Dynamic Multivariate Simplex Splines For Volume Representation And Modeling

Volume representation and modeling of heterogeneous objects acquired from real world are very challenging research tasks and playing fundamental roles in many potential applications, e.g., volume reconstruction, volume simulation and volume registration. In order to accurately and efficiently represent and model the real-world objects, this dissertation proposes an integrated computational framework based on dynamic multivariate simplex splines (DMSS) that can greatly improve the accuracy and efficacy of modeling and simulation of heterogenous objects. The framework can not only reconstruct with high accuracy geometric, material, and other quantities associated with heterogeneous real-world models, but also simulate the complicated dynamics precisely by tightly coupling these physical properties into simulation. The integration of geometric modeling and material modeling is the key to the success of representation and modeling of real-world objects. The proposed framework has been successfully applied to multiple research areas, such as volume reconstruction and visualization, nonrigid volume registration, and physically based modeling and simulation

Modelado jerárquico de objetos 3D con superficies de subdivisión

Las SSs (Superficies de Subdivisión) son un potente paradigma de modelado de objetos 3D (tridimensionales) que establece un puente entre los dos enfoques tradicionales a la aproximación de superficies, basados en mallas poligonales y de parches alabeados, que conllevan problemas uno y otro. Los esquemas de subdivisión permiten definir una superficie suave (a tramos), como las más frecuentes en la práctica, como el límite de un proceso recursivo de refinamiento de una malla de control burda, que puede ser descrita muy compactamente. Además, la recursividad inherente a las SSs establece naturalmente una relación de anidamiento piramidal entre las mallas / NDs (Niveles de Detalle) generadas/os sucesivamente, por lo que las SSs se prestan extraordinariamente al AMRO (Análisis Multiresolución mediante Ondículas) de superficies, que tiene aplicaciones prácticas inmediatas e interesantísimas, como la codificación y la edición jerárquicas de modelos 3D. Empezamos describiendo los vínculos entre las tres áreas que han servido de base a nuestro trabajo (SSs, extracción automática de NDs y AMRO) para explicar como encajan estas tres piezas del puzzle del modelado jerárquico de objetos de 3D con SSs. El AMRO consiste en descomponer una función en una versión burda suya y un conjunto de refinamientos aditivos anidados jerárquicamente llamados "coeficientes ondiculares". La teoría clásica de ondículas estudia las señales clásicas nD: las definidas sobre dominios paramétricos homeomorfos a R" o (0,1)n como el audio (n=1), las imágenes (n=2) o el vídeo (n=3). En topologías menos triviales, como las variedades 2D) (superficies en el espacio 3D), el AMRO no es tan obvio, pero sigue siendo posible si se enfoca desde la perspectiva de las SSs. Basta con partir de una malla burda que aproxime a un bajo ND la superficie considerada, subdividirla recursivamente y, al hacerlo, ir añadiendo los coeficientes ondiculares, que son los detalles 3D necesarios para obtener aproximaciones más y más finas a la superficie original. Pasamos después a las aplicaciones prácticas que constituyen nuestros principal desarrollo original y, en particular, presentamos una técnica de codificación jerárquica de modelos 3D basada en SSs, que actúa sobre los detalles 3D mencionados: los expresa en un referencial normal loscal; los organiza según una estructura jerárquica basada en facetas; los cuantifica dedicando menos bits a sus componentes tangenciales, menos energéticas, y los "escalariza"; y los codifica dinalmente gracias a una técnica similar al SPIHT (Set Partitioning In Hierarchical Tress) de Said y Pearlman. El resultado es un código completamente embebido y al menos dos veces más compacto, para superficies mayormente suaves, que los obtenidos con técnicas de codificación progresiva de mallas 3D publicadas previamente, en las que además los NDs no están anidados piramidalmente. Finalmente, describimos varios métodos auxiliares que hemos desarrollado, mejorando técnicas previas y creando otras propias, ya que una solución completa al modelado de objetos 3D con SSs requiere resolver otros dos problemas. El primero es la extracción de una malla base (triangular, en nuestro caso) de la superficie original, habitualmente dada por una malla triangular fina con conectividad arbitraria. El segundo es la generación de un remallado recursivo con conectividad de subdivisión de la malla original/objetivo mediante un refinamiento recursivo de la malla base, calculando así los detalles 3D necesarios para corregir las posiciones predichas por la subdivisión para nuevos vértices

Automatic and topology-preserving gradient mesh generation for image vectorization

