Single image fog removal algorithm in spatial domain using fractional order anisotropic diffusion

Abstract

This paper presents a novel image defogging algorithm using fractional-order anisotropic diffusion equation. The proposed algorithm uses the airlight map extracted from the foggy model as the initial image in the anisotropic diffusion process. The iterative diffusion process improves this airlight map. The anisotropic diffusion process is generalized to the order of any real number between [1, 2) using the Riemann-Liouville definition of the fractional order derivatives. The formulation of the iterative process is carried out in the spatial domain to have a simple and computationally efficient implementation. Simulation results validate that the proposed algorithm is outperforming over few of the existing algorithms. The comparison study is carried out using different metrics like contrast gain, colorfulness index, contrast-to-noise ratio and visible edges ratio.

1 Introduction

Digital images captured in foggy conditions suffer from poor visibility, colors artifacts and low contrast due to the absorption and scattering of the atmospheric particles. One of the main reasons for poor visibility is the particles/water droplets in the atmosphere. These droplets reflect light which results in degradation of an image of the scene. Poor visibility is an annoying problem as it adversely affects many computer vision applications such as object detection [10, 15], image/video retrieval [9, 13, 14], and remote sensing [3, 7]. In this context, restoration of the digital images degraded due to fog will be of great importance in computer vision.

In this study, we solve this problem using a heat diffusion-based approach. In literature, several models were proposed using diffusion partial differential equation (PDE) for denoising of digital images [20, 28, 32]. In all these models, the noise is considered as the discontinuities in the depth data because of image intensity as functions of the relative depth of the object from the camera. Therefore, anisotropic diffusion acts as smooth filtering on the noisy images by preserving the edge details. The factors responsible for the degradation of an image in the presence of fog are the airlight and attenuation. The light reflected from an object is partially absorbed by the particles in the atmosphere and gets attenuated and thus reduces the contrast of the image. This process is known as attenuation. The scattered light beams play the role of an infinite number of tiny light sources floating in the atmosphere. These particles are responsible for unwanted whiteness observed in the scene, and hence, the airlight is mainly because of these light sources. In the existing literature, an airlight map is also considered a function of the distance of the scene from the camera [19]. In an outdoor environment, the depths are varied for different objects, and thus the airlight should be different for all these objects in the scene. The airlight map must be smooth except at the edges of the scene. Therefore, a similar type of anisotropic diffusion model can be used to correct airlight map degraded by the fog. Moreover, the process of removing fog from a single image is a challenging task due to its ill-posed nature. Before applying restoration process, its model must be analyzed for the well-posedness.

In recent past, several algorithms were given for restoring the fog affected images. The first category of models was based on the simple image enhancement techniques to restore the foggy images [16]. The enhancement achieved by such algorithms were found limited. Moreover, it required multiple images of the same scene under the same weather conditions. The second category of the algorithms was based on the use of polarization filters to remove the fog effect from the images [22, 23]. In this class of methods, the depth of the scene was restored using the polarization property of scattered light from multiple images captured with varying degrees of polarization. In [19], the algorithm makes use of the multiple reference images of the scene in bright weather condition. Therefore, such a method needs a particular imaging sensor to build a database of reference images under the same weather conditions. It was one of the drawbacks of these methods for not befitting in real-time applications.

The methods related to single image fog removal depend on the characteristics and textural information of the original (fog-free) image. In [25], the algorithm maximizes the local contrast of the image using Markov-Random-Field (MRF) model. The results were found impressive except few blocking artifacts around the depth discontinuities (edges). The algorithm proposed in [5] used Independent Component Analysis (ICA). The results obtained with this ICA-based approach were found satisfactory. However, an issue related to the computational complexity was noticed. In the third category of the algorithms, Dark-Channel-Prior(DCP) was used to address the defogging of images. Mainly, DCP depends on the statistics of the images of an outdoor scene [8]. Such images use to have at least one color channel in which pixels have intensities that are very low or close to zero and termed them dark-pixels. In the presence of fog, the intensities of these dark pixels rise due to the airlight artifact. Therefore correction of DCP results in the restoration of the fog affected image. Several models were proposed by making use of DCP-based restoration of the airlight map those include [6, 29] etc. The paper [24] lists a good review of the existing fog removal algorithms. Recently, few algorithms use learnable filters to design governing PDEs and associated boundary conditions [2, 17, 18]. These approaches are data-driven and contain very few manually chosen parameters.

