Geometric observers for dynamically evolving curves
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Updated: Jul 5, 2026

Three-Dimensional Mapping of the Rotation of Interactive Virtual Objects with Eye-Tracking Data
Published on: October 18, 2024
Marc Niethammer1, Patricio A Vela, Allen Tannenbaum
1Department of Computer Science, University of North Carolina, Chapel Hill, NC 27599-3175, USA. mn@cs.unc.edu
Abstract:
This paper proposes a deterministic observer framework for visual tracking based on non-parametric implicit (level-set) curve descriptions. The observer is continuous-discrete, with continuous-time system dynamics and discrete-time measurements. Its state-space consists of an estimated curve position augmented by additional states (e.g., velocities) associated with every point on the estimated curve. Multiple simulation models are proposed for state prediction. Measurements are performed through standard static segmentation algorithms and optical-flow computations. Special emphasis is given to the geometric formulation of the overall dynamical system. The discrete-time measurements lead to the problem of geometric curve interpolation and the discrete-time filtering of quantities propagated along with the estimated curve. Interpolation and filtering are intimately linked to the correspondence problem between curves. Correspondences are established by a Laplace-equation approach. The proposed scheme is implemented completely implicitly (by Eulerian numerical solutions of transport equations) and thus naturally allows for topological changes and subpixel accuracy on the computational grid.
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