Figure 3.

Above, in (a), we see four binary images, ellipses, in a synthetic dataset with known radii, R₁ and R₂ and identical center. The unbiased template optimization initializes the template appearance by averaging these images, given the image in the upper right, (b), which has four gray levels, 0, 0.25, 0.5, 0.75 and 1. The geometric ground truth is shown in (c). The SyGN algorithm result, in (d), converges—up to interpolation error—to the expected shape and appearance. Error between SyGN and the ground truth is shown in (g). If the shape update step is removed—and we use only an optimal appearance (OA) template–the algorithm converges to a result with the wrong shape, shown in (e) and (f). The implication is that methods without explicit shape optimization will be more sensitive to initialization and are thus less likely to find the optimal minimum shape distance image. Theoretically, methods such as congealing (Learned-Miller, 2006) and (Joshi et al., September 2004), neither of which use explicit shape optimization, would converge to this type of reasonable, but geometrically less than optimal, solution. This is because the optimal solution for the problem above, with a matching criterion related to intensity difference, is to map all images to the 0.5 level set of the initial image shown in (b).