Analysis of a nonconvex problem related to signal selective smoothing
Chipot M
;
March R
;
Rosati M
;
Vergara Caffarelli G
We study some properties of a nonconvex variational problem.
The infimum of the functional that has to be minimized fails to be attained.
Instead, minimizing sequences develop gradient oscillations which allow them
to decrease the value of the functional. We show an existence result
for a perturbed nonconvex version of the problem, and we study the qualitative
properties of the corresponding minimizer.
The pattern of the gradient oscillations for the original
non perturbed problem is analyzed numerically.
Variational problems; nonconvex; parametrized measures; signal processing
Variational methods for image segmentation try to recover a piecewise smooth function together with a discontinuity set which represents the boundaries of the segmentation. This paper deals with a variational method that constrains the formation of discontinuities along smooth contours. The functional to be minimized, which involves the computation of the geometrical properties of the boundaries, is approximated by a sequence of functionals which can be discretized in a straightforward way. Computer examples of real images are presented to illustrate the feasibility of the method.
Active contour models; Discontinuity detection; Image segmentation
A compression tool for satellite based imagery of the earth's land and atmosphere is presented. Orthogonal wavelet filters are adopted for this purpose. The ability of the technique to compress images has been applied to AVHRR images of cloud fields and the earth's surface. Compression by a factor as large as 50 is possible in clear sky condition without any significant loss of information, whereas for frontal cloud fields, which are the most interesting in terms of meteorological studies, factors of ten are easily achieved
The problems of smoothing data through a transform in the Fourier domain and of retrieving a function from its Fourier coefficients are analyzed in the present paper. For both of them a solution, based on regularization tools, is known. Aim of the paper is to prove strong results of convergence of the regularized solution and optimality of the Generalized Cross Validation criterion for choosing the regularization parameter.
Convergence of numerical methods
Inverse problems
Numerical analysis
Fourier series
Generalized cross validation
Smoothing data
Fourier transforms
We consider the problem of determining quantitative information about corrosion
occurring on an inaccesible part of a specimen. The data for the problem consist of prescribed
current flux and voltage measurements on an accessible part of the specimen boundary. The
problem is modelled by Laplace's equation with an unknown term
in the boundary conditions.
Our goal is recovering
from the data. We prove uniqueness under certain regularity
assumptions and construct a regularized numerical method for obtaining approximate solutions
to the problem. The numerical method, which is based on the assumption that the specimen is
a thin plate, is tested in numerical experiments using synthetic data.
inverse problems
Robin problem
Laplace's equation
corrosion detection
