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Objective Function

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 $I_{m}$ and $I_{m}'$ to be the images before and after warping respectively. Let us also define a warping function $f_{w}(x)$ to be $f_{w}(I_{m},<parameters>)=I_{m}'$. For a similarity³⁶ function $f_{sim}$, the objective function can then take the form:

$$f_{objective}=f_{sim}(f_{w}(I_{m},<params>),I_{m}')+<norm-terms>.$$ (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].