Derivation of Mean Value Coordinates Using Interior Distance and Their Application on Mesh Deformation

The deformation methods based on cage controls became a subject of considerable interest due its simplicity and intuitive results. In this technique, the model is enclosed within a simpler mesh (the cage) and its points are expressed as function of the cage elements. Then, by manipulating the cage, the respective deformation is obtained on the model in its interior.In this direction, in the last years, extensions of barycentric coordinates, such as Mean Value coordinates, Positive Mean Value Coordinates, Harmonic coordinates and Green's coordinates, have been proposed to write the points of the model as a function of the cage elements.The Mean Value coordinates, proposed by Floater in two dimensions and extended later to three dimensions by Ju et al. and also by Floater, stands out from the other coordinates because of their simple derivation. However the existence of negative coordinates in regions bounded by non-convex cage control results in a unexpected behavior of the deformation in some regions of the model.In this work, we propose a modification in the derivation of Mean Value Coordinates proposed by Floater. Our derivation maintains the simplicity of the construction of the coordinates and eliminates the undesired behavior in the deformation by diminishing the negative influence of a control vertex on regions ofthe model not related to it. We also compare the deformation generated with our coordinates and the deformations obtained with the original Mean Value coordinates and Harmonic coordinates

On some interactive mesh deformations

Techniques devoted to deform 3D models are an important research field in Computer Graphics. They can be used in differentstages: the modelling phase, the animation process and also during some special simulations. Additionally, some applications may require the manipulation of 3D models under certain restrictions to preserve the volume of the modified object. Hence, thepresent PhD Dissertation explores new algorithms to perform flexible, robust and efficient 3D deformations. Apart from this, it also researches on a new methodology to restrict these deformations so that the volume of the manipulated model remains constant. Some of the most used methods to achieve smooth deformations are those included in the Cage-Based Deformation paradigm. Cage-based deformations enclose the model to be deformed in a coarse polyhedron, the cage. Then, they usually rely on Generalized Barycentric Coordinates to relate the model with the vertices, and other geometric elements, of this cage, which are the control points or the deformation handles. Finally, every time that one of these handles is dragged, the model is deformed accordingly. Although this paradigm is simple, elegant and performs efficient deformations, some cage-free space deformation techniques have recently appeared. They increase the flexibility of the deformation handles, which do not need to be connected, and define powerful tools that make the deformation process more versatile and intuitive. In this context, the Dissertation introduces new Generalized Barycentric Coordinate systems specially designed to be used in a cage-free environment. Any user who wants to use the presented schemes only needs to locate a set of control points in the vicinity of the model that he or she wants to deform. These handles can be placed wherever he or she considers mode suitable and the only requirement is that the model has to be enclosed in their convex hull. Up to now, there are few techniques to produce volume-preserving space deformations. However, in recent years there has been a growing interest in performing constrained deformations due to their more realistic and physically plausible results. Our contribution to this research line consists in a deformation framework that preserves the volume of the 3D models by means of its gradient and a control surface to restrict the movement of the handles. Moreover, the proposed methodology is not restricted to the cage-based schemes, but it can also be used in a cage-free environment. Finally, our research can be specially useful for spatial deformations of biological and medical models. This kind of