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### Open Access Journals

IPI

Detecting optical flow means to find the apparent displacement field in a
sequence of images.
As starting point for many optical flow methods serves the so called
optical flow constraint (OFC), that is the assumption that the gray value of
a moving point does not change over time. Variational methods are amongst
the most popular tools to compute the optical flow field. They compute the
flow field as minimizer of an energy functional that consists of a data term
to comply with the OFC and a smoothness term to obtain uniqueness of this
underdetermined problem. In this article we replace the smoothness term by
projecting the solution to a finite dimensional, affine subspace in the
spatial variables which leads to a smoothing and gives a unique solution as
well. We explain the mathematical details for the quadratic and nonquadratic
minimization framework, and show how alternative model assumptions such as
constancy of the brightness gradient can be incorporated.
As basis functions we consider tensor products of B-splines.
Under certain smoothness assumptions for the global
minimizer in Sobolev scales, we prove optimal convergence rates in terms
of the energy functional. Experiments are presented that demonstrate the
feasibility of our approach.

IPI

The relations between wavelet shrinkage and nonlinear diffusion for
discontinuity-preserving signal denoising are fairly well-understood
for single-scale wavelet shrinkage, but not for the practically
relevant multiscale case. In this paper we show that 1-D multiscale
continuous wavelet shrinkage can be linked to novel integrodifferential
equations. They differ from nonlinear diffusion filtering and corresponding
regularisation methods by the fact that they involve smoothed derivative
operators and perform a weighted averaging over all scales. Moreover,
by expressing the convolution-based smoothed derivative operators by
power series of differential operators, we show that multiscale wavelet
shrinkage can also be regarded as averaging over pseudodifferential
equations.

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