Optimised enhancement scheme for low contrast underwater images

Image acquisition in an underwater is a daunting task owning to the peculiarities in the properties of the environment involved. Underwater environment is characterized by low light penetration, varying media density and conductivity among others. Thus, images captured in underwater environments are most times affected with various kinds of degradation which include low visibility, unequal distribution of colour cast, blurriness, haze, and image noise [1]. All these impairments contribute to the poor contrast that is usually associated with underwater images. Apart from the acquisition of images of underwater landscapes and scenes, underwater imaging is important in many scientific investigations such as monitoring and inspection of underwater communication cables [2], control of underwater vehicles [3, 4], detection of sub-sea devices [5], remote sensing and marine biology [6], to name a few. Each of the aforementioned scientific efforts is geared towards the provision of basis for making informed decision from images taken in underwater environment, which are usually low in contrast. Thus it becomes necessary to increase the visual quality of images that are acquired in this environment to enhance their usability in specific areas of application.

Images taken in an underwater environment are distinguishable from those that are captured above the water because the amount of light intensity that is required to produce an image in the two media is different. During the propagation of light in an underwater environment, three varieties of light are received by the camera; these are the incident light, forward scattering light and back scattering light. The incident light is affected by attenuation which leads to loss of information in underwater images. The contribution of scattering light to the blurring of image features is insignificant. The backscattering light strongly depletes the contrast of these underwater images such that fine details and characteristics patterns are suppressed [7].

From the foregoing, the propagated light while taking underwater images is heavily attenuated, causing the resultant image to be visibly poor unlike image that are taken in a clear non-water environment. Light attenuation in water has been attributed into two main phenomena: absorption and scattering [2]. The absorption effect greatly diminishes the strength of light energy while the scattering alters the direction of propagating light wave. These two phenomena combined to make underwater captured images blurred and diminished the visual appearance of images. These threatened the achievements in underwater scene understanding and computer vision application in the specific areas of applications that were earlier mentioned. The resultant effect is poor readability of the image and inaccurate deductions [7]. Furthermore, these two effects constrained underwater visibility, typically to 20 meters approximately in clear water and about 5 meters in turbid water [8]. From the foregoing, in order, to extract useful information from the captured images in underwater environment, there is need for enhancement of the images, especially in terms of contrast.

Enhancement of the contrast of an image is a fundamental task in image signal processing. Traditionally, Contrast Stretching (CS), Contrast Transformation (CT) and Histogram Equalisation (HE) are the three fundamental techniques often adopted in the enhancement of the image contrast [9]. The HE has gained much popularity among these techniques, which culminated into incursion of different variants of it in the image processing literature. Moreover, a meta-analysis of some widely used contrast measures for assessment of image enhancement schemes that are based on the HE is provided in [10].

Global Histogram Equalisation (GHE) for instance, adopts cumulative histogram of the input image as the transformation function. This makes GHE to easily adapt to the information content of the input image. However, GHE suffers two major shortcomings. First, it has no in-built mechanism for the control of degree of enhancement needed by the input image, which often leads to over enhancement in some cases. Secondly, the brightness level of the enhanced output image from GHE is poor, compared with that of the input image.

Many investigators have come up with different methods to address these drawbacks of GHE, especially preservation of the brightness level of the input image. It is important to take note of the Bi-Histogram Equalisation (BHE) method [11, 12], where the image to be enhanced is split into two based on the value of the mean grey level. The resulting histograms from the two sub-groups are individually equalised. Weighted Threshold Histogram Equalisation (WTHE) [13, 14] is another modification to the HE. This involves the application of a Power-Law Transformation (PLT) to the original histogram of the input image prior the computation of cumulative probability density function that will be used in the enhancement process. Apart from those two approaches that are global in terms of the histogram used, another method which is based on local enhancement scheme is termed Contrast Limited Adaptive Histogram Equalisation (CLAHE) [15–17]. In CLAHE, the input image to be enhanced is sliced into small non-overlying chunks of pixels and each chunk is then equalised separately. It has been established that a good choice of clip limit and distribution should provide optimum compromise between the extent of contrast enhancement and feature preservation. The work of [17] explores the optimal selection of clip limit value in CLAHE and the appropriate distribution on which the image region to be enhanced is to be approximated for magnetic resonance images. Some important operational parameters of CLAHE were investigated with respect to clip-limit and selected probability distributions.