In this study, the proposed algorithm corrects the airlight map by using a fractional order anisotropic diffusion (Fr-AD) process. The Fr-AD algorithm is a generalization of classical anisotropic diffusion (AD) [27]. Also, it is a pseudo-PDE-based algorithm between Perona-Malik model and fourth-order anisotropic diffusion equations [32]. However, use of fractional anisotropic diffusion is made first time to the best of our knowledge for restoring foggy images. The computations of the fractional derivatives are carried out in the spatial domain instead of the existing Fourier domain-based methods [11]. Additionally, very few parameters are selected manually in the proposed algorithm. These manually tuned parameters do not depend on the image under restoration and remain fixed. The iterative algorithm takes the initial airlight map as the input, and then subsequent iterations derive towards the refined airlight map. The convergence analysis is carried out of the numerical scheme. The presented numerical results validate the better performance of the Fr-AD over some of the existing algorithms.

2 Preliminaries

2.1 Fractional order derivative

Let f(z) be an analytic function in a simple connected region of the complex z- plane $\mathbb {C}$ containing origin. The fractional integral $I_{z}^{\alpha }f(z)$ of an order α of the function f(z) is defined in [30]. The generalization of the fractional integral can be determined by taking into account the natural $n \in \mathbb {N}$ and real μ in the n-fold integral, and then operating the Cauchy’s formula for iterated kernel’s n − 1 times. With this process, the fractional integral operator is given by

$$ I_{z}^{\alpha, \mu}f(z) = \frac{({\mu + 1})^{1-\alpha}} {{\Gamma}(\alpha)} {{{\int}_{0}^{z}} (z^{\mu+ 1} - \xi^{\mu+ 1})^{\alpha-1} {\xi^{\mu}} f(\xi) d{\xi}}. $$

(1)

where $\alpha , \mu (\neq -1) \in \mathbb {R}$.

When μ = 0, we get the classic Riemann-Liouville fractional integral. Further, we will define Riemann-Liouville fractional derivatives using this fractional integral.

The generalized differential operator of order α resembling to the generalized fractional integral is defined in the subsequent manner:

$$ D_{z}^{\alpha, \mu}f(z) = \frac{({\mu + 1})^{\alpha}} {{\Gamma}(1- \alpha)} \frac{d}{dz} {{{{\int}_{0}^{z}} \frac {\xi^{\mu}} {(z^{\mu+ 1} - \xi^{\mu+ 1})^{\alpha} } f(\xi) d{\xi}}}, \ \ 0 \leq \alpha < 1 $$

(2)

$$ D_{z}^{\alpha}f(z) = \frac{d}{dz} \left( {\frac{z^{-\alpha}} {{{\Gamma}(1- \alpha)}}}* f(z) \right) = \frac{d}{dz} \left( {\frac{z^{-\alpha}} {{{\Gamma}(1- \alpha)}}} \right)* f(z) $$

(3)

The partial fractional derivatives with respect to x and y variables are calculated by convoluting the initial airlight map with fixed small size matrices $\left [ \begin {array}{cc} -1 & 1 \\ -1 & 1 \end {array} \right ]$ and $\left [ \begin {array}{cc} -1 & -1 \\ 1 & 1 \end {array} \right ]$ respectively weighted by using α and a constant viz. 0.25. The derivatives keep refining as per the refinement in airlight map with increasing iterations. The convolution method we followed returns the central part of the convolution that is the same size of the airlight map. Figure 1 illustrates the norm of the gradient vector for different fractional orders.

