A Generalised RBF Finite Difference Approach to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

Radial Basis Functions have traditionally been used to provide a continuous interpolation of scattered data sets. However, this interpolation also allows for the reconstruction of partial derivatives throughout the solution field, which can then be used to drive the solution of a partial differential equation. Since the interpolation takes place on a scattered dataset with no local connectivity, the solution is essentially meshless. RBF-based methods have been successfully used to solve a wide variety of PDEs in this fashion. Such full-domain RBF methods are highly flexible and can exhibit spectral convergence rates Madych & Nelson (1990). However, in their traditional implementation the fully-populated matrix systems which are produced lead to computational complexities of at least order-N2 with datasets of size N. In addition, they suffer fromincreasingly poor numerical conditioning as the size of the dataset grows, and also with increasingly flat interpolating functions. This is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as demonstrated in Driscoll & Fornberg (2002)), and is described by Robert Schaback Schaback (1995) as the uncertainty relation; better conditioning is associated with worse accuracy, and worse conditioning is associated with improved accuracy. Many techniques have been developed to reduce the effect of the uncertainty relation in the traditional RBF formulation, such as RBF-specific preconditioners Baxter (2002); Beatson et al. (1999); Brown (2005); Ling & Kansa (2005), or adaptive selection of data centres Ling et al. (2006); Ling & Schaback (2004). However, at present the only reliable methods of controlling numerical ill-conditioning and computational cost as problem size increases are domain decomposition Hernandez Rosales & Power (2007); Wong et al. (1999); Zhang (2007); Zhou et al. (2003), or the use of locally supported basis functions Fasshauer (1999); Schaback (1997); Wendland (1995); Wu (1995). In this work the domain decomposition principle is applied, forming a large number of heavily overlapping systems that cover the solution domain. A small RBF collocation system is formed around each global data centre, with each collocation system used to approximate the governing PDE at its centrepoint, in terms of the solution value at surrounding collocation points. This leads to a sparse global linear system which may be solved using a variety of standard solvers. In this way, the proposed formulation emulates a finite difference method, with the RBF collocation systems replacing the polynomial interpolation functions used in traditional finite difference methods. However, unlike such polynomial functions RBF collocation is well suited to scattered data, and the method may be applied to both structured and unstructured datasets without modification. The method is applied here to solve the nonlinear heat conduction equation. In order to reduce the nonlinearity in the governing equation the Kirchhoff integral transformation is applied, and the transformed equation is solved using a Picard iterative process. The application of the Kirchhoff transform necessitates that the thermal property functions be transformed to Kirchhoff space also. If the thermal properties are a known and integrable function of temperature then the transformation may be performed analytically. Otherwise, an integration-interpolation procedure can be performed using 1D radial basis functions, as described in Stevens & Power (2010). In recent years a number of local RBF collocation techniques have been proposed, and applied a wide variety of problems (for example; Divo & Kassab (2007); Lee et al. (2003); Sarler & Vertnik (2006); Wright & Fornberg (2006)). A more comprehensive review of such methods is given in Stevens et al. (2009). Unlike most local RBF collocation methods that are used in the literature, the technique described here utilises the Hermitian RBF collocation formulation (see section 2 for more details), and allows both the PDE-boundary and PDE-governing operators to be included within in the local collocation systems. This inclusion of the governing PDE within the basis functions is shown in Stevens et al. (2009) to significantly improve the accuracy and stability of solutions obtained for linear transport problems. Additionally, the incorporation of information about the convective velocity field into the basis functionswas shown to have a stabilising effect, similar to traditional upwinding methods but without the requirement to alter the stencil configuration based on the local convective field. The standard approach to the solution of linear and nonlinear heat conduction problems is the use of finite difference and finite volume methods with simple polynomial interpolants Bejan (1993); Holman (2002); Kreith & Bohn (2000). Due to the dominance of diffusion in most cases, central differencing techniques are commonly used to compute the heat fluxes. However, limiter methods (such as the unconditionally stable TVD schemes) may be used for nonlinear heat conduction problems where the effective convection term, which results from the non-zero variation of thermal conductivity with temperature, can be expected to approach the magnitude of the diffusive term (see, for example, Shen & Han (2002)). Full-domain RBF methods have also been examined for use with nonlinear heat conduction problems (see Chantasiriwan (2007)), however such methods are restricted to small dataset sizes, due to the computational cost and numerical conditioning experienced by full-domain RBF techniques on large datasets. The present work demonstrates how local RBF collocation may be used as an alternative to traditional finite difference and finite volume methods, for nonlinear heat conduction problems. The described method retains freedom from a volumetric mesh, while allowing solution over unstructured datasets. A central stencil configuration is used in each case, and the solution is stabilised via the inclusion of the governing and boundary PDEs within the local collocation systems (\u201cimplicit upwinding\u201d), rather than by adjusting the stencil configuration based on the local solution field (\u201ctraditional upwinding\u201d). The method is validated using a transient numerical example with a known analytical solution (see section 4), and the ability of the formulation to handle strongly nonlinear problems is demonstrated in the solution of a food freezing problem (see section 5)

