With the energy based approach one minimizes the energy of the deformable
mesh. The energy function is defined for every mesh in
the set of admissible meshes and the resulting surface mesh is the
minimum argument of this function. The energy is
where 
. The purpose of the internal energy
is to control the shape of mesh in order to the resulting
surface mesh to be smooth. The external energy 
couples the mesh with
salient image features.
The internal energy of 
is defined as
where 
is a measure of area of 
.
Note that
the internal energy is invariant to scalings, translations and
rotations of the mesh.
A comparison of (3) and (6) reveals
that (3) is the negative of the gradient
of the numerator of 
for each i. However, similar
result does not hold if we consider the numerator of the internal energy of
as whole because also 
, depend on .
The external energy can be defined as
where I is again the pre-processed version of the original image
with intensity values scaled from 0 to 1.
The minimization of (5) is a difficult task due to numerous local minima and a large number of variables. Furthermore, the minimization has to be constrained; Otherwise a global minimum of (5) is easily obtained by finding the voxel of the highest intensity in I and forming a mesh inside that voxel (and this is not what we want).
Methods for the minimization of the energy of the deformable surface meshes are various. Many of these are based on the iteration of some local optimization algorithm. These include multi-resolution methods (two quite different algorithms based on this idea are presented in [6] and in [11]) and methods that try to force minimization process stuck in a local energy minimum out of the minimum; two variants are presented in [17] and [14] (see also [10]). Somewhat different optimization routine based on genetic algorithms is presented in [12]. Its usability is limited by its high computational cost, but it offers a great generality when designing the energy function of the meshes.