On the application of projection methods for computing optical flow fields
Thomas Schuster Joachim Weickert
Inverse Problems & Imaging 2007, 1(4): 673-690 doi: 10.3934/ipi.2007.1.673
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.
keywords: projection methods variational methods tensor product B-spline. Optical flow
Integrodifferential equations for continuous multiscale wavelet shrinkage
Stephan Didas Joachim Weickert
Inverse Problems & Imaging 2007, 1(1): 47-62 doi: 10.3934/ipi.2007.1.47
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.
keywords: nonlinear diffusion filtering. integrodifferential equations image processing wavelet shrinkage

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