On Vesicles and Amoebae or Modeling Constrained Shape Deformations

Modeling the deformation of shapes under constraints on both perimeter

and area is a challenging task due to the highly nontrivial interaction 

between the need for flexible local rules for manipulating the boundary 

and the global constraints. We propose several methods to address this problem

 and generate "random walks" in the  space of shapes obeying quite general possibly

 time varying constraints on their perimeter and area. Design of perimeter and area

 preserving deformations are an interesting and useful special case of this problem. 

The resulting deformation models are employed in annealing processes that evolve 

original shapes toward shapes that are optimal in terms of boundary bending-energy 

or other functionals. Furthermore, such models may find applications in the analysis 

of sequences of real images of deforming objects obeying global constraints as building

 blocks for registration and tracking algorithms.