Correcting curvature-density effects in the Hamilton-Jacobi skeleton

The Hainilton-Jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two-dimensional (2-D) shapes. The approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero, while the flux is negative at the locations of skeletal points. Nonetheless, the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and, hence, unbounded near endpoints. This makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter. This problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision. However, here, we develop a method for 2-D skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error. This is done by taking into account variations of density due to boundary curvature. This yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton-Jacobi method

Coarse-to-fine skeleton extraction for high resolution 3D meshes

This paper presents a novel algorithm for medial surfaces extraction that is based on the density-corrected Hamiltonian analysis of Torsello and Hancock [1]. In order to cope with the exponential growth of the number of voxels, we compute a first coarse discretization of the mesh which is iteratively refined until a desired resolution is achieved. The refinement criterion relies on the analysis of the momentum field, where only the voxels with a suitable value of the divergence are exploded to a lower level of the hierarchy. In order to compensate for the discretization errors incurred at the coarser levels, a dilation procedure is added at the end of each iteration. Finally we design a simple alignment procedure to correct the displacement of the extracted skeleton with respect to the true underlying medial surface. We evaluate the proposed approach with an extensive series of qualitative and quantitative experiments

An adaptive hierarchical approach to the extraction of high resolution medial surfaces

We introduce a novel algorithm for medial surfaces extraction that is based on the density-corrected Hamiltonian analysis. The approach extracts the skeleton directly from a triangulated mesh and adopts an adaptive octree-based approach in which only skeletal voxels are refined to a lower level of the hierarchy, resulting in robust and accurate skeletons at extremely high resolution. The quality of the extracted medial surfaces is confirmed by an extensive set of experiments

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

Matching hierarchical structures for shape recognition

In this thesis we aim to develop a framework for clustering trees and rep- resenting and learning a generative model of graph structures from a set of training samples. The approach is applied to the problem of the recognition and classification of shape abstracted in terms of its morphological skeleton. We make five contributions. The first is an algorithm to approximate tree edit-distance using relaxation labeling. The second is the introduction of the tree union, a representation capable of representing the modes of structural variation present in a set of trees. The third is an information theoretic approach to learning a generative model of tree structures from a training set. While the skeletal abstraction of shape was chosen mainly as a exper- imental vehicle, we, nonetheless, make some contributions to the fields of skeleton extraction and its graph representation. In particular, our fourth contribution is the development of a skeletonization method that corrects curvature effects in the Hamilton-Jacobi framework, improving its localiza- tion and noise sensitivity. Finally, we propose a shape-measure capable of characterizing shapes abstracted in terms of their skeleton. This measure has a number of interesting properties. In particular, it varies smoothly as the shape is deformed and can be easily computed using the presented skeleton extraction algorithm. Each chapter presents an experimental analysis of the proposed approaches applied to shape recognition problems

A triangulation-invariant method for anisotropic geodesic map computation on surface meshes

pre-printThis paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner

An Efficient Semi-Real-Time Algorithm for Path Planning in the Hamilton-Jacobi Formulation

We present a semi-real-time algorithm for minimal-time optimal path planning based on optimal control theory, dynamic programming, and Hamilton-Jacobi (HJ) equations. Partial differential equation (PDE) based optimal path planning methods are well-established in the literature, and provide an interpretable alternative to black-box machine learning algorithms. However, due to the computational burden of grid-based PDE solvers, many previous methods do not scale well to high dimensional problems and are not applicable in real-time scenarios even for low dimensional problems. We present a semi-real-time algorithm for optimal path planning in the HJ formulation, using grid-free numerical methods based on Hopf-Lax formulas. In doing so, we retain the intepretablity of PDE based path planning, but because the numerical method is grid-free, it is efficient and does not suffer from the curse of dimensionality, and thus can be applied in semi-real-time and account for realistic concerns like obstacle discovery. This represents a significant step in averting the tradeoff between interpretability and efficiency. We present the algorithm with application to synthetic examples of isotropic motion planning in two-dimensions, though with slight adjustments, it could be applied to many other problems.Comment: 6 pages, 2 figures, submitted to American Control Conference 202

Image-based effective medium approximation for fast permeability evaluation of porous media core samples

An image-based effective medium approximation (EMA) is developed so as to permit very fast transport properties evaluations of 3D porous media. From an image-based porous network (IBPN) built upon digital image processing of 3D binary images, we focus on throat’s local geometrical properties at the pore scale, for being the most sensible structural units which build up the local pressure. This approach is a 3D image–based extension of the critical point approach proposed in 2D fractures. We show, from analyzing various core rock samples available in the literature, that the asymptotic assumptions associated with the preeminence of critical points in throats are indeed geometrically relevant. We then describe how the image-based EMA evaluated from the conductances computed from the discrete IBPN can be reliably evaluated. The proposed method is evaluated upon the estimation of core sample permeability from binarized image obtained using X-ray tomography. Since it combines digital image treatments with statistical data post-processing without the need of computational fluid dynamics (CFD) computation, it is extremely cost efficient. The results are compared with a micro-scale Stokes flow computation in various rock samples. The sensitivity to the pore discretization also is discussed and illustrated

Robust Feature Detection and Local Classification for Surfaces Based on Moment Analysis

The stable local classification of discrete surfaces with respect to features such as edges and corners or concave and convex regions, respectively, is as quite difficult as well as indispensable for many surface processing applications. Usually, the feature detection is done via a local curvature analysis. If concerned with large triangular and irregular grids, e.g., generated via a marching cube algorithm, the detectors are tedious to treat and a robust classification is hard to achieve. Here, a local classification method on surfaces is presented which avoids the evaluation of discretized curvature quantities. Moreover, it provides an indicator for smoothness of a given discrete surface and comes together with a built-in multiscale. The proposed classification tool is based on local zero and first moments on the discrete surface. The corresponding integral quantities are stable to compute and they give less noisy results compared to discrete curvature quantities. The stencil width for the integration of the moments turns out to be the scale parameter. Prospective surface processing applications are the segmentation on surfaces, surface comparison, and matching and surface modeling. Here, a method for feature preserving fairing of surfaces is discussed to underline the applicability of the presented approach.