This sequence provides an interpolation, in some appropriate space of images, between the first image and the last one. These were the input of the algorithm, and intermediate images are computer generated.

The same principles can be applied to a whole variety of deformable objects -- not only images. A general framework involves a Lie group of "deformations" acting on a Riemannian manifold of shapes or images, and evolution processes on the manifold that continuously deform a time-dependent object, called template. Optimal trajectories are then obtained by penalizing large deformations and template variation.

The following links open videos that provide more examples of image evolution for optimal metamorphosis. In all these examples, the algorithm started with the first and last images of the sequence, and estimated a best interpolation using metamorphosis. In some cases, the interpolation results in an apparent motion of the image. In other cases, more deformation and image intensity changes are involved. The last examples provides metamorphosis in the space of densities, for which the transformation multiplies the deformed image by a Jacobian determinant, like in the change of variable formula.

Faces (same person)

Faces (same person, adding glasses)

Faces (different persons)

Matching Densities

Computing Metamorphoses between Discrete Measures, C.L. Richardson, L. Younes, J. Geom. Mech. 5 (1), 131-150, 2013

The Euler-Poincaré theory of metamorphosis, D.D. Holm, A. Trouvé, L. Younes, Quarterly of Applied Mathematics 97, 661-685, 2009

Geodesic matching with free extremities, L. Garcin, L. Younes, Journal of Mathematical Imaging and Vision 25 (3), 329-340, 2006

Geodesic image matching: a wavelet based energy minimization scheme, L. Garcin, L. Younes, Energy Minimization Methods in Computer Vision and Pattern Recognition, 349-364, 2005

Metamorphoses through lie group action, A. Trouvé, L. Younes, Foundations of Computational Mathematics 5 (2), 173-198, 2005

Local geometry of deformable templates, A. Trouvé, L. Younes, SIAM J. Math. Anal., 37(1), 17–59, 2005

Geodesic interpolating splines, V. Camion, L. Younes, Energy Minimization Methods in Computer Vision and Pattern Recognition, 513-527, 2005

Group actions, homeomorphisms, and matching: A general framework, M.I. Miller, L. Younes, International Journal of Computer Vision 41 (1-2), 61-84, 2001