1,429 research outputs found
An approximation scheme for an Eikonal Equation with discontinuous coefficient
We consider the stationary Hamilton-Jacobi equation where the dynamics can
vanish at some points, the cost function is strictly positive and is allowed to
be discontinuous. More precisely, we consider special class of discontinuities
for which the notion of viscosity solution is well-suited. We propose a
semi-Lagrangian scheme for the numerical approximation of the viscosity
solution in the sense of Ishii and we study its properties. We also prove an
a-priori error estimate for the scheme in an integral norm. The last section
contains some applications to control and image processing problems
Analysis and approximation of some Shape-from-Shading models for non-Lambertian surfaces
The reconstruction of a 3D object or a scene is a classical inverse problem
in Computer Vision. In the case of a single image this is called the
Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a
simplified version like the vertical light source case. A huge number of works
deals with the orthographic SfS problem based on the Lambertian reflectance
model, the most common and simplest model which leads to an eikonal type
equation when the light source is on the vertical axis. In this paper we want
to study non-Lambertian models since they are more realistic and suitable
whenever one has to deal with different kind of surfaces, rough or specular. We
will present a unified mathematical formulation of some popular orthographic
non-Lambertian models, considering vertical and oblique light directions as
well as different viewer positions. These models lead to more complex
stationary nonlinear partial differential equations of Hamilton-Jacobi type
which can be regarded as the generalization of the classical eikonal equation
corresponding to the Lambertian case. However, all the equations corresponding
to the models considered here (Oren-Nayar and Phong) have a similar structure
so we can look for weak solutions to this class in the viscosity solution
framework. Via this unified approach, we are able to develop a semi-Lagrangian
approximation scheme for the Oren-Nayar and the Phong model and to prove a
general convergence result. Numerical simulations on synthetic and real images
will illustrate the effectiveness of this approach and the main features of the
scheme, also comparing the results with previous results in the literature.Comment: Accepted version to Journal of Mathematical Imaging and Vision, 57
page
user's guide to viscosity solutions of second order partial differential equations
The notion of viscosity solutions of scalar fully nonlinear partial
differential equations of second order provides a framework in which startling
comparison and uniqueness theorems, existence theorems, and theorems about
continuous dependence may now be proved by very efficient and striking
arguments. The range of important applications of these results is enormous.
This article is a self-contained exposition of the basic theory of viscosity
solutions.Comment: 67 page
Continuous dependence results for Non-linear Neumann type boundary value problems
We obtain estimates on the continuous dependence on the coefficient for
second order non-linear degenerate Neumann type boundary value problems. Our
results extend previous work of Cockburn et.al., Jakobsen-Karlsen, and
Gripenberg to problems with more general boundary conditions and domains. A new
feature here is that we account for the dependence on the boundary conditions.
As one application of our continuous dependence results, we derive for the
first time the rate of convergence for the vanishing viscosity method for such
problems. We also derive new explicit continuous dependence on the coefficients
results for problems involving Bellman-Isaacs equations and certain quasilinear
equation
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