New medical imaging technologies encode human anatomy in a wide variety of data structures. Computational Anatomy (CA) offers an approach to synthesize this plethora of data by comparison of anatomical features using smooth invertible transformations specific to the data structure.This proposal is for work to develop new mathematical and numerical methods for image analysis in the framework of CA, aimed at meeting the following five challenges in image analysis for the comparison and interpolation of shapes in biomedical images of the heart and the brain.1. Data structure: Develop a unified approach for registering images encoded in a wide variety of data structures. The unifying concept in our approach is the momentum map, a fundamental concept from the theory of Lie group transformations.2. Data fusion: After placing the transformations of the variety of data structures into the same conceptual framework using momentum maps, synthesise (fuse) their multiple modalities of information by accounting for the different transformation properties of the different data structures under smooth invertible maps.3. Multiple resolutions: Develop the momentum map framework to enable registration of data at multiple resolutions by concatenating the Lie group transformations that define the momentum maps.4. Time-varying (4D) images: Treat time-varying images in this geometric framework by matching snapshots in time using geodesic splines that interpolate the image snapshot from one time to another. Quantity the effects of noise and uncertainty in 4D image analysis.5. Changes in image topology: Extend the transformative approach to allow changes in topology in passing between images by using the method of metamorphosis.Simply put, the five challenges are: to 1. register images of different data structures and 2. combine them, even at 3. different resolutions; then do the same things with 4. splines and 5. metamorphosis, including noise.