Vulkan View Depth from Faramebuffer Depth

Matthew Wellings 18-Nov-2016

This note gives the mathematical derivation of the equation that recovers view space depth in the fragment shaders of the blog post Depth Peeling Pseudo-Volumetric Rendering.

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& \frac{1}{2}& \frac{1}{2}\\ 0& 0& 0& 1\end{array}\right]$ $\left[\begin{array}{cccc}\frac{2n}{r-l}& 0& \frac{r+l}{r-l}& 0\\ 0& \frac{2n}{t-b}& \frac{t+b}{t-b}& 0\\ 0& 0& \frac{-(f+n)}{f-n}& -\frac{2nf}{f-n}\\ 0& 0& -1& 0\end{array}\right]$ = $\left[\begin{array}{cccc}\frac{2n}{r-l}& 0& \frac{r+l}{r-l}& 0\\ 0& \frac{-2n}{t-b}& -\frac{t+b}{t-b}& 0\\ 0& 0& -\frac{2nf}{2(f-n)}& -\frac{nf}{f-n}\\ 0& 0& -1& 0\end{array}\right]$

This resulting matrix is then effectively pre-multiplied by the view space coordinates to give clip space:
$\left[\begin{array}{c}{x}_c\\ y_c\\ z_c\\ w_c\end{array}\right]=\left[\begin{array}{cccc}\frac{2n}{r-l}& 0& \frac{r+l}{r-l}& 0\\ 0& \frac{-2n}{t-b}& -\frac{t+b}{t-b}& 0\\ 0& 0& -\frac{2nf}{2(f-n)}& -\frac{nf}{f-n}\\ 0& 0& -1& 0\end{array}\right]\left[\begin{array}{c}{x}_v\\ x_v\\ z_v\\ 1\end{array}\right]$

This leads to the following equations for the view space to homogeneous clip space conversion:
$x_c=x_v\left(\frac{2n}{r-l}\right)+x_v\left(\frac{r+l}{r-l}\right)$

$y_c=y_v\left(\frac{-2n}{t-b}\right)+y_v\left(\frac{t+b}{t-b}\right)$

$z_c=z_v\left(-\left(\frac{f+n}{2(f-n)}\right)-\frac{1}{2}\right)-\frac{nf}{f-n}$

$w_c=-z_v$

After the conversion to Euclidean NDC space the equation for $z_c$ becomes:
$z_d=\frac{z_v}{-z_v}\left(-\left(\frac{f+n}{2(f-n)}\right)-\frac{1}{2}\right)-\frac{1}{-z_v}\frac{nf}{f-n}$
or
$z_d=\frac{f+n}{2(f-n)}+\frac{1}{2}+\frac{1}{z_v}\frac{nf}{f-n}$

Then after applying the conversion to window coordinates the transformation remains the same (if you have your minDepth and maxDepth to 0 and 1 respectively):
$z_f=\frac{f+n}{2(f-n)}+\frac{1}{2}+\frac{1}{z_v}\frac{nf}{f-n}$

Rearranging for $z_v$ this becomes:
$z_v={\left(\raisebox{1ex}{$z_f-\left(\frac{f+n}{2(f-n)}+\frac{1}{2}\right)$}\!\left/ \!\raisebox{-1ex}{$\frac{nf}{f-n}$}\right.\right)}^{-1}$

The terms $\frac{f+n}{2(f-n)}+\frac{1}{2}$ and $\frac{nf}{f-n}$ are both constant within a shader with fixed projection matrix allowing us to simplify the equation to be resolved in the shader by taking $a=\frac{nf}{f-n}$ and $b=\frac{f+n}{2(f-n)}+\frac{1}{2}$ to give $z_v=\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$(z_f-b)$}\right.$

Matthew Wellings - Blog

Depth Peeling Pseudo-Volumetric Rendering 25-Sept-2016
Depth Peeling Order Independent Transparency in Vulkan 27-Jul-2016
The new Vulkan Coordinate System 20-Mar-2016
Improving VR Video Quality with Alternative Projections 10-Feb-2016
Playing VR Videos in Cardboard Apps 6-Feb-2016
Creating VR Video Trailers for Cardboard games 2-Feb-2016
Playing Stereo 3D Video in Cardboard Apps 19-Jan-2015
Adding Ray Traced Explosions to the Bullet Physics Engine 8-Oct-2015