Although CLAHE is particularly useful in contrast enhancement in dissimilar regions in an image, its major drawback is that it does not have the capacity to sharpen high frequency regions of the image [18]. Other modifications that can as well be found in the literature include Non-parametric Modified Histogram Equalisation (NMHE) [19], Exposure-based Sub-Image Histogram Equalisation (ESIHE) [20, 21], Median-Mean Based Sub-Image Clipped Histogram Equalisation (MM-SICHE) [22], and Dominant Orientation-based Texture Histogram Equalisation (DOTHE) [23]. However, it is worthy of note here that BHE, WTHE and CLAHE represent the three basic modified variants of the HE. Furthermore, both WTHE and CLAHE have internal structures that make the manual setting of the parameters of the method for contrast control and brightness error possible: they are tuned in order to attain the desired results but this is a laborious process [9, 24]. To alleviate this problem, [9] proposed Parameter-Free Fuzzy-histogram Equalisation (PFHE); a global mapping scheme that allows for achieving application specific enhancement performance. In this work, Texture Fuzzy Subset (TFS) are evaluated with respect to the similarity of the fuzzy values in the test image along with their connected neighbours. Furthermore, the output estimate is obtained by using the transformation that is identical with histogram equalization on the Fuzzy textural histogram that was realized from (TFS). The non-linear combination of the estimated output and the input yield the final output. Other algorithms are mere extensions of the three basic techniques modifying the HE. Due to the attendant problem of discontinuities in artifacts, too many operational parameters that characterize the performance of these improved enhancement algorithms have not positioned them for optimal utilization especially in medical imaging. Thus, in [18] a non-linear edge sharpening algorithm which appropriates a very few number of parameters with no discontinuities and saturation was proposed. Here, an inverse problem approach is employed such that the amplified first derivative is used to obtain the sharpened image. Then, the specific value of the amplification factor is treated as a non-linear function of the local average of the total directional gradient in order to hold down the noise amplification. In terms of conceptual formulation, ESIHE is a hybrid of BHE and CLAHE; the procedure adopted in arriving at MM-SICHE is the opposite of that adopted in the formulation of ESIHE. In the case of DOTHE, the image to be enhanced is first partitioned into smooth and dominant orientation spots, using singular value decomposition. Thereafter, the probability of occurrence of grey levels from the dominant-orientation spots is computed and used to construct the needed histogram for equalisation as against the entire image that is used in GHE.

While GHE and its numerous variants find applications in image contrast enhancement (underwater or above-water), quite a number of other proposals exist in the literature specifically for the processing of underwater images. They can be deployed in both hardware and software approaches. The hardware method involves the use of high resolution camera lens to capture underwater scenes. However, the high cost associated with acquisition of high resolution camera makes the hardware approach less popular among researchers, when compared to the software methods. Among the software-based underwater image enhancement methods are colour balancing and fusion technique [25], combined histogram equalization and fusion techniques [26], wavelength compensation and de-hazing [27], contrast stretching and modification of colour space [28], fuzzy-histogram equalisation method [29]. In [25], contrast enhancement of the image is realized via the blend of two images that are derived directly from a colour compensated and white-balanced version of the original degraded image. In [26]; hazy underwater image is enhanced via combined adaptive histogram equalization and fusion method. Chiang et al [27] address the problem associated with artificial light and light scattering in underwater images via the use of a method called Wavelength Compensation and Image De-hazing. Meanwhile, the method for the enhancement of underwater images proposed in [28] employed the combination of contrast stretching and HIS colour space. Combined fuzzy and histogram based method for the enhancement of colour image was proposed in [29] to address colour artifacts and reduce intensity that were associated with transform domain-based image contrast enhancement methods. The approach has been successfully evaluated using different kinds of images such as underwater, remote sensing, medical images and so on.

