WIAS Preprint No. 762, (2002)

On a nonlocal model of image segmentation


  • Gajewski, Herbert
  • Gärtner, Klaus

2010 Mathematics Subject Classification

  • 35K45 35K65 35B40 80A22 74N25


  • Cahn-Hilliard equation, initial boundary value problem, Perrona-Malik model, a priori estimates, Lyapunov function, equilibria, asymptotic behaviour, classical thermodynamics, nonlocal phase separation model, image reconstruction and separation.


We understand an image as binary grey 'alloy' of a black and a white component and use a nonlocal phase separation model to describe image segmentation. The model consists in a degenerate nonlinear parabolic equation with a nonlocal drift term additionally to the familiar Perona-Malik model. We formulate conditions for the model parameters to guarantee global existence of a unique solution that tends exponentially in time to a unique steady state. This steady state is solution of a nonlocal nonlinear elliptic boundary value problem and allows a variational characterization. Numerical examples demonstrate the properties of the model.

Appeared in

  • ZAMP Z. Angew. Math. Phys., 56 (2005) pp. 572--591.

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