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Two-Plane Parametrization
Assuming the scene is contained between two parallel planes, the camera plane and the object plane, we can parametrize each ray by a point on the camera plane and a point on the object plane. The radiance along that ray is given by


Light Field Generation
1. Re-projection / Rectification of input images
2. Re-binning to produce uniform sampling on camera and object plane

Rectification
Each input image was taken at a camera position with a projection matrix projecting a point to a point on the camera plane.
is a transformation from 3D homogenous coordinates to 2D homogenous coordinates. Thus,

We want to project from the same camera position onto a different projection plane given by another projection matrix which has the same layout as

Since the coordinate of is 1, we can write



We now multiply the first equation by on both sides:




Thus, we can transform projected points from the input image into projected points of the output image by applying the homography


Re-Binning
When rectifying several images from different viewpoints, we obtain an uneven sampling on the camera / object plane. We get a uniform sampling by discretizing the light field and applying the Pull-Push algorithm to provide a color value for every pixel.

Pull-Push algorithm
Explained in 2D here. For Light Fields, we need to do this in 4D. Given: 2D grid of cells and unevenly distributed color samples which are assigned to cells .
For each cell, store a color value and a weight .

Initialization
For each cell, compute an average color and an initial weight



Pull
Average colors and sum weights onto higher levels



Push
For each cell on the base level, walk up the hierarchy until a cell with weight > 0 is found. Use the color of that cell.