608 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
Enhanced Lasso Recovery on Graph
This work aims at recovering signals that are sparse on graphs. Compressed
sensing offers techniques for signal recovery from a few linear measurements
and graph Fourier analysis provides a signal representation on graph. In this
paper, we leverage these two frameworks to introduce a new Lasso recovery
algorithm on graphs. More precisely, we present a non-convex, non-smooth
algorithm that outperforms the standard convex Lasso technique. We carry out
numerical experiments on three benchmark graph datasets
Filling in CMB map missing data using constrained Gaussian realizations
For analyzing maps of the cosmic microwave background sky, it is necessary to
mask out the region around the galactic equator where the parasitic foreground
emission is strongest as well as the brightest compact sources. Since many of
the analyses of the data, particularly those searching for non-Gaussianity of a
primordial origin, are most straightforwardly carried out on full-sky maps, it
is of great interest to develop efficient algorithms for filling in the missing
information in a plausible way. We explore practical algorithms for filling in
based on constrained Gaussian realizations. Although carrying out such
realizations is in principle straightforward, for finely pixelized maps as will
be required for the Planck analysis a direct brute force method is not
numerically tractable. We present some concrete solutions to this problem, both
on a spatially flat sky with periodic boundary conditions and on the pixelized
sphere. One approach is to solve the linear system with an appropriately
preconditioned conjugate gradient method. While this approach was successfully
implemented on a rectangular domain with periodic boundary conditions and
worked even for very wide masked regions, we found that the method failed on
the pixelized sphere for reasons that we explain here. We present an approach
that works for full-sky pixelized maps on the sphere involving a kernel-based
multi-resolution Laplace solver followed by a series of conjugate gradient
corrections near the boundary of the mask.Comment: 22 pages, 14 figures, minor changes, a few missing references adde
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