Theory and applications of bijective barycentric mappings

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle's vertices, and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications it is desirable to extend the concept of barycentric coordinates from triangles to polygons. Several variants of such generalized barycentric coordinates have been proposed in recent years. An important application of barycentric coordinates consists of barycentric mappings, which allow to naturally warp a source polygon to a corresponding target polygon, or more generally, to create mappings between closed curves or polyhedra. The principal practical application is image warping, which takes as input a control polygon drawn around an image and smoothly warps the image by moving the polygon vertices. A required property of image warping is to avoid fold-overs in the resulting image. The problem of fold-overs is a manifestation of a larger problem related to the lack of bijectivity of the barycentric mapping. Unfortunately, bijectivity of such barycentric mappings can only be guaranteed for the special case of warping between convex polygons or by triangulating the domain and hence renouncing smoothness. In fact, for any barycentric coordinates, it is always possible to construct a pair of polygons such that the barycentric mapping is not bijective. In the first part of this thesis we illustrate three methods to achieve bijective mappings. The first method is based on the intuition that, if two polygons are sufficiently close, then the mapping is close to the identity and hence bijective. This suggests to ``split'' the mapping into several intermediate mappings and to create a composite barycentric mapping which is guaranteed to be bijective between arbitrary polygons, polyhedra, or closed planar curves. We provide theoretical bounds on the bijectivity of the composite mapping related to the norm of the gradient of the coordinates. The fact that the bound depends on the gradient implies that these bounds exist only if the gradient of the coordinates is bounded. We focus on mean value coordinates and analyse the behaviour of their directional derivatives and gradient at the vertices of a polygon. The composition of barycentric mappings for closed planar curves leads to the problem of blending between two planar curves. We suggest to solve it by linearly interpolating the signed curvature and then reconstructing the intermediate curve from the interpolated curvature values. However, when both input curves are closed, this strategy can lead to open intermediate curves. We present a new algorithm for solving this problem, which finds the closed curve whose curvature is closest to the interpolated values. Our method relies on the definition of a suitable metric for measuring the distance between two planar curves and an appropriate discretization of the signed curvature functions. The second method to construct smooth bijective mappings with prescribed behaviour along the domain boundary exploits the properties of harmonic maps. These maps can be approximated in different ways, and we discuss their respective advantages and disadvantages. We further present a simple procedure for reducing their distortion and demonstrate the effectiveness of our approach by providing examples. The last method relies on a reformulation of complex barycentric mappings, which allows us to modify the ``speed'' along the edges to create complex bijective mappings. We provide some initial results and an optimization procedure which creates complex bijective maps. In the second part we provide two main applications of bijective mapping. The first one is in the context of finite elements simulations, where the discretization of the computational domain plays a central role. In the standard discretization, the domain is triangulated with a mesh and its boundary is approximated by a polygon. We present an approach which combines parametric finite elements with smooth bijective mappings, leaving the choice of approximation spaces free. This approach allows to represent arbitrarily complex geometries on coarse meshes with curved edges, regardless of the domain boundary complexity. The main idea is to use a bijective mapping for automatically warping the volume of a simple parametrization domain to the complex computational domain, thus creating a curved mesh of the latter. The second application addresses the meshing problem and the possibility to solve finite element simulations on polygonal meshes. In this context we present several methods to discretize the bijective mapping to create polygonal and piece-wise polynomial meshes

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Geometry–aware finite element framework for multi–physics simulations: an algorithmic and software-centric perspective

In finite element simulations, the handling of geometrical objects and their discrete representation is a critical aspect in both serial and parallel scientific software environments. The development of codes targeting such envinronments is subject to great development effort and man-hours invested. In this thesis we approach these issues from three fronts. First, stable and efficient techniques for the transfer of discrete fields between non matching volume or surface meshes are an essential ingredient for the discretization and numerical solution of coupled multi-physics and multi-scale problems. In particular L2-projections allows for the transfer of discrete fields between unstructured meshes, both in the volume and on the surface. We present an algorithm for parallelizing the assembly of the L2-transfer operator for unstructured meshes which are arbitrarily distributed among different processes. The algorithm requires no a priori information on the geometrical relationship between the different meshes. Second, the geometric representation is often a limiting factor which imposes a trade-off between how accurately the shape is described, and what methods can be employed for solving a system of differential equations. Parametric finite-elements and bijective mappings between polygons or polyhedra allow us to flexibly construct finite element discretizations with arbitrary resolutions without sacrificing the accuracy of the shape description. Such flexibility allows employing state-of-the-art techniques, such as geometric multigrid methods, on meshes with almost any shape.t, the way numerical techniques are represented in software libraries and approached from a development perspective, affect both usability and maintainability of such libraries. Completely separating the intent of high-level routines from the actual implementation and technologies allows for portable and maintainable performance. We provide an overview on current trends in the development of scientific software and showcase our open-source library utopia

