Algorithm 1: 3D Generalized Intersection Over Union Loss

Input: The information of the predicted ${\mathrm{B}}_{\mathrm{p}}$ and ground truth ${\mathrm{B}}_{\mathrm{g}}$ bounding box:     ${\mathrm{B}}_{\mathrm{p}}=\left({\mathrm{x}}_{\mathrm{p}},{\mathrm{y}}_{\mathrm{p}},{\mathrm{z}}_{\mathrm{p}},{\mathrm{l}}_{\mathrm{p}},{\mathrm{w}}_{\mathrm{p}},{\mathrm{h}}_{\mathrm{p}},{\mathsf{\theta }}_{\mathrm{p}}\right),{\mathrm{B}}_{\mathrm{g}}=\left({\mathrm{x}}_{\mathrm{g}},{\mathrm{y}}_{\mathrm{g}},{\mathrm{z}}_{\mathrm{g}},{\mathrm{l}}_{\mathrm{g}},{\mathrm{w}}_{\mathrm{g}},{\mathrm{h}}_{\mathrm{g}},{\mathsf{\theta }}_{\mathrm{g}}\right)$  Output: ${\mathrm{L}}_{3\mathrm{D}\_\mathrm{GIoU}}$ 1.Calculating projections ${\mathrm{B}}_{\mathrm{p}}^{\prime }$ and ${\mathrm{B}}_{\mathrm{g}}^{\prime }$ of box ${\mathrm{B}}_{\mathrm{p}}$ and ${\mathrm{B}}_{\mathrm{g}}$ on the bird’s eye view, respectively. ${\mathrm{B}}_{\mathrm{p}}^{\prime }=\left({\mathrm{x}}_{\mathrm{p}}^1,{\mathrm{y}}_{\mathrm{p}}^1,{\mathrm{x}}_{\mathrm{p}}^2,{\mathrm{y}}_{\mathrm{p}}^2,{\mathsf{\theta }}_{\mathrm{p}}^{\prime }\right),{\mathrm{B}}_{\mathrm{g}}^{\prime }=\left({\mathrm{x}}_{\mathrm{g}}^1,{\mathrm{y}}_{\mathrm{g}}^1,{\mathrm{x}}_{\mathrm{g}}^2,{\mathrm{y}}_{\mathrm{g}}^2,{\mathsf{\theta }}_{\mathrm{g}}^{\prime }\right)$ 2.${\mathrm{B}}_{\mathrm{c}}^{\prime }=\left({\mathrm{x}}_{\mathrm{c}}^1,{\mathrm{y}}_{\mathrm{c}}^1,{\mathrm{x}}_{\mathrm{c}}^2,{\mathrm{y}}_{\mathrm{c}}^2,{\mathsf{\theta }}_{\mathrm{c}}^{\prime }\right)\leftarrow$ the information of smallest enclosing box; 3.${\mathrm{A}}_{\mathrm{p}}\leftarrow$ the area of the 2D box ${\mathrm{B}}_{\mathrm{p}}^{\prime }$; 4.${\mathrm{A}}_{\mathrm{g}}\leftarrow$ the area of the 2D box ${\mathrm{B}}_{\mathrm{g}}^{\prime }$; 5.${\mathrm{A}}_{\mathrm{c}}\leftarrow$ the area of the 2D box ${\mathrm{B}}_{\mathrm{c}}^{\prime }$; 6.${\mathrm{I}}_{2\mathrm{D}}\leftarrow$ intersection between ${\mathrm{B}}_{\mathrm{p}}^{\prime }$ and ${\mathrm{B}}_{\mathrm{g}}^{\prime }$; 7.${\mathrm{U}}_{2\mathrm{D}}\leftarrow$ union between ${\mathrm{B}}_{\mathrm{p}}^{\prime }$ and ${\mathrm{B}}_{\mathrm{g}}^{\prime }$; 8.${\mathrm{I}}_{\mathrm{h}}\leftarrow$ the height of the intersection of ${\mathrm{B}}_{\mathrm{p}}$ and ${\mathrm{B}}_{\mathrm{g}};$ 9.${\mathrm{U}}_{\mathrm{h}}\leftarrow$ the height of the union of ${\mathrm{B}}_{\mathrm{p}}$ and ${\mathrm{B}}_{\mathrm{g}};$ 10.${\mathrm{V}}_{\mathrm{p}}\leftarrow$ the volume of the 3D box ${\mathrm{B}}_{\mathrm{p}}$; 11.${\mathrm{V}}_{\mathrm{g}}\leftarrow$ the volume of the 3D box ${\mathrm{B}}_{\mathrm{g}}$; 12.${\mathrm{V}}_{\mathrm{c}}\leftarrow$ the volume of the 3D box ${\mathrm{B}}_{\mathrm{c}}$, where ${\mathrm{B}}_{\mathrm{c}}$ represents the smallest 3D enclosing box; 13.Calculating intersection ${\mathrm{I}}_{\mathrm{v}}$ of ${\mathrm{B}}_{\mathrm{p}}$ and ${\mathrm{B}}_{\mathrm{g}}$: if${\mathrm{I}}_{2\mathrm{D}}\le 0:$  ${\mathrm{I}}_{\mathrm{v}}=0;$ else: if${\mathrm{I}}_{\mathrm{h}}\le 0$:  ${\mathrm{I}}_{\mathrm{v}}=0;$   else:     ${\mathrm{I}}_{\mathrm{v}}={\mathrm{I}}_{2\mathrm{D}}\times {\mathrm{I}}_{\mathrm{h}};$ 14.${\mathrm{IoU}}_{3\mathrm{D}}=\frac{{\mathrm{I}}_{\mathrm{v}}}{{\mathrm{U}}_{\mathrm{v}}}$, where ${\mathrm{U}}_{\mathrm{v}}={\mathrm{V}}_{\mathrm{p}}+{\mathrm{V}}_{\mathrm{g}}-{\mathrm{I}}_{\mathrm{v}}$; 15.${\mathrm{GIoU}}_{3\mathrm{D}}={\mathrm{IoU}}_{3\mathrm{D}}-\frac{\left({\mathrm{V}}_{\mathrm{c}}-{\mathrm{U}}_{\mathrm{v}}\right)}{{\mathrm{V}}_{\mathrm{c}}}$; 16.${\mathrm{L}}_{3\mathrm{D}\_\mathrm{GIoU}}=1-{\mathrm{GIoU}}_{3\mathrm{D}};$