TABLE I.

Glossary of terms used throughout the paper.

F, M	Fixed image F, moving image M.
Σ	Typically a diagonal matrix that models variability of feature values at a particular vertex.
σx, σT	Parameters of Demons cost function in Eq. (3).
Γ, ϒ	Transformations from S2 to S2. Γ is the transformation we are seeking. ϒ is the smooth hidden transformation close to Γ.
Γ⃗ ≜ {Γ⃗n}, ϒ⃗ ≜ {ϒ⃗n}	Discrete tangent vector representation of the deformations (see Fig. 1 and Eq. (5)). For example, given the tangent vector Γ⃗n at xn ∈ S2, one can compute Γ(xn).
υ⃗ ≜ {υ⃗n}	We parameterize diffeomorphic transformations from S2 to S2 by a composition of diffeomorphisms, each parameterized by a stationary velocity field υ⃗. υ⃗n is the velocity vector at xn.
u(·) ≜ exp(υ⃗)(·)	The diffeomorphism parameterized by the stationary velocity field υ⃗ is the solution of a stationary ODE at time 1.
En ≜ [e⃗n1 e⃗n2]	e⃗n1 and e⃗n2 are orthonormal vectors tangent to the sphere at xn
Ψn	Coordinate chart defined in Eq.(10): ${\Psi }_n(x^{\prime })=\frac{x_n+E_n{x}^{\prime }}{\Vert x_n+E_n{x}^{\prime }\Vert }$. Ψn is a diffeomorphism between ℝ2 and a hemisphere centered at xn ∈ S2.
z⃗n	z⃗n is an arbitrary tangent vector at the origin of ℝ2. At xn, the velocity vector υ⃗n = Enz⃗n via the coordinate chart Ψn (see Eq. (14)).