This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with ${L}^{2}\times {L}^{\infty}$ initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point singularities in the space-time, when the initial data are in ${H}^{1}\times {H}^{1}\cap {L}^{\infty}$. A family of fully discrete approximation...

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter $\epsilon $, and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our...

This paper studies the gradient flow of a regularized Mumford-Shah functional
proposed by Ambrosio and Tortorelli (1990, 1992) for image
segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting.
It is shown that the gradient flow with initial data
possesses a global weak solution, and it has a unique global in time
strong solution, which has at most finite number of point singularities
in the space-time, when the initial data are in .
A family of fully discrete
approximation...

We study the gradient flow for the total variation
functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow,
and establish well-posedness of the problem by the energy method.
The main idea of our approach is to exploit the relationship between
the regularized gradient flow (characterized by a small positive parameter
, see (1.7)) and the minimal surface flow [21]
and the prescribed mean curvature flow [16].
Since...

We consider a degenerate parabolic system which models
the evolution of nematic liquid crystal with variable degree of orientation.
The system
is a slight modification
to that proposed in [Calderer ,
(2002) 1033–1047], which is a special case of
Ericksen's general continuum model in [Ericksen,
(1991) 97–120].
We prove the global existence
of weak solutions by passing to the limit in a regularized system.
Moreover, we
propose a practical fully discrete finite...

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