of a high spatial frequency, we can recover a standard (low resolution) RGB facial image that can be fed into state-of-the-art 2D face recognition methods. (b) We avoid the complicated task of 3D facial reconstruction and instead, extract local 3D features from the local deformation of the projected pattern. In that sense our ideas can be implemented exploiting existing and future 2D solutions. In addition, our approach is different from those that hallucinate 3D information. As discussed before and illustrated in Figure \[fig:3dillustration\] this task requires a strong prior of the scene which is ineffective, for example, if a spoofing attack is presented (see the example provided in Figure \[fig:3dmm\] in the supplementary material).

Proposed Approach {#sec:ProposedApproach}
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#### Notation.

Let ${\mathcal{I}}\subset{\mathbb{R}}^{H\times W\times C}$ denote the space of images with $H\times W$ pixels and $C$ color channels, and ${\mathcal{X}}_{n}\subset{\mathbb{R}}^{n}$ a space of n-dimensional column vectors (in the context of this work associated to a facial feature embedding). ${\mathcal{I}}_{rgb}$ denotes the set of RGB images ($C=3$), while ${\mathcal{I}}_{\nabla z}$ is used to denote the space of two channel images ($C=2$) associated to the gradient of a single-channel image $z\in{\mathbb{R}}^{H\times W \times 1}$. (The first/second channel represents the partial derivative with respect 