Table 1.

Summary of variables in the automatic planning method.

Variable	Description
$P=\{{\mathit{p}}_k\in {ℝ}^3\},k=1,\dots ,|P|$	Input surface (patient-specific segmentation from CT or MRI) with vertices pk where k is a vertex index
$i=1\dots n\in \mathbb{N}$	Atlas member index referring to a set of n similar shapes, e.g. vertebrae of level 3, or of level 7 merged with level 8
$j\in \{\text{e},\text{a},\text{p}\}$	Annotation categories including entry point (e), anterior-most extent of a non-breaching trajectory (a), and pedicle center (p)
$V_i=\{{\mathit{v}}_{\mathit{k}}\in {ℝ}^3\},k=1,\dots ,|V_i|$	Surface for atlas member i
${\mathit{x}}_i^{(j)}\in {ℝ}^3$	Annotated point in a category j of an atlas member i
${\mathit{s}}_i^{(j)}\in {ℝ}^{{\kappa }_i}$	Mean value coordinate of annotated point in a category j of an atlas member i
$T_i=\text{cpd}(V_i,P)\in {ℝ}^{{\kappa }_i\times 3}$	Non-rigid displacement transformation mapping atlas member Vi to surface P
${\tilde{\mathit{x}}}_i^{(j)}\in {ℝ}^3$	Annotated point in a catergory j, transformed using the deformation field
${\overline{\mathit{x}}}^{(j)}\in {ℝ}^3$	Mean (over i) of transformed annotations
$t_i=({\mathit{x}}_i^{(\text{e})},{\mathit{x}}_i^{(\text{a})})$	Non-breaching transpedicle trajectory comprising an entry point ${\mathit{x}}_i^{(\text{e})}$ and anterior-most extent ${\mathit{x}}_i^{(\text{a})}$ of the trajectory in atlas member i
$C_i=\{{\mathit{c}}_k\in {ℝ}^3\}$	Cylinder surface modelling the volumetric extent of atlas trajectory ti
${\tilde{C}}_i=\{{\tilde{\mathit{c}}}_k\in {ℝ}^3\}$	Deformed cylinder surface, with each vertex transformed using the deformation Ti
$B_i\in {ℝ}^{l\times w\times h}$	Binary image of deformed cylinder surface for a padded bounding box described by length (l), width (w), and height (h)
$D=\{{\mathit{d}}_k\in {ℝ}^3\}$	Cylinder surface based on mean of transformed annotations
$A\in {ℝ}^{l\times w\times h}$	Accumulation map for a padded bounding box described by length (l), width (w), and height (h)
$z\in {ℝ}^6$	Rigid transform parameters: translation and rotation
$W\in {ℝ}^{3x4}$	Rigid transform matrix based on z
${\mathit{y}}^{(j)}\in {ℝ}^3,j\in \{\text{e},\text{a}\}$	Point locations (entry, anterior) on the estimated trajectory relative to P
${\mathit{x}}_{\text{proj}}^{(j)}\in {ℝ}^3$	Trajectory projected on reference (evaluation of anterior and entry point)
${\mathit{y}}_{\text{proj}}^{(\text{p})}\in {ℝ}^3$	Reference projected on trajectory (evaluation at pedicle center)
α ∈ [0°, 180°)	Angle between reference and planned trajectory