97 research outputs found
Optimising Spatial and Tonal Data for PDE-based Inpainting
Some recent methods for lossy signal and image compression store only a few
selected pixels and fill in the missing structures by inpainting with a partial
differential equation (PDE). Suitable operators include the Laplacian, the
biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The
quality of such approaches depends substantially on the selection of the data
that is kept. Optimising this data in the domain and codomain gives rise to
challenging mathematical problems that shall be addressed in our work.
In the 1D case, we prove results that provide insights into the difficulty of
this problem, and we give evidence that a splitting into spatial and tonal
(i.e. function value) optimisation does hardly deteriorate the results. In the
2D setting, we present generic algorithms that achieve a high reconstruction
quality even if the specified data is very sparse. To optimise the spatial
data, we use a probabilistic sparsification, followed by a nonlocal pixel
exchange that avoids getting trapped in bad local optima. After this spatial
optimisation we perform a tonal optimisation that modifies the function values
in order to reduce the global reconstruction error. For homogeneous diffusion
inpainting, this comes down to a least squares problem for which we prove that
it has a unique solution. We demonstrate that it can be found efficiently with
a gradient descent approach that is accelerated with fast explicit diffusion
(FED) cycles. Our framework allows to specify the desired density of the
inpainting mask a priori. Moreover, is more generic than other data
optimisation approaches for the sparse inpainting problem, since it can also be
extended to nonlinear inpainting operators such as EED. This is exploited to
achieve reconstructions with state-of-the-art quality.
We also give an extensive literature survey on PDE-based image compression
methods
A meshless, integration-free, and boundary-only RBF technique
Based on the radial basis function (RBF), non-singular general solution and
dual reciprocity method (DRM), this paper presents an inherently meshless,
integration-free, boundary-only RBF collocation techniques for numerical
solution of various partial differential equation systems. The basic ideas
behind this methodology are very mathematically simple. In this study, the RBFs
are employed to approximate the inhomogeneous terms via the DRM, while
non-singular general solution leads to a boundary-only RBF formulation for
homogenous solution. The present scheme is named as the boundary knot method
(BKM) to differentiate it from the other numerical techniques. In particular,
due to the use of nonsingular general solutions rather than singular
fundamental solutions, the BKM is different from the method of fundamental
solution in that the former does no require the artificial boundary and results
in the symmetric system equations under certain conditions. The efficiency and
utility of this new technique are validated through a number of typical
numerical examples. Completeness concern of the BKM due to the only use of
non-singular part of complete fundamental solution is also discussed
Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique
Based on the radial basis function (RBF), non-singular general solution and
dual reciprocity principle (DRM), this paper presents an inheretnly meshless,
exponential convergence, integration-free, boundary-only collocation techniques
for numerical solution of general partial differential equation systems. The
basic ideas behind this methodology are very mathematically simple and
generally effective. The RBFs are used in this study to approximate the
inhomogeneous terms of system equations in terms of the DRM, while non-singular
general solution leads to a boundary-only RBF formulation. The present method
is named as the boundary knot method (BKM) to differentiate it from the other
numerical techniques. In particular, due to the use of non-singular general
solutions rather than singular fundamental solutions, the BKM is different from
the method of fundamental solution in that the former does no need to introduce
the artificial boundary and results in the symmetric system equations under
certain conditions. It is also found that the BKM can solve nonlinear partial
differential equations one-step without iteration if only boundary knots are
used. The efficiency and utility of this new technique are validated through
some typical numerical examples. Some promising developments of the BKM are
also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email:
[email protected] or [email protected]
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