All the aforementioned variants of histogram-based image enhancement methods have one thing in common: the image contrast enhancement is done via the modification of the input image intensity. With the use of fuzzy logic in the formulation of image contrast enhancement technique, grey tone in an image is modified through the use of membership function to obtain a fuzzy set. The image is viewed as an array of fuzzy singletons having a membership value. This value indicates the degree of image property in the array, which can be modified only through the modification of the membership functions. The main task in this approach is the choice of membership function.

The fuzzy-based histogram equalization approach is however proposed in this paper. Our method is optimised for image brightness preservation unlike what obtains in [29]. The proposed scheme is meant for the enhancement of the contrast of underwater image. We are inspired by the Teaching-Learning-Based Optimisation (TLBO) technique in [30] to propose this Brightness Optimisation and enhancement strategy for underwater images. Section 2 of the paper presents overview of fuzzy model as well as histogram equalisation adopted while in section 3, details of the proposed method are presented. Sections 4 and 5 present experimental results and concluding remark, respectively.

Since the method adopted for underwater image contrast enhancement in this paper comprises the combination of fuzzy logic and histogram equalization, basic information about the two models are considered pertinent. These are presented below.

2.1. Fuzzy model

2.2. Histogram equalization

Enhancement of contrast is important in image processing tasks for both human and machine vision. It has been extensively applied in the processing of medical images and often represents the first stage in many other signal processing applications such as speech, texture synthesis, and diverse image/video processing. For this purpose, the Dynamic Histogram Equalization (DHE) technique is applied.

Histogram equalisation is the technique of contrast enhancement of an image through the adjustment in image pixels' intensities. The technique involves mapping of one intensity distribution to another distribution or from a given histogram to a wider or uniformly distributed histogram. Standard histogram equalisation is based on the assumption that the best visible contrast is obtainable from uniform distribution of grey level of an image as posited in [29].

Suppose f is an image whose pixel intensities matrix representation has its entries defined between 0 and L-1, where L denotes the highest value that pixel intensity can assume, which is often equal to 256 in practice. If for each possible value of the intensity, P represents the normalized histogram value of f, then one can write

$$\begin{eqnarray}&&{P}_{n}={I}_{n}{\left(\displaystyle \sum _{n=0}^{L-1}{I}_{n}\right)}^{-1}\end{eqnarray} \tag{ 1 }$$

where ${I}_{n}$ denotes pixels having value of intensity $n,$ histogram equalised image is defined as

$$\begin{eqnarray}&&{g}_{i,j}=floor\left(\left(L-1\right)\displaystyle \sum _{n=0}^{{f}_{i,j}}{P}_{n}\right)\end{eqnarray} \tag{ 2 }$$

where $floor(* )$ approximates down to the nearest integer. Expression specifies by (2) implies pixel intensities, $k(f)$ transformation by the function

$$\begin{eqnarray}&&T\left(k\right)=floor\left(\left(L-1\right)\displaystyle \sum _{n=0}^{k}{P}_{n}\right)\end{eqnarray} \tag{ 3 }$$

Usually, small histogram distributions depict low contrast images while wide histogram distributions indicate images of high contrast. This method is suitable for images that have large regions that are related in terms of tone because it shows the hidden details of an image through stretching out of the contrast of local regions.

The enhancement algorithm proposed here is an improvement on the technique of fuzzy-histogram equalisation framework presented in [31] for the enhancement of digital images enhancement. The FHEOBP addresses grey-levels imprecision through the use of appropriate fuzzy membership functions to remove effects of random fluctuations, missing intensity levels and exhibits smoothness. This ensures meaningful partitioning for the required brightness that preserves equalisation. GHE introduces undesirable artefacts during re-mapping of local maxima (histogram peaks), and distortion in the image mean brightness level. The approach presented in this paper adjusts the image histogram such that values of grey-level are re-distributed within two successive peaks instead of re-mapping the local maxima (histogram peaks). The stages involved in the FHEOBP scheme for underwater image enhancement are shown in figure 1. Each of these stages is briefly described, beginning with the image fuzzy-histogram computation.

