A K-MEANS CLUSTERING BASED SHAPE RETRIEVAL TECHNIQUE FOR 3D MESH MODELS

Due to the large size of shape databases, importance of effective and robust method in shape retrieval has been increased. Researchers mainly focus on finding descriptors which is suitable for rigid models. Retrieval of non-rigid models is a still challenging field which needs to be studied more. For non-rigid models, descriptors that are designed should be insensitive to different poses. For non-rigid model retrieval, we propose a new method which first divides a model into clusters using geodesic distance metric and then computes the descriptor using these clusters. Mesh segmentation is performed using a skeleton-based K-means clustering method.  Each cluster is represented by an area based descriptor which is invariant to scale and orientation. Finally, similar objects for the input model are retrieved. Articulated objects from human to animals are used for this study’s experiments for the validation of the proposed retrieval algorithm

On Multilevel Methods Based on Non-Nested Meshes

This thesis is concerned with multilevel methods for the efficient solution of partial differential equations in the field of scientific computing. Further, emphasis is put on an extensive study of the information transfer between finite element spaces associated with non-nested meshes. For the discretization of complicated geometries with a finite element method, unstructured meshes are often beneficial as they can easily be adjusted to the shape of the computational domain. Such meshes, and thus the corresponding discrete function spaces, do not allow for straightforward multilevel hierarchies that could be exploited to construct fast solvers. In the present thesis, we present a class of "semi-geometric" multilevel iterations, which are based on hierarchies of independent, non-nested meshes. This is realized by a variational approach such that the images of suitable prolongation operators in the next (finer) space recursively determine the coarse level spaces. The semi-geometric concept is of very general nature compared with other methods relying on geometric considerations. This is reflected in the relatively loose relations of the employed meshes to each other. The specific benefit of the approach based on non-nested meshes is the flexibility in the choice of the coarse meshes, which can, for instance, be generated independently by standard methods. The resolution of the boundaries of the actual computational domain in the constructed coarse level spaces is a characteristic feature of the devised class of methods. The flexible applicability and the efficiency of the presented solution methods is demonstrated in a series of numerical experiments. We also explain the practical implementation of the semi-geometric ideas and concrete transfer concepts between non-nested meshes. Moreover, an extension to a semi-geometric monotone multigrid method for the solution of variational inequalities is discussed. We carry out the analysis of the convergence and preconditioning properties, respectively, in the framework of the theory of subspace correction methods. Our technical considerations yield a quasi-optimal result, which we prove for general, shape regular meshes by local arguments. The relevant properties of the operators for the prolongation between non-nested finite element spaces are the H1-stability and an L2-approximation property as well as the locality of the transfer. This thesis is a contribution to the development of fast solvers for equations on complicated geometries with focus on geometric techniques (as opposed to algebraic ones). Connections to other approaches are carefully elaborated. In addition, we examine the actual information transfer between non-nested finite element spaces. In a novel study, we combine theoretical, practical and experimental considerations. A thourough investigation of the qualitative properties and a quantitative analysis of the differences of individual transfer concepts to each other lead to new results on the information transfer as such. Finally, by the introduction of a generalized projection operator, the pseudo-L2-projection, we obtain a significantly better approximation of the actual L2-orthogonal projection than other approaches from the literature.Nicht-geschachtelte Gitter in Multilevel-Verfahren Diese Arbeit beschäftigt sich mit Multilevel-Verfahren zur effizienten Lösung von Partiellen Differentialgleichungen im Bereich des Wissenschaftlichen Rechnens. Dabei liegt ein weiterer Schwerpunkt auf der eingehenden Untersuchung des Informationsaustauschs zwischen Finite-Elemente-Räumen zu nicht-geschachtelten Gittern. Zur Diskretisierung von komplizierten Geometrien mit einer Finite-Elemente-Methode sind unstrukturierte Gitter oft von Vorteil, weil sie der Form des Rechengebiets einfacher angepasst werden können. Solche Gitter, und somit die zugehörigen diskreten Funktionenräume, besitzen im Allgemeinen keine leicht zugängliche Multilevel-Struktur, die sich zur Konstruktion schneller Löser ausnutzen ließe. In der vorliegenden Arbeit stellen wir eine Klasse "semi-geometrischer" Multilevel-Iterationen vor, die auf Hierarchien voneinander unabhängiger, nicht-geschachtelter Gitter beruhen. Dabei bestimmen in einem variationellen Ansatz rekursiv die Bilder geeigneter Prolongationsoperatoren im jeweils folgenden (feineren) Raum die Grobgitterräume. Das semi-geometrische Konzept ist sehr allgemeiner Natur verglichen mit anderen Verfahren, die auf geometrischen Überlegungen beruhen. Dies zeigt sich in der verhältnismäßig losen Beziehung der verwendeten Gitter zueinander. Der konkrete Nutzen des Ansatzes mit nicht-geschachtelten Gittern ist die Flexibilität der Wahl der Grobgitter. Diese können beispielsweise unabhängig mit Standardverfahren generiert werden. Die Auflösung des Randes des tatsächlichen Rechengebiets in den konstruierten Grobgitterräumen ist eine Eigenschaft der entwickelten Verfahrensklasse. Die flexible Einsetzbarkeit und die Effizienz der vorgestellten Lösungsverfahren zeigt sich in einer Reihe von numerischen Experimenten. Dazu geben wir Hinweise zur praktischen Umsetzung der semi-geometrischen Ideen und konkreter Transfer-Konzepte zwischen nicht-geschachtelten Gittern. Darüber hinaus wird eine Erweiterung zu einem semi-geometrischen monotonen Mehrgitterverfahren zur Lösung von Variationsungleichungen untersucht. Wir führen die Analysis der Konvergenz- bzw. Vorkonditionierungseigenschaften im Rahmen der Theorie der Teilraumkorrekturmethoden durch. Unsere technische Ausarbeitung liefert ein quasi-optimales Resultat, das wir mithilfe lokaler Argumente für allgemeine, shape-reguläre Gitterfamilien beweisen. Als relevante Eigenschaften der Operatoren zur Prolongation zwischen nicht-geschachtelten Finite-Elemente-Räumen erweisen sich die H1-Stabilität und eine L2-Approximationseigenschaft sowie die Lokalität des Transfers. Diese Arbeit ist ein Beitrag zur Entwicklung schneller Löser für Gleichungen auf komplizierten Gebieten mit Schwerpunkt auf geometrischen Techniken (im Unterschied zu algebraischen). Verbindungen zu anderen Ansätzen werden sorgfältig aufgezeigt. Daneben untersuchen wir den Informationsaustausch zwischen nicht-geschachtelten Finite-Elemente-Räumen als solchen. In einer neuartigen Studie verbinden wir theoretische, praktische und experimentelle Überlegungen. Eine sorgfältige Prüfung der qualitativen Eigenschaften sowie eine quantitative Analyse der Unterschiede verschiedener Transfer-Konzepte zueinander führen zu neuen Ergebnissen bezüglich des Informationsaustauschs selbst. Schließlich erreichen wir durch die Einführung eines verallgemeinerten Projektionsoperators, der Pseudo-L2-Projektion, eine deutlich bessere Approximation der eigentlichen L2-orthogonalen Projektion als andere Ansätze aus der Literatur