ONLINE HIERARCHICAL MODELS FOR SURFACE RECONSTRUCTION

Applications based on three-dimensional object models are today very common, and can be found in many fields as design, archeology, medicine, and entertainment. A digital 3D model can be obtained by means of physical object measurements performed by using a 3D scanner. In this approach, an important step of the 3D model building process consists of creating the object's surface representation from a cloud of noisy points sampled on the object itself. This process can be viewed as the estimation of a function from a finite subset of its points. Both in statistics and machine learning this is known as a regression problem. Machine learning views the function estimation as a learning problem to be addressed by using computational intelligence techniques: the points represent a set of examples and the surface to be reconstructed represents the law that has generated them. On the other hand, in many applications the cloud of sampled points may become available only progressively during system operation. The conventional approaches to regression are therefore not suited to deal efficiently with this operating condition. The aim of the thesis is to introduce innovative approaches to the regression problem suited for achieving high reconstruction accuracy, while limiting the computational complexity, and appropriate for online operation. Two classical computational intelligence paradigms have been considered as basic tools to address the regression problem: namely the Radial Basis Functions and the Support Vector Machines. The original and innovative aspect introduced by this thesis is the extension of these tools toward a multi-scale incremental structure, based on hierarchical schemes and suited for online operation. This allows for obtaining modular, scalable, accurate and efficient modeling procedures with training algorithms appropriate for dealing with online learning. Radial Basis Function Networks have a fast configuration procedure that, operating locally, does not require iterative algorithms. On the other side, the computational complexity of the configuration procedure of Support Vector Machines is independent from the number of input variables. These two approaches have been considered in order to analyze advantages and limits of each of them due to the differences in their intrinsic nature

A hermite radial basis functions control volume numerical method to simulate transport problems

This thesis presents a Control Volume (CV) method for transient transport problems where the cell surface fluxes are reconstructed using local interpolation functions that besides interpolating the nodal values of the field variable, also satisfies the governing equation at some auxiliary points in the interpolation stencils. The interpolation function relies on a Hermitian Radial Basis Function (HRBF) mesh less collocation approach to find the solution of auxiliary local boundary/initial value problems, which are solved using the same time integration scheme adopted to update the global control volume solution. By the use of interpolation functions that approximate the governing equation, a form of analytical upwinding scheme is achieved without the need of using predefined interpolation stencils according to the magnitude and direction of the local advective velocity. In this way, the interpolation formula retains the desired information about the advective velocity field, allowing the use of centrally defined stencils even in the case of advective dominant problems. This new CV approach, which is referred to as the CV-HRBF method, is applied to a series of transport problems characterised by high Peclet number. This method is also more flexible than the classical CV formulations because the boundary conditions are explicitly imposed in the interpolation formula, without the need for artificial schemes (e.g. utilising dummy cells). The flexibility of the local meshless character of the CVHRBF is shown in the modelling of the saturated zone of the unconfined aquifer where a mesh adapting algorithm is needed to track the phreatic surface (moving boundary). Due to the use of a local RBF interpolation, the dynamic boundary condition can be applied in an arbitrary number of points on the phreatic surface, independently from the mesh element. The robustness of the Hermite interpolation is exploited to formulate a non-overlapping non-iterative multi-domain scheme where physical matching conditions are satisfied locally, i.e. imposing the continuity of the function and flux at the sub-domain interface

