Log-domain parameterisation

The updates on the Lie algebra mapped through the exponential mapping can be seen as one of the possible adaptations of the outcomes of the work done on the stationary velocity field parameterisation of the diffeomorphism by Arsigny et al. [4]. Other approaches taking advantages of the log-domain parameterisation of the diffeomorphic transformation are the Diffeomorphic Anatomical Registration

using Exponentiated Lie Algebra (DARTEL) algorithm proposed in [5], and the log- domain Demon approach proposed in [144]. In contrast to the exponentiation of the updates on the Lie algebra exp

~ dv

, the mentioned log-domain image registration algorithms parameterise the displacement field ~u as a stationary velocity field ~v.

Then, the diffeomorphic transformation ~ϕp~xq can be defined in following way: ~

ϕp~xq  ~x ~up~xq  ~x expp~vp~xqq (3.29) The diffeomorphic update and the displacement field are calculated through the exponential mapping and can be concatenated using the composition operator  (the composition operation on the Lie group). This composition is given by:

expp~viq  expp~vi
1q  expp ~dviq (3.30)

while the current velocity field ~vi can be expressed as follows:

~ vi  log
expp~vi
1q  expp ~dviq (3.31)

The drawback of Equation 3.31 is that estimation of the velocity field ~vi is linked

with computing the principal logarithm logpq which is reported to have high com- putational burden (e.g. volume of size 256x256x60 in 60 minutes using a standard computer [4]). In order to represent the velocity field ~vi in the log-domain through

an efficient way in a reasonable time, the Baker-Campbell-Hausdorff (BCH) formula [17, 144] has to be applied to approximate the current velocity field ~vi:

~vi  BCH



~vi
1, ~dvi



(3.32)

where the BCH formula for any two velocity fields ~vi
1 and ~dvi is given by:

BCH  ~ vi
1, ~dvi   ~vi
1 dv~ i 1 2  ~vi
1, ~dvi  ₁ 12  ~vi
1,  ~vi
1, ~dvi  . . . (3.33)

The Lie bracket 

~vi
1, ~dvi



is the operation on the Lie algebra vector space defined in the following way [17, 144];

 ~vi
1p~xq, ~dvip~xq   d ¸ j1  vj_i
1p~xqB ~dvip~xq Bxj
dv j ip~xq B~vi
1p~xq Bxj  (3.34)

where ~vi
1p~xq  rvi1
1p~xq, . . . , vdi
1p~xqs and ~dvip~xq  rdvi1p~xq, . . . , dvidp~xqs. Utilisation

of the BCH formula approximates the current velocity field ~vi, thereby allows one to

way. As an illustration of this reformulation of the image registration objective, the energy function is presented by Equation 3.35.

εp~vq  » Ω SimpIfp~xq, Imp~x expp~vp~xqqqqdx α » Ω Regp~vp~xqqdx (3.35) In contrast to the general image registration problem given by Equation 2.6, the log-domain formulation estimates the velocity field, and the final transformation is returned by the exponential mapping. The additional outcome of the parameterisa- tion via the stationary velocity field is that the inverse of the transformation ~ϕ
1 can be quickly estimated through the backward integration of the velocity field [5, 144]:

~

ϕ
1p~xq  ~x expp
~vp~xqq (3.36) The example presenting the forward and backward exponentiation (integration) of the velocity field ~v is shown in Figure 3.1.

It is interesting that the algorithm proposed by Ashburner [5] treats the up- dates of the velocity field ~dvi in the additive optimisation framework (given by the

Equation 3.37) without considering whether such velocity field ~v theoretically exists.

~v  ~v0 dv~ 1 . . . dv~ i (3.37)

Considering the BCH formula, the additive way of concatenation of the updates of the velocity fields can be seen as the approximation of the velocity field using just first two elements of the BCH formula (Equation 3.33). It was shown experimentally in [17, 144], that using more than three first elements of BCH formula does not lead to a significant increase in accuracy. In addition, Bossa et al. [17] showed that the approximation using the BCH formula produces errors on the similar level to those in the case of the direct calculation of the principal logarithm.

