Co-Volume Level Set Method in Subjective Surface Based Medical Image Segmentation - Biomedical Image Analysis - page 594

Figure 11.14: Results of the segmentation process for testing image plotted in
Fig. 11.12 (left) using ε = 10 − 2 (top left) and ε = 10 − 5 (top right). The isoline
(max ( u ) + min ( u )) / 2 well represents the segmented circle (bottom red line).
For large range of ε , we get satisfactory results (color slide).
11.3 Semi-implicit Co-Volume Scheme
We present our method in discretization of Eq. (11.8), although we always use
its ε -regularization (11.2) with a specific ε> 0. The notation is simpler in the
case of (11.8) and it will be clear where regularization appears in the numerical
scheme.
First we choose a uniform discrete time step τ and a variance σ of the
smoothing kernel G σ . Then we replace time derivative in (11.8) by backward
difference. The nonlinear terms of the equation are treated from the previous
time step while the linear ones are considered on the current time level, this
means semi-implicitness of the time discretization. In the last decade, semi-
implicit schemes have become a powerful tool in image processing, we refer
e.g. to [3, 4, 25-27, 33, 37, 51, 57, 58].
Semi-implicit in time discretization. Let τ and σ be fixed numbers, I 0 be
a given image, and u 0 be a given initial segmentation function. Then, for