# SVF Framework

The SVF framework of CA was developed by Arsigny, Pennec [84, 90, 93, 101] as an alternative to the LDDMM framework.

Basic Setting

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 I_{0}, *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 *g _{t}* to deform

*I*

*to*

_{0}*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 *g _{t}I_{0}* connecting

*I*and

_{0}*I*is formulated by

*g*), which is a one-parameter subgroup of G. Different from LDDMM, which constructs

_{t}= Exp(vt*g*by integrating a time-dependent vector field

_{t}*u*specified by the Riemannian metric on G, the

_{t }*g*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 I

_{t}_{0}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 *I _{0}* 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))I*by

_{0}e G*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* [92]. 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 *I _{0} ! I*,

*I*and their tangent space representations on

_{0}! I_{n}*T*by Riemannian exponential map. (b) SVF registration between images and their tangent space representations on

_{Io}M*T*= g by Lie group exponential map, where

_{e}G*g*and

_{Ii}€ G*g*

_{Ii}I_{0}= I_{i}out much more easily than LDDMM when the Cartan connections are selected to define the parallel transport operation on the manifold *G* [93].

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*I*_{0}. - • 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.