We present a study in level set representation and evolution using radial basis functions (RBFs) for active contour and active surface models. It builds on recent works by others who introduced RBFs into level sets for structural topology optimisation. Here, we introduce the concept into deformable models and present a new level set formulation able to handle more complex topological changes, in particular perturbation away from the evolving front. In the conventional level set technique, the initial active contour/surface is implicitly represented by a signed distance function and periodically re-initialised to maintain numerical stability. We interpolate the initial distance function using RBFs on a much coarser grid, which provides great potential in modelling in high dimensional space. Its deformation is considered as an updating of the RBF interpolants, an ordinary differential equation (ODE) problem, instead of a partial differential equation (PDE) problem, and hence it becomes much easier to solve. Re-initialisation is found no longer necessary, in contrast to conventional finite difference method (FDM) based level set approaches. The proposed level set updating scheme is efficient and does not suffer from self-flattening while evolving, hence it avoids large numerical errors. Further, more complex topological changes are readily achievable and the initial contour or surface can be placed arbitrarily in the image. These properties are extensively demonstrated on both synthetic and real 2D and 3D data. We also present a novel active contour model, implemented with this level set scheme, based on multiscale learning and fusion of image primitives from vector-valued data, e.g. colour images, without channel separation or decomposition.
Xianghua Xie and Majid Mirmehdi, Radial Basis Function based Level Set Interpolation and Evolution for Deformable Modelling, Image and Vision Computing, volume 29, issues 2-3, 167 – 177, February 2011.