The single channel energy equation for photometric alignment is $ E(\xi ) = \sum\limits_{i = 1}^V {{{({{I_K}({{\bf{p}}_i}) - I(\omega ({{\bf{p}}_i},{{\bf{D}}_K}({{\bf{p}}_i}),\xi ))})}^2}} $

where ${\bf \xi}$ is a $6$-vector representing the pose of the current image $I$ with respect to the reference image $I_K$ in Lie algebra $\mathfrak{se}(3)$, and $\omega$ is the 3D projective warp function that maps the pixel location ${\bf{p}}_i$ in the reference image according to its inverse depth $D_K ({\bf{p}}_i)$ and the pose ${\bf{\xi}}$ to the pixel location in the current image.

Qulitative 3D reconstruction demonstration/comparisson to the single-channel (grascale) method video: