Consistent Intensity-based Registration

The consistent intensity-based registration (CI-TPS) algorithm [32,34,33] using thin-plate spline regularization is briefly described here. It is based on minimizing the cost function given by

$$C = \sigma \int_{\Omega} \vert T(h(x)) -S(x)\vert^2 + \vert S(g(x)) - T(x)\vert^2 dx$$ (5)

$$+ \rho \int_{\Omega} \vert\vert{\mathcal L} u(x)\vert\vert^2 + \vert\vert{\mathcal L} w(x)\vert\vert^2 dx$$

$$+ \chi \int_{\Omega} \vert\vert u(x)-\tilde{w}(x)\vert\vert^2 + \vert\vert w(x)-\tilde{u}(x)\vert\vert^2 dx$$

The intensities of $T$ and $S$ are assumed to be scaled between 0 and 1. The first integral of the cost function defines the cumulative squared error similarity cost between the transformed template $T(h(x))$ and target image $S(x)$ and between the transformed target $S(g(y))$ and the template image $T(y)$. To use this similarity function, the images $T$ and $S$ must correspond to the same imaging modality and they may require pre-processing to equalize the intensities of the image. The similarity function defines the correspondence between the template and target images as the forward and reverse transformations $h$ and $g$, respectively, that minimize the squared error intensity differences between the images. The second integral is used to regularize the forward and reverse displacement fields $u$ and $w$, respectively. This term is used to enforce the displacement fields to be smooth and continuous. The third integral is called the inverse consistency constraint and is minimized when the forward and reverse transformations $h$ and $g$, respectively, are inverses of each other. The constants $\sigma$, $\rho$, and $\chi$ define the relative importance of each term of the cost function.

The cost function in Eq. 6 is discretized to numerically minimize it. The forward and reverse transformations $h$ and $g$ and their associated displacement fields $u$ and $w$ are parameterized by the discrete Fourier series defined by

$$u_d[n] = \sum_{k \in \Omega_d} \mu[k] e^{j<n,\theta [k]>} \quad$$

$$w_d[n] = \sum_{k \in \Omega_d} \eta[k] e^{j<n,\theta [k]>}$$ (6)

for $n \in \Omega_d$ where the basis coefficients $\mu[k]$ and $\eta[k]$ are $(2 \times 1)$ complex-valued vectors and $\theta[k] = [\frac {2\pi k_1} {N_1}, \frac {2\pi k_2} {N_2}]^T$. The basis coefficients have the property that they have complex conjugate symmetry, i.e., $\mu[k] = \mu^*[N-k]$ and $\eta[k] = \eta^*[N-k]$. The notation $<\cdot, \cdot>$ denotes the dot product of two vectors such that $<n,\theta[k]> = \frac {2 \pi k_1 n_1}{N_1} + \frac {2 \pi k_2 n_2}{N_2}$. The basis coefficients $\mu[k]$ and $\eta[k]$ of the discretized forward and reverse displacement fields are then minimized using gradient descent as described in [32,34].

The intensity similarity component of the cost function is forced to register the global intensity patterns before local intensity patterns by restricting the similarity gradient to modify only the low frequencies of the displacement field parameters. Restricting the similarity cost gradient to modifying the low frequency components is analogous to filtering with a zonal low-pass filter. To mitigate the Gibbs ringing associated with zonal low-pass filters, a low-pass Butterworth filter is applied to the similarity cost gradient in the gradient decent algorithm.

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