Color Ellipsoid Framework

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.

Brief Description

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.

  • Figure 1 - Color Ellipsoids in Foggy Natural Scene

    (a)-(c) Images of tree branch in fog at 3 different distances with sample windows overlaid on the same branch in each image. (d) RGB Histogram of each sample set.

Centroid Prior

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.

  • Figure 2 - Color Ellipsoids in Foggy Natural Scene

    (a) Far left column contains the original foggy images from [2], ˜x. The columns from left to right in the odd rows are the defogged images and even rows are the transmission maps using Lemma 2 created in (xC; tC), [3] (xD; tD), [1] (xF; tF), [2] (xT ; tT ), [4] (xM; tM) [5], respectively.

Ellipsoid Prior

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.

Fireman Image Set

First Row: Original, DCP, EP
Second Row: Transmission of DCP, EP

City Image Set

First Row: Original, DCP, EP
Second Row: Transmission of DCP, EP

Foggy Trees Image Set

First Row: Original, DCP, EP
Second Row: Transmission of DCP, EP


[1] Kaiming He; Jian Sun; Xiaoou Tang; , Single image haze removal using dark channel prior, CVPR 2009, pp.1956-1963
[2] Fattal R: Single Image Dehazing. In ACM Transactions on Graphics, Volume 27 2008:1.
[3] Gibson K, Vo D, Nguyen T: An Analysis of Single Image Defogging Methods using a Color Ellipsoid Framework. EURASIP, In Submission 2012.
[4] Tarel JP, Hautiere N: Fast Visibility Restoration from a Single Color or Gray Level Image. In ICCV, Kyoto, Japan 2009:2201{220
[5] Gibson, K.B.; Vo, D.T.; Nguyen, T.Q.; , An Investigation of Dehazing Effects on Image and Video Coding, Image Processing, IEEE Transactions on , vol.21, no.2, pp.662-673, Feb. 2012