Equations (11a) and (11b) can be solved by
treating *u* and *v* as functions of time and solving
The steady-state solution (as
)
of these
linear parabolic equations is the desired solution of the Euler
equations (11a) and (11b). Note that these
equations are decoupled, and therefore can be solved as separate
scalar partial differential equations in *u* and *v*.
The equations in (12) are known as *generalized
diffusion equations*, and are known to arise in such diverse fields
as heat conduction, reactor physics, and fluid flow [30].
For us, they have
appeared from our description of desirable properties of external fields
for active contours. Diffusion is a natural outcome given the desired
``filling in'' property.

For convenience, we rewrite Equation (12) as follows
where

Any digital image gradient operator (cf. [31]) can be used to calculate

To set up the iterative solution, let the indices *i*, *j*, and
*n* correspond to *x*, *y*, and *t*, respectively, and let the spacing
between pixels be
and
and the time step for each
iteration be .
Then the required partial derivatives can be
approximated as

Substituting these approximations into (13) gives our iterative solution to GVF: where

Convergence of the above iterative process is guaranteed by a standard
result in the theory of numerical methods (cf. [32]). Provided
that *b*, *c*^{1}, and *c*^{2} are bounded, (14) is
stable whenever the Courant-Friedrichs-Lewy step-size restriction
is maintained. Since normally ,
,
and
are fixed, using the definition of *r* in (15) we find
that the following restriction on the time-step
must be
maintained in order to guarantee convergence of GVF:

The intuition behind this condition is revealing. First, convergence can be made to be faster on coarser images -- i.e., when and are larger. Second, when is large and the GVF is expected to be a smoother field, the convergence rate will be slower (since must be kept small).