Understanding and controlling leakage in machine learning

Machine learning models are being increasingly adopted in a variety of real-world scenarios. However, the privacy and confidentiality implications introduced in these scenarios are not well understood. Towards better understanding such implications, we focus on scenarios involving interactions between numerous parties prior to, during, and after training relevant models. Central to these interactions is sharing information for a purpose e.g., contributing data samples towards a dataset, returning predictions via an API. This thesis takes a step toward understanding and controlling leakage of private information during such interactions. In the first part of the thesis we investigate leakage of private information in visual data and specifically, photos representative of content shared on social networks. There is a long line of work to tackle leakage of personally identifiable information in social photos, especially using face- and body-level visual cues. However, we argue this presents only a narrow perspective as images reveal a wide spectrum of multimodal private information (e.g., disabilities, name-tags). Consequently, we work towards a Visual Privacy Advisor that aims to holistically identify and mitigate private risks when sharing social photos. In the second part, we address leakage during training of ML models. We observe learning algorithms are being increasingly used to train models on rich decentralized datasets e.g., personal data on numerous mobile devices. In such cases, information in the form of high-dimensional model parameter updates are anonymously aggregated from participating individuals. However, we find that the updates encode sufficient identifiable information and allows them to be linked back to participating individuals. We additionally propose methods to mitigate this leakage while maintaining high utility of the updates. In the third part, we discuss leakage of confidential information during inference time of black-box models. In particular, we find models lend themselves to model functionality stealing attacks: an adversary can interact with the black-box model towards creating a replica `knock-off' model that exhibits similar test-set performances. As such attacks pose a severe threat to the intellectual property of the model owner, we also work towards effective defenses. Our defense strategy by introducing bounded and controlled perturbations to predictions can significantly amplify model stealing attackers' error rates. In summary, this thesis advances understanding of privacy leakage when information is shared in raw visual forms, during training of models, and at inference time when deployed as black-boxes. In each of the cases, we further propose techniques to mitigate leakage of information to enable wide-spread adoption of techniques in real-world scenarios.Modelle für maschinelles Lernen werden zunehmend in einer Vielzahl realer Szenarien eingesetzt. Die in diesen Szenarien vorgestellten Auswirkungen auf Datenschutz und Vertraulichkeit wurden jedoch nicht vollständig untersucht. Um solche Implikationen besser zu verstehen, konzentrieren wir uns auf Szenarien, die Interaktionen zwischen mehreren Parteien vor, während und nach dem Training relevanter Modelle beinhalten. Das Teilen von Informationen für einen Zweck, z. B. das Einbringen von Datenproben in einen Datensatz oder die Rückgabe von Vorhersagen über eine API, ist zentral für diese Interaktionen. Diese Arbeit verhilft zu einem besseren Verständnis und zur Kontrolle des Verlusts privater Informationen während solcher Interaktionen. Im ersten Teil dieser Arbeit untersuchen wir den Verlust privater Informationen bei visuellen Daten und insbesondere bei Fotos, die für Inhalte repräsentativ sind, die in sozialen Netzwerken geteilt werden. Es gibt eine lange Reihe von Arbeiten, die das Problem des Verlustes persönlich identifizierbarer Informationen in sozialen Fotos angehen, insbesondere mithilfe visueller Hinweise auf Gesichts- und Körperebene. Wir argumentieren jedoch, dass dies nur eine enge Perspektive darstellt, da Bilder ein breites Spektrum multimodaler privater Informationen (z. B. Behinderungen, Namensschilder) offenbaren. Aus diesem Grund arbeiten wir auf einen Visual Privacy Advisor hin, der darauf abzielt, private Risiken beim Teilen sozialer Fotos ganzheitlich zu identifizieren und zu minimieren. Im zweiten Teil befassen wir uns mit Datenverlusten während des Trainings von ML-Modellen. Wir beobachten, dass zunehmend Lernalgorithmen verwendet werden, um Modelle auf umfangreichen dezentralen Datensätzen zu trainieren, z. B. persönlichen Daten auf zahlreichen Mobilgeräten. In solchen Fällen werden Informationen von teilnehmenden Personen in Form von hochdimensionalen Modellparameteraktualisierungen anonym verbunden. Wir stellen jedoch fest, dass die Aktualisierungen ausreichend identifizierbare Informationen codieren und es ermöglichen, sie mit teilnehmenden Personen zu verknüpfen. Wir schlagen zudem Methoden vor, um diesen Datenverlust zu verringern und gleichzeitig die hohe Nützlichkeit der Aktualisierungen zu erhalten. Im dritten Teil diskutieren wir den Verlust vertraulicher Informationen während der Inferenzzeit von Black-Box-Modellen. Insbesondere finden wir, dass sich Modelle für die Entwicklung von Angriffen, die auf Funktionalitätsdiebstahl abzielen, eignen: Ein Gegner kann mit dem Black-Box-Modell interagieren, um ein Replikat-Knock-Off-Modell zu erstellen, das ähnliche Test-Set-Leistungen aufweist. Da solche Angriffe eine ernsthafte Bedrohung für das geistige Eigentum des Modellbesitzers darstellen, arbeiten wir auch an einer wirksamen Verteidigung. Unsere Verteidigungsstrategie durch die Einführung begrenzter und kontrollierter Störungen in Vorhersagen kann die Fehlerraten von Modelldiebstahlangriffen erheblich verbessern. Zusammenfassend lässt sich sagen, dass diese Arbeit das Verständnis von Datenschutzverlusten beim Informationsaustausch verbessert, sei es bei rohen visuellen Formen, während des Trainings von Modellen oder während der Inferenzzeit von Black-Box-Modellen. In jedem Fall schlagen wir ferner Techniken zur Verringerung des Informationsverlusts vor, um eine weit verbreitete Anwendung von Techniken in realen Szenarien zu ermöglichen.Max Planck Institute for Informatic

