Parametric Segmentation

Short description

This segmentation method relies on a parametric representation of the object contour. The solution is expressed as the minimizer of an energy which general form is a linear combination of a line integral along the contour, a domain integral on the contour interior, and a domain integral on the contour exterior. The energy is therefore a function of the parameters defining the contour, e.g., center and radius for a circle. \mathcal E(\Gamma = \Gamma(x,y,r))
             =
             \alpha
             \oint_\Gamma
                v_c(\Gamma, s)
             \ \mbox ds
             \ \dots
             \left\{
                \begin{array}{@{}l@{}l@{}l@{}l@{}}
                   \ +\ \beta_1 &
                   \displaystyle
                   \ \int_{D(\Gamma)} &
                      v_{r_1}(D(\Gamma), u,v) &
                   \ \mbox du
                   \mbox dv
                \\[2em]
                   \ +\ \beta_2 &
                   \displaystyle
                   \ \int_{\bar D(\Gamma)} &
                      v_{r_2}(\bar D(\Gamma), u,v) &
                   \ \mbox du
                   \mbox dv
                \end{array}
             \right. The solution is computed with a gradient descent, noting that the energy gradient with respect to the contour parameters writes as a chain rule with the parameter-related contour deformation fields and the shape derivative of the energy (see ICIP 2006 on the page Conferences for complete reference and preprint).

Illustrative result

The picture below used to illustrate the method was downloaded from WikipediA, The Free Encyclopedia. It was taken by Lee R. Berger at Palau, Micronesia, and shows a stingless jellyfish. The original picture and its licensing information can be found at:
http://en.wikipedia.org/wiki/File:Palau_stingless_jellyfish.jpg

Initialization
Jellyfish segmentation: Initialization
— circle →

ellipse
          ↘
     spline with 15
     control points
                       ↘
Jellyfish segmentation with a circle

Jellyfish segmentation with an ellipse Jellyfish segmentation with a spline with 15 cntrol points