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We design signatures for curves defined on genus zero surfaces. The signature classifies curves according to the conformal geometry of the given curves and their embedded surface. Based on Teichmüller theory, our signature describes not only the curve shape but also the intrinsic relationship between the curve and its embedded surface. Furthermore, the signature metric is stable, it is close to identity between surfaces sharing similar Riemannian geometry metrics. Based on this, we propose a surface matching framework: first, with curve signatures, we match the partitioning of two surfaces defined by simple closed curves on them; second, the segmented subregions are pairwisely matched and then compared on canonical planar domains.
This paper examines the asymptotic stabilizability of discrete-time linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable able and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open loop system is not exponentially unstable (i.e., all the open loop poles are on or inside the unit circle.) It is further shown that such a system, when subject to actuator saturation, is semi-globally asymptotically stabilizable by linear state or output feedback.
Surface parameterization establishes bijective maps from a surface onto a topologically equivalent standard domain. It is well known that the spherical parameterization is limited to genus-zero surfaces. In this work, we design a new parameter domain, two-layered sphere, and present a framework for mapping high genus surfaces onto sphere. This setup allows us to trans- fer the existing applications based on general spherical parameterization to the field of high genus surfaces, such as remeshing, consistent parameterization, shape analysis, and so on. Our method is based on Riemann surface theory. We construct meromorphic functions on surfaces: for genus one surfaces, we apply Weierstrass $P$-functions; for high genus surfaces, we compute the quotient between two holomorphic one-forms. Our method of spherical parameterization is theoretically sound and practically efficient. It makes the subsequent applications on high genus surfaces very promising.
Recursive Least Squares (RLS) algorithms have wide-spread applications in many areas, such as real-time signal processing, control and communications. This paper shows that the unique solutions to linear-equality constrained and the unconstrained LS problems, respectively, always have exactly the same recursive form. Their only difference lies in the initial values. Based on this, a recursive algorithm for the linear-inequality constrained LS problem is developed. It is shown that these RLS solutions converge to the true parameter that satisfies the constraints as the data size increases. A simple and easily implementable initialization of the RLS algorithm is proposed. Its convergence to the exact LS solution and the true parameter is shown. The RLS algorithm, in a theoretically equivalent form by a simple modification, is shown to be robust in that the constraints are always guaranteed to be satisfied no matter how large the numerical errors are. Numerical examples are provided to demonstrate the validity of the above results.