Spatial Multiresolution

The minimization problem is discretized so it can be implemented on a digital computer. The higher the sampling rate the more accurate the discrete approximation is to the continuous case. An advantage of discretizing a continuous formulation is that the problem can be solved at different spatial sampling rates. The approach that is taken is to solve the minimization at a course resolution initially to approximate the solution. The advantage of solving the problem on a course grid is that the algorithm requires fewer computations per iteration that a finer grid. This results in reduced computation time at low resolution. Each time the resolution of the grid is increase by a factor of two in each dimension, the computation time increases by a factor of eight. The drawback of solving the problem at low resolution is that there can be significant registration errors due to the loss of detail in the down sampling procedure. The trade-off between quicker execution times at low resolution and more accurate registration at higher resolution can exploited by solving the registration problem at low spatial resolution during the initial iterations to approximate the result and then increasing the spatial resolution to get a more accurate result at the later iterations.

The spatial multiresolution approach works well with the frequency multiresolution approach provided by increasing the number of harmonics used to represent the displacement fields. The number of harmonics used to represent the displacement fields is initially set small and then increased as the number of iterations are increased. A low-frequency registration result is an approximation of the desired high-frequency registration result. Computing the gradient descent for a low-frequency basis coefficient at low spatial resolution gives approximately the same answer as using high spatial resolution but the computational burden is much less.

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