docs/normal.md

MoGe-2 Normal Estimation

Qualitative comparison of normal estimation with Marigold and Metric3D V2

NOTE: Normal estimation was implemented after the submission of the MoGe-2 paper and is therefore not included in the original publication. This feature required minimal additional effort, and we do not claim any novel technical contribution.

We added a lightweight convolutional head and trained the normal output using a squared angular loss:

$${\mathcal{L}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}=\frac{1}{\mathrm{\mid }\mathcal{M}\mathrm{\mid }}\sum _{i\in \mathcal{M}}\mathrm{\angle }({\widehat{\mathbf{n}}}_i,{\mathbf{n}}_i{)}^2\backslash mathcal\; L\_\{\backslash rm\; normal\}\; =\; \{1\backslash over\; |\backslash mathcal\; M|\}\backslash sum\_\{i\backslash in\backslash mathcal\; M\}\; \backslash angle\; (\backslash hat\{\backslash mathbf\; n\}\_i,\backslash mathbf\; n\_i)^2$$

where $\hat{\mathbf{n}}_i$ is the predicted normal, $\mathbf{n}_i$ is the ground-truth normal, and $\mathcal{M}$ denotes the set of valid pixels. For convenience, we did not collect ground-truth normal maps for training. Instead, we derived surface normals from the depth map and camera intrinsics. The resulting estimates are visually and numerically satisfactory.