Fig. 1

Norm of derivatives in x and y direction on initial airlight map of Tomb image for: (b) α = 1; (c) α = 1.4; (d) α = 1.8

2.2 Anisotropic diffusion

Images consist of structures with a large collection of scales. Many times it is not known prior that which scale is suitable for the described information. Therefore, it is needed to model the image at different scales. In 1990, Perona and Malik introduced anisotropic diffusion [20] which simulates the process that generates scale space. Anisotropic diffusion is a generalization of the linear diffusion (heat equation). Perona-Malik model diffuses the intra-region prohibiting cross region boundaries in the images. Mathematically, linear anisotropic diffusion (in divergence form) is given as the following elliptic PDE

$$ {\partial_{t}{u}} = div{\left( \textbf{D} \nabla{u} \right)} $$

(4)

where, div is the divergence operator, and ∇ is the gradient operator. Here, D is a collection of symmetric and positive definite diffusion tensors. The above equation is complimented with an initial condition as u(x,0) = u₀, and the Neumann’s type boundary conditions [28]. If we consider D = I_d, the d-dimensional identity diffusion tensor, then the above equation is converted into standard heat equation. The standard heat equation is appropriate to remove the artifacts like noise and fog from the image. However, it blurs the sharp details of the output image.

In the nonlinear anisotropic diffusion, the diffusion tensor D is defined in terms of the filtered image u. Mathematically, the equation for nonlinear anisotropic diffusion is given as

$${\partial_{t}{u}} = div{\left( \textbf{D}_{u} \nabla{u} \right)} $$

together with again an initial and Neumann’s type boundary conditions. In [20], following tensors were used to define the above nonlinear anisotropic diffusion equation

$$\textbf{D}_{u}(x,t)=c_{u}(x,t)I_{d}~~~~~~\text{with}~~~~~~~~~~~~~c_{u}(x,t)=\frac{1}{\sqrt{1+\|\nabla u(x,t)\|{~}^{2} \lambda^{-2}}}$$

where λ > 0 is a user specified constant. The above equation reveals that the filtering (smooth) will act with high impact in the uniform region, however it will be having weak impact when the gradient ∇u is having higher response, i.e. near and on the edges (boundaries of the uniform regions). This avoids the presence of blurry edges in the output image.

3 Proposed algorithm

3.1 Model description

An illustration of the proposed algorithm is given in Fig. 2. In the existing literature, several models have been proposed for the image with foggy information [19, 22, 26]. However, the most general model to describe an image degraded with fog effect can be expressed as

$$ I(x,y) = I_{0}(x,y)e^{-kd(x,y)} + I_{\infty} (1-e^{-kd(x,y)}) $$

(5)

where I₀(x,y) is the original image (intensity at pixel location (x,y) in the absence of fog), k is the coefficient of the scattering related to the atmosphere and d(x,y) is the distance of the scene from camera. The array $I_{\infty }$ is atmospheric light or sky intensity and I(x,y) is the observed foggy image of the scene. Here, e^−kd(x,y) is often represented as transmission map and is given by t(x,y) = e^−kd(x,y). In clear weather conditions, we have k ≈ 0. However, k becomes non-negligible in foggy images. In (5), the first term I₀(x,y)e^−kd(x,y) is referred as direct attenuation and second term $I_{\infty } (1-e^{-kd(x,y)})$ is the airlight map. The decay of the scene radiance in the medium is occurred due to the presence of this direct attenuation term. Generally, scene color is changed due to the diversion in the airlight map. Attenuation is an decreasing function at an exponential rate. This reduces the contrast of the scene. The airlight map A(x,y) produces whiteness in the scene. The (5) can be modified as

$$ I(x,y) = I_{0}(x,y)\left( 1 - \frac{A(x,y)}{I_{\infty}}\right) + A(x,y) $$

(6)

Normalization is performed on foggy image for simulation. As effect of the fog is pure white so the sky intensity may be taken as 1 throughout the array, and hence, (6) becomes

$$ I(x,y) = I_{0}(x,y)(1 - A(x,y)) + A(x,y) $$

(7)

To restore I₀(x,y) from its degraded version I(x,y), we need to estimate A(x,y) accurately. The airlight map A(x,y) is a positive scalar map. Generally, outdoor images are colorful(contains trees, purple red plants and blue water), hence, the assumption of using DCP is true in case of natural images.

