Suppose we work on the diffeomorphism group on a shape space (for example the shape of human organs). We can regard all different shapes are achieved by applying diffeomorphic deformations on a template shape. The 'distances' between two shapes is then the length of the geodesic connecting these shapes. With the help of the distance, we can do statistics/classification on shapes. The question is:
Given N+1 different shapes S_1(0),.....,S_N(0), S_N+1(0) from N+1 individuals at time 0, the shapes will change to S_i(t) at time t. If we already have S_1(t),.....,S_N(t), we can build the 'growth_track' of each shape by the geodesic between S_i(0) and S_i(t). Then how can we predict/interpolate the S_N+1(t) from these information (if we assume the shapes may follow similar tracks which is quite reasonable).
Two possible ways: (1) Compute the geodesics from S_N+1(0) to S_1(0),....,S_N(0) and then 'parallel translate ' the initial tangent vector of the 'growth_track'(geodesic) of each shape S_1,...S_N back to the shape S_N+1, we can then work on the tangent space of S_N+1 to interpolate the initial tangent vector of the growth_track of S_N+1. This seems implementable but lack of physical meaning . The advantage of this solution is that we can achieve statistics on a linear space, which may be preferable if N is big.
(2) Compute the geodesics from S_1(0),....,S_N(0) to S_N+1(0) and 'parallel translate' the initial tangent vector to S_1(t),.....,S_N(t) along the 'growth_track's of S_1,...,S_N. We can then follow Riemannian exponential map from S_i(t) with the correspondent parallel translated vectors to get S_1_N+1(t),.....,S_N_N+1(t) as estimates of S_N+1(t) from each shape's point of view, then a final 'averaged' interpolation can be achieved by a Karcher mean of these estimates. (computational costly if N is big and I have no idea how to simplify it)
Can anybody provide other solutions? If anyway we need to do statistics on the shapes, is the tangent space based solution the only realistic solution? But since it use linear space to describe the nonlinear deformation, maybe it's only an approximation when the deformation is minor.
Another question is that how can we compute a 'mean' growth_track of the shapes? How can we classify the growth_tracks? In the case if the growth tracks are geodesics, maybe we can still use tangent space to describe it. what if the tracks are piece-wise geodesics?
Or maybe there is already a solution somewhere on carrying out statistics on manifolds? Thx.