Structured meshes: composition and remeshing guided by the Curve-Skeleton

Virtual sculpting is currently a broadly used modeling metaphor with rising popularity especially in the entertainment industry. While this approach unleashes the artists' inspiration and creativity and leads to wonderfully detailed and artistic 3D models, it has the side effect, purely technical, of producing highly irregular meshes that are not optimal for subsequent processing. Converting an unstructured mesh into a more regular and struc- tured model in an automatic way is a challenging task and still open prob- lem. Since structured meshes are useful in different applications, it is of in- terest to be able to guarantee such property also in scenarios of part based modeling, which aim to build digital objects by composition, instead of modeling them from a scratch. This thesis will present methods for obtaining structured meshes in two different ways. First is presented a coarse quad layout computation method which starts from a triangle mesh and the curve-skeleton of the shape. The second approach allows to build complex shapes by procedural composition of PAM's. Since both quad layouts and PAMs exploit their global struc- ture, similarities between the two will be discussed, especially how their structure has correspondences to the curve-skeleton describing the topology of the shape being represented. Since both the presented methods rely on the information provided by the skeleton, the difficulties of using automat- ically extracted curve-skeletons without processing are discussed, and an interactive tool for user-assisted processing is presented

Structured meshes: composition and remeshing guided by the Curve-Skeleton

Virtual sculpting is currently a broadly used modeling metaphor with rising popularity especially in the entertainment industry. While this approach unleashes the artists' inspiration and creativity and leads to wonderfully detailed and artistic 3D models, it has the side effect, purely technical, of producing highly irregular meshes that are not optimal for subsequent processing. Converting an unstructured mesh into a more regular and struc- tured model in an automatic way is a challenging task and still open prob- lem. Since structured meshes are useful in different applications, it is of in- terest to be able to guarantee such property also in scenarios of part based modeling, which aim to build digital objects by composition, instead of modeling them from a scratch. This thesis will present methods for obtaining structured meshes in two different ways. First is presented a coarse quad layout computation method which starts from a triangle mesh and the curve-skeleton of the shape. The second approach allows to build complex shapes by procedural composition of PAM's. Since both quad layouts and PAMs exploit their global struc- ture, similarities between the two will be discussed, especially how their structure has correspondences to the curve-skeleton describing the topology of the shape being represented. Since both the presented methods rely on the information provided by the skeleton, the difficulties of using automat- ically extracted curve-skeletons without processing are discussed, and an interactive tool for user-assisted processing is presented