models represent real organs and tissues, which are often soft and lack an internal rigid structure. In addition, they are elastic and incompressible. Any application designed to deal with this group of models and to train or assist doctors must be flexible, robust, efficient and user-friendly. The combination of the proposed cage-free systems with the presented volume-preserving deformation framework satisfiesLes deformacions de models 3D s'utilitzen en diverses etapes de la generació de continguts digitals: durant la fase de modelatge, durant el procés d'animació i en alguns tipus de simulacions. A més a més, hi ha aplicacions que necessiten que la manipulació dels models 3D es faci tenint en compte certes restriccions que permeten la conservació del volum de l'objecte modificat. Tot plegat fa que les tècniques de deformació 3D siguin un camp d'estudi molt important dins del món dels Gràfics. Per aquesta raó, aquesta Tesi Doctoral estudia nous algorismes que permetin realitzar deformacions 3D de manera flexible, robusta i eficient i que, a més a més, permetin conservar el volum dels objectes modificats. Un dels paradigmes més utilitzats per tal de realitzar deformacions suaus és el conegut amb el nom de Deformacions Basades en un Poliedre Englobant. Aquesta família de mètodes embolcalla el model que es vol deformar, normalment representat com una malla de triangles, dins d'un poliedre simple, amb poques cares. Un cop fet això, estableix un sistema de Coordenades Baricèntriques Generalitzades per tal de definir els vèrtexs del model a partir dels vèrtexs del poliedre englobant, els quals s'anomenen punts de control o controls de la deformació. D'aquesta manera, cada cop que s'arrossega o es modifica un d'aquests punts de control, el model que es troba dins del poliedre englobant es deforma segons el sistema de coordenades que s'ha definit. Tot i que aquest paradigma és simple, elegant i eficient, des de fa ja uns anys han començat a aparèixer noves tècniques que no necessiten el poliedre englobant per tal de realitzar la deformació. El seu principal objectiu és augmentar la flexibilitat dels controls de la deformació i definir eines que facin que el procés de deformació sigui més versàtil i intuïtiu. Tenint en compte aquest factor, aquesta Tesi també estudia sistemes de Coordenades Baricèntriques Generalitzades dissenyats per realitzar deformacions sense la necessitat de definir el poliedre englobant. D'aquesta manera, qualsevol usuari que vulgui utilitzar els mètodes que es presenten en aquesta Dissertació només s'ha d'encarregar de definir un conjunt de punts de control al voltant del model que vol deformar, podent-los posar allí on consideri més oportú segons la deformació que vulgui obtenir. L'únic requeriment necessari és que el model ha de quedar dins de l'envolupant convexa d'aquests punts de control. Actualment existeixen pocs mètodes que realitzin deformacions 3D amb preservació del volum. No obstant això, d'un temps ençà ha augmentat l'interès per realitzar deformacions subjectes a certes restriccions que fan que el resultat sigui més realista i físicament versemblant. La contribució d'aquesta Tesi dins d'aquesta línia de recerca consisteix en un sistema de deformació que preserva el volum dels objectes 3D gràcies a còmput del seu gradient i a una superfície de control que restringeix el moviment dels punts de control. Aquest mètode es pot aplicar tant als sistemes de deformació que necessiten un poliedre englobant com als que no el necessiten. Finalment, i ja per acabar, la recerca realitzada pot ser especialment útil per tal de realitzar deformacions de models mèdics i biològics. Aquests tipus de models poden representar òrgans i teixits reals, els quals, normalment, són tous, mancats d'una estructura rígida interna, elàstics i incompressibles. Qualsevol aplicació dissenyada per treballar amb aquest tipus de models i per entrenar i donar assistència a usuaris mèdics hauria de ser flexible, robusta, eficient i fàcil d'utilitzar. La combinació dels mètodes de deformació proposats conjuntament amb el sistema de preservació de volum satisfà totes aquestes condicions. Per aquesta raó es creu que les contribucions realitzades poden esdevenir eines importants per produir deformacions mèdiques.Postprint (published version

