a point of P. Any set of pairwise intersecting unit disks can be pierced by 3 points (Hadwiger and ...Debrunner (1955) 7). Stachó and independently Danzer established that any set of pairwise intersecting arbitrary disks can be pierced by 4 points (Stachó (1981–1984) 16. Danzer (1986) 4). Existing linear-time algorithms for finding a set of 4 or 5 points that pierce pairwise intersecting disks of arbitrary radius use the LP-type problem as a subroutine. We present simple linear-time algorithms for finding 3 points for piercing pairwise intersecting unit disks, and 5 points for piercing pairwise intersecting disks of arbitrary radius. Our algorithms use simple geometric transformations and avoid heavy machinery. We also show that 3 points are sometimes necessary for piercing pairwise intersecting unit disks.
Este artigo é um relato de experiência de um projeto desenvolvido com alunos de turmas de uma escola estadual da Paraíba. O projeto teve duração de dois bimestres em 2021, no qual além das aulas ...remotas para formação dos conceitos de Geometria Analítica, Transformações e Projeções Cartográficas foram propostas atividades com o uso de smartphones, utilizando o aplicativo GeoGebra. Tal aplicação metodológica teve como ponto de destaque o fato de oportunizar ao estudante o protagonismo na sua aprendizagem, com o professor atuando apenas no seu real papel de criador, incentivador e orientador nestas atividades em prol do alcance dos objetivos propostos. De modo geral, observou-se uma maior dedicação dos estudantes nestas atividades tendo em vista a dinamicidade do processo de ensino na busca duma aprendizagem significativa.
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