Image restoration has been a significant problem in the fields of vehicle conjunction monitoring on the road [1], airborne photography [2], army applications [3], and related purposes [4]. Imaging in adverse conditions has a major impact on the restoration process and has become a challenge for various applications [5]. Unfavorable atmospheric conditions create an environment where varying density particles, such as haze and smog, are present [6]. The ambient light is typically scattered differently by these dissolved particles, which causes images taken by the camera to be distorted at that moment [7]. Low brightness and poor contrast are issues with these photos, which have a big impact on various activities of image processing applications like segmenting images [8], identifying targets [9], tracing objects of desire, etc. [10,11]. Furthermore, since these foreign particles generate a varying scattering environment inside a single image, degraded photos have different haze densities [12]. Therefore, recovering a perfect image from a murky one is a challenging task. Defogging the distorted images for use in computer vision applications requires an effective haze removal algorithm that considers the variation of scattering.

The atmospheric scattering model described in the literature serves as the foundation for existing single-image dehazing methods. This method produces better restoration results by estimating depth, transmission, and ambient light. The formation of an unclear image is explained by the atmospheric scattering model in the following way: (1)If(x)=Rd(x)tr(x)+A(1−tr(x))where I_f(x) is the intensity of input at position x, R_d(x) is the glow of the reinstated image, t_r(x) is the transmission map, and A is atmospheric light. Additionally, transmissivity is determined as: (2)tr(x)=e−βd(x)where β and d(x) are the scattering coefficient and scene depth, respectively. Therefore, two parameters, β and d(x), are the only ones used in the calculation of the transmission map [5,9]. Over the previous two decades, numerous researchers [13–17], have presented their works on the calculation of depth d(x). It explains how the haze changes linearly in the image itself. The bottom and top sides of a 2-D projection of a 3-D scene determine the image’s objects, both close and far away, respectively [18–21]. Similarly, the top side of any hazy image will always have more fog than the bottom since fog density is always higher for distant things. For precise transmission map calculations, a rectilinear depth model may be helpful because the amount of fog grows linearly with depth [22]. In the previous decade, numerous scholars created a linear model by bearing these factors in mind, as is explained in the succeeding paragraph. Nayar et al. [13] have given a method in which, by comparing two images taken under various atmospheric situations, it was possible to determine the scene depth. Additionally, to acquire the scene depth required by the transmission function, Oakley et al. [23] explained the technique for minimizing this degradation in conditions where the scene geometry is known. Afterward, Narasimhan et al. [24] offered a technique to locate the sky zone manually in an image to determine the scene depth. The aforementioned techniques, meanwhile, are frequently ineffective in actual practice. For dehazing an image, He et al. [25] developed the dark channel prior method. Due to its simplicity and efficacy compared to the prior methods, this method attracted a lot of attention. Further, in order to recover the densely hazy images, Zhu et al. [15] have given a technique that is based on color attenuation, w.r.t. the difference between saturation and intensity value, the prior color attenuation characterizes the fog changes. Real-time defogging for surveillance applications is made possible by using color channel prior knowledge, which enables the rapid computation of the transmission map for a piece of pixel in the unique image. Raikwar et al. [16] introduced an enhanced linear depth system that integrates a hue factor into the depth calculation formula, which is dependent on the variation between saturation and the combination of hue and intensity. Assuming homogeneous scattering in the local area, it has been discovered that the depth maps created up to this point compute the transmissivity with constant values of the scattering coefficient. For calculating the transmission map, the scattering coefficient plays a role that is nearly as important as that of the depth map, as variations in the scattering medium lead to changes in haze thickness. Therefore, it is not appropriate to take the value of β as constant over the entire image. Selecting various values of β, such as β > 1, β = 1, and β < 1, will generally allow us to see how β affects the restored image, and that effect can be easily observed in Fig. 1. Fig. 1a shows two hazy input images of different locations, though the restored images are shown in Fig. 1b using β > 1. Through Fig. 1b, it is clear that the fog in the background has been removed, causing the foreground to appear darker or disturbed (as indicated by the red line in the figure). Contrarily, for β < 1, the fog has a minimal defogging effect in the background and is primarily removed from the foreground of photographs, as highlighted by the red line in Fig. 1d. Consequently, as can be seen in Fig. 1c, β = 1 produced uniform defogging in the backgrounds and foregrounds. As far as we know, the variable scattering concept has not yet been taken into account by any de-fogging algorithms. To significantly improve defogging techniques, it is necessary to develop a scattering model that incorporates changeable values of β. With this information in mind, this article presents a model created using the linear regression technique, which accounts for variations in β based on the fog density in the image. An improved transmission map that serves the intended restoration objective is obtained using the proposed approach. In contrast, the depth map is calculated using the color attenuation prior method from [15,16], the atmospheric light is approximated employing the technique from [25], and the computation of R_d(x) is performed using (1). The rest of the article is structured as follows: Section 2 discusses the color attenuation prior-based restoration approach. In Section 3 suggested technique is described for calculating β. The experimental evaluation is discussed in Section 4, and lastly, Section 5 provides the concluding remarks.

