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Principles

The objective function is the actual function which needs to be minimised (fundamentally by finding a set of values for all parameters) 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 was briefly explained earlier, but it allows this measure to be extended in some way. For example, it can be helpful to include the 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 uncomplicated ones that perform the task equally well or even better. This cost is often considered a regularisation term [] which penalises a sequence of warps that form large trajectories in space. It will seek a solution that is simple rather than finding an odd trajectory in space that gives a similar solution. This is the case since the solutions are often not unique.

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' (or black box) 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 the two images $\mathbf{I}_{m}$ and $\mathbf{I}_{m}'$ be defined as the images before and after warping respectively. Let a warping function $f_{w}(x)$ also be defined to be $f_{w}(\mathbf{I}_{m},<parameters>)=\mathbf{I}_{m}'$. For a similarity^5.3 function $f_{sim}$, the objective function can then take the form:

$$f_{objective}=f_{sim}(f_{w}(\mathbf{I}_{m},<params>),\mathbf{I}_{r})+<reg-terms>.$$ (5.1)

The function then attempts to find a series of 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 (or difference minimised)^5.4.

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 [].