Tangential curved planar reformation for topological and orientation invariant visualization of vascular trees

Recently, extensions to curved planar reformation (CPR) were proposed to improve vascular visualization of medical images. While these projective transformations provide enhanced visualization of vascular trees, non-planar alignment and arbitrary topology can cause visualization artifacts. Vascular trees in medical images are not aligned to planar cross-sections of volumetric image slices and thus aggravate simultaneous visualization of diagnostic features. Complex tree topology and non-planar alignment requires the need for an adaptive projection scheme to prevent visualization artifacts while preserving correctness of anatomical information. In this paper, we present algorithmic details for topological and orientation invariant visualization of vascular trees. Vascular high-level description of the medial axis guides the reformation process by flattening the vascular tree interior to successive image planes for respective radial sampling angles. Tree orientations are estimated from intrinsic shape properties of the vascular tree for rotation invariant projection. Radial sampling planes perpendicular to the medial axis tangents are the basis for topological invariant visualization of complete vascular interiors. We present experimental results on two different vascular tree topologies and demonstrate that our method is able to produce artifact free visualization of vascular interiors

Doctor of Philosophy

dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges

Computing a Compact Spline Representation of the Medial Axis Transform of a 2D Shape

We present a full pipeline for computing the medial axis transform of an arbitrary 2D shape. The instability of the medial axis transform is overcome by a pruning algorithm guided by a user-defined Hausdorff distance threshold. The stable medial axis transform is then approximated by spline curves in 3D to produce a smooth and compact representation. These spline curves are computed by minimizing the approximation error between the input shape and the shape represented by the medial axis transform. Our results on various 2D shapes suggest that our method is practical and effective, and yields faithful and compact representations of medial axis transforms of 2D shapes.Comment: GMP14 (Geometric Modeling and Processing

Medial/skeletal linking structures for multi-region configurations

We consider a generic configuration of regions, consisting of a collection of distinct compact regions $\{\Omega_i\}$ in $\mathbb{R}^{n+1}$ which may be either smooth regions disjoint from the others or regions which meet on their piecewise smooth boundaries $\mathcal{B}_i$ in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the collection. The linking structure extends in a minimal way the individual "skeletal structures" on each of the regions, allowing us to significantly extend the mathematical methods introduced for single regions to the configuration. We prove for a generic configuration of regions the existence of a special type of Blum linking structure which builds upon the Blum medial axes of the individual regions. This requires proving several transversality theorems for certain associated "multi-distance" and "height-distance" functions for such configurations. We show that by relaxing the conditions on the Blum linking structures we obtain the more general class of skeletal linking structures which still capture the geometric properties. In addition to yielding geometric invariants which capture the shapes and geometry of individual regions, the linking structures are used to define invariants which measure positional properties of the configuration such as: measures of relative closeness of neighboring regions and relative significance of the individual regions for the configuration. These invariants, which are computed by formulas involving "skeletal linking integrals" on the internal skeletal structures, are then used to construct a "tiered linking graph," which identifies subconfigurations and provides a hierarchical ordering of the regions.Comment: 135 pages, 36 figures. Version to appear in Memoirs of the Amer. Math. So

A Multivariate Surface-Based Analysis of the Putamen in Premature Newborns: Regional Differences within the Ventral Striatum

Many children born preterm exhibit frontal executive dysfunction, behavioral problems including attentional deficit/hyperactivity disorder and attention related learning disabilities. Anomalies in regional specificity of cortico-striato-thalamo-cortical circuits may underlie deficits in these disorders. Nonspecific volumetric deficits of striatal structures have been documented in these subjects, but little is known about surface deformation in these structures. For the first time, here we found regional surface morphological differences in the preterm neonatal ventral striatum. We performed regional group comparisons of the surface anatomy of the striatum (putamen and globus pallidus) between 17 preterm and 19 term-born neonates at term-equivalent age. We reconstructed striatal surfaces from manually segmented brain magnetic resonance images and analyzed them using our in-house conformal mapping program. All surfaces were registered to a template with a new surface fluid registration method. Vertex-based statistical comparisons between the two groups were performed via four methods: univariate and multivariate tensor-based morphometry, the commonly used medial axis distance, and a combination of the last two statistics. We found statistically significant differences in regional morphology between the two groups that are consistent across statistics, but more extensive for multivariate measures. Differences were localized to the ventral aspect of the striatum. In particular, we found abnormalities in the preterm anterior/inferior putamen, which is interconnected with the medial orbital/prefrontal cortex and the midline thalamic nuclei including the medial dorsal nucleus and pulvinar. These findings support the hypothesis that the ventral striatum is vulnerable, within the cortico-stiato-thalamo-cortical neural circuitry, which may underlie the risk for long-term development of frontal executive dysfunction, attention deficit hyperactivity disorder and attention-related learning disabilities in preterm neonates. © 2013 Shi et al

Towards Persistence-Based Reconstruction in Euclidean Spaces

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes of compact subsets of Euclidean space $\R^d$. However, the complexities of these methods scale up exponentially with d, which makes them impractical in medium or high dimensions, even for handling low-dimensional submanifolds. In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Specifically, when the data points are sufficiently densely sampled from a smooth $m$-submanifold of $\R^d$, our method retrieves the homology of the submanifold in time at most $c(m)n^5$, where $n$ is the size of the input and $c(m)$ is a constant depending solely on $m$. It can also provably well handle a wide range of compact subsets of $\R^d$, though with worse complexities. Along the way to proving the correctness of our algorithm, we obtain new results on \v{C}ech, Rips, and witness complex filtrations in Euclidean spaces