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3.1. Fuzzy-histogram computation

A fuzzy-histogram is defined as $h\left(i\right),\,i\in \{0,\,1,\,\cdots L-1\}$ where $h\left(i\right),$ is a sequence of real numbers and denotes the rate at which grey-levels occur near $i.$ Suppose grey-level $X\left(x,y\right)$ is considered as a fuzzy number $\overline{X}\left(x,y\right),$ then the fuzzy-histogram for the system is computed as:

$$\begin{eqnarray}&&h\left(i\right)\leftarrow h\left(i\right)+\displaystyle \displaystyle \sum _{x}\displaystyle \displaystyle \sum _{y}{\mu }_{\overline{X}\left(x,y\right)i}\end{eqnarray} \tag{ 4 }$$

provided ${\mu }_{\overline{X}\left(x,y\right)i}$ denotes fuzzy membership function.

Through this representation, the exactness of grey-levels is better handled when related to hard histogram, thus producing a better and smooth histogram.

3.2. Histogram partitioning

Sub-histograms are obtained through partitioning of the histogram local maxima computed using (4). Each valley section between two successive local maxima constitutes a partition. Fuzzy-histogram local maxima are detected using the first and second derivatives of the fuzzy-histogram. Central difference approximation is used to accomplish evaluation of derivatives since fuzzy-histogram data sequence is discrete-valued. Consequently, the first order derivative is defined as:

$$\begin{eqnarray}&&h^{\prime} \left(i\right)=\displaystyle \frac{dh\left(i\right)}{di}\cong \displaystyle \frac{h\left(i+1\right)-h\left(i-1\right)}{2}\end{eqnarray} \tag{ 5 }$$

where $h^{\prime} \left(i\right)$ indicates the first-order derivative of the fuzzy-histogram $h\left(i\right)$ at the ith level of the intensity.

In order to minimize error that is inherent from the evaluation of the second-order derivative of the fuzzy-histogram directly from (5), the central difference operator is applied as follows:

$$\begin{eqnarray}&&h^{\prime\prime} \left(i\right)=\displaystyle \frac{{d}^{2}h\left(i\right)}{d{i}^{2}}\cong h\left(i+1\right)-2h\left(i\right)+h\left(i-1\right)\end{eqnarray} \tag{ 6 }$$

where $h^{\prime\prime} \left(i\right)$ stands for the second-order derivative of the fuzzy-histogram $h\left(i\right)$ at its ith level of intensity. Using (5) and (6), values of intensity level where the first-order derivatives are zero and the second-order derivative has negative value are the local maxima points. That is,

$$\begin{eqnarray}&&{i}_{\max }=i\forall \left\{h^{\prime} \left(i+1\right)\times h^{\prime} \left(i-1\right)\lt 0,h^{\prime\prime} \left(i\right)\lt 0\right\}\end{eqnarray} \tag{ 7 }$$

Perfect zero crossings of intensity levels take place at non-integral values, which may lead to ambiguity. To avert such situations, two neighbouring points are generally taken as points of maxima; the points having the highest count among the neighbouring pairs are ultimately taken as the maxima. Once the maxima of the fuzzy-histogram are known, the partitions are then constructed as follows: suppose $\left\{{m}_{0},{m}_{1},{m}_{2},\cdots {m}_{n}\right\}$ represents a set of detected local maxima having $\left(n+1\right)$ intensity levels. If the initial fuzzy-histogram is distributed in the range $\left[{X}_{\min },{X}_{\max }\right],$ it follows that $\left(n+1\right)$ partitions/sub-histograms are given as $\left\{\left[{X}_{\min },{m}_{0}\right],\right.$ $\left[{m}_{0}+1,{m}_{1}\right],$ $\cdots \left.\left[{m}_{n}+1,{X}_{\max }\right]\right\}.$

3.3. Dynamic Equalisation (DE) of the sub-histograms

Dynamically equalising the histogram alleviates the shortcomings of the traditional histogram equalization. This enhances the image without preserving the image details. DE partitions the local minima image histogram into portions and thereafter allocates particular grey level span for each partition prior to their independent equalization. These earlier partitions are subsequently subjected to re-partitioning test in order to ascertain the absence of any dominating portions.

