Unidirectional Linear-elastic Registration algorithm

Unidirectional image registration algorithms[2,23,24,25,26,27,28,29,25] estimate a single transformation that registers a template image with a target image. Registering the same images while reversing the roles of the template and target images produces a second transformation that ideally should be the inverse transformation of the first transformation.

The unidirectional linear-elastic registration algorithm used in this work is described by the following minimization expression:

$$\hat{h} = \mathop {\rm arg\,min}_{h} \sigma \int_{\Omega} \vert T... ...)) - T_B(x)\vert^2 dx + \rho \int_{\Omega} \vert\vert Lu_{AB}(x)\vert\vert^2 dx$$ (5)

where image $T_A(x)$ is transformed by $h_{AB}(x)=x+u_{AB}(x)$ to match the shape of image $T_B(x)$. The first integral is called the similarity function and describes the difference in shape of the deformed image $T_A(h_{AB}(x))$ and the target image $T_B(x)$. This term is minimized for transformations that deform $T_A$ such that it matches the target $T_B$ as measured by squared intensity difference between the two images. The second term is a regularization term that constrains the transformation to satisfy a linear-elastic solid deformation model. The linear-elastic regularization model helps to smooth and prevent folding of the transformation. For our implementation the linear-elastic differential operator $L$ has the form

$$Lu(x)=-\alpha \nabla^{2} u(x)-\beta \nabla (\nabla \cdot u(x))+\gamma u(x).$$ (6)

Note that the squared intensity error similarity function of Eq. 5 can be replaced by other similarity functions such as mutual information[30,31], demons[15], or an intensity variance cost function[14].

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