Warping cubes: better triangles from marching cubes

National Science Foundatio

Multi-Dimensional Medial Geometry: Formulation, Computation, and Applications

Medial axis is a classical shape descriptor. It is a piece of geometry that lies in the middle of the original shape. Compared to the original shape representation, the medial axis is always one dimension lower and it carries many intrinsic shape properties explicitly. Therefore, it is widely used in a large amount of applications in various fields. However, medial axis is unstable to the boundary noise, often referred to as its instability. A small amount of change on the object boundary can cause a dramatic change in the medial axis. To tackle this problem, a significance measure is often associated with the medial axis, so that medial points with small significance are removed and only the stable part remains. In addition to this problem, many applications prefer even lower dimensional medial forms, e.g., shape centers of 2D shapes, and medial curves of 3D shapes. Unfortunately, good significance measures and good definitions of lower dimensional medial forms are still lacking. In this dissertation, we extended Blum\u27s grassfire burning to the medial axis in both 2D and 3D to define a significance measure as a distance function on the medial axis. We show that this distance function is well behaved and it has nice properties. In 2D, we also define a shape center based on this distance function. We then devise an iterative algorithm to compute the distance function and the shape center. We demonstrate usefulness of this distance function and shape center in various applications. Finally we point out the direction for future research based on this dissertation

Novel approaches to statistical shape modelling of bone

The femur is the longest bone in the human body and serves the important purposes of load-bearing and allowing bipedal locomotion. Accurate modelling of the variation in shape within the healthy adult population can be useful for a variety of applications: from the mere anatomical description of its features, in order to better understand its function, to more complex tasks such as pathology detection or surgical planning. Statistical Shape Modelling (SSM) is a well-established technique that enables to capture the variability within a set of training shapes and describes it with a reduced set of variables. The aim of this thesis is to evaluate the performance of a SSM based on a point cloud representation of shape, and introduce and test subsequent improvements to the modelling process that can increase its clinical relevance and scope of application. The standard approach to SSM employs a dimension-reducing technique, generally by means of Principal Component Analysis (PCA). However, this approach favours the compactness of the model, thus not focusing on other aspects that may be more relevant to clinical practice. Although rotation of the principal components is commonly performed as a post-processing step in statistical analysis involving PCA, it is not routinely applied in SSM. By applying this class of rotation, the components' effects are more localised, allowing a better interpretation, understanding and classification of pathological deformities. Among other possible representations, the Medial Axis Transform (MAT) could offer a further insight into shape modelling, since it allows the information about thickness to be decoupled from the rest of the shape. SSMs based on this representation can lead to a di erent perspective on the understanding of femoral anatomy and function,and can also enable the reconstruction of the complete anatomy starting from a reduced set of features, with diverse applications in the elds of surgical planning, forensic science and paleontology

Skeletal representations of orthogonal shapes

Skeletal representations are important shape descriptors which encode topological and geometrical properties of shapes and reduce their dimension. Skeletons are used in several fields of science and attract the attention of many researchers. In the biocad field, the analysis of structural properties such as porosity of biomaterials requires the previous computation of a skeleton. As the size of three-dimensional images become larger, efficient and robust algorithms that extract simple skeletal structures are required. The most popular and prominent skeletal representation is the medial axis, defined as the shape points which have at least two closest points on the shape boundary. Unfortunately, the medial axis is highly sensitive to noise and perturbations of the shape boundary. That is, a small change of the shape boundary may involve a considerable change of its medial axis. Moreover, the exact computation of the medial axis is only possible for a few classes of shapes. For example, the medial axis of polyhedra is composed of non planar surfaces, and its accurate and robust computation is difficult. These problems led to the emergence of approximate medial axis representations. There exists two main approximation methods: the shape is approximated with another shape class or the Euclidean metric is approximated with another metric. The main contribution of this thesis is the combination of a specific shape and metric simplification. The input shape is approximated with an orthogonal shape, which are polygons or polyhedra enclosed by axis-aligned edges or faces, respectively. In the same vein, the Euclidean metric is replaced by the L infinity or Chebyshev metric. Despite the simpler structure of orthogonal shapes, there are few works on skeletal representations applied to orthogonal shapes. Much of the efforts have been devoted to binary images and volumes, which are a subset of orthogonal shapes. Two new skeletal representations based on this paradigm are introduced: the cube skeleton and the scale cube skeleton. The cube skeleton is shown to be composed of straight line segments or planar faces and to be homotopical equivalent to the input shape. The scale cube skeleton is based upon the cube skeleton, and introduces a family of skeletons that are more stable to shape noise and perturbations. In addition, the necessary algorithms to compute the cube skeleton of polygons and polyhedra and the scale cube skeleton of polygons are presented. Several experimental results confirm the efficiency, robustness and practical use of all the presented methods