The algorithm for the restoration of a foggy image is created using the atmospheric scattering model that is given in Eqs. (1) and (2). According to (1), the image restitution process involves four key factors: scene depth, transmissivity, atmospheric light, and picture radiance retrieval. In the next subsections, these four elements are briefly discussed.

As discussed in previous sections, the transmission map can be enhanced by developing it with a variable scattering coefficient β(x) for better recovery of the hazy pictures. Consequently, this section introduces a pattern-based linear model for calculating the scattering coefficients of the input foggy image. In any given hazy image, the fog density generally increases from the bottom to the top. Fog is primarily caused by foreign particles, such as moisture, dust, smog, and murk, suspended in the air. The scattering of these particles is affected by their properties, such as dimension, dispersion, or positioning, which are never entirely uniform. The varying particle sizes result in a variable-density scattering medium in the atmosphere, complicating the prediction of a variable scattering coefficient model. Moreover, several linear models have been developed in the literature [15,16,25] for determining depth, considering that fog density varies linearly with depth. In a similar way, and taking the same factors into account, the scattering map can also be produced.

In general, three separate information planes, like HSV or RGB, are used to model color images. The scattering map is created using information from any foggy image’s HSV model. Several studies have been conducted on various foggy images to explore the potential of finding a relationship between scattering and the values of HSV planes. In essence, the tests are based on random procedures carried out on the HSV values of chosen images that are blurry. Experimental analysis statistics reveal that the ratio between the difference and the sum of saturation and brightness values is directly linked to the distribution of the scattering medium. (6)β(x)∝v(x)−s(x)v(x)+s(x)

A linear model for the variable scattering coefficient is suggested, as presented in (7), based on an approximation of (6), which indicates the concentration of scattering particles. (7)β(x)=c1+c2η(x)+c3χ(x)+ε(x)where η(x)=v(x)v(x)+s(x) and χ(x)=s(x)v(x)+s(x) are the normalization factors for the brightness and saturation of a hazy picture. Here, c₁, c₂, and c₃ are the unidentified linear constants, and ε(x) represents an arbitrary error in the scattering model, characterized as an arbitrary picture with zero mean and variance σ2 (i.e., ε(x)∼N(0, σ2)).

To reduce the square of error, the optimal values of linear coefficients are computed as follows: (8)(ε(x))2=(β(x)−c1−c2η(x)−c3χ(x))2≅0

The mathematical method of calculating linear coefficients, known as regression analysis, is predicated on the idea that the sum of the squared errors from the n samples has to be kept to a minimum. For conducting the regression examination, the linear method (7) can be rewritten in its globalized system as: (9)βi(x)=c1+c2ηi(x)+c3χi(x)+εi(x)

Likewise, stochastic errors might be globalized as follows: (10)εi(x)=βi(x)−c1−c2ηi(x)−c3χi(x)

Afterward, squares of errors are calculated and combined, which is given as: (11)E=ε12(x)+ε22(x)+ε32(x)+…+εn2(x)or (12)E=∑i=1n(εi(x))2=∑i=1n[βi(x)−(c1+c2ηi(x)+c3χi(x))]2

