Image-Based Rendering - The Magic of Computer Graphics - page 157

4D Fourier transform

Virtual refocusing in real space
(change of basis + integral projection)

Virtual refocusing in Fourier space
(change of basis + slicing)

Inverse 2D Fourier
transform
Figure 5.26 Diagram of Fourier slice photography. Taking the 4D Fourier transform of the light field,
transforming in the Fourier domain (a change of basis), taking a 2D slice, and transforming
back is equivalent to a virtual refocus in the real domain. (After [Ng et al. 05] c 2005

ACM, Inc. Included here by permission.)

phy” paper allows for theoretical justification of some of the experimental results
in the “Light Field Photography” paper. In particular, it set limits on the avail-
able range of exact refocusing and proved that the sharpness increases linearly
with the directional sampling rate. The analysis also shows that “the digitally re-
focused photograph is a 2D (low-pass) filtered version of the exact photograph”
which gives a proof for the observation that refocused photograph is less noisy.
In summary, the paper answered the question of what quality could be expected
from refocused images of acquired light fields.
The Fourier slice photography theorem leads to a more efficient digital refo-
cusing algorithm (in the sense of being faster and less dependent on the sampling
rate of the light field) called Fourier slice digital refocusing . As illustrated in Fig-
ure 5.26, the Fourier transform of a 4D light field is computed first, and then the
operations for refocusing (application of the shear and the 2D slice) are performed
in the Fourier domain. The inverse transform of the result is the final image. An-
other feature of this approach is that the Fourier transform of the 4D light field,
which is the costliest part, only has to be computed once for as many different
refocuses as desired.
Figure 5.27 contains images of the digital refocusing process using the Fourier
transform. The first row shows the light field as captured by the light field
camera. Closeups are included at the right. The second row shows the Fourier
transform of the light field, arranged into a 2D image similar to the way the light
field is stored in the 2D light field image. The third row contains slices of the 4D
transform data, after the shear transformation is applied for each of three refocus-
ing arrangements; from left to right 1. (Note that the Fourier

transform images do not correspond very well to the actual images, although
they are interesting to look at.) The bottom two rows show refocused images. The
images in the second lowest row are the inverse transforms of the 2D
Fourier transform slices shown in the images above. The images in the bottom

α > 1, α = 1, α <