The diffeomorphic transformation ~ϕ estimated during the evaluation of the proce-
dures described above is a smooth displacement field, preserving one-to-one mapping properties. Although, the smooth inverse transformation ~ϕ1 also exists (due to the determinant of the Jacobian matrix for the transformation ~ϕ being locally posi-
tive), it is usually not calculated explicitly during the registration process (except in the log-domain approaches). Intuitively, it should be easily to estimate the inverse transformation ~ϕ1 by changing the order of the input images i.e. a reference image becomes a moving image and a moving image becomes a reference image. Whereas the diffeomorphic formulation preserves that every image point in the moving image
Im is mapped onto the image point in the reference image If, there is no reason
expp0.25 ~vq expp0.5 ~vq expp0.75 ~vq expp1.0 ~vq
expp0.25 ~vq expp0.5 ~vq expp0.75 ~vq expp1.0 ~vq Figure 3.1: Examples of the forward and backward integration of the velocity field for the sample images. (Top row) The forward integration of the velocity field ~v at
different times provides the deformation field for warping the moving image, while (Bottom row) the backward integration of the velocity field ~v at different times
provides the deformation field for warping the fixed image
age points from the moving image to the corresponding image points in the fixed image. In other words, the transformation estimated between the fixed and mov- ing image (named the forward transformation) ~ϕf orw is not the same as the inverse
of the transformation calculated for the moving and fixed image ~ϕback (named the
backward transformation or reverse transformation [27, 63]). This inconsistency
~
ϕf orw ~ϕ1back (3.42)
comes partially from the fact of the ill-posed of the image registration and in some cases can also be a result of choosing a similarity criterion with asymmet- ric properties such as the Correlation Ratio [108]. The Inverse Consistency Er- ror (ICE) that describes the inconsistency between the forward transformation
~
ϕf orwp~xq ~x ~uf orwp~xq and backward transformation ~ϕbackp~xq ~x ~ubackp~xq can
be defined at every spatial position ~x as follows:
ICEp~ϕf orwp~xq, ~ϕbackp~xqq }p~x p~ϕf orw ~ϕbackqp~xqq} (3.43)
where}} is a Euclidean distance [27, 127]. An example presenting the inconsistency between the forward and backward transformation estimated using the classical
Figure 3.2: Examples of the inconsistency between forward and backward trans- formation estimated using classical Demon algorithm (top row) and results of en- suring the inverse consistency criterion during the consistent Demon registration (bottom row). (From left to right) forward transformation, backward transforma- tion,p~x p~ϕf orw ~ϕbackqp~xqq, and p~x p~ϕback ~ϕf orwqp~xqq. In perfect case of fulfilling
the inverse consistency criterion, p~x p~ϕf orw ~ϕbackqp~xqq and p~x p~ϕback ~ϕf orwqp~xqq
should be represented by the regular grid.
Demon algorithm is shown in Figure 3.2. In the perfect case of preserving the inverse consistency criterion, p~x p~ϕf orw ~ϕbackqp~xqq should be represented by the
regular grid. However, the transformations that were first estimated for the fixed and moving image and then with the swapped input images are not consistent thereby the grid is significantly distorted (top row of Figure 3.2). On the other hand, when using the consistent log-domain Demon algorithm, the obtained forward and backward transformations are close to the ideal case of the consistency. Indeed, the grid showingp~xp~ϕf orw ~ϕbackqp~xqq is almost regular (see bottom row of Figure 3.2).
In addition, the transformations estimated via consistent algorithms look smoother (for the same set of the design parameters).
