rotation and shrinkage also include random translation, and when combined with the imperfect slice correspondences represent the closest condition to the real data.

Quantitative Evaluation {#quantitative-evaluation .unnumbered}
-----------------------

The accuracy of the radiology-histopathology registration was evaluated using the Dice similarity coefficient, which assesses the overlap of the prostate outlined on T2w MRI and the outline of the prostate from the histopathology reconstruction:

$$Dice (\mathcal{H}, \mathcal{M}) = \frac{1}{D}\sum^D_{i=1} \frac{2 \times |\mathcal{H}^{Pr}_i \cap  \mathcal{M}^{Pr}_i|}{|\mathcal{H}^{Pr}_i| +  |\mathcal{M}^{Pr}_i|}   
\label{eq:Dice}$$

where $D$ is the number of slices in the histopathology specimen, $\mathcal{H}^{Pr}_i$ represents the slice $i$ in the prostate segmentation on histopathology, while $\mathcal{M}^{Pr}_i$ represents the slice $i$ in the prostate segmentation on MRI.

Additionally, we evaluated the Hausdorff distance between the prostate boundary, to asses how far the boundary is after performing the registration:

$$\begin{split}
    Hausdorff^{Pr} (\mathcal{H}, \mathcal{M}) = 
    max\{sup_{h \in \mathcal{H}^{Pr}} ~ inf_{m \in \mathcal{M}^{Pr}}d(\mathcal{H}, 
    \mathcal{M}),\\sup_{m \in \mathcal{M}^{Pr}} inf_{h \in \mathcal{H}^{Pr}} d(\mathcal{H}, \mathcal{M}) \}
\end{split}
\label{eq:HDice}$$

where $sup$ represents the supremum operator and $inf$ represents the infimum operators.

Moreover, we evaluated the landmark distance:

$$Dist (\mathcal{L}^H,\mathcal{L}^M) = \frac{1}{X}\sum^X_{j=1}|\mathcal{L}^H_j,\mathcal{L}^M_j|_2
\label{eq:Dist}$$

where $|.|_2$ represents the Euclidean distance of the center of mass of the j landmark $\mathcal{L}^H_j$ on histopathology and center of mass of the 