A hermite radial basis functions control volume numerical method to simulate transport problems

This thesis presents a Control Volume (CV) method for transient transport problems where the cell surface fluxes are reconstructed using local interpolation functions that besides interpolating the nodal values of the field variable, also satisfies the governing equation at some auxiliary points in the interpolation stencils. The interpolation function relies on a Hermitian Radial Basis Function (HRBF) mesh less collocation approach to find the solution of auxiliary local boundary/initial value problems, which are solved using the same time integration scheme adopted to update the global control volume solution. By the use of interpolation functions that approximate the governing equation, a form of analytical upwinding scheme is achieved without the need of using predefined interpolation stencils according to the magnitude and direction of the local advective velocity. In this way, the interpolation formula retains the desired information about the advective velocity field, allowing the use of centrally defined stencils even in the case of advective dominant problems. This new CV approach, which is referred to as the CV-HRBF method, is applied to a series of transport problems characterised by high Peclet number. This method is also more flexible than the classical CV formulations because the boundary conditions are explicitly imposed in the interpolation formula, without the need for artificial schemes (e.g. utilising dummy cells). The flexibility of the local meshless character of the CVHRBF is shown in the modelling of the saturated zone of the unconfined aquifer where a mesh adapting algorithm is needed to track the phreatic surface (moving boundary). Due to the use of a local RBF interpolation, the dynamic boundary condition can be applied in an arbitrary number of points on the phreatic surface, independently from the mesh element. The robustness of the Hermite interpolation is exploited to formulate a non-overlapping non-iterative multi-domain scheme where physical matching conditions are satisfied locally, i.e. imposing the continuity of the function and flux at the sub-domain interface

3D Reconstruction of Anatomical Structures Using Interpolation Techniques and local Approaches

The reconstruction of the surface is the process by which a 3D object is reproduced from a collection of discrete values that sample the shape. These values are generally called point cloud. Commonly, the reconstruction methods are based on the fundamental properties of the point clouds, which are the density samples, noise, missing data, and outliers. We aim to reconstruct the surface of anatomical structures from medical images; Consider two main problems that are missing data and the presence of noise. We resolve the missing data by generating new samples from a set of contours based on Shape Morphing techniques. If we add noise to the previous problem, we must change the focus and therefore use an implicit rebuild method that is solid for the problems presented above. Finally, we combine part of the previous proposals to solve a specific problem, that occurs when we reconstruct medical images, when a contour is bifurcated into another. The methods are evaluated with the public database from medical images and compared with the standardized algorithms of state of the art and the Hausdorff distance is used to measure the perfanceResumen La reconstrucción de superficie es el proceso mediante el cual un objeto 3D se reproduce de una colección de valores discretos que muestran la forma. Estos valores generalmente son llamados nubes de puntos. Comúnmente, los métodos de reconstrucción se basan en las propiedades básicas de las nubes de puntos, que son la densidad de muestras, ruido, datos faltantes y los valores atípicos. Nuestro objetivo es reconstruir la superficie de estructuras anatómicas a partir de imágenes médicas. Consideraremos dos problemas principales que son los datos faltantes y la presencia de ruido. Resolvemos la falta de datos generando nuevas muestras a partir de un conjunto de contornos, basándonos en técnicas de Shape Morphing (forma cambiante). Si adicionamos ruido al problema anterior, debemos cambiar de enfoque por lo tanto utilizamos un método de reconstrucción implícita que se ha demostrado que es robusto a los problemas anteriormente mencionados. Por ´ultimo combinamos parte de las propuestas anteriores para resolver un problema específico, que se presenta cuando reconstruimos imágenes médicas, que se trata, cuando un contorno se bifurca en otros contornos. Los métodos se evaluarán sobre bases de datos públicas de imágenes médicas y se compararon con dos algoritmos estándar del estado del arte y la medición de rendimiento de reconstrucción será la distancia de HausdorffMaestrí