In this paper, the approach proposed in [32] is adopted to equalize each of the sub-histograms. A spanning function, which is dependent on the number of available pixels in a division, is used to carry out the equalization. The process encompasses mapping of divisions to a dynamic range and realisation of the histogram equalization. Equations (8)–(10) describe mapping of partitions to a dynamic range.

$$\begin{eqnarray}&&{S}_{i}={H}_{i}-{L}_{i}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{F}_{i}={S}_{i}\times {\mathrm{log}}_{10}{M}_{i}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{R}_{i}=\displaystyle \frac{\left(L-1\right)\times {F}_{i}}{\displaystyle {\sum }_{k=1}^{n+1}{F}_{k}}\end{eqnarray} \tag{ 10 }$$

where ${H}_{i}$ and ${L}_{i}$ represent the largest and smallest values of the intensity level contained in the ${i}^{th}$ input sub-histogram, ${M}_{i}$ denotes total number of available pixels in that division, ${S}_{i}$ specifies the dynamic range of the input sub-histogram and ${R}_{i}$ stands for the output sub-histogram dynamic range. Thus, the dynamic range of the ith output sub-histogram is obtainable from ${R}_{i}$ as:

$$\begin{eqnarray}&&S{t}_{i}=\displaystyle \sum _{k=1}^{i-1}{R}_{k}+1\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&S{p}_{i}=\displaystyle \sum _{k=1}^{i}{R}_{k}\end{eqnarray} \tag{ 12 }$$

Exceptions to (11) and (12) occur at the two extremes, where $\left[S{t}_{1},S{p}_{1}\right]=\left[0,{R}_{1}\right]$ and $\left[S{t}_{n+1},S{p}_{n+1}\right]\,=\left[\displaystyle {\sum }_{k=1}^{n+1}{R}_{k},L-1\right].$

The method of equalising each of the partitions of the histogram is similar to that employed in GHE. The re-mapped values of the ith sub-division are generated using:

$$\begin{eqnarray}&&y\left(j\right)=S{t}_{i}+{R}_{i}\displaystyle \sum _{k=s{t}_{i}}^{j}\displaystyle \frac{h\left(k\right)}{{M}_{i}}\end{eqnarray} \tag{ 13 }$$

where $y\left(j\right)$ represents the new level of the intensity, corresponding to the jth level of the intensity in the initial image, $h\left(k\right)$ is the value of the histogram corresponding to the kth intensity level on the fuzzy-histogram, and the total population count in the ith partition of the fuzzy-histogram is given by $\displaystyle {\sum }_{k=s{t}_{i}}^{j}\left(h\left(k\right)/{M}_{i}\right).$

3.4. Normalisation of the image brightness

It is worth noting that the output image's mean brightness obtained after dynamically equalising sub-histograms is slightly different from that of the input image. This difference is removed through normalization of the output. Suppose an image obtained after dynamic histogram equalisation is denoted as $\xi ;$ assuming ${m}_{i}$ and ${m}_{o}$ are the mean of brightness levels of the input image and output image, respectively; If $\zeta$ symbolises the brightness level of the output image obtained after application of the FHEOBP technique, then the value of the grey level at the pixel location $\left(x,y\right)$ in the image $\zeta$ is expressible in the form written as:

$$\begin{eqnarray}&&\zeta \left(x,y\right)=\displaystyle \frac{{m}_{i}}{{m}_{o}}\xi \left(x,y\right)\end{eqnarray} \tag{ 14 }$$

With this brightness-preserving strategy, both the input and output images have the same value of mean intensity. It is pertinent to state here that since most electronic equipment acquires and displays colour images, technique of enhancing colour images ought to be given some attention. Most often in the literature, the conventional approach is the equalisation of the Red (R), Green (G), and Blue (B) planes on the RGB images. The main drawback of this method is the attendant problem of altering the hue of the output image. In this paper, the $Y{C}_{b}{C}_{r}$ colour space transformation is adopted, where the intensity band of the image only is equalised and the chromaticity is preserved. This approach yields improved results when compared to what is obtainable when R, G, B channels are distinctly equalised.

