Regularization Constraint

Minimizing the cost function in (3) does not ensure that the transformations $h$ and $g$ are diffeomorphic transformations except for when $C_{\text{ICC}}=0$. Continuum mechanical models such as linear elasticity [28,22] and viscous fluid [22,15] can be used to regularize the transformations. In this work, a linear-elastic constraint of the form

$$C_{\text{REG}}(u) + C_{\text{REG}}(w) = \int_{\Omega} \vert\vert L_c u_c(x)\vert\vert^2 dx + \int_{\Omega} \vert\vert L_c w_c (x)\vert\vert^2 dx$$ (5)

is used to regularize the transformations where $u_c(x) = h_c(x)-x$ and $w_c(x) = g_c(x)-x$. The linear elasticity operator $L_c$ has the form $L_c u_c(x) = - \alpha \nabla^2 u_c(x) - \beta \nabla ( \nabla \cdot u_c(x)) +\gamma u_c(x)$ where $\nabla = \left[ \frac {\partial}{\partial x_1}, \frac {\partial}{\partial x_2}, \frac {\partial}{\partial x_3} \right]$ and $\nabla^2 = \nabla \cdot \nabla = \left[ \frac {\partial^2}{\partial x_1^2} +\frac {\partial^2}{\partial x_2^2} + \frac {\partial^2}{\partial x_3^2} \right]$. In general, $L_c$ can be any nonsingular linear differential operator [29]. The limitation of using linear differential operators is that they can't prevent the transformation from folding onto itself, i.e., destroying the topology of the images under transformation [30]. This includes the linear elasticity and thin-plate spline models. The linear elasticity operator is used in this work to help prevent the Jacobian of the transformation from going negative. At each iteration the Jacobian of the transformation is checked to make sure that it is positive for all points in $\Omega _d$ which implies that the transformation preserves topology when transforming images.

The purpose of the regularization constraint is to ensure that the transformations maintain the topology of the images $T$ and $S$. Thus, the elasticity constraint can be replaced by or combined with other regularization constraints that maintain desirable properties of the template (source) and target when deformed. An example would be a constraint that prevented the Jacobian of both the forward and reverse transformations from going to zero or infinity. A constraint that penalizes small and large Jacobian values is given by $C_{\text{Jac}}(h) + C_{\text{Jac}}(g) = \int_{\Omega} (J(h_c(x)))^2 + \left( ... ...J(h_c(x))} \right)^2 + (J(g_c(x)))^2 + \left( \frac 1 {J(g_c(x))} \right)^2 dx$ where $J$ denotes the Jacobian operator. Further examples of regularization constraints that penalize large and small Jacobians can be found in Ashburner et al. [21].

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