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Light Field Duality: Concept and Applications: Difference between revisions
Light Field Duality: Concept and Applications (view source)
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Authors: George Chen, Li Hong, Kim Ng, Peter McGuinness, Christian Hofsetz, Yang Liu, Nelson Max | Authors: George Chen, Li Hong, Kim Ng, Peter McGuinness, Christian Hofsetz, Yang Liu, Nelson Max | ||
Affiliations: STMicroelectronics, UC Davis | Affiliations: STMicroelectronics, UC Davis | ||
* [https://dl.acm.org/doi/pdf/10.1145/585740.585743 Paper (escholorship mirror)] | |||
==Background== | |||
A light field is the set of all rays in a 2D space. It is a 5D function of <math>(x,y,z)</math> position and <math>(\theta, \phi)</math> angle. | |||
However for practical scenes, you can assume that the light along a ray is consistent through empty spaces (air). | |||
Thus it can be represented in 4D as either a position <math>(u, v)</math> on a fixed plane and an angle <math>(\theta, \phi)</math> or as two positions <math>(u, v)</math> and <math>(s, t)</math> on two nearby parallel planes. | |||
See [[Light field]]. | |||
==Dual Representation== | ==Dual Representation== | ||
[[File:Light field duality fig1.png|thumb|400px]] | |||
The paper forms the dual representation between points in 3D world coordinates and hyperlines in the 2-plane 4D parameterization of light fields. | The paper forms the dual representation between points in 3D world coordinates and hyperlines in the 2-plane 4D parameterization of light fields. | ||
By hyperline, they mean a 2D manifold in 4D space that is the intersection of two 4D hyperplanes. | By hyperline, they mean a 2D manifold in 4D space that is the intersection of two 4D hyperplanes. | ||
The duality is defined as follows: | |||
Given a 3D point <math>P=(x,y,z)</math>, any ray <math>(s,t,u,v)</math> going through P would have: | |||
<math> | |||
\frac{s-x}{u-x} = \frac{t-y}{v-y} = \frac{z-z_{st}}{z-z_{uv}} | |||
</math> | |||
where <math>z_{st}</math>, <math>z_{uv}</math> are fixed values. | |||
See the figure to the right. | |||
With some algebra, this constraint can be written as either: | |||
<math> | |||
\begin{bmatrix} | |||
z_{uv}-z_{st} & 0 & s-u\\ | |||
0 & z_{uv} - z_{st} & t-v | |||
\end{bmatrix} | |||
\begin{bmatrix} | |||
x \\ y \\ z | |||
\end{bmatrix} | |||
= | |||
\begin{bmatrix} | |||
sz_{uv} - uz_{st}\\ | |||
tz_{uv} - vz_{st}\\ | |||
\end{bmatrix} | |||
</math> | |||
<math> | |||
\begin{bmatrix} | |||
z-z_{uv} & 0 & z_{st}-z & 0\\ | |||
0 & z-z_{uv} & 0 & z_{st} - z | |||
\end{bmatrix} | |||
\begin{bmatrix}s \\ t \\ u \\ v \end{bmatrix} | |||
= | |||
\begin{bmatrix} | |||
x(z_{st}-z_{uv})\\ | |||
y(z_{st}-z_{uv}) | |||
\end{bmatrix} | |||
</math> | |||
Given a point, this constraint defines the hyperline in the 4D light field space. | |||
The paper has the following remarks: | |||
* Points in one space are lines in the other. | |||
* (?) A bundle of lines in word space correspond to a set of hyperlinear points in the light field. | |||
* (?) A set of colinear points in world space corresponds to a bundle of hyperlines in the light field. | |||
In the light field space, you get: | |||
* Camera hyperlines (CHL) which correspond to camera points in world space. Hyperpoints along the hyperline have heterogenous colors since they represent images. | |||
* Geometry hyperlines (GHL) which correspond to points in the scene (on surfaces) in world space. Hyperpoints along these hyperlines have homogeneous colors assuming no view dependent effects. | |||
==CHL Light Field Rendering== | |||
==GHL Light Field Rendering== | |||