In order to take advantage of the parameterisation of the displacement field via the stationary velocity field, the algorithms presented in the first section of this chapter (Section 3.1), are reformulated here to their log-domain versions. Instead of calculating the update of the displacement field ~dui, the update of the velocity field

~

dvi on the Lie algebra space is computed and then concatenated with the current

velocity field based on the BCH formula (Equation 3.33). The regularisation is performed on the velocity field (and/or update of the velocity field) rather then on the displacement field (and/or update of the displacement field). The update of the velocity field for the steepest descent approach is given by Equation 3.38, and Equation 3.39 provides the update for the Newton’s iteration approach. The Demon method’s update of the velocity field was presented in the previous section

Algorithm 5 Log-domain steepest descent approach Input: Images: If and Im

Weight of regularisation α

Design parameters (for linear elastic model: λelas, µelas)

Output: Velocity field ~v, transformation ~ϕ

1: ~v0  0, i  1

2: repeat

3: exponentiation of velocity field: ~ui  expp~viq

4: for all ~xP Ω do

5: calculate update of velocity field: ~dvip~xq (Equation 3.38)

6: update velocity field ~vip~xq  BCH

 ~ vi
1p~xq, ~dvip~xq  7: end for 8: i i 1

9: until (velocity field does not change) or (i¥ IterMax) 10: calculate transformation ~ϕ (Equation 3.29)

11: return ~v and ~ϕ

(given by Equation 3.28) and the regularisation of output is performed via Gaussian filtering of the velocity field/update of the velocity field rather than the displacement field/update of the displacement.

~ dvip~xq  pIfp~xq
Imp~ϕi
1p~xqqq∇Imp~ϕi
1p~xqq αpµ∆~vi
1p~xq pλ µq∇p∇ ~vi
1p~xqqq (3.38) ~ dvl_ip~xq 
pIfp~xq
Imp~ϕi
1p~xqqq∇ l_I mp~ϕi
1p~xqq λdif f∆vil
1p~xq λdif f p∇lImp~ϕi
1p~xqqq2 (3.39)

The overall log-domain image registration algorithm based on the steepest de- scent approach is summarised in Algorithm 5. The log-domain Newton’s iteration approach can be deduced in the similar manner, when the update of the velocity field calculated using Equation 3.38 (Line 5 of Algorithm 5) has to be replaced by the update calculated using Equation 3.39. The presented log-domain algorithms extend the original works (the Newton’s iteration method [78, 97], and the steep- est descent approach [56, 24]), and enable to take advantages of the diffeomorphic framework.

The log-domain parameterisation of the diffeomorphic transformation has in- teresting properties when utilised for calculating the statistics on the displacement fields [4, 144]. It is important to realise that performing the Euclidean statistics on

the diffeomorphic displacement fields does not preserve the diffeomorphic properties of the results of those statistics. The most compelling example can be shown easily by taking the average of the diffemorphic transformations:

~ uave  1 N N ¸ j1 ~ uj (3.40)

where N is the number of the displacement fields used for the calculation of the average. While the input displacement fields ~ui can be enforced to be diffeomorphic

(e. g. as a result of performing a diffeomorphic algorithm), the output displace- ment ~uave may no longer be diffeomorphic. This occurs due to the fact that the

displacement fields are not vector fields. The framework proposed by Arsigny et al. [4] showed that calculating the principal logarithm of the diffeomorphic transforma- tions, results in well-defined vector fields. As a consequence, the Euclidean statistics on those vector fields can be obtained. For example, when the average of the ve- locity fields is considered, the output is also a velocity field and the diffeomorphic displacement field associated with this velocity field can be obtained through the exponential mapping: ~ uave  exp  1 N N ¸ j1 logp~ujq  (3.41)

As it was outlined, calculating the principal logarithm is a time-consuming procedure [4, 17] but, when the parameterisation of the displacement field is enforced explicitly in the image registration, the output of this registration is already the velocity field. Indeed, the calculation of the principal logarithm can be avoided using the log- domain image registration approaches.