Summary
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.
This work was supported by the Director, Office of Science, Office of Advanced Scientific Research, Mathematical, Information, and Computational Sciences Division, U.S. Department of Energy under Contract No. DE-ACO3-76SF00098, and LBNL Directed Research and Development Program
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Malladi, R., Ravve, I. (2003). Fast Difference Schemes for Edge Enhancing Beltrami Flow and Subjective Surfaces. In: Hege, HC., Polthier, K. (eds) Visualization and Mathematics III. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05105-4_15
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DOI: https://doi.org/10.1007/978-3-662-05105-4_15
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