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Anatomy is undergoing a renaissance driven by availability of large digitaldata sets generated by light microscopy. A central computational task is to mapindividual data volumes to standardized templates. This is accomplished byregularized estimation of a diffeomorphic transformation between the coordinatesystems of the individual data and the template, building the transformationincrementally by integrating a smooth flow field. The canonical volume form ofthis transformation is used to quantify local growth, atrophy, or cell density.While multiple implementations exist for this estimation, less attention hasbeen paid to the variance of the estimated diffeomorphism for noisy data.Notably, there is an infinite dimensional un-observable space defined by thosediffeomorphisms which leave the template invariant. These form the stabilizersubgroup of the diffeomorphic group acting on the template. The correspondingflat directions in the energy landscape are expected to lead to increasedestimation variance. Here we show that a least-action principle used togenerate geodesics in the space of diffeomorphisms connecting the subject brainto the template removes the stabilizer. This provides reduced-varianceestimates of the volume form. Using simulations we demonstrate that theasymmetric large deformation diffeomorphic mapping methods (LDDMM), whichexplicitly incorporate the asymmetry between idealized template images andnoisy empirical images, provide lower variance estimators than theirsymmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for thevariances in the limit of small deformations. Analytical results are shown forthe Jacobian in terms of perturbations of the vector fields and divergence ofthe vector field.

Estimating Diffeomorphic Mappings between Templates and Noisy Data: Variance Bounds on the Estimated Canonical Volume Form