1,216 research outputs found
Correcting curvature-density effects in the Hamilton-Jacobi skeleton
The Hainilton-Jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two-dimensional (2-D) shapes. The approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero, while the flux is negative at the locations of skeletal points. Nonetheless, the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and, hence, unbounded near endpoints. This makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter. This problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision. However, here, we develop a method for 2-D skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error. This is done by taking into account variations of density due to boundary curvature. This yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton-Jacobi method
Electromagnetic field theory without divergence problems: 1. The Born Legacy
A fully consistent classical relativistic electrodynamics with spinless point
charges is constructed. The classical evolution of the electromagnetic fields
is governed by the nonlinear Maxwell--Born--Infeld field equations, the
classical evolution of the point charges by a many-body Hamilton--Jacobi law of
motion. The Pauli principle for bosons can be incorporated in the classical
Hamilton--Jacobi formalism. The Cauchy problem is explained and illustrated
with examples. The question of charge-free field solitons is addressed also and
it is shown that if they exist, their peak field strengths must be enormous.
The value The value of Born's constant is shown to be a subtle open issue.Comment: Minor corrections at galley stage incorporated. 66p; to appear in JSP
vol. 116, issue dedicated to Elliott H. Lieb on his 70th birthday. Part II is
math-ph/031103
Semidefinite Relaxations for Stochastic Optimal Control Policies
Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation
have led to the discovery of a formulation of the value function as a linear
Partial Differential Equation (PDE) for stochastic nonlinear systems with a
mild constraint on their disturbances. This has yielded promising directions
for research in the planning and control of nonlinear systems. This work
proposes a new method obtaining approximate solutions to these linear
stochastic optimal control (SOC) problems. A candidate polynomial with variable
coefficients is proposed as the solution to the SOC problem. A Sum of Squares
(SOS) relaxation is then taken to the partial differential constraints, leading
to a hierarchy of semidefinite relaxations with improving sub-optimality gap.
The resulting approximate solutions are shown to be guaranteed over- and
under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in
Portland, Oregon. 7 pages, colo
Semiclassical Theory and the Koopman-van Hove Equation
The phase space Koopman-van Hove (KvH) equation can be derived from the
asymptotic semiclassical analysis of partial differential equations.
Semiclassical theory yields the Hamilton-Jacobi equation for the complex phase
factor and the transport equation for the amplitude. These two equations can be
combined to form a nonlinear semiclassical version of the KvH equation in
configuration space. Every solution of the configuration space KvH equation
satisfies both the semiclassical phase space KvH equation and the
Hamilton-Jacobi constraint. For configuration space solutions, this constraint
resolves the paradox that there are two different conserved densities in phase
space. For integrable systems, the KvH spectrum is the Cartesian product of a
classical and a semiclassical spectrum. If the classical spectrum is
eliminated, then, with the correct choice of Jeffreys-Wentzel-Kramers-Brillouin
(JWKB) matching conditions, the semiclassical spectrum satisfies the
Einstein-Brillouin-Keller quantization conditions which include the correction
due to the Maslov index. However, semiclassical analysis uses different choices
for boundary conditions, continuity requirements, and the domain of definition.
For example, use of the complex JWKB method allows for the treatment of
tunneling through the complexification of phase space. Finally, although KvH
wavefunctions include the possibility of interference effects, interference is
not observable when all observables are approximated as local operators on
phase space. Observing interference effects requires consideration of nonlocal
operations, e.g. through higher orders in the asymptotic theory.Comment: 49 pages, 10 figure
Particle dynamics inside shocks in Hamilton-Jacobi equations
Characteristics of a Hamilton-Jacobi equation can be seen as action
minimizing trajectories of fluid particles. For nonsmooth "viscosity"
solutions, which give rise to discontinuous velocity fields, this description
is usually pursued only up to the moment when trajectories hit a shock and
cease to minimize the Lagrangian action. In this paper we show that for any
convex Hamiltonian there exists a uniquely defined canonical global nonsmooth
coalescing flow that extends particle trajectories and determines dynamics
inside the shocks. We also provide a variational description of the
corresponding effective velocity field inside shocks, and discuss relation to
the "dissipative anomaly" in the limit of vanishing viscosity.Comment: 15 pages, no figures; to appear in Philos. Trans. R. Soc. series
Involutive constrained systems and Hamilton-Jacobi formalism
In this paper, we study singular systems with complete sets of involutive
constraints. The aim is to establish, within the Hamilton-Jacobi theory, the
relationship between the Frobenius' theorem, the infinitesimal canonical
transformations generated by constraints in involution with the Poisson
brackets, and the lagrangian point (gauge) transformations of physical systems
- …