The objective function is the actual function which needs to be minimised
in order for an optimal choice or a solution to be picked from the
many alternatives offered. The function is most heavily based on similarity
measures as briefly explained earlier, but it allows this measure
to be extended in some way. For example, it can be helpful to include
the computational cost of the warps that are used. The reason why
the cost of the warp is sometimes an integral part of the function
is that long-winded warps are not nearly as desirable as simple ones
that perform the task equally well or even better. This cost is often
considered a *normalisation term*.

Objective functions are built to encapsulate in a concise and effective
way everything that is repeatedly evaluated. They are therefore required
to be a very efficient unit which will be invoked quite frequently.
The *speed* of the registration will directly depend on the choice
of an objective function that adds up results from warps, similarity
calculations and possibly more components, as can be seen in current
group-wise registration papers. The *quality* of the registration
will of course depend on this function, too.

Let us define two images and to be the images before
and after warping respectively. Let us also define a warping function
to be
. For a similarity^{36} function , the objective function can then take the form:

(3.2) |

The function then attempts to find the parameter values that will
lead it to a globally minimal solution. More precisely, it attempts
to find *assignments* for all parameters that describe the warps
so that similarity is maximised.

The explanation on the objective function concludes the algorithmic approach that registration takes. Non-rigid registration algorithms can be assessed by methods such as the one described by Warfield [39].