Fig. 2

An illustration of the proposed algorithm

3.2 Implementation

In general, smoothing is considered as one of the impressive technique that has been used in several applications related to restoration and enhancement of images. Here, our objective is to find a more refined/enhanced airlight map $\widehat {A}(x,y)$, so that the foggy image can be restored using this in the (7). If we perform traditional smoothing filtering approach on A(x,y) in order to get $\widehat {A}(x,y)$, it can contaminate the image features like lines, edges and textures from this initial airlight map. In order to evade the damage, smoothing has to be robustly controlled by the extent of smoothing and direction of smoothing. Anisotropic diffusion theory proposed by Perona and Malik [20] is a classical example of adaptive smoothing, where smoothing process is defined by PDEs. However, in this paper we generalize it to fractional order as the novelty. To do this, fractional order derivatives are included instead of general integer order derivatives. Let c be the diffusion constant and t denotes the time, the anisotropic diffusion is presented as

$$ {\partial_{t} A} = div(c(\nabla A)\nabla A) $$

(8)

The (8) is associated with the energy functional:

$$ \mathrm{E}(A) = {\int}_{\Omega} f(|\nabla{A}|)d{\Omega} $$

(9)

where Ω stands for image support and f(.) ≥ 0 is an increasing function associated with the diffusion coefficient defined as

$$ c(s^{2}) = \frac{f^{\prime}(s)}{ s^{2}} $$

(10)

Anisotropic diffusion is presented as an energy consuming function that explore for the energy functional minimum. We consider the following functional defined in the space of continuous images over a support of Ω

$$ \mathrm{E}(A) = {\int}_{\Omega} f(| \mathrm{D}^{\alpha}A |)d{\Omega} $$

(11)

where D^α is the fractional derivative operator defined as $\mathrm {D}^{\alpha }A = (\mathrm {D}^{\alpha }_{x}A, \mathrm {D}^{\alpha }_{y}A)$ and $|\mathrm {D}^{\alpha }A| = \sqrt {(\mathrm {D}^{\alpha }_{x})^{2} + (\mathrm {D}^{\alpha }_{y})^{2}}$. We can compute the Euler-Lagrange equation for this minimization problem as follows:

Take any test function $\eta \in \mathcal {C}^{\infty }({\Omega })$. Define

$$ {\Phi}(a) = {\int}_{\Omega} f(|\mathrm{D}^{\alpha}A + a \mathrm{D}^{\alpha}\eta|)dxdy $$

(12)

We obtain

$$\begin{array}{@{}rcl@{}} {\Phi}^{\prime}(0) &=& \frac{d}{da} {\int}_{\Omega} f(|\mathrm{D}^{\alpha}A + a \mathrm{D}^{\alpha}\eta|)dxdy \left|{~}_{a = 0}\right. \\ &=& {\int}_{\Omega} \left( f^{\prime}(|\mathrm{D}^{\alpha}A|) \frac{\mathrm{D}^{\alpha}_{x}A}{|\mathrm{D}^{\alpha}A|} {\mathrm{D}^{\alpha}_{x}\eta} \right. \\ &&\quad\quad\left. +f^{\prime}(|\mathrm{D}^{\alpha}A|) \frac{\mathrm{D}^{\alpha}_{y}A}{|\mathrm{D}^{\alpha}A|} {\mathrm{D}^{\alpha}_{y}\eta} \right) dxdy \\ &=& {\int}_{\Omega} ((\mathrm{D}^{\alpha}_{x})^{*} (c(|\mathrm{D}^{\alpha} A|{~}^{2}) \mathrm{D}^{\alpha}_{x}A) \\ &&+ (\mathrm{D}^{\alpha}_{y})^{*} (c(|\mathrm{D}^{\alpha} A|{~}^{2}) \mathrm{D}^{\alpha}_{y}A)) \eta dxdy \end{array} $$

(13)

for all $\eta \in \mathcal {C}^{\infty }({\Omega })$, where $(\mathrm {D}^{\alpha }_{x})^{*}$ and $(\mathrm {D}^{\alpha }_{y})^{*}$ are adjoint of $\mathrm {D}^{\alpha }_{x}$ and $\mathrm {D}^{\alpha }_{y}$, respectively. Hence, the Euler-Lagrange equation becomes

$$ (\mathrm{D}^{\alpha}_{x})^{*} (c(|\mathrm{D}^{\alpha} A|{~}^{2}) \mathrm{D}^{\alpha}_{x}A) + (\mathrm{D}^{\alpha}_{y})^{*} (c(|\mathrm{D}^{\alpha} A|{~}^{2}) \mathrm{D}^{\alpha}_{y}A) = 0 $$

