14,240 research outputs found
Discrete embeddings for Lagrangian and Hamiltonian systems
The general topic of the present paper is to study the conservation for some
structural property of a given problem when discretising this problem.
Precisely we are interested with Lagrangian or Hamiltonian structures and thus
with variational problems attached to a least action principle. Considering a
partial differential equation (PDE) deriving from such a variational principle,
a natural question is to know whether this structure at the continuous level is
preserved at the discrete level when discretising the PDE. To address this
question a concept of \textit{coherence} is introduced. Both the differential
equation (the PDE translating the least action principle) and the variational
structure can be embedded at the discrete level. This provides two discrete
embeddings for the original problem. In case these procedures finally provide
the same discrete problem we will say that the discretisation is
\textit{coherent}. Our purpose is illustrated with the Poisson problem.
Coherence for discrete embeddings of Lagrangian structures is studied for
various classical discretisations (finite elements, finite differences and
finite volumes). Hamiltonian structures are shown to provide coherence between
a discrete Hamiltonian structure and the discretisation of the mixed
formulation of the PDE, both for mixed finite elements and mimetic finite
differences methods.Comment: Acta Mathematica Vietnamica, Springer Singapore, A Para{\^i}tr
Estimation of vector fields in unconstrained and inequality constrained variational problems for segmentation and registration
Vector fields arise in many problems of computer vision, particularly in non-rigid registration. In this paper, we develop coupled partial differential equations (PDEs) to estimate vector fields that define the deformation between
objects, and the contour or surface that defines the segmentation of the objects as well.We also explore the utility of inequality constraints applied to variational problems in vision such as estimation of deformation fields in non-rigid registration and tracking. To solve inequality constrained vector
field estimation problems, we apply tools from the Kuhn-Tucker theorem in optimization theory. Our technique differs from recently popular joint segmentation and registration algorithms, particularly in its coupled set of PDEs derived from the same set of energy terms for registration and
segmentation. We present both the theory and results that demonstrate our approach
A Variational Stereo Method for the Three-Dimensional Reconstruction of Ocean Waves
We develop a novel remote sensing technique for the observation of waves on the ocean surface. Our method infers the 3-D waveform and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are given by minimizers of a composite energy functional that combines a photometric matching term along with regularization terms involving the smoothness of the unknowns. The desired ocean surface shape and radiance are the solution of a system of coupled partial differential equations derived from the optimality conditions of the energy functional. The proposed method is naturally extended to study the spatiotemporal dynamics of ocean waves and applied to three sets of stereo video data. Statistical and spectral analysis are carried out. Our results provide evidence that the observed omnidirectional wavenumber spectrum S(k) decays as k-2.5 is in agreement with Zakharov's theory (1999). Furthermore, the 3-D spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents
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