Solution of Anisotropic Diffusion Equations
We suggest and analyze the computational method based on Rothe's approximation in time and finite element method in space for solving the modified Perona-Malik anisotropic diffusion equation
u_t - div ( g( |grad G_sigma * u|) grad u ) = f (u_0 - u)
together with zero Neumann boundary conditions and initial condition representing the processed image. Here, g(s) tends to 0 for s tends to infinity. It causes the selective smoothing of the image regions and keeping of the edges on which the 'gaussian gradient' is large (G_sigma is smoothing kernel of the convolution). Such image analysis is included in the so-called nonlinear scale space theory.
We present some results of image reconstructions. The successive coarsening finite element grid strategy which decreases the computational costs is added to the method.
Figure 1a. Original and noisy images.
Figure 1b. Reconstruction of the original and final computational finite element grid.
Figure 2a. Original and noisy images.
Figure 2b. Reconstruction of the original and final computational finite element grid.
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