Consider the well-known morphing videos in which a person's face smoothly transforms into someone else's. In such videos, facial features from the original image progressively deform to reach their target, but at the same time, these feature may change color, as illustrated by the strip below, which was created by Laurent Garcin during his Ph. D. Thesis. Here, you see eyes, mouth or hair moving, but also the color progressively changing when transitioning from the first image to the last. The first image's background fades out and is replaced by the last image's one.

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.


Some examples.
The following 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.

References:
[1] Group actions, homeomorphisms, and matching: A general framework, M.I. Miller, L. Younes, International Journal of Computer Vision 41 (1-2), 61-84, 2001
[2] Geodesic interpolating splines, V. Camion, L. Younes, Energy Minimization Methods in Computer Vision and Pattern Recognition, 513-527, 2005
[3] Local geometry of deformable templates, A. Trouvé, L. Younes, SIAM J. Math. Anal., 37(1), 17-59, 2005
[4] Metamorphoses through lie group action, A. Trouvé, L. Younes, Foundations of Computational Mathematics 5 (2), 173-198, 2005
[5] 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
[6] Geodesic matching with free extremities, L. Garcin, L. Younes,  Journal of Mathematical Imaging and Vision 25 (3), 329-340, 2006
[7] The Euler-Poincaré theory of metamorphosis, D.D. Holm, A. Trouvé, L. Younes, Quarterly of Applied Mathematics 97, 661-685, 2009
[8] Computing Metamorphoses between Discrete Measures, C.L. Richardson, L. Younes, J. Geom. Mech. 5 (1), 131-150, 2013
[9] Metamorphosis of images in reproducing kernel Hilbert spaces , C.L. Richardson, L. Younes, Adv. Comp. Math. 42, 573-603, 2016