the $x$-axis with a deformation given by $p(x+\phi(x,y),y)$ ($\phi(x,y) \propto z(x,y)$). A detailed description of active stereo geometry is provided in the supplementary material Section \[sec:reviewActiveStereo\]. Let us denote $I_0(x,y)$ the image we would acquire under homogeneous illumination, and $p(x,y)$ the intensity profile of the projected light. Without loss of generality we assume the system baseline is parallel to the $x$ axis. The image acquired by the camera when the projected light is modulated with a profile $p(x,y)$ is $$\label{eq:pat_desing_1}
    I(x,y) = I_0(x,y)p(x+\phi(x,y), y).$$ We will restrict to periodic modulation patterns and let $T$ denote the pattern spatial period, we also define $f_0 \stackrel{def}{=} \frac 1 T$. To simplify the system design and analysis, lets also restrict to periodic patterns that are invariant to the $y$ coordinate. In these conditions we can express $p(x,y) = \sum_{n=-\infty}^{+\infty} a_n\, e^{i2\pi n f_0 x}$ where $a_n$ represent the coefficients of the Fourier series of $p$. (Note that because of the invariance with respect to the $y$ coordinate, the coefficients $a_n$ are constant instead of a function of $y$.) Equation  can be expressed as $$\label{eq:pat_design_2}
    I(x,y) = \sum_{n=-\infty}^{+\infty} I_0(x,y)\, a_n\, e^{i2\pi n f_0 (x+\phi(x,y))}.$$ Defining $q_n(x,y)\stackrel{def}{=}I_0(x,y)\,a_n\,e^{i2\pi n f_0 \phi(x,y)}$, Equation  can be expressed as 