(14)

The Euler-Lagrange equation can be solved using the following gradient descent method:

$$ {\partial_{t} A} = -(\mathrm{D}^{\alpha}_{x})^{*} (c(|\mathrm{D}^{\alpha} A|{~}^{2}) \mathrm{D}^{\alpha}_{x}A) - (\mathrm{D}^{\alpha}_{y})^{*} (c(|\mathrm{D}^{\alpha} A|{~}^{2}) \mathrm{D}^{\alpha}_{y}A) $$

(15)

taking initial condition as the observed image A(x,y). The solution is reached as $t \rightarrow \infty$ but the growth time can be stopped prior to attain the optimal trade off between fog removal and edge preservation.

When α = 1, the (14) is absolutely the Perona-Malik (8); when α = 2 (14) is absolutely the fourth order anisotropic diffusion in [32]; when 1 ≤ α ≤ 2, (14) leads to a “natural interpolation” between them.

3.3 Numerical algorithm

For practical applications, firstly assume that initial discrete image A is of m × m pixels. It has been sampled from its continuous version at uniformly spaced points starting at (0,0), i.e. A(x,y) = A(xΔx,yΔy) for x,y = 0,...,m − 1. The grid size Δx and Δy is chosen as Δx,Δy = 1.

Let K₁ and K₂ be purely diagonal operators in the spatial domain, defined by

$$\mathrm{K}_{1} = \mathrm{F}^{-1}{\left( diag\left( \left( 1 - exp({-j 2 \pi (\omega_{1})/m}))^{\alpha} \times exp({{j \pi \alpha \omega_{1}}/{m}}\right) \right)\right)} $$

$$\mathrm{K}_{2} = \mathrm{F}^{-1}{\left( diag\left( \left( 1 - exp({-j 2 \pi (\omega_{2})/m}))^{\alpha} \times exp({{j \pi \alpha \omega_{2}}/{m}}\right) \right)\right)} $$

where,

$$ \mathrm{K}^{\ast}_{1} = \mathrm{F}^{-1}{\left( diag\left( conj\left( \left( 1 - exp({-j 2 \pi (\omega_{1})/m}))^{\alpha} \\ \times exp({{j \pi \alpha \omega_{1}}/{m}}\right) \right) \right)\right)}, $$

(16)

and

$$ \mathrm{K}^{\ast}_{2} = \mathrm{F}^{-1}{\left( diag\left( conj\left( \left( 1 - exp({-j 2 \pi (\omega_{2})/m}))^{\alpha} \\ \times exp({{j \pi \alpha \omega_{2}}/{m}}\right) \right) \right)\right)}. $$

(17)

We compute the evolution of the initial image A , along flow (14) work in the spatial domain only. Moreover, we define

$$ h_{xn} = c \left( | {\mathrm{D}}^{\alpha}A_{n} |{~}^{2} \right) {\mathrm{D}}^{\alpha}_{x}A_{n} $$

(18)

and

$$ h_{yn} = c \left( | {\mathrm{D}}^{\alpha}A_{n} |{~}^{2} \right) {\mathrm{D}}^{\alpha}_{y}A_{n} $$

(19)

To summarize, our fog removal approach is done in following steps.

3.4 Analysis

The iterative scheme defined in the previous section needs to be converged for getting an enhanced airlight map $u=\widehat {A}(x,y)$. In this subsection, we briefly describe the convergence analysis of the scheme. The scheme defined in the Algorithm 1 converges for any choice of the initial image, if the energy function f(|∇^αu|) defined in (11) is smooth and convex [21]. Alternatively, there exists a unique solution of the diffusion PDE given in (14) if energy function defined in (11) is smooth and convex. From [31],the eigenvalues of the Hessian matrix of the energy function f(|∇^αu|) can be written as

$$ \lambda_{1}=\frac{f^{\prime}(|\nabla^{\alpha}{u}|)}{|\nabla^{\alpha}{u}|} ~~~~~~~~~~~~~\text{and} ~~~~~~~~~~~~~~~~ \lambda_{2}={f^{\prime\prime}(|\nabla^{\alpha}{u}|)} $$

