Semi-implicit finite volume scheme for image processing in 3D cylindrical geometry

Nowadays, 3D echocardiography is a well-known technique in medical diagnosis. Inexpensive echocardiographic acquisition devices are applied to scan 2D slices rotated along a prescribed direction. Then the discrete 3D image information is given on a cylindrical grid. Usually, this original discrete image intensity function is interpolated to a uniform rectangular grid and then numerical schemes for 3D image processing operations (e.g. nonlinear smoothing) in the uniform rectangular geometry are used. However, due to the generally large amount of noise present in echocardiographic images, the interpolation step can yield undesirable results. In this paper, we avoid this step and suggest a 3D finite volume method for image selective smoothing directly in the cylindrical image geometry. Specifically, we study a semi-implicit 3D cylindrical finite volume scheme for solving a Perona-Malik-type nonlinear diffusion equation and apply the scheme to 3D cylindrical echocardiographic images. The L∞-stability and convergence of the scheme to the weak solution of the regularized Perona-Malik equation is proved.