Algorithm 1: 3D manipulation algorithm

Input: index’s coordinate ${\mathit{h}}_{\mathit{i}}$, thumb’s coordinate ${\mathit{t}}_{\mathit{k}}$, encompassing lung

volume’s coordinates $\left\{\mathit{e}\right\}$

Output: lung’s matrix (rotation matrix ${\mathsf{\Theta }}_{ĩ}$, translation vector ${\mathbf{T}}_{ĩ}$)

${\mathit{h}}_{\mathit{i}}{}^{\mathit{T}}$ ← coordinateFormIndex $\left(\mathit{x}{\mathit{d}}_{\mathit{i}},\mathit{y}{\mathit{d}}_{\mathit{i}},\mathit{z}{\mathit{d}}_{\mathit{i}}\right)$

${\mathit{t}}_{\mathit{k}}{}^{\mathit{T}}$ ← coordinateFormThumb $\left(\mathit{x}{\mathit{t}}_{\mathit{k}},\mathit{y}{\mathit{t}}_{\mathit{k}},\mathit{z}{\mathit{t}}_{\mathit{k}}\right)$

${\mathit{e}}_{\mathit{j}}{}^{\mathit{T}}$ ← coordinateFormCube $\left(\mathit{x}{\mathit{c}}_{\mathit{j}},\mathit{y}{\mathit{c}}_{\mathit{j}},\mathit{z}{\mathit{c}}_{\mathit{j}}\right)$

${\mathit{e}}_{\mathit{l}}{}^{\mathit{T}}$ ← coordinateFormCube $\left(\mathit{x}{\mathit{c}}_{\mathit{l}},\mathit{y}{\mathit{c}}_{\mathit{l}},\mathit{z}{\mathit{c}}_{\mathit{l}}\right)$

while (${\mathit{h}}_{\mathit{i}}{}^{\mathit{T}}\ne \mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}$) and (${\mathit{t}}_{\mathit{k}}{}^{\mathit{T}}\ne \mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}$) do

if (${\mathit{h}}_{\mathit{i}}{}^{\mathit{T}}={\mathit{e}}_{\mathit{j}}{}^{\mathit{T}}$) and (${\mathit{t}}_{\mathit{k}}{}^{\mathit{T}}={\mathit{e}}_{\mathit{l}}{}^{\mathit{T}}$) then

${\mathit{h}}_{\mathit{i},\mathit{j}}{}^{\mathit{T}}$ ← IncidenceIndexCube (${\mathit{h}}_{\mathit{i}}{}^{\mathit{T}}$)

${\mathit{t}}_{\mathit{k},\mathit{l}}{}^{\mathit{T}}$← IncidenceThumbCube (${\mathit{t}}_{\mathit{k}}{}^{\mathit{T}}$)

if (${\mathit{h}}_{\mathit{i}+\mathbf{1},\mathit{j}+\mathbf{1}}{}^{\mathit{T}}\ne {\mathit{h}}_{\mathit{i},\mathit{j}}{}^{\mathit{T}}$) and (${\mathit{t}}_{\mathit{k}+\mathbf{1},\mathit{l}+\mathbf{1}}{}^{\mathit{T}}\ne {\mathit{t}}_{\mathit{k},\mathit{l}}{}^{\mathit{T}}$) then

${\mathsf{\Theta }}_{ĩ}$ ← (${\mathit{h}}_{\mathit{i}+\mathbf{1},\mathit{j}+\mathbf{1}}{}^{\mathit{T}}-{\mathit{t}}_{\mathit{k}+\mathbf{1},\mathit{l}+\mathbf{1}}{}^{\mathit{T}})\times {\left({\mathit{h}}_{\mathit{i},\mathit{j}}{}^{\mathit{T}}-{\mathit{t}}_{\mathit{k},\mathit{l}}{}^{\mathit{T}}\right)}^{-\mathbf{1}}$

${\displaystyle {\mathbf{T}}_{ĩ}}$ ← [${\mathit{h}}_{\mathit{i}+\mathbf{1},\mathit{j}+\mathbf{1}}{}^{\mathit{T}}\left]-\right[\left({\mathit{h}}_{\mathit{i}+\mathbf{1},\mathit{j}+\mathbf{1}}{}^{\mathit{T}}-{\mathit{t}}_{\mathit{k}+\mathbf{1},\mathit{l}+\mathbf{1}}{}^{\mathit{T}}\right)\times {\left({\mathit{h}}_{\mathit{i},\mathit{j}}{}^{\mathit{T}}-{\mathit{t}}_{\mathit{k},\mathit{l}}{}^{\mathit{T}}\right)}^{-\mathbf{1}}\times \left({\mathit{h}}_{\mathit{i},\mathit{j}}\right)]$

end

end

end