178 research outputs found
Consistent Computation of First- and Second-Order Differential Quantities for Surface Meshes
Differential quantities, including normals, curvatures, principal directions,
and associated matrices, play a fundamental role in geometric processing and
physics-based modeling. Computing these differential quantities consistently on
surface meshes is important and challenging, and some existing methods often
produce inconsistent results and require ad hoc fixes. In this paper, we show
that the computation of the gradient and Hessian of a height function provides
the foundation for consistently computing the differential quantities. We
derive simple, explicit formulas for the transformations between the first- and
second-order differential quantities (i.e., normal vector and principal
curvature tensor) of a smooth surface and the first- and second-order
derivatives (i.e., gradient and Hessian) of its corresponding height function.
We then investigate a general, flexible numerical framework to estimate the
derivatives of the height function based on local polynomial fittings
formulated as weighted least squares approximations. We also propose an
iterative fitting scheme to improve accuracy. This framework generalizes
polynomial fitting and addresses some of its accuracy and stability issues, as
demonstrated by our theoretical analysis as well as experimental results.Comment: 12 pages, 12 figures, ACM Solid and Physical Modeling Symposium, June
200
Jet_fitting_3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting
Surfaces of R^3 are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in Computer Graphics, Computer Aided Design, Computational Geometry, Computer Vision. One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpolation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, extraction of the differential properties. This paper presents a generic C++ software package solving these problems. On the theoretical side and as established in a companion paper, the interpolation and approximation methods provided achieve the best asymptotic error bounds known to date. On the implementation side and following state-of-the-art coding rules in Computational Geometry, genericity of the package is achieved thanks to three template classes accounting for (a) the type of the input points (b) the internal geometric computations and (c) the linear algebra operations. An instantiation within the Computational Geometry Algorithms Library (CGAL, version 3.3) and using CLAPACK is also provided
Discrete schemes for Gaussian curvature and their convergence
In this paper, several discrete schemes for Gaussian curvature are surveyed.
The convergence property of a modified discrete scheme for the Gaussian
curvature is considered. Furthermore, a new discrete scheme for Gaussian
curvature is resented. We prove that the new scheme converges at the regular
vertex with valence not less than 5. By constructing a counterexample, we also
show that it is impossible for building a discrete scheme for Gaussian
curvature which converges over the regular vertex with valence 4. Finally,
asymptotic errors of several discrete scheme for Gaussian curvature are
compared
Распределение значений локальной кривизны как структурный признак для off-line верификации рукописной подписи
In the paper, a new feature for describing a digital image of a handwritten signature based on the frequency distribution of the values of the local curvature of the signature contours, is proposed. The calculation of this feature on the binary image of a signature is described in detail. A normalized histogram of distributions of local curvature values for 40 bins is formed. The frequency values recorded as a 40-dimensional vector are called the local curvature code of the signature.During verification, the proximity of signature pairs is determined by correlation between curvature codes and LBP codes described by the authors in [23]. To perform the signature verification procedure, a two-dimensional feature space is constructed containing images of the proximity of signature pairs. When verifying a signature with N authentic signatures of the same person, N(N-1)/2 patterns of the proximity of pairs of genuine signatures and N images of pairs of proximity of the analyzed signature with genuine signatures are presented in the feature space. The Support Vector Machine (SVM) is used as a classifier.Experimental studies were carried out on digitized images of genuine and fake signatures from two databases. The accuracy of automatic verification of signatures on the publicly available CEDAR database was 99,77 % and on TUIT was 88,62 %.В работе предложен новый признак описания цифрового изображения рукописной подписи на базе частотного распределения значений локальной кривизны контуров этой подписи. Подробно описывается вычисление этого признака на бинарном изображении подписи. Формируется нормализованная гистограмма распределений значений локальной кривизны для 40 интервалов. Частотные значения, записанные в виде 40-мерного вектора, названы кодом локальной кривизны подписи.При верификации близость двух подписей определяется корреляцией между кодами кривизны и LBP-кодами, описанными авторами в работе [23]. Для выполнения процедуры верификации подписи строится двумерное признаковое пространство, содержащее образы корреляционной близости пар подписей. При верификации подписи с N подлинными подписями этого же человека в признаковом пространстве представлено N(N-1)/2 образов близости пар подлинных подписей и N образов пар близости анализируемой подписи с подлинными. В качестве классификатора используется машина опорных векторов (SVM).Экспериментальные исследования выполнены на оцифрованных изображениях подлинных и фальшивых подписей из двух баз. Точность автоматической верификации подписей на общедоступной базе CEDAR составила 99,77 %, а на базе TUIT 88,62 %
Robust Geometry Estimation using the Generalized Voronoi Covariance Measure
The Voronoi Covariance Measure of a compact set K of R^d is a tensor-valued
measure that encodes geometric information on K and which is known to be
resilient to Hausdorff noise but sensitive to outliers. In this article, we
generalize this notion to any distance-like function delta and define the
delta-VCM. We show that the delta-VCM is resilient to Hausdorff noise and to
outliers, thus providing a tool to estimate robustly normals from a point cloud
approximation. We present experiments showing the robustness of our approach
for normal and curvature estimation and sharp feature detection
Point Normal Orientation and Surface Reconstruction by Incorporating Isovalue Constraints to Poisson Equation
Oriented normals are common pre-requisites for many geometric algorithms
based on point clouds, such as Poisson surface reconstruction. However, it is
not trivial to obtain a consistent orientation. In this work, we bridge
orientation and reconstruction in implicit space and propose a novel approach
to orient point clouds by incorporating isovalue constraints to the Poisson
equation. Feeding a well-oriented point cloud into a reconstruction approach,
the indicator function values of the sample points should be close to the
isovalue. Based on this observation and the Poisson equation, we propose an
optimization formulation that combines isovalue constraints with local
consistency requirements for normals. We optimize normals and implicit
functions simultaneously and solve for a globally consistent orientation. Owing
to the sparsity of the linear system, an average laptop can be used to run our
method within reasonable time. Experiments show that our method can achieve
high performance in non-uniform and noisy data and manage varying sampling
densities, artifacts, multiple connected components, and nested surfaces
Распределение значений локальной кривизны как структурный признак для off-line верификации рукописной подписи
В работе предложен новый признак описания цифрового изображения рукописной подписи на базе частотного распределения значений локальной кривизны контуров этой подписи. Подробно описывается вычисление этого признака на бинарном изображении подписи. Формируется нормализованная гистограмма распределений значений локальной кривизны для 40 интервалов. Частотные значения, записанные в виде 40-мерного вектора, названы кодом локальной кривизны подписи. При верификации близость двух подписей определяется корреляцией между кодами кривизны и LBP-кодами, описанными авторами в работе [23]. Для выполнения процедуры верификации подписи строится двумерное признаковое пространство, содержащее образы корреляционной близости пар подписей. При верификации подписи с N подлинными подписями этого же человека в признаковом пространстве представлено N(N-1)/2 образов близости пар подлинных подписей и N образов пар близости анализируемой подписи с подлинными. В качестве классификатора используется машина опорных векторов (SVM). Экспериментальные исследования выполнены на оцифрованных изображениях подлинных и фальшивых подписей из двух баз. Точность автоматической верификации подписей на общедоступной базе CEDAR составила 99,77 %, а на базе TUIT 88,62 %
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