Editing Light field (section)

=== Discrete focal stack transform ===
Another way to efficiently compute 2-D photographs is to adopt discrete focal stack transform (DFST).<ref>{{Cite journal|last1=Nava|first1=F. Pérez|last2=Marichal-Hernández|first2=J.G.|last3=Rodríguez-Ramos|first3=J.M.|date=August 2008|title=The Discrete Focal Stack Transform|url=https://ieeexplore.ieee.org/document/7080334|journal=2008 16th European Signal Processing Conference|pages=1–5}}</ref> DFST is designed to generate a collection of refocused 2-D photographs, or so-called [[Focus stacking|Focal Stack]]. This method can be implemeted by fast [[Fractional Fourier transform|fractional fourier transform]] (FrFT).

The discrete photography operator <math>\mathcal{P}_{\alpha}\left[\cdot\right]</math> is defined as follows for a lightfield <math>L_{F}(\boldsymbol {s},\boldsymbol {u})</math> sampled in a 4-D grid <math>\boldsymbol {s} = \Delta s \tilde{\boldsymbol {s}},</math> <math>\tilde{\boldsymbol {s}} = -\boldsymbol {n}_{\boldsymbol {s}}, ..., \boldsymbol {n}_{\boldsymbol {s}}</math>, <math>\boldsymbol {u} = \Delta u \tilde{\boldsymbol {u}}, \tilde{\boldsymbol {u}}=-\boldsymbol {n}_{\boldsymbol {u}},...,\boldsymbol {n}_{\boldsymbol {u}}</math>:

:<math>\mathcal{P}_{q}[L](\boldsymbol{s})=
\sum_{\tilde{\boldsymbol{u}}=-\boldsymbol{n}_{\boldsymbol{u}}}^{\boldsymbol{n}_{\boldsymbol{u}}} L(\boldsymbol{u} q+\boldsymbol{s}, \boldsymbol{u}) \Delta \boldsymbol{u}, 
\quad \Delta \boldsymbol{u}=\Delta u\Delta v,
\quad q=\left(1-\frac{1}{\alpha}\right)</math>

Because <math>(\boldsymbol{u} q+\boldsymbol{s}, \boldsymbol{u}) </math> is usually not on the 4-D grid, DFST adopts [[trigonometric interpolation]] to compute the non-grid values.

The algorithm consists of these steps:

* Sample the light field <math>L_{F}(\boldsymbol {s},\boldsymbol {u})</math> with the sampling period <math>\Delta s</math> and <math>\Delta u</math> and get the discretized light field <math>L^d_{F}(\boldsymbol {s},\boldsymbol {u})</math>.
* Pad <math>L^d_{F}(\boldsymbol {s},\boldsymbol {u})</math> with zeros such that the signal length is enough for FrFT without aliasing.
* For every <math>\boldsymbol {u}</math>, compute the [[Discrete Fourier transform]] of <math>L^d_{F}(\boldsymbol {s},\boldsymbol {u})</math>, and get the result <math>R1</math>.
* For every focal length <math>\alpha F</math>, compute the [[Fractional Fourier transform|fractional fourier transform]] of <math>R1</math>, where the order of the transform depends on <math>\alpha</math>, and get the result <math>R2</math>.
* Compute the inverse Discrete Fourier transform of <math>R2</math>.
* Remove the marginal pixels of <math>R2</math> so that each 2-D photograph has the size <math>(2{n}_{\boldsymbol {s}}+1)</math> by <math>(2{n}_{\boldsymbol {s}}+1)</math>