The earliest work on overcoming the aforementioned inconsistency was done by the symmetrisation of the similarity measure Sim and explicit minimisation of the inverse consistency error via joint estimation of the forward and backward transformations [27]. The symmetrisation of the similarity criterion is defined in following manner:
SimsympIf, Im, ~ϕf orw, ~ϕbackq SimpIf, Im ~ϕf orwq SimpIf ~ϕback, Imq (3.44)
by:
P enICEp~uf orw, ~ubackq
»
Ω
}~uf orwp~xq ~u1backp~xq}
2
dx
»
Ω
}~ubackp~xq ~u1f orwp~xq}
2
dx
(3.45) In addition, the general energy function of this image registration is coupled with the regularisation terms for each displacement field Regp~uf orwq and Regp~ubackq to
preserve their smoothness and the constraints (P enJ acp~uf orwq and P enJ acp~ubackq)
that prevent determinant of the Jacobian matrix for both transformations from going below zero. The overall cost function for the consistent image registration proposed in [27] can be summarised by:
εp~uf orw, ~ubackq SimsympIf, Im, ~uf orw, ~ubackq Regp~uf orwq Regp~ubackq
P enJ acp~uf orwq P enJ acp~ubackq P enICEp~uf orw, ~ubackq (3.46)
The inverse displacement fields involved in the penalty term P enICE, are calculated
directly by the algorithm presented in the mentioned work (it is also summarised in the Section 3.4.2). Although the input images here are still differentiated by the subscripts f and m to name the fixed and moving image, this has only been done in order to keep notation consistent throughout the thesis. The order of the input images in the consistent image registration is not important due to symmetric properties and joint estimation of both forward and backward transformations. The detailed description of the framework for consistent image registration proposed by Christensen and Johnson [27] is motivated by two principal reasons. In the first place, this framework can be considered as the state-of-art method for consistent image registration. Furthermore, each condition that must be fulfilled to keep the image registration algorithm consistent is demonstrated explicitly in the energy func- tion ε (Equation 3.46). This means it can be used as a reference to assess whether the other methods are preserving all the consistency criteria.
A variational multi-modal framework of the consistent image registration was proposed in [162]. Besides the use of the different similarity criterion to enable multi-modal registration, the direct inversion of the deformation fields that is a crucial step in [27] was avoided by a different formulation of the problem. The energy function is defined there by:
εp~uf orw, ~ubackq SimsympIf, Im, ~uf orw, ~ubackq Regp~uf orwq Regp~ubackq
P enICEp~uf orw, ~ubackq (3.47)
and backward transformations. As a result of the two Euler-Lagrange equations, the PDE for both the forward and the backward transformations has to be solved. Multi- modal consistent image registration was also investigated in [136], where the Demon- like flows of the Mutual Information were calculated. In contrast to aforementioned techniques, the ICE is not included directly in the objective function but it is minimised by the additional iterative procedure inserted into the image registration framework. Although both approaches minimise the ICE in every iteration, the ensurance of the one-to-one mapping is questionable, because no restrictions on the determinant of the Jacobian matrix are introduced.
The consistent landmark-based image registration algorithms were proposed in [63]. Although the registration of the landmarks is usually consistent due to finding the forward and backward transformations (correspondences) for the same set of the features, the forward and backward dense deformation fields interpolated for the regions away from the features are not consistent. One of the proposed algorithms first registers the landmarks and then minimises the ICE between estimated forward and backward transformations by a repeatable procedure as long as the distance between landmarks and the ICE is reduced below the desired levels. The second method combines the above landmark-matching procedure with the intensity-based registration proposed in [27].
Log-domain consistent image registration
The log-domain parameterisation of the displacement field provides some interest- ing outcomes that can be incorporated straightaway into the formulation of the consistent image registration. The major advantage that is offered by the station- ary velocity field parameterisation is the fact that the estimated transformation is diffeomorphic (this provides a smooth transformation with a locally positive deter- minant of the Jacobian matrix for this transformation). The second advantage of the log-domain image registration is derived from the fact that the backward (in- verse) transformation can be quickly calculated from the velocity field obtained for the forward transformation based on the backward integration. When compared to the state-of-the-art consistent image registration [27], the missing component of the registration to be consistent is reducing the ICE between the forward and backward transformations. For this purpose, Vercauteren et al. [144] proposed the averaging procedure for the forward and backward velocity fields that guarantees the sym- metry of the results and make registration consistent. The origin of the averaging procedure applied in [144] to the velocity fields can be derived from the seminal paper for the Demon algorithm. Thirion [138] proposed procedure to compose the forward and the backward transformation and then take the half of the composition
result to add to the forward transformation and subtract from the backward trans- formation. This procedure there was performed to enforce that the transformation would be a one-to-one mapping.
