Adaptivity in 3D Image Processing
We suggest computational method based on Rothe's approximation in time and finite element method in space for solving the nonlinear image selective smoothing equation
b(u)_t - div ( g( |grad G_sigma * u|) grad u ) = f (u_0 - u).
The equation is a generalization of the 'modified' (in the sense of Catte, Lions, Morel and Coll) Perona-Malik anisotropic diffusion model. The new nonlinearity represented by an increasing function b(u) is added making the image analysis dependent on values of the intensity function.
We present some results of image reconstructions. The successive coarsening of 3D tetrahedral finite element grid (obtained by refinement of a macro-triangulation by bisection method) is documented. The coarsening decreases computational costs of the finite element method rapidly.
Figure 1. Original form which is inside a 3D image (without noise).
Figure 2. Reconstruction of the form from noisy image (top left) by our model equation. Three steps until the shape without noise is reached.
Figure 3. The extracted shape and cut of the computational finite element grid after 4 discrete scale steps.
Figure 4. The extracted shape and cut of the computational finite element grid after 10 discrete scale steps.
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