Consistent Linear-elastic Registration Algorithm

A common problem with unidirectional registration algorithms is that they rarely produce forward and reverse transformations that are inverses of one another. The consistent linear-elastic image registration algorithm[16,17] was developed to overcome this problem. In this approach the forward transformation $h_{AB}$ that maps $T_A$ to $T_B$ and reverse transformation $h_{BA}$ that maps $T_B$ to $T_A$ are jointly estimated subject to the constraint that $h_{AB}$ and $h_{BA}$ are inverses. The transformations are constructed as $h_{AB}(x)=x+u_{AB}(x)$ and $h_{BA}(x)=x+u_{BA}(x)$ where $u_{AB}(x)$ and $u_{BA}(x)$ are the associated deformation fields.

The consistent linear-elastic image registration problem can be stated as

$$\hat{h}_{AB}, \hat{h}_{BA} = \mathop {\rm arg\,min}_{h_{AB},h_{BA}} \sigma \int_{\Omega} \vert T_A(h_{AB}(x)) -T_B(x)\vert^2 +\vert T_B(h_{BA}(x)) - T_A(x)\vert^2 dx$$

$$+ \rho \int_{\Omega} \vert\vert Lu_{AB}(x)\vert\vert^2 + \vert\vert Lu_{BA}(x)\vert\vert^2 dx$$

$$+ \chi \int_{\Omega} \vert\vert h_{AB}(x)- h^{-1}_{BA}(x)\vert\vert^2 + \vert\vert h_{BA}(x)-h^{-1}_{AB}(x)\vert\vert^2 dx$$ (7)

where $h_{AB}$ and $h_{BA}$ are estimated to transform image $T_A$ to $T_B$ and $T_B$ to $T_A$, respectively. The first two integrals are equivalent to independently estimating the forward and reverse transformations between $T_A$ and $T_B$ as stated in Eq. 5. The third integral is referred to as the inverse consistency constraint. This constraint is minimized when the forward and reverse transformations are inverses of one another. This term also couples the estimation of the forward and reverse transformations together.

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