Formally, the consistent log-domain image registration can be stated as optimi- sation of the single velocity field ~v defined by the following energy term:
εp~vq SimsympIf, Im, expp~vq, expp~vqq Regp~vq (3.48)
During the optimisation of the above energy function, the log-domain consistent image registration requires calculation of the forward and the backward update of the velocity field. The forward ~dvi,f orwp~xq and backward ~dvi,backp~xq updates of the
velocity field for the steepest descent approach are given by:
~
dvi,f orwp~xq pIfp~xq Imp~ϕi1p~xqqq∇Imp~ϕi1p~xqq
αpµ∆~vi1p~xq pλ µq∇p∇ ~vi1p~xqqq
(3.49)
~
dvi,backp~xq pImp~xq Ifp~ϕi11p~xqqq∇Ifp~ϕ1i1p~xqq
αpµ∆p~vi1p~xqq pλ µq∇p∇ p~vi1p~xqqqq
The forward ~dvi,f orwp~xq and backward ~dvi,backp~xq updates of the velocity field for
Newton’s iteration approach are given by:
~ dvli,f orwlp~xq pIfp~xq Imp~ϕi1p~xqqq∇ lI mp~ϕi1p~xqq λdif f∆vil1p~xq λdif f ∇lImp~ϕi1p~xqq ∇lImp~ϕi1p~xqq (3.50) ~ dvli,f orwp~xq pImp~xq Ifp~ϕ 1 i1p~xqqq∇lIfp~ϕ1i1p~xqq λdif f∆pvli1p~xqq λdif f ∇lIfp~ϕ1i1p~xqq ∇lIfp~ϕ1i1p~xqq
The forward ~dvi,f orwp~xq and backward ~dvi,backp~xq updates of the velocity field for the
Demon approach are given by:
~ dvi,f orwp~xq pIfp~xq Imp~ϕi1p~xqqq }∇Iip~xq}2 pIfp~xq Imp~ϕi1p~xqqq2 ∇Iip~xq (3.51) ~ dvi,backp~xq pI mp~xq Ifp~ϕ1i1p~xqqq }∇Iip~xq}2 pImp~xq Ifp~ϕ1i1p~xqqq2 ∇Ip~xq
The log-domain consistent image registration algorithm based on the steepest de- scent approach is summarised in Algorithm 6. The Newton’s iteration algorithm and the Demon approach can be deduced in the same straightforward manner. Origi-
nally the consistent symmetric Demon approach was proposed in [144], where it was called the symmetric log-domain Demon approach to emphasise the symmetrisa- tion of the similarity criterion in the objective function. The consistent log-domain methods based on the steepest descent approach and Newton’s iteration approach presented in this section benefit from the log-domain parameterisation and enforce the inverse consistency criterion, thereby opening the already known algorithms to new applications.
Algorithm 6 Consistent log-domain steepest descent approach Input: Images: If and Im, α
Parameters: α, λelas, µelas
Output: Velocity field ~v, Transformations ~ϕf orw and ~ϕback
1: ~v 0, i 1
2: repeat
3: forward integration of velocity field: ~ui,f orw expp~vi1q
4: backward integration of velocity field: ~ui,back expp~vi1q
5: for all ~xP Ω do
6: calculate forward update of the velocity field:
~
dvi,f orwp~xq (Equation 3.50)
7: calculate backward update of the velocity field:
~
dvi,backp~xq (Equation 3.50)
8: calculate forward velocity field ~vf orw:
~vf orwp~xq BCH
~
vi1p~xq, ~dvi,f orwp~xq
9: calculate backward velocity field ~vback:
~vbackp~xq BCH
~vi1p~xq, ~dvi,backp~xq
10: average forward and backward velocity field:
~
dvip~xq 12p~vf orwp~xq ~vbackp~xqq (Equation 3.41)
11: update velocity field ~vip~xq:
~vip~xq ~dvip~xq
12: end for
13: i i 1
14: until (velocity field does not change)or (i ¥ IterMax)
15: calculate forward transformation ~ϕf orw (Equation 3.29)
16: calculate backward transformation ~ϕback (Equation 3.36)
Figure 3.3: Symmetric image registration scheme