A Novel Total Generalized Variation Model for Image Dehazing

Abstract

In this paper, we propose a new variational model for removing haze from a single input image. The proposed model combines two total generalized variation (TGV) regularizations, which are related to the image intensity and the transmission map, respectively, to build an optimization problem. Actually, TGV functionals are more appropriate for describing a natural color image and its transmission map with slanted plane. By minimizing the energy functional with double-TGV regularizations, we obtain the final haze-free image and the refined transmission map simultaneously instead of the general two-step framework. The existence and uniqueness of solutions to the proposed variational model are further obtained. Moreover, the variational model can be solved in a unified way by realizing a primal–dual method for associated saddle-point problems. A number of experimental results on natural hazy images are presented to demonstrate our superior performance, in comparison with some state-of-the-art methods in terms of the subjective and objective visual quality assessments. Compared with the total variation-based models, the proposed model can generate a haze-free image with less staircasing artifacts in the slanted plane and more details in the remote scene of an input image.

Appendix: Image Quality Assessment

The criteria of objective image quality assessment we used are described in detail in [17].

Yu et al. [18] presented an image visibility measurement method based on the visible edge segmentation. The IVM is defined as

$$\begin{aligned} \mathrm{IVM}=\frac{n_r}{n_{\mathrm{total}}} \log \sum _{x\in A}C(x). \end{aligned}$$

where $n_r$ is the number of visible edges, $n_{\mathrm{total}}$ is the number of edges, C(x) is the mean contrast and A denotes the image area of visible edges.

Jobson et al. [19] proposed a visual contrast measure (VCM) to quantify the degree of visibility of the image which is calculated as follows:

$$\begin{aligned} \mathrm{VCM}=100*\frac{R_v}{R_\mathrm{t}}, \end{aligned}$$

where $R_v$ is the number of local areas, the standard deviation of which is larger than the given threshold and $R_\mathrm{t}$ is the total number of local areas. The VCM uses the local standard deviation which denotes the contrast of the image to measure the visibility. In general, the higher the VCM, the clearer the enhanced image.

The contrast of a clear image is usually much higher than that of a hazy image, so image contrast can be used to compare different dehazing algorithms. The higher the contrast of the enhanced image, the better the dehazing algorithm. Ma and Wen [20] used the image global contrast to compare the performance of different dehazing algorithms. Tripathi et al. [21] used contrast gain to compare different dehazing algorithms. Contrast gain denotes the mean contrast difference between the enhanced image and original hazy image and is calculated by

$$\begin{aligned} C_{\mathrm{gain}}={\bar{C}}_J-{\bar{C}}_I, \end{aligned}$$

where ${\bar{C}}_J$ and ${\bar{C}}_I$ represent the mean contrast of the enhanced image and hazy image, respectively. And C is the local contrast of the image in a small window and is calculated by

$$\begin{aligned} C(x,y)=\frac{S(x,y)}{m(x,y)}, \end{aligned}$$

where

$$\begin{aligned} S(x,y)= & {} \frac{1}{(2r+1)^2}\sum _j\sum _i(I(x+i,y+j)-m(x,y))^2,\\ m(x,y)= & {} \frac{1}{(2r+1)^2}\sum _j\sum _i I(x+i,y+j), \end{aligned}$$

and r is the radius of the local area. The larger the contrast gain, the better the result of the dehazing algorithm.

Yu et al. thought that a good dehazing algorithm should allow the original hazy image and enhanced image to have similar histogram distributions. They used the histogram correlation coefficient (HCC) of the two color images as a criterion to assess the performance of color restoration.

Wu and Zhu used the image structural similarity (SSIM) and universal quality index (UQI) to assess the performance of the structural similarity between the original hazy image and the enhanced image.