Construction of Smooth Branching Surfaces using T-splines

The request for designing or reconstructing objects from planar cross sections arises in various applications, ranging from CAD to GIS and Medical Imaging. The present work focuses on the " one-to-many " branching problem, where one of the planes can be populated with many, possibly tortuous and densely packed, contours. The proposed method combines the proximity information offered by the Euclidean Voronoi diagram with the concept of surrounding curve, introduced in [1], and T-splines technology [2] for securing a flexible and portable representation. Our algorithm delivers a single T-spline that deviates from the given contours less than a user-specified tolerance, measured via the so-called discrete Fréchet distance [3] and is C 2 everywhere except from a finite set of point-neighborhoods. Subject to minor enrichment, the algorithm is also capable to handle the " many-to-many " configuration as well as the global reconstruction problem involving contours on several planes

Accurate geometry reconstruction of vascular structures using implicit splines

3-D visualization of blood vessel from standard medical datasets (e.g. CT or MRI) play an important role in many clinical situations, including the diagnosis of vessel stenosis, virtual angioscopy, vascular surgery planning and computer aided vascular surgery. However, unlike other human organs, the vasculature system is a very complex network of vessel, which makes it a very challenging task to perform its 3-D visualization. Conventional techniques of medical volume data visualization are in general not well-suited for the above-mentioned tasks. This problem can be solved by reconstructing vascular geometry. Although various methods have been proposed for reconstructing vascular structures, most of these approaches are model-based, and are usually too ideal to correctly represent the actual variation presented by the cross-sections of a vascular structure. In addition, the underlying shape is usually expressed as polygonal meshes or in parametric forms, which is very inconvenient for implementing ramification of branching. As a result, the reconstructed geometries are not suitable for computer aided diagnosis and computer guided minimally invasive vascular surgery. In this research, we develop a set of techniques associated with the geometry reconstruction of vasculatures, including segmentation, modelling, reconstruction, exploration and rendering of vascular structures. The reconstructed geometry can not only help to greatly enhance the visual quality of 3-D vascular structures, but also provide an actual geometric representation of vasculatures, which can provide various benefits. The key findings of this research are as follows: 1. A localized hybrid level-set method of segmentation has been developed to extract the vascular structures from 3-D medical datasets. 2. A skeleton-based implicit modelling technique has been proposed and applied to the reconstruction of vasculatures, which can achieve an accurate geometric reconstruction of the vascular structures as implicit surfaces in an analytical form. 3. An accelerating technique using modern GPU (Graphics Processing Unit) is devised and applied to rendering the implicitly represented vasculatures. 4. The implicitly modelled vasculature is investigated for the application of virtual angioscopy

An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS

We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching

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

L-systems in Geometric Modeling

We show that parametric context-sensitive L-systems with affine geometry interpretation provide a succinct description of some of the most fundamental algorithms of geometric modeling of curves. Examples include the Lane-Riesenfeld algorithm for generating B-splines, the de Casteljau algorithm for generating Bezier curves, and their extensions to rational curves. Our results generalize the previously reported geometric-modeling applications of L-systems, which were limited to subdivision curves.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Solid NURBS Conforming Scaffolding for Isogeometric Analysis

This work introduces a scaffolding framework to compactly parametrise solid structures with conforming NURBS elements for isogeometric analysis. A novel formulation introduces a topological, geometrical and parametric subdivision of the space in a minimal plurality of conforming vectorial elements. These determine a multi-compartmental scaffolding for arbitrary branching patterns. A solid smoothing paradigm is devised for the conforming scaffolding achieving higher than positional geometrical and parametric continuity. Results are shown for synthetic shapes of varying complexity, for modular CAD geometries, for branching structures from tessellated meshes and for organic biological structures from imaging data. Representative simulations demonstrate the validity of the introduced scaffolding framework with scalable performance and groundbreaking applications for isogeometric analysis

Hybrid model for vascular tree structures

This paper proposes a new representation scheme of the cerebral blood vessels. This model provides information on the semantics of the vascular structure: the topological relationships between vessels and the labeling of vascular accidents such as aneurysms and stenoses. In addition, the model keeps information of the inner surface geometry as well as of the vascular map volume properties, i.e. the tissue density, the blood flow velocity and the vessel wall elasticity. The model can be constructed automatically in a pre-process from a set of segmented MRA images. Its memory requirements are optimized on the basis of the sparseness of the vascular structure. It allows fast queries and efficient traversals and navigations. The visualizations of the vessel surface can be performed at different levels of detail. The direct rendering of the volume is fast because the model provides a natural way to skip over empty data. The paper analyzes the memory requirements of the model along with the costs of the most important operations on it.Postprint (published version