823 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
Sparse image reconstruction on the sphere: implications of a new sampling theorem
We study the impact of sampling theorems on the fidelity of sparse image
reconstruction on the sphere. We discuss how a reduction in the number of
samples required to represent all information content of a band-limited signal
acts to improve the fidelity of sparse image reconstruction, through both the
dimensionality and sparsity of signals. To demonstrate this result we consider
a simple inpainting problem on the sphere and consider images sparse in the
magnitude of their gradient. We develop a framework for total variation (TV)
inpainting on the sphere, including fast methods to render the inpainting
problem computationally feasible at high-resolution. Recently a new sampling
theorem on the sphere was developed, reducing the required number of samples by
a factor of two for equiangular sampling schemes. Through numerical simulations
we verify the enhanced fidelity of sparse image reconstruction due to the more
efficient sampling of the sphere provided by the new sampling theorem.Comment: 11 pages, 5 figure
A combined first and second order variational approach for image reconstruction
In this paper we study a variational problem in the space of functions of
bounded Hessian. Our model constitutes a straightforward higher-order extension
of the well known ROF functional (total variation minimisation) to which we add
a non-smooth second order regulariser. It combines convex functions of the
total variation and the total variation of the first derivatives. In what
follows, we prove existence and uniqueness of minimisers of the combined model
and present the numerical solution of the corresponding discretised problem by
employing the split Bregman method. The paper is furnished with applications of
our model to image denoising, deblurring as well as image inpainting. The
obtained numerical results are compared with results obtained from total
generalised variation (TGV), infimal convolution and Euler's elastica, three
other state of the art higher-order models. The numerical discussion confirms
that the proposed higher-order model competes with models of its kind in
avoiding the creation of undesirable artifacts and blocky-like structures in
the reconstructed images -- a known disadvantage of the ROF model -- while
being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
Deep spatial and tonal data optimisation for homogeneous diffusion inpainting
Difusion-based inpainting can reconstruct missing image areas with high quality from sparse data, provided that their location and their values are well optimised. This is particularly useful for applications such as image compression, where the
original image is known. Selecting the known data constitutes a challenging optimisation problem, that has so far been only
investigated with model-based approaches. So far, these methods require a choice between either high quality or high speed
since qualitatively convincing algorithms rely on many time-consuming inpaintings. We propose the frst neural network
architecture that allows fast optimisation of pixel positions and pixel values for homogeneous difusion inpainting. During
training, we combine two optimisation networks with a neural network-based surrogate solver for difusion inpainting. This
novel concept allows us to perform backpropagation based on inpainting results that approximate the solution of the inpainting equation. Without the need for a single inpainting during test time, our deep optimisation accelerates data selection by
more than four orders of magnitude compared to common model-based approaches. This provides real-time performance
with high quality results
Optimising Different Feature Types for Inpainting-based Image Representations
Inpainting-based image compression is a promising alternative to classical
transform-based lossy codecs. Typically it stores a carefully selected subset
of all pixel locations and their colour values. In the decoding phase the
missing information is reconstructed by an inpainting process such as
homogeneous diffusion inpainting. Optimising the stored data is the key for
achieving good performance. A few heuristic approaches also advocate
alternative feature types such as derivative data and construct dedicated
inpainting concepts. However, one still lacks a general approach that allows to
optimise and inpaint the data simultaneously w.r.t. a collection of different
feature types, their locations, and their values. Our paper closes this gap. We
introduce a generalised inpainting process that can handle arbitrary features
which can be expressed as linear equality constraints. This includes e.g.
colour values and derivatives of any order. We propose a fully automatic
algorithm that aims at finding the optimal features from a given collection as
well as their locations and their function values within a specified total
feature density. Its performance is demonstrated with a novel set of features
that also includes local averages. Our experiments show that it clearly
outperforms the popular inpainting with optimised colour data with the same
density
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