CNR Institute for Applied Mathematics and Information Technologies

Milano, Italy

CNR Institute for Applied Mathematics and Information Technologies

Milano, Italy
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Patane G.,CNR Institute for Applied Mathematics and Information Technologies
Computer Aided Geometric Design | Year: 2017

In engineering, geographical applications, bio-informatics, and scientific visualisation, a variety of phenomena is described by data modelled as the values of a scalar function defined on a surface or a volume, and critical points (i.e., maxima, minima, saddles) usually represent a relevant information about the input data or an underlying phenomenon. Furthermore, the distribution of the critical points is crucial for geometry processing and shape analysis; e.g., for controlling the number of patches in quadrilateral remeshing and the number of nodes of Reeb graphs and Morse-Smale complexes. In this context, we address the design of a smooth function, whose maxima, minima, and saddles are selected by the user or imported from a template (e.g., Laplacian eigenfunctions, diffusion maps). In this way, we support the selection of the saddles of the resulting function and not only its extrema, which is one of the main limitations of previous work. Then, we discuss the meshless approximation of an input scalar function by preserving its persistent critical points and its local behaviour, as encoded by the spatial distribution and shape of the level-sets. Both problems are addressed by computing an implicit approximation with radial basis functions, which is independent of the discretisation of differential operators and of assumptions on the sampling of the input domain. This approximation allows us to introduce a meshless iso-contouring and classification of the critical points, which are characterised in terms of the differential properties of the meshless approximation and of the geometry of the input surface, as encoded by its first and second fundamental form. Furthermore, the computation is performed at an arbitrary resolution by locally refining the input surface and by applying differential calculus to the meshless approximation. As main applications, we consider the approximation and analysis of scalar functions on both 3D shapes and volumes in graphics, Geographic Information Systems, medicine, and bio-informatics. © 2017 Elsevier B.V.

Patane G.,CNR Institute for Applied Mathematics and Information Technologies
Journal of Advanced Research | Year: 2015

This paper proposes an accurate, computationally efficient, and spectrum-free formulation of the heat diffusion smoothing on 3D shapes, represented as triangle meshes. The idea behind our approach is to apply a (. r, r)-degree Padé-Chebyshev rational approximation to the solution of the heat diffusion equation. The proposed formulation is equivalent to solve r sparse, symmetric linear systems, is free of user-defined parameters, and is robust to surface discretization. We also discuss a simple criterion to select the time parameter that provides the best compromise between approximation accuracy and smoothness of the solution. Finally, our experiments on anatomical data show that the spectrum-free approach greatly reduces the computational cost and guarantees a higher approximation accuracy than previous work. © 2014.

Laga H.,University of South Australia | Mortara M.,CNR Institute for Applied Mathematics and Information Technologies | Spagnuolo M.,CNR Institute for Applied Mathematics and Information Technologies
ACM Transactions on Graphics | Year: 2013

We address the problem of automatic recognition of functional parts of man-made 3D shapes in the presence of significant geometric and topological variations. We observe that under such challenging circumstances, the context of a part within a 3D shape provides important cues for learning the semantics of shapes. We propose to model the context as structural relationships between shape parts and use them, in addition to part geometry, as cues for functionality recognition. We represent a 3D shape as a graph interconnecting parts that share some spatial relationships. We model the context of a shape part as walks in the graph. Similarity between shape parts can then be defined as the similarity between their contexts, which in turn can be efficiently computed using graph kernels. This formulation enables us to: (1) find part-wise semantic correspondences between 3D shapes in a nonsupervised manner and without relying on user-specified textual tags, and (2) design classifiers that learn in a supervised manner the functionality of the shape components. We specifically show that the performance of the proposed context-aware similarity measure in finding part-wise correspondences outperforms geometry-only-based techniques and that contextual analysis is effective in dealing with shapes exhibiting large geometric and topological variations. © 2013 ACM.

Hughes T.J.R.,University of Texas at Austin | Evans J.A.,University of Colorado at Boulder | Reali A.,University of Pavia | Reali A.,CNR Institute for Applied Mathematics and Information Technologies
Computer Methods in Applied Mechanics and Engineering | Year: 2014

We study the spectral approximation properties of finite element and NURBS spaces from a global perspective. We focus on eigenfunction approximations and discover that the L2-norm errors for finite element eigenfunctions exhibit pronounced "spikes" about the transition points between branches of the eigenvalue spectrum. This pathology is absent in NURBS approximations. By way of the Pythagorean eigenvalue error theorem, we determine that the squares of the energy-norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the L2-norm eigenfunction errors. The spurious behavior of the higher eigenvalues for standard finite elements is well-known and therefore inherited by the energy-norm errors along with the spikes in the L2-norm of the eigenfunction errors. The eigenvalue pathology is absent for NURBS. The implications of these results to the corresponding elliptic boundary-value problem and parabolic and hyperbolic initial-value problems are discussed. © 2013 Elsevier B.V.