Preprocessing methods for morphometric brain analysis and quality assurance of structural magnetic resonance images

Gegenstand der Dissertation ist die Neuentwicklung und Validierung von Verfahren zur Aufbereitung von anatomischen Daten, die mittels Magnetresonanztomographie gewonnen wurden. Ziel ist dabei die Erfassung von morphometrischen Kennwerten zur Beschreibung der Struktur und Form des Gehirns, wie beispielsweise Volumen, Fläche, Dicke oder Faltung der Großhirnrinde. Die Kennwerte erlauben sowohl die Erforschung individueller gesunder und pathologischer Entwicklung als auch der evolutionären Anpassung des Gehirns. Die zur Datenanalyse notwendige Vorverarbeitung beinhaltet dabei die Angleichung von Bildeigenschaften und individueller Anatomie. Die fortlaufende Weiterentwicklung der Scanner- und Rechentechnik ermöglicht eine zunehmend genauere Bildgebung, erfordert aber die kontinuierliche Anpassung existierender Verfahren. Die Schwerpunkte dieser Dissertation lagen in der Entwicklung neuer Verfahren zur (i) Klassifikation der Hirngewebe (Segmentierung), (ii) räumlichen Abbildung des individuellen Gehirns auf ein Durchschnittsgehirn (Registrierung), (iii) Bestimmung der Dicke der Großhirnrinde und Rekonstruktion einer repräsentativen Oberfläche und (iv) Qualitätssicherung der Eingangsdaten. Die Segmentierung gleicht die Bildeigenschaften unterschiedlicher Protokolle an, während die Registrierung anatomische Merkmale normalisiert und so den Vergleich verschiedener Gehirne ermöglicht. Die Rekonstruktion von Oberflächen erlaubt wiederum die Gewinnung einer Vielzahl weiterer morphometrischer Maße zur spezifischen Charakterisierung des Gehirns und seiner Entwicklung. Anhand von simulierten und realen Daten wird die Validität der neuen Methoden belegt und mit anderen Ansätzen verglichen. Die Verfahren sind Bestandteil der Computational Anatomy Toolbox (CAT; http://dbm.neuro.uni-jena.de/cat), deren Schwerpunkt die Vorverarbeitung von strukturellen Daten ist und die Teil des Statistical Parametric Mapping (SPM) Softwarepaketes in MATLAB ist.This Ph.D. thesis focuses on the development, optimization and validation of preprocessing methods of structural magnetic resonance images of the brain. The preprocessing describes the creation of morphometric data that support a statistical analysis of brain anatomy. Image interferences have to be removed to allow a tissue classification (segmentation). In order to compare different subjects a spatial normalization to an average-shaped brain (template) is required, where atlas maps allow identification of specific brain structures and regions of interest. Beside the analysis in a voxel-grid, the cortex can be represented by surfaces that allow further measures such as the cortical thickness or folding. The derived brain features (such as volume, area, and thickness) permit the individual study of normal and pathological development during the lifespan but also of the evolutionary adaption of the brain. The ongoing progress of imaging and computing technology demands continous enhancement of preprocessing tools but also facilitates the exploration of novel approaches and models. The basis of this thesis is the development of a method that uses a tissue segmentation to estimate the cortical thickness and the central surface in one integrated step. Further essential improvements of surface reconstruction algorithms were achieved by specific refinement of processing steps such as (i) the classification of brain tissue (segmentation), (ii) the spatial mapping of the individual brain to an average brain (registration), (iii) determining the thickness of the cerebral cortex and reconstructing a representative surface and (iv) the quality assurance of input data. The validity of the new methods is proven and compared with other approaches by simulated and real data. The procedures are part of the Computational Anatomy Toolbox (CAT; http://dbm.neuro.uni-jena.de/cat), which focuses on the preprocessing of structural data and is part of the Statistical Parametric Mapping (SPM) software package in MATLAB