Underwater digital images captured by cameras cause undying noise, affecting the image with low contrast, lousy illumination, blurring, and diminished color. Numerous research studies have demonstrated the need for further enhancements in underwater image processing. In addition to underwater viewing, underwater imaging is a fascinating source of research, specifically in-depth estimation, color destination, and environment studies [1]. Image enrichment algorithms play a critical role in image enhancement’s pre-processing and post-processing phases [2]. Image pre-processing enhances the image by decreasing entropy through image filtering, noise reduction, color correction, segmentation, contrast enhancement, and feature extraction [3]. Schechner et al. [4] found that standard boosting methods like histogram equalization (HE) and gamma correction greatly enhanced visual perception. Despite their significant achievements, the practical applicability of gamma correction and HE techniques is limited. Yüksel et al. [5] presented an effective strategy for enhancing image quality using a fuzzy algorithm to filter out noise.

There have been numerous attempts to recover and enhance degraded images. Tang et al. [6] investigated the use of a solid hash for index images based on color vector angle and the canny operator, which combines interpolation with a Gaussian low-pass filter. In extreme light environments, haze and color distortions inside the water cause images to degrade significantly [7]. The scattering mode is what primarily causes the haze phenomenon. Reference [8] suggests a novel gradient-domain method for evaluating the quality of reduced reference images based on natural image statistical priors. The image derivatives describe the intricate geometric details of images that appeal to human visual perception.

Previously, colored filters applied to black-and-white pictures changed the tones of a scene file on film. A yellow filter enhances the prominence of clouds, as their white light remains unaffected by a yellow filter, in contrast to the blue sky which is altered. It significantly alters the most appealing aspect of the image. Most of these attempts focused on aerial photography, but some individuals also aimed to explore the underwater realm. This section surveys the literature on underwater image processing and improvement methods. Restoring original, precise scene contents from degraded underwater images is a challenging task. Researchers have experimented with numerous techniques to enhance the contrast of underwater photographs.

McGlamery [9] proposed a computational photo production model that simultaneously takes into account forward-scattered, backward-scattered, and non-scattered light characteristics to enhance underwater images. Schechner et al. [4,10] devised with a simpler way to make an image by leaving out the forward scattering term and employing the polarization filter technique to make up for the loss of visibility. He et al. [11] developed an enhanced dark channel prior (DCP) model to improve underwater images enrichment by demonstrating the connection between foggy or hazy conditions and backward scattering from airborne particles. The maximum intensity of purple prior (MIRP) measures scene depth by employing the highest intensity of the purple channel, based on the fact purple absorbs more light in water than other color channels [12].

Numerous scholars have addressed the color distortion issue in underwater image enhancement. To find the ratio of attenuation factors, Chiang et al. [13] grouped the ocean types and suggested that the attenuation factors of the transmission map in the photo formation model should change based on shade channels. Other researchers also addressed the color distortion problem in underwater image enhancement. After that, the study examined several underwater image enchantment conditions that it had discovered using historical global light as a guide [14]. Peng et al. [15] proposed a novel, reliable method by combining DCP and MIRP for the restoration of the underwater image. Researchers have developed gradient-based picture processing techniques [16] to address issues with dynamic variation and dehazing. Li et al. [17] talked about how the moderate rays’ absorption and scattering in the marine environment caused the degraded underwater image to have less contrast and uneven colors. Mi et al. proposed an innovative solution to low-complexity underwater image improvement based on dark channels to improve blurry images in deep-sea engineering [18]. According to Ref. [19], the dehazing and wavelength correction processes for underwater image improvement heavily rely on light scattering and image shadows. The algorithm proposed in Ref. [20] implemented a new photo-enhanced technique known as “significance degree” to address image-intensive artifacts, low contract, shade shrink, and noise in the underwater image enrichment process. We proposed an HE approach [21] to improve the rundown visibility of deep-sea pictures, addressing issues such as low image contrast, low visibility, intense artifacts, and noise.

