Resolution Adapted Powell's Method

A major component in the overall optimisation process is the local optimisation method. This local optimisation is called repeatedly and so should be efficient. Here Powell's method was chosen since it is a commonly used and efficient local optimisation method [Press et al., 1995].

The changes in transformation used by the optimisation method should also be adapted to the resolution of the volumes. For example, computing very small changes in transformation when the voxel size is large corresponds to very small sub-voxel changes and is a waste of computation. Consequently, a lower limit for the transformation step size is enforced. This step size is easily calculated by imposing the condition that the maximum voxel position shift should be less than half a voxel. More precisely, with a brain of radius R, and an origin located at the centre of mass for the brain, the maximum shifts are:

$$\Delta x_{max} = \Delta t$$ Translation: (28)

$$\Delta x_{max} = 2 R \cos \left( \frac{\Delta \theta}{2} \right)$$ Rotation: (29)

$$\Delta x_{max} = R \vert \Delta s \vert$$ Scale: (30)

$$\Delta x_{max} = R \vert \Delta k \vert.$$ Skew: (31)

For cubic voxels of side-length n mm the constraint becomes $x_{max} \le \frac{n}{2}$ which gives:

$$\Delta t \le \frac{n}{2}$$ (32)

$$\Delta \theta \le \frac{n}{2R}$$ (33)

$$\Delta s \le \frac{n}{2R}$$ (34)

$$\Delta k \le \frac{n}{2R}.$$ (35)

For n=1 mm this gives $\Delta t\le 0.5$ mm, $\Delta \theta\le 0.3^\circ$ , $\Delta s\le 0.005$ and $\Delta k\le 0.005$. However, since the voxel shift due to translation is constant across the volume, whereas all others are less towards the centre, it pays to be more conservative for $\Delta t$.

Given these lower limits on the parameter steps, Powell's algorithm is modified by changing the termination conditions so that when the 1D optimisation (Brent's method) has bounded the minimum within an interval that is less than one parameter step, then it returns the mid-point of the current interval. In this way, significant savings in computation can be had without any sacrifice in accuracy. Moreover, accuracy is largely determined on the final pass, which can have even more conservative bounds if better accuracy is required.