3.5. Fuzzy filters

3.6. Optimising the proposed FHEOBP algorithm

Several methods exist in the literature for the appropriate solution to optimisation problems, majority of which are evolutionary or swarm-intelligence-based. In addition to method-specific restrictions, most of these optimisation techniques entail joint control constraints like the size of the population, number of off-springs, best size, etc. As instances, mutation probability, crossover probability and selection operator are peculiar to Genetic Algorithm (GA) whereas Particle Swarm Optimiser (PSO) is associated with inertia weight, social, and cognitive strictures [33]. In addition to common control parameters tuning, performance of these algorithms is heavily dependent on proper regulation of respective process-definite constraints. Improper modification of these parameters either results in increased computational overhead or produces a solution that is locally optimal, which of course is not desirable. Where possible, the algorithm which requires few or no method-definite parameters is desirable. This informs the choice of teaching-learning-based optimisation algorithm [31] for this work.

The choice of the objective function as well as its parameters greatly influences the performance of the proposed FHEOBP algorithm. Since different images exhibit different brightness and contrast levels, it is necessary that parameters of objective function vary from image to image. In order to maximise the FHEOBP algorithm and avoid luminance distortion, an objective function is chosen such that the contrast level is improved while maintaining nominal brightness levels with minimal distortion. Thus, luminance distortion $\left(Q\right),$ which is a measure of closeness of the mean luminance of any two images, is chosen in this work as the objective function. It is employed here for the evaluation of the capability of FHEOBP algorithm for brightness preservation.

Consider $X=\left\{{x}_{i}| i=1,2,\cdots N\right\}$ as the reference image while $Y=\left\{{y}_{i}| i=1,2,\cdots N\right\}$ is the test image, luminance distortion between the two images is defined as:

$$\begin{eqnarray}&&Q=\displaystyle \frac{2{\mu }_{x}{\mu }_{y}}{{\mu }_{x}^{2}+{\mu }_{y}^{2}}\end{eqnarray} \tag{ 15 }$$

where ${\mu }_{x}$ and ${\mu }_{y}$ are mean luminance of $X$ and $Y,$ respectively. $Q$ takes values in the range $\left[0,1\right]$ with $Q=1,$ when the values of the mean luminance of the two images being compared are identically equal.

The method proposed in [34] for the computation of the luminance distortion is adopted in this work, where the luminance distortion in an image is expressed as:

$$\begin{eqnarray}&&{Q}_{image}=\displaystyle \frac{1}{M\times N}\displaystyle \sum _{i=0}^{M-1}\displaystyle \sum _{j=0}^{N-1}{Q}_{i,j}\end{eqnarray} \tag{ 16 }$$

where $i$ and $j$ define a location in an image such that ${Q}_{i,j}$ is the luminance distortion of the image at the location $(i,j)$ as computed via (15), when $i=1,2,\cdots M$ and $j=1,2,\cdots N.$ ${Q}_{image}$ has its value approaching unity (or 100%) when the difference in the mean brightness of the image introduced by FHEOBP algorithm becomes small. That is, the higher the numerical value of ${Q}_{image}$ as computed from (16) for a filter, the better is the filter in terms of image brightness preservation.

Fuzzy member function used for the FHEOBP algorithm is of the Gaussian kind and it is defined as:

$$\begin{eqnarray}&&\mu ={e}^{-{\alpha }^{2}/{\beta }^{2}}\end{eqnarray} \tag{ 17 }$$

where $\alpha$ and $\beta$ are parameters that define the Gaussian function.

