instead of a color one because the projected pattern is achromatic, and therefore, no 3D information is encoded in the colors of the image. In addition, we crop the input image to exclude the edges of the face. (Facial registration and cropping is performed automatically using dlib [@king2009dlib] facial landmarks.) As discussed in Section \[sec:ProposedApproach\], and in particular, in the proof of Proposition \[prop:gradient\_depth\], the deformation of the projected fringes only provide local gradient information if the norm of the gradient of the depth is bounded. In other words, where the scene present depth discontinuities, no local depth information can be extracted by our proposed approach. This is one of the main reasons why differential 3D information can be exploited for face recognition, while bypassing the more complicated task of a 3D facial reconstruction.

![Differential depth information extracted from the image with the projected pattern. The first row illustrates the input image (depth information can be extracted from a gray version of the input as the designed patter is achromatic). The second and third row show the ground truth and the retrieved $x$ and $y$ partial derivatives of the depth respectively.[]{data-label="fig:depth_gradient_descomposition"}](figures/results_i3d.jpg){width="\columnwidth"}

One of the advantages of the proposed approach is 