Manual Partial Differential Equations with Minimal Smoothness and Applications

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A coupled PDE model of nonlinear diffusion for image smoothing and segmentation Abstract: Image denoising and segmentation are fundamental problems in the field of image processing and computer vision with numerous applications. We propose a partial differential equation PDE based smoothing and segmentation framework wherein the image data are smoothed via an evolution equation that is controlled by a vector field describing a viscous fluid flow.

Image segmentation in this framework is defined by locations in the image where the fluid velocity is a local maximum. The nonlinear image smoothing is selectively achieved to preserve edges in the image. The novelty of this approach lies in the fact that the selective term is derived from a nonlinearly regularized image gradient field unlike most earlier techniques which either used a constant with respect to time selective term or a time varying nonlinearly smoothed scalar valued term.

Implementation results on synthetic and real images are presented to depict the performance of the technique in comparison to methods recently reported in literature. In other scenarios, partial differential equations different with above mentioned can be applied. For solving the Eq. Finite differential method and iterative algorithm are simple and easy to be applied in solving Diffusion Eq.

The numerical formula is denoted as Eq.

Finite element method or meshless method can be applied to solve the Laplace and Poisson equations. Figure 4a shows a random information distribution of sensornets.

What are applications of Partial differential equations?

Similar boundary conditions are set for the Laplace problem. The expected potential field is approximated by the proposed method. The result is shown in Fig. The smoothness and reservation of main feathers can be advantageous to various applications. A smooth version of the distribution is expected and it will benefit the later application. With this purpose, Diffusion, Laplace and Poisson equations are introduced to construct the information potential fields in sensornets.

They are the smooth versions of some original information distributions and keep the main features.

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These features are useful to many applications. The authors would like to thank the anonymous reviewers for their valuable comments. Subscribe Today. Science Alert.

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About the Book

Krysl, Meshless methods: An overview and recent developments. Methods Applied Mech. Haussecker and F. Zhao, Scalable information driven sensor querying and routing for ad hoc heterogeneous sensor networks. High Performance Comput. Helmy, Faruque, J. In recent years there has been a great deal of activity in both the theoretical and applied aspects of partial differential equations, with emphasis on realistic engineering applications, which usually involve lack of smoothness. On March , , the University of Chicago hosted a workshop that brought together approximately fortyfive experts in theoretical and applied aspects of these subjects.

The workshop was a vehicle for summarizing the current status of research in these areas, and for defining new directions for future progress - this volume contains articles from participants of the workshop. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

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