The SVF framework of CA was developed by Arsigny, Pennec [84, 90, 93, 101] as an alternative to the LDDMM framework.
Similar to LDDMM, SVF framework also works on a vector space of images V and a Lie group of diffeomorphic transformation G with inner products on V and g. The action of g e G on I e V is exactly as given in (2.164).
SVF Diffeomorphic Registration
The registration of two images I0, I is formulated in SVF framework as where
and Exp : g ! G is the Lie group exponential map.
The goal of the registration formulation is to find the optimal v 2 g that generates a smooth transformation gt to deform I0 to I. The distance between shapes is given by the norm < v, v >g.
SVF vs. LDDMM
By comparing (2.169) and (2.170) with (2.165) and (2.166), we can observe that:
Lie group structure The underlying space of the SVF framework is not a Rie- mannian manifold and there is no Riemannian metric, geodesic, or connection involved. Instead, SVF works on the Lie group structure of G.
One-parameter subgroup The optimal curve gtI0 connecting I0 and I is formulated by gt = Exp(vt), which is a one-parameter subgroup of G. Different from LDDMM, which constructs gt by integrating a time-dependent vector field ut specified by the Riemannian metric on G, the gt of SVF is an integral curve of a stationary vector field (the source of the term) v e g (2.169). Finding the optimal v to register I0 with I can also be achieved by gradient flow or geodesic shooting algorithms as in LDDMM, but in practice, the most commonly used algorithm is the Log-Demons algorithm [91, 92].
Lie group exponential and shape space linearization Similar to LDDMM, SVF assumes that every image I e V can be reached from a template image I0 by a Lie group exponential map Exp(v(I)). The linearization of the shape space is achieved by the Lie group exponential Exp instead of the Riemannian exponential map exp in LDDMM, i.e., representing I = Exp(v(I))I0 e G by v(I) e g.
So, to a first approximation, SVF framework can be regarded as a simplified alternative to LDDMM which (a) works on stationary vector fields on the tangent space of Diff (^), equivalently vectors in g instead of time dependent vector fields in LDDMM and (b) replaces the Riemannian exponential with the Lie group exponential as explained in Fig. 2.17a, b.
The main advantage of SVF over LDDMM is that both the Lie group exponential Exp and its inverse Log can be computed with a higher efficiency as matrix operations than the Riemannian exp and log . Another advantage of using SVF is that the parallel transport operation for longitudinal data analysis can also be carried
Fig. 2.17 Geometry of diffeomorphic registration in LDDMM and SVF frameworks (a) LDDMM registration between images I0 ! I, I0 ! In and their tangent space representations on TIo M by Riemannian exponential map. (b) SVF registration between images and their tangent space representations on TeG = g by Lie group exponential map, where gIi € G and gIiI0 = Ii
out much more easily than LDDMM when the Cartan connections are selected to define the parallel transport operation on the manifold G .
The main difficulty of SVF appears when we consider the existence, and properties of the optimal transformation between images are considered, as explained in the LDDMM case. For SVF framework, we know that generally:
- • The Exp map is usually not subjective, which means that not all images in G can be reached by an Exp(v) from a template I0.
- • The optimal transformation is not smooth with regard to the images, so that a small change in images may lead to a large change of the curve connecting them.
But even with these known theoretical obstacles, SVF framework still shows promising performance in practical applications and plays an important role in CA computation [101, 102]. The reader is referred to [100, 103] for more information about the mathematical foundation of SVF.