327 research outputs found
Convergent numerical approximation of the stochastic total variation flow
We study the stochastic total variation flow (STVF) equation with linear
multiplicative noise. By considering a limit of a sequence of regularized
stochastic gradient flows with respect to a regularization parameter
we obtain the existence of a unique variational solution of the
STVF equation which satisfies a stochastic variational inequality. We propose
an energy preserving fully discrete finite element approximation for the
regularized gradient flow equation and show that the numerical solution
converges to the solution of the unregularized STVF equation. We perform
numerical experiments to demonstrate the practicability of the proposed
numerical approximation
Regularized Interpolation for Noisy Images
Interpolation is the means by which a continuously defined model is fit to discrete data samples. When the data samples are exempt of noise, it seems desirable to build the model by fitting them exactly. In medical imaging, where quality is of paramount importance, this ideal situation unfortunately does not occur. In this paper, we propose a scheme that improves on the quality by specifying a tradeoff between fidelity to the data and robustness to the noise. We resort to variational principles, which allow us to impose smoothness constraints on the model for tackling noisy data. Based on shift-, rotation-, and scale-invariant requirements on the model, we show that the Lp-norm of an appropriate vector derivative is the most suitable choice of regularization for this purpose. In addition to Tikhonov-like quadratic regularization, this includes edge-preserving total-variation-like (TV) regularization. We give algorithms to recover the continuously defined model from noisy samples and also provide a data-driven scheme to determine the optimal amount of regularization. We validate our method with numerical examples where we demonstrate its superiority over an exact fit as well as the benefit of TV-like nonquadratic regularization over Tikhonov-like quadratic regularization
Trends in Mathematical Imaging and Surface Processing
Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments
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