In images, geometric structures are instantiated with luminance (gray-level) information due to lighting variation and/or shading. To compare two such images, displacements of the structures and variation of the gray levels must be jointly considered. Metamorphoses implement this by measuring and minimizing the cost of an evolution process which deforms a given image into another one, and varies its gray levels at the same time. This analysis also provides a metric distance on the space of images.

The mathematical tools which are required to study these processes belong to differential geometry and the theory of in.nite dimensional Lie groups. They have interesting connections with .uid mechanics, exhibiting the same singular solutions known as solitons.

This method provides and quanti.es an optimal correspondence between images or shapes, in 2D or 3D. It may naturally be applied for medical imaging problems, related in particular to computational anatomy.

Images (extracted from L. Garcin's Ph.D. dissertation) on the left show examples of the interpolation achieved using a multigrid, wavelet-based optimization. The algorithm provides an optimal .ow of diffeomorphisms inducing a transition between the starting point and the end point. The bottom image is purely synthetic.