define the wrapping function $\mathcal{W}(u)={\mbox{atan}}({\mbox{tan}}(u))$. This function wraps the real set into the interval $\left(-\pi/2,\pi/2\right]$ [@ghiglia1998unwrapping]. This definition can be extended to vector inputs wrapping the modulus of the vector field while keeping its direction unchanged, i.e., $\mathcal{W}(\vec{u})=\frac{\mathcal{W}(\|\vec{u}\|)}{\|\vec{u}\|}\vec{u}$ if $\|\vec{u}\|\neq 0$ and $\mathcal{W}(\vec{u})=\vec{0}$ if $\|\vec{u}\| = 0$. From $q_1(x,y)$ and $q_0(x,y)$ we can compute[^2] $$\label{eq:prop_prof_1}
    \phi_{\mathcal{W}}(x,y) = \frac{1}{2\pi f_0}\, {\mbox{atan}}\left(\frac{\mbox{Im}\{\frac{q_1(x,y)}{q_0(x,y)}\}}{\mbox{Re}\{\frac{q_1(x,y)}{q_0(x,y)}\}}\right)$$ where $\phi_{\mathcal{W}}$ denotes the wrapped version of $\phi$. Moreover, $\phi_{\mathcal{W}}(x,y)=\phi(x,y) + \pi k(x,y)$ with $k(x,y)\in\mathbb{N}$ (wrapping introduces shifts of magnitude multiple of $\pi$). Computing the gradient both sides leads to $\nabla\phi_{\mathcal{W}}(x,y)=\nabla\phi(x,y) + \pi \nabla k(x,y)$ where $\|\nabla k(x,y)\|\in\mathbb{N}$. Assuming the magnitude of the gradient of $\phi(x,y)$ is bounded by $\pi/2$ and considering that $\|\nabla k(x,y)\|\in\mathbb{N}$, we can apply the wrapping function both sides of the previous equality to obtain $\mathcal{W}(\nabla\phi_{\mathcal{W}})(x,y)=\nabla\phi(x,y)$ which proves (recall Equation ) that the gradient of $\phi$ can be extracted from the components $q_0$ and $q_1$. To conclude the proof, we use the property of linearity of the gradient operation and the fact that $\phi(x,y)$ is proportional to the depth map of the scene (see Equation  and Section \[sec:reviewActiveStereo\] in the supplementary material).

![Faces average spectral content. The first column illustrates the mean luminance and depth map 