  Averaging diffeomorphisms is a challenging problem, and it has great
applications in areas like medical image atlases. The simple Euclidean average
can neither guarantee the averaged transformation is a diffeomorphism, nor get
reasonable result when there is a local rotation. The goal of this paper is to
propose a new approach to averaging diffeomorphisms based on the Jacobian
determinant and the curl vector of the diffeomorphisms. Instead of averaging
the diffeomorphisms directly, we average the Jacobian determinants and the curl
vectors, and then construct a diffeomorphism based on the averaged Jacobian
determinant and averaged curl vector as the average of diffeomorphisms.
Numerical examples with convincible results are presented to demonstrate the
method.
