The large number of the applications where it was used makes the Demon algorithm [138] probably one of the best known image registration algorithm in computer vision. In contrast to the other approaches that have been discussed previously, originally the concept of the Demon method was neither established in any optimi- sation framework nor linked into any deformation model. The method was derived from the physics analogy with the Maxwell’s demon in thermodynamics (detailed discussion in the seminal paper by Thirion [138]). The lack of the underlying math- ematical proof of the possibility of estimating the optimal transformation have led to many variants of this methods, dependent on the experimental results. Besides the intuitive way of presenting this algorithm, some possible frameworks have been developed to put the Demon algorithm as an optimisation problem [90, 145, 163].
The Demon algorithm is derived from the basic assumption of the optical flow equation [59], where it is assumed that the intensity (brightness) of the moving image is constant over time (so-called brightness consistency criterion):
If Imp~ϕq (3.15)
For small changes of the displacement field ~du, the Taylor expansion of the images’
intensity difference pIfp~xq Imp~x du~ p~xqqq holds:
pIfp~xq Imp~x du~ p~xqqq pIfp~xq Imp~xqq ∇Imp~xq ~dup~xq (3.16)
providing the mentioned optical flow equation [59]. The unique solution of the Equation 3.16 can not be determined directly, thus Thirion [138] proposed to select the direction of the minimum length vector ~du towards the fixed image If:
~
dup~xq Ifp~xq Imp~ϕp~xqq
}∇Ifp~xq}2
∇Ifp~xq (3.17)
The value of the above equation tends to infinity when the gradient of the fixed image }∇Ifp~xq}2 is close to zero. For the purpose of stabilising it, an additional
component κ has to be considered:
~
dup~xq Ifp~xq Imp~ϕp~xqq
}∇Ifp~xq}2 κ
∇Ifp~xq (3.18)
In the seminal paper of the Demon approach [138] the additional component was chosen to be κ pIfp~xq Imp~ϕip~xqqq2. Due to the fact that calculation of the
the displacement field can be computed in an iterative manner from the following formula: ~ duip~xq $ & % pIfp~xqI ~ ϕi1 m p~xqq }∇Iip~xq}2 pIfp~xqI ~ ϕi1 m p~xqq2 ∇Iip~xq if }∇Iip~xq}2 pIfp~xq Imϕ~i1p~xqq2 ¡ 0 0 otherwise (3.19) where ∇Iip~xq is chosen to be ∇Ifp~xq for the originally proposed method, and
Iϕ~i1
m p~xq Imp~x ~ui1p~xqq. Using general notation of gradient ∇Iip~xq in Equation
3.19 was motivated to emphasise that in principle other regularisation techniques of Equation 3.16 can be considered. It is mainly due to the shortcomings from the lack of any underlying optimisation framework in the original work. For this reason ∇Ii can be calculated in many ways: the so-called fixed Thirion’s gradient (a static
gradient) that was originally proposed in [138]:
∇Iip~xq ∇Ifp~xq (3.20)
a moving Thirion’s gradient (or an active gradient) taken from the assumption that at the end of the registration Ifp~xq Imp~ϕp~xqq and similarly the gradient of images
∇Ifp~xq ∇Imp~ϕp~xqq:
∇Iip~xq ∇Imp~x ~ui1p~xqq (3.21)
or the combination of both of them - the symmetric gradient [145]:
∇Iip~xq
1
2p∇Ifp~xq ∇Imp~x ~ui1p~xqqq (3.22) The update of the displacement field (Equation 3.19) can be calculated for each image point, the contours, or the labels of the segmented objects [138]. The regular- isation of Demon is done by smoothing via Gaussian filtering either for the update of displacement (denoted by Gf luid) or directly displacement (denoted by Gf dif f)
[145].
Due to very high efficiency and widespread use in the multiple applications [138, 148, 21], several attempts to interpret and theoretically justify the Demon have been made. Modersitzki [90] showed the Demon as a variational problem of solving the SSD and the diffusive model as similarity criterion and regularisation term respectively. Another approach was given by Vercauteren et al. [145], where the Demon was presented as minimisation of a global energy function including the sum of the squared difference with additional hidden variable so-called correspondence, optimised using the Newton-like strategy. Recently, Zikic et al. [163] presented the general preconditioning system based on the steepest descent strategy to exploit dif-
Algorithm 3 Demon approach Input: Images: If and Im
Parameters: Gf luid, Gdif f
Output: Transformation ~ϕ
1: ~u0 ~0, i 1
2: repeat
3: for all ~xP Ω do
4: calculate update ~dup~xq (Equation 3.19)
5: smooth update of the deformation field using Gaussian filter Gf luid:
~
duip~xq Gf luid p ~duip~xqq
6: update deformation field ~uip~xq ~ui1p~xq du~ ip~xq
7: smooth deformation field ~uip~xq using Gaussian filter Gdif f:
~
uip~xq Gdif f p~uip~xqq
8: end for 9: i i 1
10: until (deformation field does not change) or (i¥ IterMax)
11: return ~u
ferent similarity criteria using the Demon approach. The proposed preconditioning system modifies the magnitude of ∇Ip~xq.