A Parametrization-Based Surface Reconstruction System for Triangular Mesh Simplification with Application to Large Scale Scenes

The laser scanner is nowadays widely used to capture the geometry of art, animation maquettes, or large architectural, industrial, and land form models. It thus poses specific problems depending on the model scale. This thesis provides a solution for simplification of triangulated data and for surface reconstruction of large data sets, where feature edges provide an obvious segmentation structure. It also explores a new method for model segmentation, with the goal of applying multiresolution techniques to data sets characterized by curvy areas and the lack of clear demarcation features. The preliminary stage of surface segmentation, which takes as input single or multiple scan data files, generates surface patches which are processed independently. The surface components are mapped onto a two-dimensional domain with boundary constraints, using a novel parametrization weight coefficient. This stage generates valid parameter domain points, which can be fed as arguments to parametric modeling functions or surface approximation schemes. On this domain, our approach explores two types of remeshing. First, we generate points in a regular grid pattern, achieving multiresolution through a flexible grid step, which nevertheless is designed to produce a globally uniform resampling aspect. In this case, for reconstruction, we attempt to solve the open problem of border reconciliation across adjacent domains by retriangulating the border gap between the grid and the fixed irregular border. Alternatively, we straighten the domain borders in the parameter domain and coarsely triangulate the resulting simplified polygons, resampling the base domain triangles in a 1-4 subdivision pattern, achieving multiresolution from the number of subdivision steps. For mesh reconstruction, we use a linear interpolation method based on the original mesh triangles as control points on local planes, using a saved triangle correspondence between the original mesh and the parametric domain. We also use a region-wide approximation method, applied to the parameter grid points, which first generates data-trained control points, and then uses them to obtain the reconstruction values at the resamples. In the grid resampling scheme, due to the border constraints, the reassembly of the segmented, sequentially processed data sets is seamless. In the subdivision scheme, we align adjacent border fragments in the parameter space, and use a region-to-fragment map to achieve the same border reconstruction across two neighboring components. We successfully process data sets up to 1,000,000 points in one pass of our program, and are capable of assembling larger scenes from sequential runs. Our program consists of a single run, without intermediate storage. Where we process large input data files, we fragment the input using a nested application of our segmentation algorithm to reduce the size of the input scenes, and our pipeline reassembles the reconstruction output from multiple data files into a unique view

Variational Methods and Numerical Algorithms for Geometry Processing

In this work we address the problem of shape partitioning which enables the decomposition of an arbitrary topology object into smaller and more manageable pieces called partitions. Several applications in Computer Aided Design (CAD), Computer Aided Manufactury (CAM) and Finite Element Analysis (FEA) rely on object partitioning that provides a high level insight of the data useful for further processing. In particular, we are interested in 2-manifold partitioning, since the boundaries of tangible physical objects can be mathematically defined by two-dimensional manifolds embedded into three-dimensional Euclidean space. To that aim, a preliminary shape analysis is performed based on shape characterizing scalar/vector functions defined on a closed Riemannian 2-manifold. The detected shape features are used to drive the partitioning process into two directions – a human-based partitioning and a thickness-based partitioning. In particular, we focus on the Shape Diameter Function that recovers volumetric information from the surface thus providing a natural link between the object’s volume and its boundary, we consider the spectral decomposition of suitably-defined affinity matrices which provides multi-dimensional spectral coordinates of the object’s vertices, and we introduce a novel basis of sparse and localized quasi-eigenfunctions of the Laplace-Beltrami operator called Lp Compressed Manifold Modes. The partitioning problem, which can be considered as a particular inverse problem, is formulated as a variational regularization problem whose solution provides the so-called piecewise constant/smooth partitioning function. The functional to be minimized consists of a fidelity term to a given data set and a regularization term which promotes sparsity, such as for example, Lp norm with p ∈ (0, 1) and other parameterized, non-convex penalty functions with positive parameter, which controls the degree of non-convexity. The proposed partitioning variational models, inspired on the well-known Mumford Shah models for recovering piecewise smooth/constant functions, incorporate a non-convex regularizer for minimizing the boundary lengths. The derived non-convex non-smooth optimization problems are solved by efficient numerical algorithms based on Proximal Forward-Backward Splitting and Alternating Directions Method of Multipliers strategies, also employing Convex Non-Convex approaches. Finally, we investigate the application of surface partitioning to patch-based surface quadrangulation. To that aim the 2-manifold is first partitioned into zero-genus patches that capture the object’s arbitrary topology, then for each patch a quad-based minimal surface is created and evolved by a Lagrangian-based PDE evolution model to the original shape to obtain the final semi-regular quad mesh. The evolution is supervised by asymptotically area-uniform tangential redistribution for the quads