Implicit muscle models for interactive character skinning

En animation de personnages 3D, la déformation de surface, ou skinning, est une étape cruciale. Son rôle est de déformer la représentation surfacique d'un personnage pour permettre son rendu dans une succession de poses spécifiées par un animateur. La plausibilité et la qualité visuelle du résultat dépendent directement de la méthode de skinning choisie. Sa rapidité d'exécution et sa simplicité d'utilisation sont également à prendre en compte pour rendre possible son usage interactif lors des sessions de production des artistes 3D. Les différentes méthodes de skinning actuelles se divisent en trois catégories. Les méthodes géométriques sont rapides et simples d'utilisation, mais leur résultats manquent de plausibilité. Les approches s'appuyant sur des exemples produisent des résultats réalistes, elles nécessitent en revanche une base de données d'exemples volumineuse, et le contrôle de leur résultat est fastidieux. Enfin, les algorithmes de simulation physique sont capables de modéliser les phénomènes dynamiques les plus complexes au prix d'un temps de calcul souvent prohibitif pour une utilisation interactive. Les travaux décrits dans cette thèse s'appuient sur Implicit Skinning, une méthode géométrique corrective utilisant une représentation implicite des surfaces, qui permet de résoudre de nombreux problèmes rencontrés avec les méthodes géométriques classiques, tout en gardant des performances permettant son usage interactif. La contribution principale de ces travaux est un modèle d'animation qui prend en compte les effets des muscles des personnages et de leur interactions avec d'autres éléments anatomiques, tout en bénéficiant des avantages apportés par Implicit Skinning. Les muscles sont représentés par une surface d'extrusion le long d'axes centraux. Les axes des muscles sont contrôlés par une méthode de simulation physique simplifiée. Cette représentation permet de modéliser les collisions des muscles entre eux et avec les os, d'introduire des effets dynamiques tels que rebonds et secousses, tout en garantissant la conservation du volume, afin de représenter le comportement réel des muscles. Ce modèle produit des déformations plus plausibles et dynamiques que les méthodes géométriques de l'état de l'art, tout en conservant des performances suffisantes pour permettre son usage dans une session d'édition interactive. Elle offre de plus aux infographistes un contrôle intuitif sur la forme des muscles pour que les déformations obtenues se conforment à leur vision artistique.Surface deformation, or skinning is a crucial step in 3D character animation. Its role is to deform the surface representation of a character to be rendered in the succession of poses specified by an animator. The quality and plausiblity of the displayed results directly depends on the properties of the skinning method. However, speed and simplicity are also important criteria to enable their use in interactive editing sessions. Current skinning methods can be divided in three categories. Geometric methods are fast and simple to use, but their results lack plausibility. Example-based approaches produce realistic results, yet they require a large database of examples while remaining tedious to edit. Finally, physical simulations can model the most complex dynamical phenomena, but at a very high computational cost, making their interactive use impractical. The work presented in this thesis are based on, Implicit Skinning, is a corrective geometric approach using implicit surfaces to solve many issues of standard geometric skinning methods, while remaining fast enough for interactive use. The main contribution of this work is an animation model that adds anatomical plausibility to a character by representing muscle deformations and their interactions with other anatomical features, while benefiting from the advantages of Implicit Skinning. Muscles are represented by an extrusion surface along a central axis. These axes are driven by a simplified physics simulation method, introducing dynamic effects, such as jiggling. The muscle model guarantees volume conservation, a property of real-life muscles. This model adds plausibility and dynamics lacking in state-of-the-art geometric methods at a moderate computational cost, which enables its interactive use. In addition, it offers intuitive shape control to animators, enabling them to match the results with their artistic vision

