[@takeda1983fourier] $$\label{eq:pat_design_3}
    I(x,y) = \sum_{n=-\infty}^{+\infty} q_n(x,y) e^{i2\pi n f_0 x}.$$ Applying the 2D Fourier Transform (FT) in both sides of Equation  and using standard properties of the FT [@distributions] we obtain $$\label{eq:pat_design_4}
    \tilde{I}(f_x,f_y) = \sum_{n=-\infty}^{+\infty} \tilde{q_n}(f_x - nf_0, f_y).$$ We denote as $\tilde{I}$ the FT of $I$ and use $(f_x,f_y)$ to represent the 2D frequency domain associated to $x$ and $y$ axis respectively.

Equation  shows that the FT of the acquired image can be decomposed into the components $\tilde{q_n}$ centered at $(nf_0,0)$. In the context of this section, we refer to a function $h(x,y)$ being smooth if $$\frac{\|\tilde{h}(f_x,f_y)\|}{\|\tilde{h}(0,0)\|}<10^{-3}\ \ \  \forall \ \ \ |f_x|>\frac{f_0}{2} .$$ Assuming $I_0(x,y)$ and $\phi(x,y)$ are smooth (we empirically validate this hypothesis below), the components $\tilde{q_n}$ can be isolated as illustrated in Figure \[fig:fringesDescomposition\]. The central component is of particular interest, $q_0(x,y) = a_0\, I_0(x,y)$ captures the facial texture information and can be recovered from $I(x,y)$ if $f_0$ is large enough (we provide a more precise quantitative analysis in what follows). On the other hand, relative (gradient) 3D information can be retrieved from the components $\{q_0, q_1\}$ as we show in Proposition \[prop:gradient\_depth\].

\[prop:gradient\_depth\] Gradient depth information is encoded in the components $\{q_0(x,y), q_1(x,y)\}$.

We 