(20)

This gives λ₁ = c(.). Here, the value of λ₁ is positive due to our choice of the function c(s²) = s^− 1, since $f^{\prime }(s)=s^{2}.c(s^{2})$. To make the anisotropic diffusion problem as a well-posed, the value of second eigenvalue of the hessian matrix is quite important. In this scheme, we have

$$\lambda_{2}=\frac{d}{ds}f^{\prime}(s)=f^{\prime\prime}(s)= 1>0$$

due to the choice of the edge preserving function c(s). This indicates that the Hessian matrix associated with the integrand of energy function f is positive definite and hence the function is strictly convex. In this case, the function f attains a unique global minima for any choice of initial image u₀ = A(x,y) which is the airlight map extracted from the degraded image.

3.5 Post-processing

After estimating airlight map $\widehat {A}$ then de-foggy image I₀(x,y) can be restored as

$$ I_{0}(x,y,c) = \frac{(I(x,y,c)- \widehat{A}(x,y))}{(1 - (\widehat{A}(x,y)/{I_{\infty}(c)}))} $$

(21)

where c ∈ (r,g,b). The above method can also be applied for gray-scale images. The only difference will be the initial airlight map.

Histogram stretching is applied as a post-processing step of the proposed algorithm. The restored image I₀(x,y,c) obtained from the improved airlight map may be of poor contrast. Thus, the restored image may be visually dim because of the difference in the brightness of the scene radiance and the atmospheric light. The later one finds a bit brighter. In general, histogram equalization is a favorite technique for contrast enhancement. However, it may not be that worthy due to the saturated output image of the fog removal step. Moreover, histogram specification technique cannot be applied due to the absence of a standard reference image. Here, we adopted the histogram stretching in our algorithm for the task of contrast enhancement.

4 Results and discussions

This section presents the numerical results obtained with the proposed algorithm to enhance the digital images degraded by the fog effect. The proposed algorithm tests the performance on seven different images, namely Forest, Wheat, Tomb, Ny17, Swan, N6 and Train. These images are having different textural details and degraded with varying amount of the fog. In this study, the fractional order parameter α takes a real value in the interval [1, 2). For the computational purpose, we choose five fixed values of α = {1.0, 1.2, 1.4, 1.6, 1.8} as taken in many references [1]. It is worth to mention that at α = 1, the proposed scheme behaves similarly to AD scheme [27]. In existing literature, different strategies were adopted to stop the iterative process of anisotropic diffusion. We follow the stopping criteria as in [1]. Four different metricscontrast gain (CG), colorfulness index (CI), contrast-to-noise ratio (CNR) and visible edges ratio (VER) are used to measure the performance of the proposed algorithm (See Table 1). The stopping criterion is chosen to be 30 iterations of the diffusion process based on all these metrics. The proposed algorithm uses the airlight map extracted from the degraded image as the initial image of the diffusion process. We compare the performance of the proposed algorithm with some of the existing algorithms such as DCP-based approach [8], Kernel regression model with DCP [29], Two-Layer Gaussian process regression (GPR) [4] and Integer order AD [27].

Table 1 Details of the metrics used for quantitative analysis in this study

Table 2 lists the numerical results in terms of contrast gain (CG). In case of Wheat, Swan and N6 images, fractional order α = 1.8 gives the best CG value. For Forest, Ny17 and Train images, α = 1.4 is giving a slight better CG value. For Tomb image, the best CG value is obtained at α = 1.2. It is also worth to mention that the results are significantly improved with fractional orders as compared to the case of α = 1 (AD [27]).

Table 2 Contrast gain (CG) obtained with the proposed algorithm with different fractional orders α with 30 iterations as stopping criteria

The numerical results in terms of colorfulness index (CI) are listed in Table 3. The best CI values for all the images are obtained for the fractional orders α = 1.6 & 1.8. This is due to the textural details of the image. Unlike the CG value, best CI values are obtained with α = 1.6 for Forest and Wheat images. For rest of the images, the best performance is found at α = 1.8. However, in all the these images, Fr-AD gives better results when compared to the AD.

