Fast Difference Schemes for Edge Enhancing Beltrami Flow and Subjective Surfaces

  • Ravi Malladi
  • Igor Ravve
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Numerical integration of space-scale PDE is one of the most time consuming operation of image processing. The scale step is limited by conditional stability of explicit schemes. In this work, we introduce the unconditionally stable semi-implicit linearized difference scheme for the Beltrami flow. The Beltrami flow [14, 15] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the Beltrami flow equation can be arranged in a reaction-diffusion form. This reveals the edge-enhancing properties of the equation and suggests the application of additive operator split (AOS) methods [4, 5] for faster convergence. As we show with numerical simulations, the AOS method results in an unconditionally stable semi-implicit linearized difference scheme in 2D and 3D. The values of the edge indicator function are used from the previous step in scale, while the pixel values of the next step are used to approximate the flow. The optimum ratio between the reaction and diffusion counterparts of the governing PDE is studied. We then apply this approach to fast subjective surface computation. The computational time decreases by a factor of 20 and more, as compared to the explicit scheme.


Beltrami flow subjective surfaces unconditionally stable scheme segmentation 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ravi Malladi
    • 1
  • Igor Ravve
    • 1
  1. 1.Lawrence Berkeley National Laboratory, Computing Science DepartmentUniversity of CaliforniaBerkeleyUSA

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