3.4 Image registration with symmetric warping
3.4.2 Direct inverse update estimation approach
The direct inverse deformation field approach proposed by Papiez and Matuszewski [96] is inspired by the need to check how good the approximation of the inverse update given by the Equation 3.59 is in practice. To assess this, the direct inverse approach is built on the previous approach by directly inverting the update of the deformation field in each iteration. The proposed update scheme is defined for the half-way forward displacement field as follows:
~ui,hfp~xq Gdif f ~ ui1,hfp~xq Gf luid ~ dui,hfp~xq (3.60)
and for the half-way backward displacement field by;
~ui,hbp~xq Gdif f ~ ui1,hbp~xq Gf luid ~ dui,hbp~xq (3.61)
where the update of the half-way backward transformation ~dui,hbp~xq is calculated as
the inverse of the update of the half-way forward transformation ~dui,hfp~xq:
~
dui,hbp~xq ~du
1
i,hfp~xq (3.62)
The technique applied to the update scheme in Equation 3.61 and in Equation 3.62 are in contrast with the approximation given by Equation 3.59 (compare formula for updating ~ui,hb given in Equation 3.58) as it uses the direct inverse update of the
deformation which does not suffer from the limitations of the small-step multiple
pass approach (because ~dui p ~duiq1p~xq ~x). The direct inverse deformation based
algorithm is summarised in Algorithm 7.
For the small-step multiple pass approach and the direct inverse approach, there is a need to estimate the inverse deformation fields ~ϕhf and ~ϕhb. This has to be done
accurately and fast, especially for the direct inverse approach, where the inverse is also calculated in each iteration. In the small-step multiple pass approach, the inverse transformations of ~ϕhf and ~ϕhbare calculated using the method proposed by
[7]. Here, a novel method is presented (based on the method proposed by [27]) that is more accurate then the previously reported methods.
Algorithm 7 Direct inverse deformation field approach to image registration with
symmetric warping
Input: Images: If and Im
Parameters: Gf luid, Gdif f
Output: Forward transformation ~ϕf orw, backward transformation ~ϕback
1: ~uf orw ~0, ~uback ~0, i 1
2: repeat
3: for all ~xP Ω do
4: calculate forward update ~dui,hfp~xq (Equation 3.55 )
5: smooth update of the deformation field using Gaussian filter Gf luid:
~
dui,hfp~xq Gf luid p ~dui,hfp~xqq
6: calculate backward update ~du1i,hbp~xq using the Algorithm 8
7: update forward displacement field ~ui,hfp~xq ~ui1,hfp~xq ~dui,hfp~xq
8: smooth forward displacement field ~ui,hfp~xq using Gaussian filter Gdif f:
~
ui,hfp~xq Gdif f p~ui,hfp~xqq
9: update backward deformation field ~ui,hbp~xq ~ui1,hbp~xq ~dui,hbp~xq
10: smooth backward deformation field ~ui,hbp~xq using Gaussian filter Gdif f:
~
ui,hbp~xq Gdif f p~ui,hbp~xqq
11: end for
12: i i 1
13: until (deformation fields do not change) or (i¥ IterMax) 14: calculate the inverse of the ~ϕhf and the inverse of the ~ϕhb
using the Algorithm 8
15: calculate forward transformation (Equation 3.53) and backward transformation
(Equation 3.54) by composition of the half-way transformations
16: return ~ϕf orw and ~ϕback
Christensen’s method
The procedure used to compute the inverse transformation proposed in [27] assumes that an input transformation ~ϕ is a continuously differentiable mapping from ΩÑ Ω
with a positive determinant of the Jacobian detpJp~ϕp~xqqq for all spatial position
~
x P Ω. An inverse deformation field can be found by selecting a point at a spatial
position ~y ry1, . . . , yds P Ω and carrying out an iterative process to search for a
iterations defining the inverse transformation are given by: ~ xi 1 ~xi ~ y ~ϕp~xiq 2 (3.63)
The initially selected point ~x0 should not be far from the final estimate ~x. The
drawback of this method is that it is not established via a formal mathematical scheme. Although the method has been shown to converge to good results when the minimum value of the determinant of Jacobian is greater than zero, the method has been validated for relatively small deformations fields (i.e. the CT and MRI brain scans [27], [63]).
Proposed deformation field inversion model
In the proposed method the inversion of the deformation field is achieved by using a Newton-Raphson like method. Let a point misalignment function ~fp~xq be defined
as:
~
fp~xq ~y ~ϕp~xq (3.64) For each ~yP Ω, the aim is to find a corresponding ~x which will make ~fp~xq as close
to zero as possible. This is achieved in an iterative fashion:
~
xi 1 ~xi dx~ i (3.65)
Approximating the misalignment function using a first order Taylor series expansion gives:
~
fp~x dx~ q ~fp~xq Jp ~fp~xqq ~dx (3.66) where Jp ~fp~xqq denotes the Jacobian matrix Assuming ~fp~x ~dxq 0 and introducing
regularisation, the updates ~dxi can be calculated from a set of linear equations:
J ~ fp~xq ρD ~dx ~fp~xq (3.67) where D is an d-dimensional diagonal matrix, and ρ is a position dependent regu- larisation parameter. The regularisation is introduced only for the areas where the determinant of Jp ~fp~xqq is close to zero.