Proceedings of the NASA Workshop on Density Estimation and Function Smoothing

Statistical model identification techniques being developed to provide workable solutions to problems in density estimation and function smoothing are examined

Isogeometric Analysis for High Order Geometric Partial Differential Equations with Applications

In this thesis, we consider the numerical approximation of high order geometric Partial Differential Equations (PDEs). We first consider high order PDEs defined on surfaces in the 3D space that are represented by single-patch tensor product NURBS. Then, we spatially discretize the PDEs by means of NURBS-based Isogeometric Analysis (IGA) in the framework of the Galerkin method. With this aim, we consider the construction of periodic NURBS function spaces with high degree of global continuity, even on closed surfaces. As benchmark problems for the proposed discretization, we propose Laplace-Beltrami problems of the fourth and sixth orders, as well as the corresponding eigenvalue problems, and we analyze the impact of the continuity of the basis functions on the accuracy as well as on computational costs. The numerical solution of two high order phase field problems on both open and closed surfaces is also considered: the fourth order Cahn-Hilliard equation and the sixth order crystal equation, both discretized in time with the generalized-alpha method. We then consider the numerical approximation of geometric PDEs, derived, in particular, from the minimization of shape energy functionals by L^2-gradient flows. We analyze the mean curvature and the Willmore gradient flows, leading to second and fourth order PDEs, respectively. These nonlinear geometric PDEs are discretized in time with Backward Differentiation Formulas (BDF), with a semi-implicit formulation based on an extrapolation of the geometry, leading to a linear problem to be solved at each time step. Results about the numerical approximation of the two geometric flows on several geometries are analyzed. Then, we study how the proposed mathematical framework can be employed to numerically approximate the equilibrium shapes of lipid bilayer biomembranes, or vesicles, governed by the Canham-Helfrich curvature model. We propose two numerical schemes for enforcing the conservation of the area and volume of the vesicles, and report results on benchmark problems. Then, the approximation of the equilibrium shapes of biomembranes with different values of reduced volume is presented. Finally, we consider the dynamics of a vesicle, e.g. a red blood cell, immersed in a fluid, e.g. the plasma. In particular, we couple the curvature-driven model for the lipid membrane with the incompressible Navier-Stokes equations governing the fluid. We consider a segregated approach, with a formulation based on the Resistive Immersed Surface method applied to NURBS geometries. After analyzing benchmark fluid simulations with immersed NURBS objects, we report numerical results for the investigation of the dynamics of a vesicle under different flow conditions

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstracts

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstract

A comparison of the CAR and DAGAR spatial random effects models with an application to diabetics rate estimation in Belgium

When hierarchically modelling an epidemiological phenomenon on a finite collection of sites in space, one must always take a latent spatial effect into account in order to capture the correlation structure that links the phenomenon to the territory. In this work, we compare two autoregressive spatial models that can be used for this purpose: the classical CAR model and the more recent DAGAR model. Differently from the former, the latter has a desirable property: its ρ parameter can be naturally interpreted as the average neighbor pair correlation and, in addition, this parameter can be directly estimated when the effect is modelled using a DAGAR rather than a CAR structure. As an application, we model the diabetics rate in Belgium in 2014 and show the adequacy of these models in predicting the response variable when no covariates are available

A Statistical Approach to the Alignment of fMRI Data

Multi-subject functional Magnetic Resonance Image studies are critical. The anatomical and functional structure varies across subjects, so the image alignment is necessary. We define a probabilistic model to describe functional alignment. Imposing a prior distribution, as the matrix Fisher Von Mises distribution, of the orthogonal transformation parameter, the anatomical information is embedded in the estimation of the parameters, i.e., penalizing the combination of spatially distant voxels. Real applications show an improvement in the classification and interpretability of the results compared to various functional alignment methods