Research on Calculation of Sobolev Active Contour Model

During the last decade, Active Contour Models had been greatly developed and were used in image edge detection, image segmentation and classification, image registration, 3D reconstruction and many other areas. In the meantime, Active Contour Models themselves had also been enriched and completed. Various typed Active Contour Models had emerged, such as Sobolev Active Contour Model(SACM). SACM was considered as a global model and showed great promise in application. It out-performs the traditional model for the same energy in many cases. SACM is a new model, so there are few cases of its application was reported. But it was believed that SACM will be a very important tool for image segmentation. SACM covers vast areas of study such as Soloblev space, Steepest decent methods, Variation of functionals, Riesz representing theorem, Riemannian geometry, PDE, and Image procession. This paper tries to clarify the relationships among those different areas of studies. Especially, the paper had introduced new formulae for the calculation of Soloblev gradients, because the formulae initiated by Sundaramoorthi are not strict mathematically.