Problem Statement

Traditionally, the image registration problem has been stated as: Find the transformation $h$ that maps the template image volume $T$ into correspondence with the target image volume $S$. Alternatively, the problem can be stated as: Find the transformation $g$ that transforms $S$ into correspondence with $T$. For this paper, the previous two statements are combined into a single problem and restated as:

Problem Statement: Jointly estimate the transformations $h$ and $g$ such that $h$ maps $T$ to $S$ and $g$ maps $S$ to $T$ subject to the constraint that $h = g^{-1}$.

The image registration algorithm is formulated on the continuum and is discretized for implementation. Let $T$ and $S$ represent 3D image volumes of voxels dimension $N_1 \times N_2 \times N_3$ corresponding to medical imaging modalities such as MRI, fMRI, CT, cryosection imagery, etc. Let $\Omega_d = \{ (n_1, n_2, n_3) \vert 0 \leq n_1 < N_1; 0 \leq n_2 < N_2; 0 \leq n_3 < N_3 ;$    and $n_1, n_2, n_3$    are integers$\}$ correspond to the set voxel lattice coordinates of the discrete images $T$ and $S$ and let $\Omega_c = [0,1)^3$. Continuous functions will be denoted with a subscript c and parentheses while discrete functions will be denoted with a subscript d and square brackets. The subscripts are dropped when a statement is true for both continuous and discrete functions or where the context is clear due to the use of parentheses or square brackets. Let the continuous image $T_c(x)$ for $x \in \Omega_c$ be related to the discrete image $T_d[n]$ for $n \in \Omega_d$ in the normal multi-dimensional sampling sense $T_d[n] = T_c(\frac n N)$ where the notation $\frac n N$ is defined as the $3 \times 1$ column vector $[\frac {n_1}{N_1}, \frac {n_2}{N_2}, \frac {n_3}{N_3}]^T$. Likewise, let $S_d[n] = S_c(\frac n N)$ for $n \in \Omega_d$. The discrete images are extended to the continuum using trilinear interpolation.

The transformations are vector-valued functions that map the image coordinate system $\Omega_c$ to itself, i.e., $h_c: \Omega_c \mapsto \Omega_c$ and $g_c: \Omega_c \mapsto \Omega_c$. Regularization constraints are placed on $h$ and $g$ so that they preserve topology. Throughout it is assumed that $h_c(x) = x+u_c(x)$, $h_c^{-1}(x) = x+\tilde{u}_c(x)$, $g_c(x) = x+w_c(x)$ and $g_c^{-1}(x) = x+\tilde{w}_c(x)$ where $h_c(h_c^{-1}(x))=x$ and $g_c(g_c^{-1}(x))=x$. The vector-valued functions $u$, $w$, $\tilde{u}$, and $\tilde{w}$ are called displacement fields since they define the transformation in terms of a displacement from a location $x$. All of the functions $h_c$, $g_c$, $h_c^{-1}$, $g_c^{-1}$, $u_c$, $\tilde{u}_c$, $w_c$, and $\tilde{w}_c$ are $(3 \times 1)$ vector-valued functions defined on the $\Omega_c$. The continuous transformations and displacement fields are extended to the continuum from their corresponding discrete representations using trilinear interpolation. For example, $h_d[n] = h_c(\frac n N) = \frac n N + u_c(\frac n N) = \frac n N + u_d[n]$ .

Registration is defined using a symmetric similarity cost function that describes the distance between the transformed template $T \circ h$ and target $S$, and the distance between the transformed target $S \circ g$ and template $T$. To ensure the desired properties, the transformations $h$ and $g$ are jointly estimated by minimizing the similarity cost function while satisfying regularization constraints and inverse transformation consistency constraints. The regularization constraints are enforced on the transformations by constraining the them to satisfy the laws of continuum mechanics [24].

Copyright © 2002 • The University of Iowa. All rights reserved. Iowa City, Iowa 52242
Questions or Comments: gary-christensen@uiowa.edu

Copyright © 2002 • The University of Iowa. All rights reserved. Iowa City, Iowa 52242
Questions or Comments: gary-christensen@uiowa.edu