Ian H. Jermyn

Shape Analysis of Elastic Curves in Euclidean Spaces

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL2 metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses.

Globally optimal regions and boundaries as minimum ratio weight cycles

We describe a new form of energy functional for the modeling and identification of regions in images. The energy is defined on the space of boundaries in the image domain and can incorporate very general combinations of modeling information both from the boundary (intensity gradients, etc.) and from the interior of the region (texture, homogeneity, etc.). We describe two polynomial-time digraph algorithms for finding the global minima of this energy. One of the algorithms is completely general, minimizing the functional for any choice of modeling information. It runs in a few seconds on a 256/spl times/256 image. The other algorithm applies to a subclass of functionals, but has the advantage of being extremely parallelizable. Neither algorithm requires initialization.

A Novel Representation for Riemannian Analysis of Elastic Curves in Rn

We propose a novel representation of continuous, closed curves in Rn that is quite efficient for analyzing their shapes. We combine the strengths of two important ideas-elastic shape metric and path-straightening methods - in shape analysis and present a fast algorithm for finding geodesies in shape spaces. The elastic metric allows for optimal matching of features while path-straightening provides geodesies between curves. Efficiency results from the fact that the elastic metric becomes the simple L2 metric in the proposed representation. We present step-by-step algorithms for computing geodesies in this framework, and demonstrate them with 2-D as well as 3-D examples.

Higher Order Active Contours

We introduce a new class of active contour models that hold great promise for region and shape modelling, and we apply a special case of these models to the extraction of road networks from satellite and aerial imagery. The new models are arbitrary polynomial functionals on the space of boundaries, and thus greatly generalize the linear functionals used in classical contour energies. While classical energies are expressed as single integrals over the contour, the new energies incorporate multiple integrals, and thus describe long-range interactions between different sets of contour points. As prior terms, they describe families of contours that share complex geometric properties, without making reference to any particular shape, and they require no pose estimation. As likelihood terms, they can describe multi-point interactions between the contour and the data. To optimize the energies, we use a level set approach. The forces derived from the new energies are non-local however, thus necessitating an extension of standard level set methods. Networks are a shape family of great importance in a number of applications, including remote sensing imagery. To model them, we make a particular choice of prior quadratic energy that describes reticulated structures, and augment it with a likelihood term that couples the data at pairs of contour points to their joint geometry. Promising experimental results are shown on real images.

Removing shape-preserving transformations in square-root elastic (SRE) framework for shape analysis of curves

This paper illustrates and extends an efficient framework, called the square-root-elastic (SRE) framework, for studying shapes of closed curves, that was first introduced in [2]. This framework combines the strengths of two important ideas - elastic shape metric and path-straightening methods - for finding geodesics in shape spaces of curves. The elastic metric allows for optimal matching of features between curves while path-straightening ensures that the algorithm results in geodesic paths. This paper extends this framework by removing two important shape preserving transformations: rotations and re-parameterizations, by forming quotient spaces and constructing geodesics on these quotient spaces. These ideas are demonstrated using experiments involving 2D and 3D curves.

Looking for Shapes in Two-Dimensional Cluttered Point Clouds

We study the problem of identifying shape classes in point clouds. These clouds contain sampled points along contours and are corrupted by clutter and observation noise. Taking an analysis-by-synthesis approach, we simulate high-probability configurations of sampled contours using models learned from training data to evaluate the given test data. To facilitate simulations, we develop statistical models for sources of (nuisance) variability: 1) shape variations within classes, 2) variability in sampling continuous curves, 3) pose and scale variability, 4) observation noise, and 5) points introduced by clutter. The variability in sampling closed curves into finite points is represented by positive diffeomorphisms of a unit circle. We derive probability models on these functions using their square-root forms and the Fisher-Rao metric. Using a Monte Carlo approach, we simulate configurations from a joint prior on the shape-sample space and compare them to the data using a likelihood function. Average likelihoods of simulated configurations lead to estimates of posterior probabilities of different classes and, hence, Bayesian classification.

Unsupervised image segmentation via Markov trees and complex wavelets

The goal in image segmentation is to label pixels in an image based on the properties of each pixel and its surrounding region. Recently content-based image retrieval (CBIR) has emerged as an application area in which retrieval is attempted by trying to gain unsupervised access to the image semantics directly rather than via manual annotation. To this end, we present an unsupervised segmentation technique in which colour and texture models are learned from the image prior to segmentation, and whose output (including the models) may subsequently be used as a content descriptor in a CBIR system. These models are obtained in a multiresolution setting in which hidden Markov trees (HMT) are used to model the key statistical properties exhibited by complex wavelet and scaling function coefficients. The unsupervised mean shift iteration (MSI) procedure is used to determine a number of image regions which are then used to train the models for each segmentation class.

Elastic shape matching of parameterized surfaces using square root normal fields

In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple

Incorporating Generic and Specific Prior Knowledge in a Multiscale Phase Field Model for Road Extraction From VHR Images

\ensuremath{\mathbb{L}^2}

Extended Phase Field Higher-Order Active Contour Models for Networks

metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods.