Sebastian Kurtek

Parameterization-Invariant Shape Comparisons of Anatomical Surfaces

We consider 3-D brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized. Past comparisons of such surfaces involve either volume deformations or non-rigid matching under fixed parameterizations of surfaces. We propose a new mathematical representation of surfaces, called q-maps, such that L2 distances between such maps are invariant to re-parameterizations. This property allows for removing the parameterization variability by optimizing over the re-parameterization group, resulting in a proper parameterization-invariant distance between shapes of surfaces. We demonstrate this method in shape analysis of multiple brain structures, for 34 subjects in the Detroit Fetal Alcohol and Drug Exposure Cohort study, which results in a 91% classification rate for attention deficit hyperactivity disorder cases and controls. This method outperforms some existing techniques such as spherical harmonic point distribution model (SPHARM-PDM) or iterative closest point (ICP).

Elastic Geodesic Paths in Shape Space of Parameterized Surfaces

This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces.

A novel riemannian framework for shape analysis of 3D objects

In this paper we introduce a novel Riemannian framework for shape analysis of parameterized surfaces. We derive a distance function between any two surfaces that is invariant to rigid motion, global scaling, and re-parametrization. It is the last part that presents the main difficulty. Our solution to this problem is twofold: (1) we define a special representation, called a q-map, to represent each surface, and (2) we develop a gradient-based algorithm to optimize over different re-parameterizations of a surface. The second step is akin to deforming the mesh on a fixed surface to optimize its placement. (This is different from the current methods that treat the given meshes as fixed.) Under the chosen representation, with the L2 metric, the action of the re-parametrization group is by isometries. This results in, to our knowledge, the first Riemannian distance between parameterized surfaces to have all the desired invariances. We demonstrate this framework with several examples using some toy shapes, and real data with anatomical structures, and cropped facial surfaces. We also successfully demonstrate clustering and classification of these objects under the proposed metric.

Statistical Modeling of Curves Using Shapes and Related Features

Motivated by the problems of analyzing protein backbones, diffusion tensor magnetic resonance imaging (DT-MRI) fiber tracts in the human brain, and other problems involving curves, in this study we present some statistical models of parameterized curves, in , in terms of combinations of features such as shape, location, scale, and orientation. For each combination of interest, we identify a representation manifold, endow it with a Riemannian metric, and outline tools for computing sample statistics on these manifolds. An important characteristic of the chosen representations is that the ensuing comparison and modeling of curves is invariant to how the curves are parameterized. The nuisance variables, including parameterization, are removed by forming quotient spaces under appropriate group actions. In the case of shape analysis, the resulting spaces are quotient spaces of Hilbert spheres, and we derive certain wrapped truncated normal densities for capturing variability in observed curves. We demonstrate these models using both artificial data and real data involving DT-MRI fiber tracts from multiple subjects and protein backbones from the Shape Retrieval Contest of Non-rigid 3D Models (SHREC) 2010 database.

Landmark-Guided Elastic Shape Analysis of Spherically-Parameterized Surfaces

We argue that full surface correspondence (registration) and optimal deformations (geodesics) are two related problems and propose a framework that solves them simultaneously. We build on the Riemannian shape analysis of anatomical and star‐shaped surfaces of Kurtek et al. and focus on articulated complex shapes that undergo elastic deformations and that may contain missing parts. Our core contribution is the re‐formulation of Kurtek et al.s approach as a constrained optimization over all possible re‐parameterizations of the surfaces, using a sparse set of corresponding landmarks. We introduce a landmark‐constrained basis, which we use to numerically solve this optimization and therefore establish full surface registration and geodesic deformation between two surfaces. The length of the geodesic provides a measure of dissimilarity between surfaces. The advantages of this approach are: (1) simultaneous computation of full correspondence and geodesic between two surfaces, given a sparse set of matching landmarks (2) ability to handle more comprehensive deformations than nearly isometric, and (3) the geodesics and the geodesic lengths can be further used for symmetrizing 3D shapes and for computing their statistical averages. We validate the framework on challenging cases of large isometric and elastic deformations, and on surfaces with missing parts. We also provide multiple examples of averaging and symmetrizing 3D models.

A Riemannian Elastic Metric for Shape-Based Plant Leaf Classification

The shapes of plant leaves are of great importance to plant biologists and botanists, as they can help to distinguish plant species and measure their health. In this paper, we study the performance of the Squared Root Velocity Function (SRVF) representation of closed planar curves in the analysis of plant-leaf shapes. We show that it provides a joint framework for computing geodesics (registration) and similarities between plant leaves, which we use for their automatic classification. We evaluate its performance using standard databases and show that it outperforms significantly the state-of-the-art descriptor-based techniques. Additionally, we show that it enables the computation of shape statistics, such as the average shape of a leaf population and its principal directions of variation, suggesting that the representation is suitable for building generative models of plant- leaf shapes.

Landmark-free statistical analysis of the shape of plant leaves

The shapes of plant leaves are important features to biologists, as they can help in distinguishing plant species, measuring their health, analyzing their growth patterns, and understanding relations between various species. Most of the methods that have been developed in the past focus on comparing the shape of individual leaves using either descriptors or finite sets of landmarks. However, descriptor-based representations are not invertible and thus it is often hard to map descriptor variability into shape variability. On the other hand, landmark-based techniques require automatic detection and registration of the landmarks, which is very challenging in the case of plant leaves that exhibit high variability within and across species. In this paper, we propose a statistical model based on the Squared Root Velocity Function (SRVF) representation and the Riemannian elastic metric of Srivastava et al. (2011) to model the observed continuous variability in the shape of plant leaves. We treat plant species as random variables on a non-linear shape manifold and thus statistical summaries, such as means and covariances, can be computed. One can then study the principal modes of variations and characterize the observed shapes using probability density models, such as Gaussians or Mixture of Gaussians. We demonstrate the usage of such statistical model for (1) efficient classification of individual leaves, (2) the exploration of the space of plant leaf shapes, which is important in the study of population-specific variations, and (3) comparing entire plant species, which is fundamental to the study of evolutionary relationships in plants. Our approach does not require descriptors or landmarks but automatically solves for the optimal registration that aligns a pair of shapes. We evaluate the performance of the proposed framework on publicly available benchmarks such as the Flavia, the Swedish, and the ImageCLEF2011 plant leaf datasets.

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

Parameterization-invariant shape statistics and probabilistic classification of anatomical surfaces

\ensuremath{\mathbb{L}^2}

On advances in differential-geometric approaches for 2D and 3D shape analyses and activity recognition

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.