Visibility enhancement using an image filtering approach

Visibility is the ability to see through air, irrelevant to the sunlight or the moonlight. Clear clean air has a better visibility than air polluted with dust particles or water droplets. There are a number of factors affecting visibility, including precipitation, fog, mist, haze, smoke, and in coastal areas sea spray, and they are generally composed principally of water droplets or the particles whose size cannot be ignored for the wavelength. The difference between fog, mist, and haze can be quantified as the visibility distance. Visibility degradation is caused by the absorption and scattering of light by particles and gases in the atmosphere. Scattering by particulate, impairs visibility more severe than absorption. Visibility is mainly reduced by scattering from particles between an observer and a distant object. Particles scatter light from the sun and the rest of the sky through the line of sight of the observer, thereby decreasing the contrast between the object and the background sky.

Images or videos acquired are often affected by visibility in surveillance, traffic, and remote sensing systems, due to light scattering and absorbtion by the atmospheric particles and water droplets. For the rest of the paper, the the atmospheric particles and water droplets from the mist, haze, fog, smog, and cloud are not distinguished for convenience. Visibility enhancement methods of degraded outdoor images fall into two main categories. The first category is non-model-based methods, such as histogram equalization[1, 2], Retinex theory[3], and Wavelet transform[4]. However, the shortcomings of these methods are that they have less effectiveness on maintaining the color fidelity, and also seriously affect the clear region. The second category is model-based methods, who can achieve better results by modeling from the scattering and absorbtion, but usually need additional assumptions of imaging environment or imaging system, such as scene depth[5–7] or multiple images[8–11]. Nonetheless, when their assumptions are not accurate, the effectiveness is greatly compromised. Consequently, visibility enhancement using a single image for haze removal has become a key focus of ongoing studies in image restoration in recent years. Very recently, Sun et al.[12] provided a de-fogging method based on Poisson matting which uses the boosting brush to tune the transmission coefficient in the selected region to generate the fogless image by solving Poisson equations, which combines the local operations including channel selection and local filtering. In this respect, the least square filters[13, 14] and the patch-based method[15] attract great attention in visibility enhancement. But this de-fogging method needs the user to manually adjust the local scattering coefficient and scene depth. During the subsequent years,the four approaches in[16–21] were proposed for single image dehazing without the need of any extra information. The algorithm in[16] is based on color information under the assumption that the surface shading and transmission functions are locally uncorrelated. This method can only be applied to color images. Those algorithms in[17, 18, 20] can be applied to both gray and color images, but they are computationally intensive. The results of the algorithm in[17] depend on the scene saturation, and both of the algorithms in[18, 20] introduce dark channel prior to remove haze, but sometimes have very serious color distortion and poor results. Then, Tarel et al.[19, 21] proposed the visibility restoration method with low complexity for gray and color images, which adopted white balance, gamma correction, and tone mapping to maintain color fidelity. But, the results of some degraded images processed by the algorithm in[19, 21] still show high level residual haze.

Inspired by a multilayer image formed from several types of measurement on the same detection area covered by a single pixel[11, 22], the hazy image can be seen as a combined one in the presence of a multilayer structure consisting of both the clean attenuated component image and the haze layer, which mainly result from the medium absorbtion and atmospheric scattering, respectively. Since the atmospheric veil almost has no specific edges or textures, we may adopt image filtering approach to estimate the haze layer from the hazy image. In this article, different from the existing visibility restoration methods in the previous studies, with the help of dimension reduction technique, the proposed method utilizes the image filtering approach consisting of the median filter and the truncated singular value decomposition to estimate atmospheric veil with dark channel prior to restore the haze-free image. And a comparison of the proposed approach with the state-of-the-arts is also presented.

The rest of this article is organized as follows. In “Image degradation model” section, we briefly review the outdoor degraded bilayer image model. We adopt this model for visibility enhancement from a single hazy image in “The image filtering approach” section. The proposed scheme of the image filtering approach using low-rank technique that produces the haze layer approximation is described here. The experimental results and analysis of the comparison between developed approach and the state-of-the-arts are shown in “Results and analysis” section, and the conclusion and future work are given at the last section.

where x denotes pixel position, I∈ C^M×Nis an observed image, J is scene radiance, A is global atmosphere light, and t is medium transmission. J(x)t(x) is direct attenuation term, representing the scene radiation decay effect in the medium; A(1−t(x)) is airlight term, describing the light scattering from atmosphere particles inducing color distortion.

As a consequence, instead of estimating the medium transmission coefficient t, the haze removal algorithm can be decomposed into several steps: inference of B(x) from I(x), estimation of A, and derivation of J(x) after inverting Equation (3).

where Ω denotes the local neighborhood at each pixel.

It is observed that the haze layer is smooth, while retaining the depth changes. To remove the redundant image texture affecting the haze layer, we first explain how the low-rank technique can be applied to the coarse haze layer F(x) to extract the corrected haze layer. Let F _i be the i th sample in F and F be a column stacked representation of F(x), i.e., F is a matrix of size MN × L, whose each row contains the$\sqrt{L}\times \sqrt{L}$ patch around the location of F _i in the image F.

where U is the matrix of eigenvectors derived from${\bar{\mathbf{f}}}^T\bar{\mathbf{f}}$,$\mathrm{\Sigma }=\text{diag}\left\{{\sigma }_1,\begin{array}{l}{\sigma }_2,\dots \end{array},{\sigma }_r\right\},$$r=\text{rank}\left({\bar{\mathbf{f}}}^T\bar{\mathbf{f}}\right)$, and its diagonal singular values σ₁ ≥ σ₂ ≥ ⋯ ≥ σ _r > 0.

where σ _p and α _p are the singular values of the raw image$\bar{\mathbf{f}}$ and the simulated data, respectively. The intuition is that α _p is a threshold for σ _p below which the p th component is judged to have occurred due to chance. Currently, it is recommended to use the singular values that corresponds to a given percentile, such as the 95th of the distribution of singular values derived from the random data. From the diagonal singular values, K(K ≪ r) principal components is truncated as σ _i (i = 1,2,…,K).

where N _x is the number of a pixel used in patch stacks for the whole image; and the variables p,L are defined as above.

where the parameter q ∈ (0,1).

Through substituting Equation (14) and (15) in Equation (4), the final haze-free image J(x) after contrast enhancement is acquired from a single hazy image.

We analyze and compare the experimental results in visual effects, speed, and objective evaluation criteria. Though comparing the results, we demonstrate the advantage and disadvantage of these methods. We have proposed simple but powerful algorithm based on median filtering using low-rank technique for visibility enhancement from a single hazy image. Since the computational complexity of the low-rank technique is low, it is shown that the proposed approach for haze removal is fast, and can even achieve better results than the state-of-the-art methods in a single image dehazing.

However, the proposed approach maybe not work well for the far scenes with heavy fog and great depth jump. The restored image has the halos or residual haze at depth discontinuities that can be observed in these experimental results. And another shortcoming is unable to obtain the actual value of global atmosphere light A. To overcome these constraints of our current method, we intend to incorporate better edge-preserving image filtering method with low complexity and other techniques. This is our future research.