Demonska vjera polazi od toga da se ne nalazi isključivo u nevjeri, nego je ona onkraj nevjere. Ovim radom smo prvo željeli prikazati glavne postavke najbitnijih elemenata teologalne vjere da bismo kasnije kroz filozofske teze F. Hadjadja prikazali glavna obilježja vjere zlih demona koja su duhovnog karaktera i predstavljaju veliku opasnost za čovjeka i njegovu vjeru. Unatoč različitim tumačenjima, Hadjadj želi upozoriti na to da se zli demoni ne nalaze samo među ateistima, agnosticima i ostalim pokretima koji, na neki način, niječu Božu opstojnost, nego ti zli demoni zapravo vjeruju, pa zato za polazište uzima redak iz Jakovljeve poslanice u kojoj piše da oni „vjeruju i dršću”. Na kraju smo ovim radom prikazali važnost sinteze duše i tijela, odnosno utjelovljenja u kršćanstvu jer je misterij katoličke vjere misterij utjelovljenja i nemoguće je odbacivati tjelesnost kao što je to slučaj u različitim spiritualističkim dimenzijama.Demonic faith is not strictly associated with infidelity itself, but rather exists beyond it. This work aims to present the main postulates of the most crucial elements of the theological faith in order to show, using F. Hadjadj's philosophical thesis, the main characteristics associated with evil demons, which are of the spiritual nature and present great danger for man and his faith. Despite various interpretations, Hadjadj wishes to point out that evil demons do not dwel lonly within atheists, agnostics and other movements which, in a way, negate the existence of God, but that these evil demons actually do believe, and therefore he uses as a starting point the line from the Epistle of Jacob that states that demons „believe and tremble”. In the end, this work presented the importance of the body-soul synthesis, which is the incarnation in Christianity, since the mistery of the Catholic faith is the mistery of the incarnation, and it is impossible to discard corporeality, which is the case in various spiritual dimensions

Numerical Methods in Shape Spaces and Optimal Branching Patterns

The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functional—subject to appropriate constraints—can be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Fréchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Müller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound

New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes

With the advent of powerful 3D acquisition technology, there is a growing demand for the modeling, processing, and visualization of surfaces and volumes. The proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression. This thesis presents several novel solutions to these problems for surfaces (Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples

Compression of 3D models with NURBS

With recent progress in computing, algorithmics and telecommunications, 3D models are increasingly used in various multimedia applications. Examples include visualization, gaming, entertainment and virtual reality. In the multimedia domain 3D models have been traditionally represented as polygonal meshes. This piecewise planar representation can be thought of as the analogy of bitmap images for 3D surfaces. As bitmap images, they enjoy great flexibility and are particularly well suited to describing information captured from the real world, through, for instance, scanning processes. They suffer, however, from the same shortcomings, namely limited resolution and large storage size. The compression of polygonal meshes has been a very active field of research in the last decade and rather efficient compression algorithms have been proposed in the literature that greatly mitigate the high storage costs. However, such a low level description of a 3D shape has a bounded performance. More efficient compression should be reachable through the use of higher level primitives. This idea has been explored to a great extent in the context of model based coding of visual information. In such an approach, when compressing the visual information a higher level representation (e.g., 3D model of a talking head) is obtained through analysis methods. This can be seen as an inverse projection problem. Once this task is fullled, the resulting parameters of the model are coded instead of the original information. It is believed that if the analysis module is efficient enough, the total cost of coding (in a rate distortion sense) will be greatly reduced. The relatively poor performance and high complexity of currently available analysis methods (except for specific cases where a priori knowledge about the nature of the objects is available), has refrained a large deployment of coding techniques based on such an approach. Progress in computer graphics has however changed this situation. In fact, nowadays, an increasing number of pictures, video and 3D content are generated by synthesis processing rather than coming from a capture device such as a camera or a scanner. This means that the underlying model in the synthesis stage can be used for their efficient coding without the need for a complex analysis module. In other words it would be a mistake to attempt to compress a low level description (e.g., a polygonal mesh) when a higher level one is available from the synthesis process (e.g., a parametric surface). This is, however, what is usually done in the multimedia domain, where higher level 3D model descriptions are converted to polygonal meshes, if anything by the lack of standard coded formats for the former. On a parallel but related path, the way we consume audio-visual information is changing. As opposed to recent past and a large part of today's applications, interactivity is becoming a key element in the way we consume information. In the context of interest in this dissertation, this means that when coding visual information (an image or a video for instance), previously obvious considerations such as decision on sampling parameters are not so obvious anymore. In fact, as in an interactive environment the effective display resolution can be controlled by the user through zooming, there is no clear optimal setting for the sampling period. This means that because of interactivity, the representation used to code the scene should allow the display of objects in a variety of resolutions, and ideally up to infinity. One way to resolve this problem would be by extensive over-sampling. But this approach is unrealistic and too expensive to implement in many situations. The alternative would be to use a resolution independent representation. In the realm of 3D modeling, such representations are usually available when the models are created by an artist on a computer. The scope of this dissertation is precisely the compression of 3D models in higher level forms. The direct coding in such a form should yield improved rate-distortion performance while providing a large degree of resolution independence. There has not been, so far, any major attempt to efficiently compress these representations, such as parametric surfaces. This thesis proposes a solution to overcome this gap. A variety of higher level 3D representations exist, of which parametric surfaces are a popular choice among designers. Within parametric surfaces, Non-Uniform Rational B-Splines (NURBS) enjoy great popularity as a wide range of NURBS based modeling tools are readily available. Recently, NURBS has been included in the Virtual Reality Modeling Language (VRML) and its next generation descendant eXtensible 3D (X3D). The nice properties of NURBS and their widespread use has lead us to choose them as the form we use for the coded representation. The primary goal of this dissertation is the definition of a system for coding 3D NURBS models with guaranteed distortion. The basis of the system is entropy coded differential pulse coded modulation (DPCM). In the case of NURBS, guaranteeing the distortion is not trivial, as some of its parameters (e.g., knots) have a complicated influence on the overall surface distortion. To this end, a detailed distortion analysis is performed. In particular, previously unknown relations between the distortion of knots and the resulting surface distortion are demonstrated. Compression efficiency is pursued at every stage and simple yet efficient entropy coder realizations are defined. The special case of degenerate and closed surfaces with duplicate control points is addressed and an efficient yet simple coding is proposed to compress the duplicate relationships. Encoder aspects are also analyzed. Optimal predictors are found that perform well across a wide class of models. Simplification techniques are also considered for improved compression efficiency at negligible distortion cost. Transmission over error prone channels is also considered and an error resilient extension defined. The data stream is partitioned by independently coding small groups of surfaces and inserting the necessary resynchronization markers. Simple strategies for achieving the desired level of protection are proposed. The same extension also serves the purpose of random access and on-the-fly reordering of the data stream

