306 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
Exact asymptotics of the uniform error of interpolation by multilinear splines
The question of adaptive mesh generation for approximation by splines has
been studied for a number of years by various authors. The results have
numerous applications in computational and discrete geometry, computer aided
geometric design, finite element methods for numerical solutions of partial
differential equations, image processing, and mesh generation for computer
graphics, among others. In this paper we will investigate the questions
regarding adaptive approximation of C2 functions with arbitrary but fixed
throughout the domain signature by multilinear splines. In particular, we will
study the asymptotic behavior of the optimal error of the weighted uniform
approximation by interpolating and quasi-interpolating multilinear splines
Anisotropic Mesh Adaptation for Image Representation
Triangular meshes have gained much interest in image representation and have
been widely used in image processing. This paper introduces a framework of
anisotropic mesh adaptation (AMA) methods to image representation and proposes
a GPRAMA method that is based on AMA and greedy-point removal (GPR) scheme.
Different than many other methods that triangulate sample points to form the
mesh, the AMA methods start directly with a triangular mesh and then adapt the
mesh based on a user-defined metric tensor to represent the image. The AMA
methods have clear mathematical framework and provides flexibility for both
image representation and image reconstruction. A mesh patching technique is
developed for the implementation of the GPRAMA method, which leads to an
improved version of the popular GPRFS-ED method. The GPRAMA method can achieve
better quality than the GPRFS-ED method but with lower computational cost.Comment: 25 pages, 15 figure
Exact asymptotics of the optimal Lp-error of asymmetric linear spline approximation
In this paper we study the best asymmetric (sometimes also called penalized
or sign-sensitive) approximation in the metrics of the space , , of functions with nonnegative
Hessian by piecewise linear splines , generated by given
triangulations with elements. We find the exact asymptotic
behavior of optimal (over triangulations and splines error of such approximation as
Graph Wedgelets: Adaptive Data Compression on Graphs based on Binary Wedge Partitioning Trees and Geometric Wavelets
We introduce graph wedgelets - a tool for data compression on graphs based on
the representation of signals by piecewise constant functions on adaptively
generated binary graph partitionings. The adaptivity of the partitionings, a
key ingredient to obtain sparse representations of a graph signal, is realized
in terms of recursive wedge splits adapted to the signal. For this, we transfer
adaptive partitioning and compression techniques known for 2D images to general
graph structures and develop discrete variants of continuous wedgelets and
binary space partitionings. We prove that continuous results on best m-term
approximation with geometric wavelets can be transferred to the discrete graph
setting and show that our wedgelet representation of graph signals can be
encoded and implemented in a simple way. Finally, we illustrate that this
graph-based method can be applied for the compression of images as well.Comment: 12 pages, 10 figure
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