Subdivision Shell Elements with Anisotropic Growth

A thin shell finite element approach based on Loop's subdivision surfaces is proposed, capable of dealing with large deformations and anisotropic growth. To this end, the Kirchhoff-Love theory of thin shells is derived and extended to allow for arbitrary in-plane growth. The simplicity and computational efficiency of the subdivision thin shell elements is outstanding, which is demonstrated on a few standard loading benchmarks. With this powerful tool at hand, we demonstrate the broad range of possible applications by numerical solution of several growth scenarios, ranging from the uniform growth of a sphere, to boundary instabilities induced by large anisotropic growth. Finally, it is shown that the problem of a slowly and uniformly growing sheet confined in a fixed hollow sphere is equivalent to the inverse process where a sheet of fixed size is slowly crumpled in a shrinking hollow sphere in the frictionless, quasi-static, elastic limit.Comment: 20 pages, 12 figures, 1 tabl

Geometric modeling of non-rigid 3D shapes : theory and application to object recognition.

One of the major goals of computer vision is the development of flexible and efficient methods for shape representation. This is true, especially for non-rigid 3D shapes where a great variety of shapes are produced as a result of deformations of a non-rigid object. Modeling these non-rigid shapes is a very challenging problem. Being able to analyze the properties of such shapes and describe their behavior is the key issue in research. Also, considering photometric features can play an important role in many shape analysis applications, such as shape matching and correspondence because it contains rich information about the visual appearance of real objects. This new information (contained in photometric features) and its important applications add another, new dimension to the problem\u27s difficulty. Two main approaches have been adopted in the literature for shape modeling for the matching and retrieval problem, local and global approaches. Local matching is performed between sparse points or regions of the shape, while the global shape approaches similarity is measured among entire models. These methods have an underlying assumption that shapes are rigidly transformed. And Most descriptors proposed so far are confined to shape, that is, they analyze only geometric and/or topological properties of 3D models. A shape descriptor or model should be isometry invariant, scale invariant, be able to capture the fine details of the shape, computationally efficient, and have many other good properties. A shape descriptor or model is needed. This shape descriptor should be: able to deal with the non-rigid shape deformation, able to handle the scale variation problem with less sensitivity to noise, able to match shapes related to the same class even if these shapes have missing parts, and able to encode both the photometric, and geometric information in one descriptor. This dissertation will address the problem of 3D non-rigid shape representation and textured 3D non-rigid shapes based on local features. Two approaches will be proposed for non-rigid shape matching and retrieval based on Heat Kernel (HK), and Scale-Invariant Heat Kernel (SI-HK) and one approach for modeling textured 3D non-rigid shapes based on scale-invariant Weighted Heat Kernel Signature (WHKS). For the first approach, the Laplace-Beltrami eigenfunctions is used to detect a small number of critical points on the shape surface. Then a shape descriptor is formed based on the heat kernels at the detected critical points for different scales. Sparse representation is used to reduce the dimensionality of the calculated descriptor. The proposed descriptor is used for classification via the Collaborative Representation-based Classification with a Regularized Least Square (CRC-RLS) algorithm. The experimental results have shown that the proposed descriptor can achieve state-of-the-art results on two benchmark data sets. For the second approach, an improved method to introduce scale-invariance has been also proposed to avoid noise-sensitive operations in the original transformation method. Then a new 3D shape descriptor is formed based on the histograms of the scale-invariant HK for a number of critical points on the shape at different time scales. A Collaborative Classification (CC) scheme is then employed for object classification. The experimental results have shown that the proposed descriptor can achieve high performance on the two benchmark data sets. An important observation from the experiments is that the proposed approach is more able to handle data under several distortion scenarios (noise, shot-noise, scale, and under missing parts) than the well-known approaches. For modeling textured 3D non-rigid shapes, this dissertation introduces, for the first time, a mathematical framework for the diffusion geometry on textured shapes. This dissertation presents an approach for shape matching and retrieval based on a weighted heat kernel signature. It shows how to include photometric information as a weight over the shape manifold, and it also propose a novel formulation for heat diffusion over weighted manifolds. Then this dissertation presents a new discretization method for the weighted heat kernel induced by the linear FEM weights. Finally, the weighted heat kernel signature is used as a shape descriptor. The proposed descriptor encodes both the photometric, and geometric information based on the solution of one equation. Finally, this dissertation proposes an approach for 3D face recognition based on the front contours of heat propagation over the face surface. The front contours are extracted automatically as heat is propagating starting from a detected set of landmarks. The propagation contours are used to successfully discriminate the various faces. The proposed approach is evaluated on the largest publicly available database of 3D facial images and successfully compared to the state-of-the-art approaches in the literature. This work can be extended to the problem of dense correspondence between non-rigid shapes. The proposed approaches with the properties of the Laplace-Beltrami eigenfunction can be utilized for 3D mesh segmentation. Another possible application of the proposed approach is the view point selection for 3D objects by selecting the most informative views that collectively provide the most descriptive presentation of the surface

