15,147 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
Analysis and approximation of some Shape-from-Shading models for non-Lambertian surfaces
The reconstruction of a 3D object or a scene is a classical inverse problem
in Computer Vision. In the case of a single image this is called the
Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a
simplified version like the vertical light source case. A huge number of works
deals with the orthographic SfS problem based on the Lambertian reflectance
model, the most common and simplest model which leads to an eikonal type
equation when the light source is on the vertical axis. In this paper we want
to study non-Lambertian models since they are more realistic and suitable
whenever one has to deal with different kind of surfaces, rough or specular. We
will present a unified mathematical formulation of some popular orthographic
non-Lambertian models, considering vertical and oblique light directions as
well as different viewer positions. These models lead to more complex
stationary nonlinear partial differential equations of Hamilton-Jacobi type
which can be regarded as the generalization of the classical eikonal equation
corresponding to the Lambertian case. However, all the equations corresponding
to the models considered here (Oren-Nayar and Phong) have a similar structure
so we can look for weak solutions to this class in the viscosity solution
framework. Via this unified approach, we are able to develop a semi-Lagrangian
approximation scheme for the Oren-Nayar and the Phong model and to prove a
general convergence result. Numerical simulations on synthetic and real images
will illustrate the effectiveness of this approach and the main features of the
scheme, also comparing the results with previous results in the literature.Comment: Accepted version to Journal of Mathematical Imaging and Vision, 57
page
On the optimality of shape and data representation in the spectral domain
A proof of the optimality of the eigenfunctions of the Laplace-Beltrami
operator (LBO) in representing smooth functions on surfaces is provided and
adapted to the field of applied shape and data analysis. It is based on the
Courant-Fischer min-max principle adapted to our case. % The theorem we present
supports the new trend in geometry processing of treating geometric structures
by using their projection onto the leading eigenfunctions of the decomposition
of the LBO. Utilisation of this result can be used for constructing numerically
efficient algorithms to process shapes in their spectrum. We review a couple of
applications as possible practical usage cases of the proposed optimality
criteria. % We refer to a scale invariant metric, which is also invariant to
bending of the manifold. This novel pseudo-metric allows constructing an LBO by
which a scale invariant eigenspace on the surface is defined. We demonstrate
the efficiency of an intermediate metric, defined as an interpolation between
the scale invariant and the regular one, in representing geometric structures
while capturing both coarse and fine details. Next, we review a numerical
acceleration technique for classical scaling, a member of a family of
flattening methods known as multidimensional scaling (MDS). There, the
optimality is exploited to efficiently approximate all geodesic distances
between pairs of points on a given surface, and thereby match and compare
between almost isometric surfaces. Finally, we revisit the classical principal
component analysis (PCA) definition by coupling its variational form with a
Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can
handle cases that go beyond the scope defined by the observation set that is
handled by regular PCA
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