Foggy images taken in the bad weather inevitably suffer from contrast loss and color distortion. Existing defogging methods merely resort to digging out an accurate scene transmission in ignorance of their unpleasing distortion and high complexity. Different from previous works, we propose a simple but powerful method based on histogram equalization and the physical degradation model. By revising two constraints in a variational histogram equalization framework, the intensity component of a fog-free image can be estimated in HSI color space, since the airlight is inferred through a color attenuation prior in advance. To cut down the time consumption, a general variation filter is proposed to obtain a numerical solution from the revised framework. After getting the estimated intensity component, it is easy to infer the saturation component from the physical degradation model in saturation channel. Accordingly, the fog-free image can be restored with the estimated intensity and saturation components. In the end, the proposed method is tested on several foggy images and assessed by two no-reference indexes. Experimental results reveal that our method is relatively superior to three groups of relevant and state-of-the-art defogging methods.

To perceive natural scenes from captured images means a lot to the computer vision applications such as image retrieval, video analysis, object recognition, and car navigation. In the foggy weather, the visual quality of images is impaired by the distribution of atmospheric particles, resulting in a loss of image contrast and color fidelity. This kind of degradation undoubtedly makes a great discount on the effectiveness of those applications. Therefore, the technique of removing foggy image degradation, namely, “image defogging,” has attracted much attention in the image processing field [

Since the fog degradation is largely dependent on the distant from the object to the camera, many existing approaches rely on the physical degradation model [

Traditional image enhancement algorithms like intensity transformation functions and histogram equalization which do not have to evaluate the scene transmission will not suffer from those problems mentioned above. However, they are likely to produce distorted results, due to their ignorance of the physical degradation mechanism. Nowadays, people change their minds by combining image enhancement algorithms with the physical degradation model. In 2014, Arigela and Asari design a sine nonlinear function to refine a rough scene transmission [

Here, we propose an improved variational histogram equalization framework for single color image defogging. Similar to the previously presented methods in [

The remainder of this paper is organized as follows. In the next section, a physical degradation model is expressed in HSI color space and our strategy for color image defogging is then illustrated in brief; in Section

Generally, because of suspended particles’ absorption and scattering in the foggy weather, the scene-reflected light

Histogram equalization is one of the most representative methods in the image enhancement field, but the original one is limited, since a mean brightness constraint is not considered. In 2007, Jafar and Ying proposed a constrained variational framework of histogram equalization for image contrast enhancement [

The flowchart of our proposed algorithm.

As is well known, a foggy image possesses a high mean brightness, so the mean brightness constraint in the framework should be improved through the physical degradation model. Due to

Comparison of mean local values in both rough and refined

First emerging in image restoration field, TV and

Here, we suppose

First of all, we would like to decompose the divergence term into two orthotropic components along the level set curve, as is shown in the following expression:

Based on those two rules of pointed diffuse behavior mentioned above, a satisfactory function is designed and turns out to be

It is easy to examine whether

Apparently, the function

Combined with a mean brightness constraint in formula (

From formula (

As to the solution of the proposed framework, we can learn from Wang’s algorithm. According to the alternating direction method of multipliers (ADMM) [

With formula (

If formula (

Now, it is available to describe the expression of

Therefore, a general variation filter is formed to get a numerical solution from the energy functional framework precisely and promptly. In particular,

To recover the fog-free scene without yielding color shifting, the airlight is another important factor which is often neglected. It is simply inferred by selecting the brightest pixel of the entire image in [

To recover the local airlight, the first step aims at measuring the fog density. We introduce a color attenuation prior [

Because

Since there is one-to-one correspondence between

According to formula (

In order to perform a qualitative and quantitative analysis of the proposed method, we do some simulation experiments on color foggy images in comparison with three pairs of state-of-the-art defogging approaches. The first pair is Ranota and Kaur’s [

Before the comparison, we ought to inform the experimental condition and the parameter selection. All the mentioned approaches are carried out in the MATLAB R2014a environment on a 3.5 GHz computer with 4 GB RAM. On the simulation platform, the parameters utilized in our method are set to be

Synthetic images named as L08005 and L08010: in the columns, foggy images and ground-truth images.

Firstly, we might as well fix

Results of our method initialized by different

Image quality of defogging results on L08005 and L08010 images is assessed by (a) AMBE index and (b) MSE index, respectively.

Secondly,

Results of our method initialized by different

Image quality of defogging results on L08005 and L08010 images is assessed by (a) EI index and (b) MSE index, respectively.

Since our method is based on a variational histogram equalization framework, it is acceptable to be compared with Ranota and Kaur’s [

Comparison of results on house image obtained by Ranota’s method, Wang’s method, and ours. In the columns, an original foggy image, fog-removal results processed by Ranota’s method (up), Wang’s method (middle), and ours (down), corresponding zoom-in view of red or blue boxes.

Comparison of results on trees image obtained by Ranota’s method, Wang’s method, and ours. In the columns, an original foggy image, fog-removal results processed by Ranota’s method (up), Wang’s method (middle), and ours (down), corresponding zoom-in view of red or blue boxes.

Histogram equalization and intensity-transformation-based methods are similar enhancement algorithms, so we launch a comparison among Arigela’s method [

Comparison of results on street image obtained by Arigela’s method, Liu’s method, and ours. In the columns, an original foggy image, fog-removal results processed by Arigela’s method (up), Liu’s method (middle), and ours (down), corresponding zoom-in view of red or blue boxes.

Comparison of results on train image obtained by Arigela’s method, Liu’s method, and ours. In the columns, an original foggy image, fog-removal results processed by Arigela’s method (up), Liu’s method (middle), and ours (down), corresponding zoom-in view of red or blue boxes.

To make the performance of the proposed method more persuasive and convincing, it is a must to compare our method with several classical and representative ones such as He et al.’s [

Comparison of results on cliff image obtained by He’s method, Nishino’s method, and ours. In the columns, an original foggy image, fog-removal results processed by He’s method (up), Nishino’s method (middle), and ours (down), corresponding zoom-in view of red or blue boxes.

Comparison of results on tower image obtained by He’s method, Nishino’s method, and ours. In the columns, an original foggy image, fog-removal results processed by He’s method (up), Nishino’s method (middle), and ours (down), corresponding zoom-in view of red or blue boxes.

In order to strengthen the qualitative analyses mentioned above, two no-reference assessment indexes are introduced, including EI index and no-reference image quality evaluator index (NIQE) [

Image quality of defogging results on house and trees images obtained by Ranota’s method, Wang’s method, and ours is assessed by (a) EI index and (b) NIQE index.

Image quality of defogging results on street and train images obtained by Arigela’s method, Liu’s method, and ours is assessed by (a) EI index and (b) NIQE index.

Image quality of defogging results on cliff and tower images obtained by He’s method, Nishino’s method, and ours is assessed by (a) EI index and (b) NIQE index.

The time consumption needs to be taken into consideration, if the method is put into the practice. Figure

Time consumption for three groups of defogging methods including Ranota’s method, Wang’s method, Arigela’s method, Liu’s method, He’s method, Nishino’s method, and ours.

In the paper, we propose an image defogging method using a variational histogram equalization framework. A previous variational framework on image enhancement inspires us to establish a constrained energy functional that contains histogram equalization and the physical degradation model. The mean brightness constraint in the framework is revised to preserve the brightness of a fog-free image while the regularization term is redesigned for avoiding manual intervention. To pursue the processing efficiency, a general variation filter is proposed to solve the constrained framework promptly. As to another important unknown quantity

The authors declare that they have no competing interests.