The computational tool used in this work for modelling, implementation and simulation is the Matlab/Simulink 2017a software. Specifically, the Fuzzy Inference System (FIS) editor toolbox is used. Additionally, three different standard underwater images are used as test images for the evaluation of the performance of the proposed FHEOBP for underwater image contrast enhancement. Figure 2 depicts the test images. The first image describes underwater environment showing costal landscape below the water. The second shows a typical sea bed while in the third; an image of a diver exploring underwater environment is illustrated.

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Performance of the proposed FHEOBP scheme is compared with those of two different histogram equalisation methods, which are Local Histogram Equalisation (LHE) and Global Histogram Equalisation (GHE) algorithms. Using each of the three images that formed the test image set for this work as input; output filtered images are generated by the filters. The exercise begins with the application of 3 by 3 LHE windows; followed by 10 by 20 LHE windows per image. The third algorithm that is used for the evaluation purpose is the GHE scheme. Performances of above-mentioned three filters are compared with those of the two variants of the proposed algorithm in this work: that is, non-optimised FHEOBP and the optimized FHEOBP algorithms, respectively. Obtained spectrographs for test images along with corresponding image outputs under different filtering window sizes are shown in figures 3–5, for test images 2(a), 2(b) and 2(c), respectively, while output images are depicted in figure 6, for the three test images.

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The LHE algorithms show a very poor performance compared to the GHE algorithm. It can be observed that the performance of LHE improves with increase in the window size from 3 by 3 to 10 by 20. The GHE has a much better performance compared to the LHE. Although it is similar to the LHE, its window dimension is identical to the image proportion. The implication of this window dimension in GHE is to avoid leaving out of any of the intensity level in the entire image out in the equalisation, unlike in LHE. The output of the non-optimised FHEOBP shows great improvement over that of GHE algorithm. Optimising the FHEOBP with the TLBO algorithm shows the overall best performance as can be inferred from figures 3–6, which illustrate the obtained results for the three test images utilized in this study.

Apart from the generation of output images from different variants of histogram equalization employed, the level of luminance distortion is computed for each variants of histogram equalization using (15)–(17). The numerical values of parameters α and β are shown in table 1. These define the Gaussian function, for the three test images employed in this paper.

Table 1. Variation of Gaussian parameters with test images.

Test Image	$\alpha$	$\beta$
A	3.501 618 616	0.144 425 114
B	7.683 185 763	0.023 033 965
C	0.460 007 098	0.036 805 173

Table 2 presents the computed luminance distortion of different filters used for the three test images.

It is obvious from the results of brightness preservation presented in table 2 that FHEOBP algorithms (both optimised and non-optimised) out-perform both variants of LHE and GHE algorithms, for each of the three test images. In fact, computed luminance distortion figures of optimised FHEOBP are 16.4%, 28.3% and 20.1% higher than those of corresponding GHE, for test images figures 2(a)–(c), respectively. Between the optimised and non-optimised, luminance distortion figures of optimised FHEOBP are 8%, 6.8% and 9.8% more than those of non-optimised FHEOBP for test images figures 2(a)–(c), respectively. With these results, it is revealed that not only is FHEOBP algorithm suitable for the enhancement of underwater image contrast, it also preserves the brightness level.

Underwater images suffer from non-uniform lighting, thus, causing degraded images, and consequently, details are hidden. Image contrast enhancement improves the interpretability and readability of information content of underwater images. The proposed FHEOBP algorithm in this paper ensures the preservation of the brightness of the underwater image to be preserved while enhancing the contrast. The optimised FHEOBP has been shown to exhibit superior performance over LHE and GHE algorithms. This study has demonstrated the effectiveness of FHEOBP scheme on a set of three standard underwater images. This illustrates its effectiveness for contrast level enhancement and preservation of the image brightness.

Authors wish to acknowledge the effort of Mr Ifeta Adekunle during testing and debugging of MATLAB codes used in the generation of results for this work.