8,901 research outputs found
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
Strain Analysis by a Total Generalized Variation Regularized Optical Flow Model
In this paper we deal with the important problem of estimating the local
strain tensor from a sequence of micro-structural images realized during
deformation tests of engineering materials. Since the strain tensor is defined
via the Jacobian of the displacement field, we propose to compute the
displacement field by a variational model which takes care of properties of the
Jacobian of the displacement field. In particular we are interested in areas of
high strain. The data term of our variational model relies on the brightness
invariance property of the image sequence. As prior we choose the second order
total generalized variation of the displacement field. This prior splits the
Jacobian of the displacement field into a smooth and a non-smooth part. The
latter reflects the material cracks. An additional constraint is incorporated
to handle physical properties of the non-smooth part for tensile tests. We
prove that the resulting convex model has a minimizer and show how a
primal-dual method can be applied to find a minimizer. The corresponding
algorithm has the advantage that the strain tensor is directly computed within
the iteration process. Our algorithm is further equipped with a coarse-to-fine
strategy to cope with larger displacements. Numerical examples with simulated
and experimental data demonstrate the very good performance of our algorithm.
In comparison to state-of-the-art engineering software for strain analysis our
method can resolve local phenomena much better
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