for the faces in the dataset ND-2006. The second column shows the mean Fourier Transform of the faces luminance and depth respectively. The third column shows the profile across different sections of the 2D Fourier domain. Columns two and three represent the absolute value of the Fourier transform in logarithmic scale. Faces are registered using the eyes landmarks and the size normalized to $480\times480$ pixels.[]{data-label="fig:face_spectrum"}](figures/face_spectrum.png){width=".6\columnwidth"}

#### Analytic versus data-driven texture and gradient depth extraction

The previous analysis shows that closed forms can be obtained to extract texture and depth gradient information. However, to compute these expressions is necessary to isolate different spectral components $\tilde{q_n}$. To that end, filters need to be carefully designed. The design of these filters is challenging, e.g., one need to control over-smoothing versus introducing ringing artifact which are drastically amplified by a posterior gradient computation [@DiMartino2015one; @zhang2006high]. To overcome these challenges, we chose to perform a depth (gradient) and texture decomposition in a data-driven fashion, which as we show in Section \[sec:Experiments\], provides an efficient and effective solution.

#### Bounds on $f_0$ and optimal spectral orientation.

As discussed above, the projected pattern $p(x,y)$ should have a large fundamental frequency $f_0$. In addition, the orientation of the 