The goal of this work is to explain how several single image defogging methods work using a color ellipsoid framework. The foundation of the framework is the atmospheric dichromatic model which is analogous to the reflectance dichromatic model. A key step in single image defogging is the ability to estimate relative depth. Therefore properties of the color ellipsoids are tied to depth queues within an image.

In Fig. 1, there are 3 images of the same tree on
a foggy day at 3 different distances. A sample
window *Omega 1* is located on the same tree
branch in each image. For each *Omega i*, the
densities are plotted in Fig. 3d. Note that the densities
are ellipsoidal in shape. Also, for the tree branch
positioned closer to the camera, the ellipsoid is larger
in size and positioned closer to the RGB cube origin
(*Omega 3*). For the tree branch positioned
farthest away (*Omega 1*), the ellipsoid is
smaller in size and positioned farther away from the
RGB origin.

The *Centroid Prior* was developed using a
cost function based on the CEF properties. The
solution to the cost function was in closed form
oweing to a simple approach to fog removal. Figure 2
compares this method with other single image
defogging methods.

Ellipsoid Prior
An *Ellipsoid Prior* (EP) is introduced to
demonstrate geometrically how the Dark Channel
Prior (DCP) [1] works. Below are more comparisons.
Note the DCP and EP are not refined in these
examples therefore halo artifacts exist at
occlusion edges.

First Row: Original, DCP, EP

Second Row: Transmission of DCP, EP

First Row: Original, DCP, EP

Second Row: Transmission of DCP, EP

First Row: Original, DCP, EP

Second Row: Transmission of DCP, EP

[2] Fattal R:

[3] Gibson K, Vo D, Nguyen T:

[4] Tarel JP, Hautiere N:

[5] Gibson, K.B.; Vo, D.T.; Nguyen, T.Q.; ,