Disconnected Skeleton: Shape at its Absolute Scale

We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.Comment: The work excluding {\S}V and {\S}VI has first appeared in 2005 ICCV: Aslan, C., Tari, S.: An Axis-Based Representation for Recognition. In ICCV(2005) 1339- 1346.; Aslan, C., : Disconnected Skeletons for Shape Recognition. Masters thesis, Department of Computer Engineering, Middle East Technical University, May 200

The modular geometry of Random Regge Triangulations

We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N punctures. Such an analysis allows us to associate a Weil-Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity.Comment: 36 pages corrected typos, enhanced introductio

Computerized Analysis of Magnetic Resonance Images to Study Cerebral Anatomy in Developing Neonates

The study of cerebral anatomy in developing neonates is of great importance for the understanding of brain development during the early period of life. This dissertation therefore focuses on three challenges in the modelling of cerebral anatomy in neonates during brain development. The methods that have been developed all use Magnetic Resonance Images (MRI) as source data. To facilitate study of vascular development in the neonatal period, a set of image analysis algorithms are developed to automatically extract and model cerebral vessel trees. The whole process consists of cerebral vessel tracking from automatically placed seed points, vessel tree generation, and vasculature registration and matching. These algorithms have been tested on clinical Time-of- Flight (TOF) MR angiographic datasets. To facilitate study of the neonatal cortex a complete cerebral cortex segmentation and reconstruction pipeline has been developed. Segmentation of the neonatal cortex is not effectively done by existing algorithms designed for the adult brain because the contrast between grey and white matter is reversed. This causes pixels containing tissue mixtures to be incorrectly labelled by conventional methods. The neonatal cortical segmentation method that has been developed is based on a novel expectation-maximization (EM) method with explicit correction for mislabelled partial volume voxels. Based on the resulting cortical segmentation, an implicit surface evolution technique is adopted for the reconstruction of the cortex in neonates. The performance of the method is investigated by performing a detailed landmark study. To facilitate study of cortical development, a cortical surface registration algorithm for aligning the cortical surface is developed. The method first inflates extracted cortical surfaces and then performs a non-rigid surface registration using free-form deformations (FFDs) to remove residual alignment. Validation experiments using data labelled by an expert observer demonstrate that the method can capture local changes and follow the growth of specific sulcus

Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary set consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface spanning the boundary.Comment: 32 page

The geometry of dynamical triangulations

We discuss the geometry of dynamical triangulations associated with 3-dimensional and 4-dimensional simplicial quantum gravity. We provide analytical expressions for the canonical partition function in both cases, and study its large volume behavior. In the space of the coupling constants of the theory, we characterize the infinite volume line and the associated critical points. The results of this analysis are found to be in excellent agreement with the MonteCarlo simulations of simplicial quantum gravity. In particular, we provide an analytical proof that simply-connected dynamically triangulated 4-manifolds undergo a higher order phase transition at a value of the inverse gravitational coupling given by 1.387, and that the nature of this transition can be concealed by a bystable behavior. A similar analysis in the 3-dimensional case characterizes a value of the critical coupling (3.845) at which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil

Shape Analysis Using Spectral Geometry

Shape analysis is a fundamental research topic in computer graphics and computer vision. To date, more and more 3D data is produced by those advanced acquisition capture devices, e.g., laser scanners, depth cameras, and CT/MRI scanners. The increasing data demands advanced analysis tools including shape matching, retrieval, deformation, etc. Nevertheless, 3D Shapes are represented with Euclidean transformations such as translation, scaling, and rotation and digital mesh representations are irregularly sampled. The shape can also deform non-linearly and the sampling may vary. In order to address these challenging problems, we investigate Laplace-Beltrami shape spectra from the differential geometry perspective, focusing more on the intrinsic properties. In this dissertation, the shapes are represented with 2 manifolds, which are differentiable. First, we discuss in detail about the salient geometric feature points in the Laplace-Beltrami spectral domain instead of traditional spatial domains. Simultaneously, the local shape descriptor of a feature point is the Laplace-Beltrami spectrum of the spatial region associated to the point, which are stable and distinctive. The salient spectral geometric features are invariant to spatial Euclidean transforms, isometric deformations and mesh triangulations. Both global and partial matching can be achieved with these salient feature points. Next, we introduce a novel method to analyze a set of poses, i.e., near-isometric deformations, of 3D models that are unregistered. Different shapes of poses are transformed from the 3D spatial domain to a geometry spectral one where all near isometric deformations, mesh triangulations and Euclidean transformations are filtered away. Semantic parts of that model are then determined based on the computed geometric properties of all the mapped vertices in the geometry spectral domain while semantic skeleton can be automatically built with joints detected. Finally we prove the shape spectrum is a continuous function to a scale function on the conformal factor of the manifold. The derivatives of the eigenvalues are analytically expressed with those of the scale function. The property applies to both continuous domain and discrete triangle meshes. On the triangle meshes, a spectrum alignment algorithm is developed. Given two closed triangle meshes, the eigenvalues can be aligned from one to the other and the eigenfunction distributions are aligned as well. This extends the shape spectra across non-isometric deformations, supporting a registration-free analysis of general motion data