2,843 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
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
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