Differentiate (12) partially about c₁, c₂, and c₃, and equate it to zero to reduce the value of error E. (13)∂E∂c1=0=−2[β1(x)−(c1+c2η1(x)+c3χ1(x))]−2[β2(x)−(c1+c2η2(x)+c3χ2(x))]−…−2[βn(x)−(c1+c2ηn(x)+c3χn(x))] (14)∂E∂c2=0=−2η1(x)[β1(x)−(c1+c2η1(x)+c3χ1(x))]−2η2(x)[β2(x)−(c1+c2η2(x)+c3χ2(x))]−…−2ηn(x)[βn(x)−(c1+c2ηn(x)+c3χn(x))] (15)∂E∂c3=0=−2χ1(x)[β1(x)−(c1+c2η1(x)+c3χ1(x))]−2χ2(x)[β2(x)−(c1+c2η2(x)+c3χ2(x))]−…−2χn(x)[βn(x)−(c1+c2ηn(x)+c3χn(x))]

Eqs. (13)–(15) are simplified as: (16)∑i=1nβi(x)=nc1+c2∑i=1nηi(x)−c3∑i=1nχi(x) (17)∑i=1nβi(x)×ηi(x)=c1∑i=1nηi(x)+c2∑i=1nηi2(x)−c3∑i=1nχi(x)×ηi(x) (18)∑i=1nβi(x)×χi(x)=c1∑i=1nχi(x)+c2∑i=1nηi(x)×χi(x)−c3∑i=1nχi2(x)

To determine the optimal values for scattering coefficients c₁, c₂, and c₃, above mentioned equation could be used. However, solving these equations requires the values of the variables ∑i=1nβi(x),∑i=1nηi(x)and∑i=1nχi(x). Hence, these variables could be derived using a dataset which have hazy pictures along with ground truth pictures. This dataset can be generated using 500 clear pictures taken from the Google Images dataset. After that, hazy pictures are developed using the model given in (1), which produces the i-th hazy picture, I_f(x) for each recovered picture R_d(x). Once I_f(x) is obtained for each picture, the hue, saturation, and brightness components are extracted. Subsequently, the necessary matrices for these three variables are computed. The hazy picture developing model (1) is utilized to create the i-th hazy picture, I_f(x), for each lucid picture R_d(x). Afterward, obtaining I_f(x) for each picture, the hue, saturation, and brightness components are separated. Subsequently, the necessary matrices for the three variables are computed. The three matrices previously mentioned are used to create the sample space for fitting the curve of (12) using the least squares regression approach, as explained in (16)–(18). This allows for the determination of the values of the linear coefficients. Algorithm 1 can be utilized to compute the required values of variables c₁, c₂, and c₃ as delineated below:

The linear coefficient values obtained using Algorithm 1 are c₁ = 0.8612, c₂ = 0.8059, and c₃ = −0.65525. To generate a transmission map, the scattering map is modeled using both the estimated linear coefficients and (7). As a result, the redeveloped form of the transmission map could be expressed as follows: (19)tr(x)=exp⁡(−β(x)×d0(x)) (20)tr(x)=exp⁡(−(c1×d0(x)+c2η(x)×d0(x)+c3χ(x)×d0(x)+ε(x)×d0(x)))

Additionally, the outcomes of current approaches have improved with the help of (20), and the efficiency of the suggested model is evaluated in the following section.

To compare the performance of the suggested technique with existing methods, both qualitative and quantitative analyses were conducted in MATLAB 19 on a 64-bit Core i7 Processor with 6 GB RAM [4,15,16,25]. The test images were selected from available datasets, including Frida2 [26] and the Waterloo IVC Dehazed Image Database [27]. The dataset included real-world foggy photos captured at various times of the day, morning, afternoon, and night.

Removing fog from a single image presents a significant challenge for most consumer and computer vision applications. The effectiveness of the fog removal process depends on the accuracy of estimating the transmission map, which in turn relies on precise scene depth assessment and a constant scattering coefficient, β. Furthermore, the current technique only aims to increase scene depth while maintaining a constant β. However, the haze removal method might perform more effectively with a transmission map based on variable scattering. As a result, an updated transmission map model that accounts for fluctuations in scattering information in foggy images is proposed. For effective image restoration, this updated transmission map is generated using the proposed scattering model combined with scene depth from existing approaches. Experimental findings show that the suggested model outperforms current fog removal methods in both qualitative and quantitative analyses. Moreover, the results suggest that the revised transmission map addresses established issues such as edge preservation and chromatic constancy.

A significant limitation of the proposed work is the increased computational complexity required to compute β(x), which scales with n×m compared to using a constant β. Additionally, it raises hardware complexity when implemented in hardware. However, these limitations are acceptable given the substantial improvement in image restoration quality.