Reali A.,University of Pavia | Reali A.,CNR Institute for Applied Mathematics and Information Technologies | Gomez H.,University of La Coruña
Computer Methods in Applied Mechanics and Engineering | Year: 2015

In this paper, IGA collocation methods are for the first time introduced for the solution of thin structural problems described by the Bernoulli-Euler beam and Kirchhoff plate models. In particular, a precise description of the proposed methods, of the relevant implementation details, and of the strategy to efficiently deal with different combinations of boundary conditions is given. Finally, several numerical experiments confirm that the proposed formulations represent an efficient and geometrically flexible tool for the simulation of thin structures. © 2014 Elsevier B.V.

Patane G.,CNR Institute for Applied Mathematics and Information Technologies
Fuzzy Sets and Systems | Year: 2011

Investigating the relations between the least-squares approximation techniques and the Fuzzy Transform, in this paper we show that the Discrete Fuzzy Transform is invariant with respect to the interpolating and least-squares approximation. Additionally, the Fuzzy Transform is evaluated at any point by simply resampling the continuous approximation underlying the input data. Using numerical linear algebra, we also derive new properties (e.g., stability to noise, additivity with respect to the input data) and characterizations (e.g., radial and dual membership maps) of the Discrete Fuzzy Transform. Finally, we define the geometry- and confidence-driven Discrete Fuzzy Transforms, which take into account the intrinsic geometry and the confidence weights associated to the data. © 2010 Elsevier B.V. All rights reserved.

Fiaschi A.,CNR Institute for Applied Mathematics and Information Technologies
Networks and Heterogeneous Media | Year: 2010

A quasistatic evolution problem for a phase transition model with nonconvex energy density is considered in terms of Young measures. We focus on the particular case of a finite number of phases. The new feature consists in the usage of suitable regularity arguments in order to prove an existence result for a notion of evolution presenting some improvements with respect to the one defined in [13], for infinitely many phases. © American Institute of Mathematical Sciences.

Biasotti S.,CNR Institute for Applied Mathematics and Information Technologies
3DOR'10 - Proceedings of the 2010 ACM Workshop on 3D Object Retrieval, Co-located with ACM Multimedia 2010 | Year: 2010

Spectral analysis provides a library of shape description elements intrinsically defined by the shape itself. Among all, the eigenfunctions of the Laplace-Beltrami operator can be thought as a set of real valued functions that implicitly abstract and code the shape. In this scenario, this paper introduces a new shape signature derived from the mutual distances between couples of Laplace-Beltrami eigenfunctions. This signature can be seen as a feature vector that acts as an intrinsic shape pattern. Experiments show that it can be effectively used for shape retrieval and its robustness with respect to changes in topology, model resampling, small perturbations and pose variations.

Rotondi R.,CNR Institute for Applied Mathematics and Information Technologies
Geophysical Journal International | Year: 2013

In this paper, some methods for scoring the performances of an earthquake forecasting probabilitymodel are applied retrospectively for different goals. The time-dependent occurrenceprobabilities of a renewal process are tested against earthquakes of Mw = 5.3 recorded inItaly according to decades of the past century. An aim was to check the capability of themodel to reproduce the data by which the model was calibrated. The scoring procedures usedcan be distinguished on the basis of the requirement (or absence) of a reference model and ofprobability thresholds. Overall, a rank-based score, information gain, gambling scores, indicesused in binary predictions and their loss functions are considered. The definition of variousprobability thresholds as percentages of the hazard functions allows proposals of the valuesassociated with the best forecasting performance as alarm level in procedures for seismic riskmitigation. Some improvements are then made to the input data concerning the completenessof the historical catalogue and the consistency of the composite seismogenic sources with thehypotheses of the probability model. Another purpose of this study was thus to obtain hints onwhat is the most influential factor and on the suitability of adopting the consequent changes ofthe data sets. This is achieved by repeating the estimation procedure of the occurrence probabilitiesand the retrospective validation of the forecasts obtained under the new assumptions.According to the rank-based score, the completeness appears to be the most influential factor,while there are no clear indications of the usefulness of the decomposition of some compositesources, although in some cases, it has led to improvements of the forecast. © The Authors 2013 Published by Oxford University Press on behalf of The Royal Astronomical Society.

Patane G.,CNR Institute for Applied Mathematics and Information Technologies
Computer Aided Geometric Design | Year: 2013

Recent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the heat diffusion kernel. In this paper, we focus our attention on the properties (e.g.; scale-invariance, semi-group property, robustness to noise) of the wFEM heat kernel, recently proposed in Patanè and Falcidieno (2010), and its application to shape comparison and feature-driven approximation. After proving that the wFEM heat kernel is intrinsically scale-covariant (i.e.; without shape or kernel normalization) and scale-invariant through a normalization of the Laplacian eigenvalues, we experimentally verify that the wFEM heat kernel descriptors are more robust against shape/scale changes and provide better matching performances with respect to previous work. In the space F(M) of piecewise linear scalar functions defined on a triangle mesh M, we introduce the wFEM heat kernel Kt, which is used to increase the degree of flexibility in the design of geometry-aware basis functions. Furthermore, we efficiently compute scale-based representations of maps on M by specializing the Chebyshev method through the solution of a set of sparse linear systems, thus avoiding the spectral decomposition of the Laplacian matrix. Finally, the scalar product induced by Kt makes F(M) a Reproducing Kernel Hilbert Space, whose (reproducing) kernel is the linear FEM heat kernel, and induces the FEM diffusion distances on M. © 2013 Elsevier B.V.

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