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Published online by Cambridge University Press: 08 December 2022
In this chapter, we study in detail the (weak) L^2-metric on spaces of smooth mappings. Its importance stems from the fact that this metric and its siblings, the Sobolev H^s -metrics are prevalent in shape analysis. It will be essential for us that geodesics with respect to the L^2-metric can explicitely be computed. Let us clarify what we mean here by shape and shape analysis. Shape analysis seeks to classify, compare and analyse shapes. In recent years there has been an explosion of applications of shape analysis to diverse areas such as computer vision, medical imaging, registration of radar images and many more. Another typical feature in (geometric) shape analysis is that one wants to remove superfluous information from the data. For example, in the comparison of shapes, rotations, translations, scalings and reflections are typically disregarded as being inessential differences. Conveniently, these inessential differences can mostly be described by actions of suitable Lie groups (such as the rotation and the diffeomorphism groups).
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