An ALE method for simuations of elastic surfaces in flow

Die Dynamik von elastischen Membranen, Kapseln und Schalen hat sich zu einem aktiven Forschungsgebiet in der simulationsgestützten Physik und Biologie entwickelt. Die dünne Oberfläche dieser elastischen Materialien ermöglicht es, sie effizient als Hyperfläche zu approximieren. Solche Oberflächen reagieren auf Dehnungen in Oberflächenrichtung und Verformungen in Normalenrichtung mit einer elastischen Kraft. Zusätzlich können Oberflächenspannungskräfte auftreten. In dieser Arbeit präsentieren wir eine neuartige Arbitrary Lagrangian-Eulerian (ALE) Methode um solche in (Navier-Stokes) Fluiden eingebetteten elastischen Schalen zu simulieren. Dadurch, dass das Gitter an die elastische Oberfläche angepasst ist, kombiniert die vorgeschlagene Methode hohe Genauigkeit mit Effizienz in der Berechnung der Lösungen. Folglich kann man die Simulationen mit einer verhältnismäßig geringen Gitterauflösung durchführen. Der Fokus dieser Arbeit liegt bei achsensymmetrischen Formen und Strömungen, wie sie bei vielen biophysikalischen Anwendungen zu finden sind. Neben einer allgemeinen dreidimensionalen Beschreibung formulieren wir achsensymmetrische Kräfte auf der Oberfläche, für welche wir eine Diskretisierung mit der Finite Differenzen Methode vorschlagen, welche an eine Finite-Elemente Methode für die umgebenden Fluide gekoppelt ist. Weiterhin entwickeln wir eine Strategie zur impliziten Kopplung der Kräfte, um Zeitschrittrestriktionen zu reduzieren. In verschiedenen numerischen Tests werden wir zeigen, dass akkurate Ergebnisse schon in einer Größenordnung von Minuten auf einer Single-Core CPU erreicht werden können. Die Methode wurde in drei aktuellen Anwendungen verwendet, wobei mindestens zwei davon nach unserer Kenntnis im Moment mit keiner anderen numerischen Methode simuliert werden können: Zunächst präsentieren wir Simulationen von biologischen Zellen, die im Zuge eines RT-DC (Real-Time Deformability Cytometry) Experiments durch einen schmalen mikrofluidischen Kanal advektiert und dabei verformt werden. Danach zeigen wir die Ergebnisse erster Simulationen der uniaxialen Kompression biologischer Zellen zwischen zwei parallelen Platten im Zuge eines AFM Experiments. Schließlich präsentieren wir Resultate erster Simulationen von neuartigen mikroschwimmenden Schalen, welche lediglich durch äußere Einflüsse (wie z.B. Ultraschall), zum Schwimmen angeregt werden können.The dynamics of membranes, shells, and capsules in fluid flow has become an active research area in computational physics and computational biology. The small thickness of these elastic materials enables their efficient approximation as a hypersurface, which exhibits an elastic response to in-plane stretching and out-of-plane bending, possibly accompanied by a surface tension force. In this work, we present a novel arbitrary Lagrangian-Eulerian (ALE) method to simulate such elastic surfaces immersed in Navier-Stokes fluids. The method combines high accuracy with computational efficiency, since the grid is matched to the elastic surface and can therefore be resolved with relatively few grid points. The focus of this work is on axisymmetric shapes and flow conditions, which are present in a wide range of biophysical problems. Next to a general three-dimensional description, we formulate axisymmetric elastic surface forces and propose a discretization with surface finite-differences coupled to evolving finite elements. We further develop an implicit coupling strategy to reduce time step restrictions. Several numerical test cases show that accurate results can be achieved at computational times on the order of minutes on a single core CPU. Three state-of-the-art applications are demonstrated, where to our knowledge at least two of them cannot be simulated with any other numerical method so far. First, simulations of biological cells being advected through a microfluidic channel and therefore being deformed during an RT-DC (Real-Time Deformability Cytometry) experiment are presented. Then, the uniaxial compression of the cortex of a biological cell during an AFM experiment is investigated. Finally, we present the results of first simulations of the observed shape oscillations of novel microswimming shells which can be locomoted by exterior influences (e.g. ultrasound waves) only

Creep crack-growth: A new path-independent integral (T sub c), and computational studies

The development of valid creep fracture criteria is considered. Two path-independent integral parameters which show some degree of promise are the C* and (Delta T)sub c integrals. The mathematical aspects of these parameters are reviewed by deriving generalized vector forms of the parameters using conservation laws which are valid for arbitrary, three dimensional, cracked bodies with crack surface tractions (or applied displacements), body forces, inertial effects, and large deformations. Two principal conclusions are that (Delta T)sub c has an energy rate interpretation whereas C* does not. The development and application of fracture criteria often involves the solution of boundary/initial value problems associated with deformation and stresses. The finite element method is used for this purpose. An efficient, small displacement, infinitesimal strain, displacement based finite element model is specialized to two dimensional plane stress and plane strain and to power law creep constitutive relations. A mesh shifting/remeshing procedure is used for simulating crack growth. The model is implemented with the quartz-point node technique and also with specially developed, conforming, crack-tip singularity elements which provide for the r to the n-(1+n) power strain singularity associated with the HRR crack-tip field. Comparisons are made with a variety of analytical solutions and alternate numerical solutions for a number of problems

Analysis and new constructions of generalized barycentric coordinates in 2D

Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer graphics and computational mechanics. There exist many coordinate functions that satisfy all the basic properties of barycentric coordinates, but they differ by a number of other properties. We start by providing an extensive comparison of all existing coordinate functions and pointing out which important properties of generalized barycentric coordinates are not satisfied by these functions. This comparison shows that not all of existing coordinates have fully investigated properties, and we complete such a theoretical analysis for a particular one-parameter family of generalized barycentric coordinates for strictly convex polygons. We also perform numerical analysis of this family and show how to avoid computational instabilities near the polygon’s boundary when computing these coordinates in practice. We conclude this analysis by implementing some members of this family in the Computational Geometry Algorithm Library. In the second half of this dissertation, we present a few novel constructions of non-negative and smooth generalized barycentric coordinates defined over any simple polygon. In this context, we show that new coordinates with improved properties can be obtained by taking convex combinations of already existing coordinate functions and we give two examples of how to use such convex combinations for polygons without and with interior points. These new constructions have many attractive properties and perform better than other coordinates in interpolation and image deformation applications