Subdivision Surface based One-Piece Representation

Subdivision surfaces are capable of modeling and representing complex shapes of arbi-trary topology. However, methods on how to build the control mesh of a complex surfaceare not studied much. Currently, most meshes of complicated objects come from trian-gulation and simplification of raster scanned data points, like the Stanford 3D ScanningRepository. This approach is costly and leads to very dense meshes.Subdivision surface based one-piece representation means to represent the final objectin a design process with only one subdivision surface, no matter how complicated theobject\u27s topology or shape. Hence the number of parts in the final representation isalways one.In this dissertation we present necessary mathematical theories and geometric algo-rithms to support subdivision surface based one-piece representation. First, an explicitparametrization method is presented for exact evaluation of Catmull-Clark subdivisionsurfaces. Based on it, two approaches are proposed for constructing the one-piece rep-resentation of a given object with arbitrary topology. One approach is to construct theone-piece representation by using the interpolation technique. Interpolation is a naturalway to build models, but the fairness of the interpolating surface is a big concern inprevious methods. With similarity based interpolation technique, we can obtain bet-ter modeling results with less undesired artifacts and undulations. Another approachis through performing Boolean operations. Up to this point, accurate Boolean oper-ations over subdivision surfaces are not approached yet in the literature. We presenta robust and error controllable Boolean operation method which results in a one-piecerepresentation. Because one-piece representations resulting from the above two methodsare usually dense, error controllable simplification of one-piece representations is needed.Two methods are presented for this purpose: adaptive tessellation and multiresolutionanalysis. Both methods can significantly reduce the complexity of a one-piece represen-tation and while having accurate error estimation.A system that performs subdivision surface based one-piece representation was im-plemented and a lot of examples have been tested. All the examples show that our ap-proaches can obtain very good subdivision based one-piece representation results. Eventhough our methods are based on Catmull-Clark subdivision scheme, we believe they canbe adapted to other subdivision schemes as well with small modifications

Fast Visualization by Shear-Warp using Spline Models for Data Reconstruction

This work concerns oneself with the rendering of huge three-dimensional data sets. The target thereby is the development of fast algorithms by also applying recent and accurate volume reconstruction models to obtain at most artifact-free data visualizations. In part I a comprehensive overview on the state of the art in volume rendering is given. Part II is devoted to the recently developed trivariate (linear,) quadratic and cubic spline models defined on symmetric tetrahedral partitions directly obtained by slicing volumetric partitions of a three-dimensional domain. This spline models define piecewise polynomials of total degree (one,) two and three with respect to a tetrahedron, i.e. the local splines have the lowest possible total degree and are adequate for efficient and accurate volume visualization. The following part III depicts in a step by step manner a fast software-based rendering algorithm, called shear-warp. This algorithm is prominent for its ability to generate projections of volume data at real time. It attains the high rendering speed by using elaborate data structures and extensive pre-computation, but at the expense of data redundancy and visual quality of the finally obtained rendering results. However, to circumvent these disadvantages a further development is specified, where new techniques and sophisticated data structures allow combining the fast shear-warp with the accurate ray-casting approach. This strategy and the new data structures not only grant a unification of the benefits of both methods, they even easily admit for adjustments to trade-off between rendering speed and precision. With this further development also the 3-fold data redundancy known from the original shear-warp approach is removed, allowing the rendering of even larger three-dimensional data sets more quickly. Additionally, real trivariate data reconstruction models, as discussed in part II, are applied together with the new ideas to onward the precision of the new volume rendering method, which also lead to a one order of magnitude faster algorithm compared to traditional approaches using similar reconstruction models. In part IV, a hierarchy-based rendering method is developed which utilizes a wavelet decomposition of the volume data, an octree structure to represent the sparse data set, the splines from part II and a new shear-warp visualization algorithm similar to that presented in part III. This thesis is concluded by the results centralized in part V