Dynamic Estimation of Rigid Motion from Perspective Views via Recursive Identification of Exterior Differential Systems with Parameters on a Topological Manifold

We formulate the problem of estimating the motion of a rigid object viewed under perspective projection as the identification of a dynamic model in Exterior Differential form with parameters on a topological manifold. We first describe a general method for recursive identification of nonlinear implicit systems using prediction error criteria. The parameters are allowed to move slowly on some topological (not necessarily smooth) manifold. The basic recursion is solved in two different ways: one is based on a simple extension of the traditional Kalman Filter to nonlinear and implicit measurement constraints, the other may be regarded as a generalized "Gauss-Newton" iteration, akin to traditional Recursive Prediction Error Method techniques in linear identification. A derivation of the "Implicit Extended Kalman Filter" (IEKF) is reported in the appendix. The ID framework is then applied to solving the visual motion problem: it indeed is possible to characterize it in terms of identification of an Exterior Differential System with parameters living on a C0 topological manifold, called the "essential manifold". We consider two alternative estimation paradigms. The first is in the local coordinates of the essential manifold: we estimate the state of a nonlinear implicit model on a linear space. The second is obtained by a linear update on the (linear) embedding space followed by a projection onto the essential manifold. These schemes proved successful in performing the motion estimation task, as we show in experiments on real and noisy synthetic image sequences

Poisson integrators

An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. Numerical integrators using generating functions, Hamiltonian splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems and the discrete gradient methods are also presented.Comment: 30 page

From Darboux-Egorov system to bi-flat FF-manifolds

Motivated by the theory of integrable PDEs of hydrodynamic type and by the generalization of Dubrovin's duality in the framework of $F$-manifolds due to Manin [22], we consider a special class of $F$-manifolds, called bi-flat $F$-manifolds. A bi-flat $F$-manifold is given by the following data $(M, \nabla_1,\nabla_2,\circ,*,e,E)$, where $(M, \circ)$ is an $F$-manifold, $e$ is the identity of the product $\circ$, $\nabla_1$ is a flat connection compatible with $\circ$ and satisfying $\nabla_1 e=0$, while $E$ is an eventual identity giving rise to the dual product *, and $\nabla_2$ is a flat connection compatible with * and satisfying $\nabla_2 E=0$. Moreover, the two connections $\nabla_1$ and $\nabla_2$ are required to be hydrodynamically almost equivalent in the sense specified in [2]. First we show that, similarly to the way in which Frobenius manifolds are constructed starting from Darboux-Egorov systems, also bi-flat $F$-manifolds can be built from solutions of suitably augmented Darboux-Egorov systems, essentially dropping the requirement that the rotation coefficients are symmetric. Although any Frobenius manifold possesses automatically the structure of a bi-flat $F$-manifold, we show that the latter is a strictly larger class. In particular we study in some detail bi-flat $F$-manifolds in dimensions n=2, 3. For instance, we show that in dimension 3 bi-flat $F$-manifolds are parametrized by solutions of a two parameters Painlev\'e VI equation, admitting among its solutions hypergeometric functions. Finally we comment on some open problems of wide scope related to bi-flat $F$-manifolds.Comment: 32 pages, eliminated a remark at the end of proof of Theorem 6.

On a computational approach for the approximate dynamics of averaged variables in nonlinear ODE systems: toward the derivation of constitutive laws of the rate type

A non-perturbative approach to the time-averaging of nonlinear, autonomous ODE systems is developed based on invariant manifold methodology. The method is implemented computationally and applied to model problems arising in the mechanics of solids.Comment: 34 pages PD