Table 3 Colorfulness index (CI) value obtained with the proposed algorithm with different fractional orders α with 30 iterations as stopping criteria

Similar type of results are obtained in case of contrast-to-noise ratio. Table 4 lists the values of CNR at different fractional orders in case of all these images. It is noticeable that the best results are obtained with fractional orders and not with α = 1.0. It is also noticeable from Table 5 that the Fr-AD based approach gives better visible edges ratio (VER) than the AD. From these observations, the significance of fractional order derivatives in anisotropic diffusion can be observed especially in the restoration of fog affected images.

Table 4 Contrast-to-noise ratio (CNR) obtained with the proposed algorithm with different fractional orders α with 30 iterations as stopping criteria

Table 5 Visible edges ratio (VER) obtained with the proposed algorithm with different fractional orders α with 30 iterations as stopping criteria

The performance of the proposed algorithm is compared with four different algorithms. Table 6 lists the results of this comparison study. We make the following remarks based on this comparison study.

• 1. In case of contrast gain, the proposed algorithm performs better than other all the four algorithms for Forest and Tomb images. The DCP algorithm performs better than the proposed algorithm in case of rest images. For Ny17, Swan and Train images, the proposed algorithm performs better than the three algorithms KRM, GPR and AD.

• 2. The proposed algorithm performs better in case of colorfulness index for Ny17, Swan and Train images. The KRM algorithm performs better than all algorithms for Forest, Tomb and N6 images. The GPR algorithm performs best for Wheat image.

• 3. In case of metric CNR, the KRM algorithm performs better for Tomb, Ny17 and Swan images. The proposed algorithm gives best results for Forest, N6 and Train images. The GPR algorithm performs best for Wheat image. For Swan image, the proposed algorithm performs better than the three methods DCP, GPR and AD.

• 4. The proposed algorithm gives better VER value than DCP, KRM, GPR and AD for images Forest, Swan and N6. The KRM algorithm [29] performs better for other images. In case of visible edges ratio, the proposed algorithm performs better than the three methods DCP, GPR and AD for the images Wheat, Tomb, Ny17 and Train.

Based on the overall comparison of the results, the proposed algorithm is a good choice to restore the foggy images. The computational time taken in the implementation of the proposed diffusion process is compared with a Fourier domain-based algorithm [1]. On a single processor, the proposed algorithm takes less than half of the computational time taken by other algorithms (See Table 7). In case of multi-chip, the Fourier domain based implementation may be a better choice.

Table 6 Comparison of the results obtained with existing algorithms DCP [8], KRM [29], GPR [4], AD [27] and the proposed algorithm (with α)

Table 7 Computational time (in seconds) obtained in Fourier domain and by the proposed algorithm in spatial domain

Figure 3 gives a qualitative representation of the results obtained with DCP, KRM, GPR, AD and the proposed algorithm. These qualitative results affirm that the restored images with the proposed algorithm are visibly better than other four existing algorithms. The results obtained with KRM algorithm [29] has a whiteness along the edges which is smooth in case of the results obtained with the proposed algorithm. Hence, this apparently makes the proposed algorithm a better choice for image restoration where the fog degrades the original images.

Fig. 3

The original foggy images are given in first column. Restored images with DCP [8], KRM [29], GPR [4], AD [27] and the proposed algorithm (at α = 1.8) are shown in second, third, fourth, fifth columns and sixth columns, respectively

5 Conclusions

This paper addresses the issue of defogging of the digital images by using a fractional order generalization of the anisotropic diffusion. The quality of restored images in the proposed algorithm depends on the diffused airlight map. The proposed algorithm diffuses airlight map visually better when compared to classical anisotropic diffusion. The proposed algorithm gives restored images better than some of the benchmark algorithms based on different metrics. The computation of the fractional order derivatives in the proposed algorithm takes lower computational efforts when compared to a Fourier domain-based implementation. In this study, five fixed values between [1, 2) are considered for computing fractional order derivatives. An adaptive choice of the fractional orders and use of learnable PDEs might be possible future directions to extend the proposed work.