Joan Lasenby

New Geometric Methods for Computer Vision: An Application toStructure and Motion Estimation

We discuss a coordinate-free approach to the geometry of computer vision problems. The technique we use to analyse the three-dimensional transformations involved will be that of geometric algebra: a framework based on the algebras of Clifford and Grassmann. This is not a system designed specifically for the task in hand, but rather a framework for all mathematical physics. Central to the power of this approach is the way in which the formalism deals with rotations; for example, if we have two arbitrary sets of vectors, known to be related via a 3D rotation, the rotation is easily recoverable if the vectors are given. Extracting the rotation by conventional means is not as straightforward. The calculus associated with geometric algebra is particularly powerful, enabling one, in a very natural way, to take derivatives with respect to any multivector (general element of the algebra). What this means in practice is that we can minimize with respect to rotors representing rotations, vectors representing translations, or any other relevant geometric quantity. This has important implications for many of the least-squares problems in computer vision where one attempts to find optimal rotations, translations etc., given observed vector quantities. We will illustrate this by analysing the problem of estimating motion from a pair of images, looking particularly at the more difficult case in which we have available only 2D information and no information on range. While this problem has already been much discussed in the literature, we believe the present formulation to be the only one in which least-squares estimates of the motion and structure are derived simultaneously using analytic derivatives.

FABRIK: A fast, iterative solver for the Inverse Kinematics problem

Inverse Kinematics is defined as the problem of determining a set of appropriate joint configurations for which the end effectors move to desired positions as smoothly, rapidly, and as accurately as possible. However, many of the currently available methods suffer from high computational cost and production of unrealistic poses. In this paper, a novel heuristic method, called Forward And Backward Reaching Inverse Kinematics (FABRIK), is described and compared with some of the most popular existing methods regarding reliability, computational cost and conversion criteria. FABRIK avoids the use of rotational angles or matrices, and instead finds each joint position via locating a point on a line. Thus, it converges in few iterations, has low computational cost and produces visually realistic poses. Constraints can easily be incorporated within FABRIK and multiple chains with multiple end effectors are also supported.

Applications of conformal geometric algebra in computer vision and graphics

This paper introduces the mathematical framework of conformal geometric algebra (CGA) as a language for computer graphics and computer vision. Specifically it discusses a new method for pose and position interpolation based on CGA which firstly allows for existing interpolation methods to be cleanly extended to pose and position interpolation, but also allows for this to be extended to higher-dimension spaces and all conformal transforms (including dilations). In addition, we discuss a method of dealing with conics in CGA and the intersection and reflections of rays with such conic surfaces. Possible applications for these algorithms are also discussed.

High angular resolution submillimeter observations of Sagittarius B2

Continuum observations of the Sgr B2 molecular cloud have been carried out. The data offer the first detailed description of the submillimeter radiation from this complex source. The emission is dominated by the middle and northern compact sources, but the far-infrared radiation associated with the Northeast H II region L and from the southern H II region/molecular maser source H have been detected for the first time. The data, together with existing infrared measurements, indicate that the optical depth within Sgr B2(N) is significantly greater than that of Sgr B2(M) and that it has a noticeable effect for lambda = 800 microns or less. The envelope of Sgr B2(M) becomes optically thick for the radiation from Sgr B2(N), located within it, for wavelengths 100 microns or less. From radiative transfer modeling, a luminosity of 10 million solar is obtained for the middle source and 2 million solar for the northern source. 30 refs.

Motion capture with constrained inverse kinematics for real-time hand tracking

Articulated hand tracking systems have been commonly used in virtual reality applications, including systems with human-computer interaction or interaction with game consoles. However, building an effective real-time hand pose tracker remains challenging. In this paper, we present a simple and efficient methodology for tracking and reconstructing 3d hand poses using a markered optical motion capture system. Markers were positioned at strategic points, and an inverse kinematics solver was incorporated to fit the rest of the joints to the hand model. The model is highly constrained with rotational and orientational constraints, allowing motion only within a feasible set. The method is real-time implementable and the results are promising, even with a low frame rate.

Geometric algebra: a framework for computing point and line correspondences and projective structure using n uncalibrated cameras

We present geometric algebra as system for analysing the geometry of multiple-view images. The power of this approach is illustrated by giving purely geometric derivations of the constraints, for point and line correspondences in n-views and via a discussion of projective structure.

Real-time marker prediction and CoR estimation in optical motion capture

Optical motion capture systems suffer from marker occlusions resulting in loss of useful information. This paper addresses the problem of real-time joint localisation of legged skeletons in the presence of such missing data. The data is assumed to be labelled 3d marker positions from a motion capture system. An integrated framework is presented which predicts the occluded marker positions using a Variable Turn Model within an Unscented Kalman filter. Inferred information from neighbouring markers is used as observation states; these constraints are efficient, simple, and real-time implementable. This work also takes advantage of the common case that missing markers are still visible to a single camera, by combining predictions with under-determined positions, resulting in more accurate predictions. An Inverse Kinematics technique is then applied ensuring that the bone lengths remain constant over time; the system can thereby maintain a continuous data-flow. The marker and Centre of Rotation (CoR) positions can be calculated with high accuracy even in cases where markers are occluded for a long period of time. Our methodology is tested against some of the most popular methods for marker prediction and the results confirm that our approach outperforms these methods in estimating both marker and CoR positions.

Real-Time Estimation of Missing Markers in Human Motion Capture

This paper considers the problem of taking marker locations from optical motion capture data to identify and parameterise the underlying human skeleton structure and motion over time. It is concerned with real-time algorithms suitable for use within a visual feedback system. A common problem in motion capture is marker occlusion. Most current methods are only useful for offline processing or become ineffective when a significant portion of markers are missing for a long period of time. This paper presents a prediction algorithm, using a Kalman filter approach in combination with inferred information from neighbouring markers, to provide a continuous flow of data. The results are accurate and reliable even in cases where all markers on a limb are occluded, or one or two markers are not visible for a large sequence of frames. Pre-defined models are not required and skeleton fitting to this complete data can then be updated in real-time.

A new methodology for computing invariants in computer vision

We present geometric algebra, as a new, framework for the theory and computation of invariants in computer vision and compare it with the currently popular Grassmann-Cayley algebra. We also discuss the formation, of 3D projective invariants in terms of image coordinates.

Guide to Geometric Algebra in Practice

This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.