  We present Gradient-SDF, a novel representation for 3D geometry that combines
the advantages of implict and explicit representations. By storing at every
voxel both the signed distance field as well as its gradient vector field, we
enhance the capability of implicit representations with approaches originally
formulated for explicit surfaces. As concrete examples, we show that (1) the
Gradient-SDF allows us to perform direct SDF tracking from depth images, using
efficient storage schemes like hash maps, and that (2) the Gradient-SDF
representation enables us to perform photometric bundle adjustment directly in
a voxel representation (without transforming into a point cloud or mesh),
naturally a fully implicit optimization of geometry and camera poses and easy
geometry upsampling. Experimental results confirm that this leads to
significantly sharper reconstructions. Since the overall SDF voxel structure is
still respected, the proposed Gradient-SDF is equally suited for (GPU)
parallelization as related approaches.