MIPS-Fusion: Multi-Implicit-Submaps for Scalable and Robust Online Neural RGB-D Reconstruction

We introduce MIPS-Fusion, a robust and scalable online RGB-D reconstruction method based on a novel neural implicit representation -- multi-implicit-submap. Different from existing neural RGB-D reconstruction methods lacking either flexibility with a single neural map or scalability due to extra storage of feature grids, we propose a pure neural representation tackling both difficulties with a divide-and-conquer design. In our method, neural submaps are incrementally allocated alongside the scanning trajectory and efficiently learned with local neural bundle adjustments. The submaps can be refined individually in a back-end optimization and optimized jointly to realize submap-level loop closure. Meanwhile, we propose a hybrid tracking approach combining randomized and gradient-based pose optimizations. For the first time, randomized optimization is made possible in neural tracking with several key designs to the learning process, enabling efficient and robust tracking even under fast camera motions. The extensive evaluation demonstrates that our method attains higher reconstruction quality than the state of the arts for large-scale scenes and under fast camera motions

Spiking neurons in 3D growing self-organising maps

In Kohonen’s Self-Organising Maps (SOM) learning, preserving the map topology to simulate the actual input features appears to be a significant process. Misinterpretation of the training samples can lead to failure in identifying the important features that may affect the outcomes generated by the SOM model. Nonetheless, it is a challenging task as most of the real problems are composed of complex and insufficient data. Spiking Neural Network (SNN) is the third generation of Artificial Neural Network (ANN), in which information can be transferred from one neuron to another using spike, processed, and trigger response as output. This study, hence, embedded spiking neurons for SOM learning in order to enhance the learning process. The proposed method was divided into five main phases. Phase 1 investigated issues related to SOM learning algorithm, while in Phase 2; datasets were collected for analyses carried out in Phase 3, wherein neural coding scheme for data representation process was implemented in the classification task. Next, in Phase 4, the spiking SOM model was designed, developed, and evaluated using classification accuracy rate and quantisation error. The outcomes showed that the proposed model had successfully attained exceptional classification accuracy rate with low quantisation error to preserve the quality of the generated map based on original input data. Lastly, in the final phase, a Spiking 3D Growing SOM is proposed to address the surface reconstruction issue by enhancing the spiking SOM using 3D map structure in SOM algorithm with a growing grid mechanism. The application of spiking neurons to enhance the performance of SOM is relevant in this study due to its ability to spike and to send a reaction when special features are identified based on its learning of the presented datasets. The study outcomes contribute to the enhancement of SOM in learning the patterns of the datasets, as well as in proposing a better tool for data analysis

A Survey of Surface Reconstruction from Point Clouds

International audienceThe area of surface reconstruction has seen substantial progress in the past two decades. The traditional problem addressed by surface reconstruction is to recover the digital representation of a physical shape that has been scanned, where the scanned data contains a wide variety of defects. While much of the earlier work has been focused on reconstructing a piece-wise smooth representation of the original shape, recent work has taken on more specialized priors to address significantly challenging data imperfections, where the reconstruction can take on different representations – not necessarily the explicit geometry. We survey the field of surface reconstruction, and provide a categorization with respect to priors, data imperfections, and reconstruction output. By considering a holistic view of surface reconstruction, we show a detailed characterization of the field, highlight similarities between diverse reconstruction techniques, and provide directions for future work in surface reconstruction