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A new approach is presented for elastic registration of medical images, and is applied to magnetic resonance images of the brain. Experimental results demonstrate very high accuracy in superposition of images from different subjects. There are two major novelties in the proposed algorithm. First, it uses an attribute vector, i.e., a set of geometric moment invariants (GMIs) that are defined on each voxel in an image and are calculated from the tissue maps, to reflect the underlying anatomy at different scales. The attribute vector, if rich enough, can distinguish between different parts of an image, which helps establish anatomical correspondences in the deformation procedure; it also helps reduce local minima, by reducing ambiguity in potential matches. This is a fundamental deviation of our method, referred to as the hierarchical attribute matching mechanism for elastic registration (HAMMER), from other volumetric deformation methods, which are typically based on maximizing image similarity. Second, in order to avoid being trapped by local minima, i.e., suboptimal poor matches, HAMMER uses a successive approximation of the energy function being optimized by lower dimensional smooth energy functions, which are constructed to have significantly fewer local minima. This is achieved by hierarchically selecting the driving features that have distinct attribute vectors, thus, drastically reducing ambiguity in finding correspondence. A number of experiments demonstrate that the proposed algorithm results in accurate superposition of image data from individuals with significant anatomical differences.