The analysis above maximizes the likelihood. For a full MAP estimation, we include a prior term. Because we are working with the logarithm of the likelihood, the effect of the prior is additive:
Thus in addition to the noise model, we can incorporate some knowledge about the kinds of shapes that give rise to the measurements. With appropriately fashioned priors, we can push the solution toward desirable shapes or density values, or penalize certain shape properties, such as roughness or complexity. The choice of prior is intimately related to the choice of surface representation and the specific application, but is independent of the formulation that describes the relationship between the estimate and the data, given in Eq. (8.37).
Because the data is noisy and incomplete it is useful to introduce a simple, low-level prior on the surface estimate. We therefore use a prior that penalizes surface area, which introduces a second-order smoothing term in the surface motion. That term introduces a free parameter C, which controls the relative influence of the smoothing term. The general question of how best to smooth surfaces remains an important, open question. However, if we restrict ourselves to curvature-based geometric flows, there are several reasonable options in the literature [7,31,97]. The following subsection, which describes the surface representation used for our application, gives a more precise description of our smoothing methods.
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