George Kamberov

Bonnet pairs and isothermic surfaces

In this note we classify all Bonnet pairs on a simply connected domain. Our main intent was to apply what we call a quaternionic function theory to a concrete problem in differential geometry. The ideas are simple: conformal immersions into quaternions or imaginary quaternions take the place of chart maps for a Riemann surface. Starting from a reference immersion we construct all conformal immersions of a given (simply connected) Riemann surface (up to translational periods) by spin transformations. With this viewpoint in mind we discuss how to construct all Bonnet pairs on a simply connected domain from isothermic surfaces and vice versa. Isothermic surfaces are solutions to a certain soliton equation and thus a simple dimension count tells us that we obtain Bonnet pairs which are not part of any of the classical Bonnet families. The corresponcence between Bonnet pairs and isothermic surfaces is explicit and to each isothermic surface we obtain a 4-parameter family of Bonnet pairs.

An Adaptive Background Model for Camshift Tracking with a Moving Camera

Continuously Adaptive Mean shift (CAMSHIFT) is a popular algorithm for visual tracking, providing speed and robustness with minimal training and computational cost. While it performs well with a fixed camera and static background scene, it can fail rapidly when the camera moves or the background changes since it relies on static models of both the background and the tracked object. Furthermore it is unable to track objects passing in front of backgrounds with which they share significant colours. We describe a new algorithm, the Adaptive Background CAMSHIFT (ABCshift), which addresses both of these problems by using a background model which can be continuously relearned for every frame with minimal additional computational expense. Further, we show how adaptive background relearning can occasionally lead to a particular mode of instability which we resolve by comparing background and tracked object distributions using a metric based on the Bhattacharyya coefficient.

3D geometry from uncalibrated images

We present an automatic pipeline for recovering the geometry of a 3D scene from a set of unordered, uncalibrated images. The contributions in the paper are the presentation of the system as a whole, from images to geometry, the estimation of the local scale for various scene components in the orientation-topology module, the procedure for orienting the cloud components, and the method for dealing with points of contact. The methods are aimed to process complex scenes and non-uniformly sampled, noisy data sets.

Quaternions, Spinors, and Surfaces

Conformal immersions via quaternions: Quaternionic calculus and immersions Applications Surfaces and Dirac spinors: Spinor algebra Dirac spinors and conformal immersions Bibliography Glossary of symbols Index.

Topology and geometry of unorganized point clouds

We present a new method for defining neighborhoods, and assigning principal curvature frames, and mean and Gauss curvatures to the points of an unorganized oriented point-cloud. The neighborhoods are estimated by measuring implicitly the surface distance between points. The 3D shape recovery is based on conformal geometry, works directly on the cloud, does not rely on the generation of polygonal, or smooth models. Test results on publicly available synthetic data, as ground truth, demonstrate that the method compares favorably to the established approaches for quantitative 3D shape recovery. The proposed method is developed to serve applications involving point based rendering and reliable extraction of differential properties from noisy unorganized point-clouds.

Prescribing mean curvature: existence and uniqueness problems

The paper presents results on the extent to which mean curvature data can be used to determine a surface in R3 or its shape. The emphasis is on Bonnet’s problem: classify and study the surface immersions in R3 whose shape is not uniquely determined by the first fundamental form and the mean curvature function. These immersions are called Bonnet immersions. A local solution of Bonnet’s problem for umbilic-free immersions follows from papers by Bonnet, Cartan, and Chern. The properties of immersions with umbilics and global rigidity results for closed surfaces are presented in the first part of this paper. The second part of the paper outlines an existence theory for conformal immersions based on Dirac spinors along with its immediate applications to Bonnet’s problem. The presented existence paradigm provides insight into the topology of the moduli space of Bonnet immersions of a closed surface, and reveals a parallel between Bonnet’s problem and Pauli’s exclusion principle.

Conformal method for quantitative shape extraction: performance evaluation

We evaluate our developed conformal method for quantitative shape extraction from unorganized 3D oriented point clouds. The conformal method has been tested previously on real, noisy, 3D data. Here we focus on the empirical evaluation of its performance on synthetic, ground truth data, and comparisons with other methods for quantitative extraction of mean and Gauss curvatures presented in the literature.

Designing an Experiential Learning Environment for Logistics and Systems Engineering

Systems engineering increasingly addresses the system lifecycle, as opposed to its more traditional role focusing on design and development. This new situation results in part from the recognition that upstream design and deployment decisions have potentially significant cost and performance implications post-deployment. For military systems, the role that typically addresses post-deployment issues is the logistician. Over the system lifecycle, it is important that the traditional roles of systems engineer and logistician understand issues faced by one another, as well as joint cost and performance implications. This paper presents the design of a role-based experiential learning environment for logisticians involved in military sustainment. This design leverages the generic components of an existing single-learner technology base, the Experience Accelerator, for presenting and controlling the learner experience, plus simulating program outcomes resulting from learner decisions. This technology base has been used to create a learning experience for a lead systems engineer in charge of designing and developing a new unmanned aerial vehicle (UAV) system. In this new environment, the logistician learner interacts with systems engineers during UAV system acquisition and sustainment, learns about systems engineering issues and their effect on logistics, tries to influence upstream systems engineering decisions, and also performs logistics functions.

3D shape from unorganized 3d point clouds

We present a framework to automatically infer topology and geometry from an unorganized 3D point cloud obtained from a 3D scene. If the cloud is not oriented, we use existing methods to orient it prior to recovering the topology. We develop a quality measure for scoring a chosen topology/orientation. The topology is used to segment the cloud into manifold components and later in the computation of shape descriptors.

Recovering Surfaces from the Restoring Force

We present a new theoretical method and experimental results for direct recovery of the curvatures, the principal curvature directions, and the surface itself by exolicit integration of the Gauss map. The method does not rely on polygonal approximations, smoothing of the data, or model fitting. It is based on the observation that one can recover the surface restoring force from the Gauss map, and (i) applies to orientable surfaces of arbitrary topology (not necessarily closed); (ii) uses only first order linear differential equations; (iii) avoids the use of unstable computations; (iv) provides tools for filtering noise from the sampled data. The method can be used for stable extraction of surfaces and surface shape invariants, in particular, in applications requiring accurate quantitative measurements.