A physics-based approach was developed by Li et al. [22] when they examined background light using quad-tree subdivision and graph-based complete segmentation. Moreover, the method verifies the medium transmission map by utilizing optical properties and an underwater imaging-based minimal information loss foundation. Utilizing the primary difference between the three color water channels that is established from attenuation, we can estimate the scene depth in a straightforward and effective manner [13]. The new, more realistic way of using the fusion method [23] makes it easier to define the inputs and weight maps that accompany them, which can improve the quality of the image. To enhance the contrast of the underwater image more effectively, HE [24] implemented a combination of differential grey-level histogram equalization and the adaptive Grow World technique. This combination not only improved the overall clarity of the image but also addressed and removed any unwanted color cast, ensuring that the true colors of the underwater environment were accurately represented. The differential grey-level HE specifically targets variations in intensity, while the adaptive Grow World method adjusts the processing based on local image characteristics, resulting in a more balanced and vibrant visual outcome. Liu et al. [25] introduced an unsupervised multi-expert learning framework that analyzes the red, green, and blue (RGB) channels of underwater images independently. This approach employs a multi-expert encoder coupled with a corresponding matching multi-expert discriminator to enhance feature extraction and classification accuracy in challenging aquatic environments. In addition, multi-level fusion should be considered to balance the appearance of the entire image. An enhanced network approach [26] based on the convolutional neural network (CNN) technique for real-time image dehazing requires less processing time without compromising performance.

Research studies on underwater image restoration frequently tackle the challenging shortcomings of improving the visibility and color accuracy of images captured underwater, according to the literature survey. Approaches that rely on the information loss principle and underwater imaging’s optical properties typically have some common limitations.

1) Uniform lighting conditions: Many algorithms assume uniform lighting conditions, which is rarely true in underwater environments due to light scattering and absorption. This can lead to inaccurate color correction and contrast enhancement.

2) Challenges with depth estimation: Correcting color distortion due to different wavelengths absorbing at different rates requires accurate depth estimation. However, estimating depth from a single underwater image is difficult and often leads to errors in restoration.

3) Lack of ground truth data: Collecting ground truth data for underwater scenes is challenging. Without accurate reference images, it is challenging to evaluate the performance of restoration algorithms objectively.

4) Light scattering and absorption complexity: Underwater light scattering and absorption are complex phenomena influenced by water type, turbidity, and particle presence. Simplified models may not capture these effects accurately, leading to suboptimal restoration.

5) Computational complexity: Advanced algorithms that model optical properties and minimize information loss often require significant computational resources, making them impractical for real-time applications or devices with limited processing power.

6) Water conditions: Various environments (e.g., ocean, lake, and river), and algorithms may not generalize well across these diverse conditions. We may need to customize our algorithms for specific conditions.

7) Handling dynamic scenes: Moving objects and dynamic scenes introduce additional challenges, such as motion blur and varying light conditions, which are difficult to address with static image restoration techniques.

8) Color constancy problems: Maintaining color consistency under varying underwater lighting conditions is difficult. Many algorithms struggle to maintain accurate colors, leading to unnatural-looking restored images.

9) Noise amplification: Restoration techniques can sometimes amplify noise present in the original image, especially in low-light conditions, leading to degraded image quality.

10) Limited benchmarking and standardization: There is a lack of standardized benchmarks and evaluation metrics for underwater image restoration, making it difficult to objectively compare the performance of different algorithms.

Understanding these limitations is crucial for researchers and practitioners to improve existing methods and develop more robust underwater image restoration techniques.

The proposed model evaluation presented in this article is a unique method of suppressing noise in underwater images taken with conventional digital cameras. The proposed method consists of three steps to eliminate noise in underwater images. The initial stage removes haziness, the second stage adjusts the image’s illumination, and the final stage uses adaptive filtering to enhance the underwater image. Fig. 1 depicts the overall process of the proposed model.

The spanning data investigated over nearly two decades was used to analyze the optical nature of ocean water [27], which is similar to the model of a hazy image. The distance-dependent model [28] proposed two methods to enhance quality of underwater image by compensating for distortion caused by color loss and haze. The first method is the light attenuation adopted from the Beer-Lambert law. The second method uses path radiance, also known as backscattering to improve the quality of images affected by ambient light. The overall underwater imaging equation is as follows:

$ {I}_{\lambda }\left(x{\mathrm{,}}\; y\right)=J\left(x{\mathrm{,}}\; y\right){t}_{\lambda }\left(x{\mathrm{,}}\;y\right)+\left(1-{t}_{\lambda }\left(x{\mathrm{,}}\;y\right)\right){A}_{\lambda } $ (1)

where I(x, y) is the pixel position, λ represents different wavelength channels, I is the captured image, J is the scene radiance, A is the homogeneous background light, and t is the residual energy ratio. From the Beer-Lambert law, the residual energy ratio is a function of λ and the distance between the object and camera, which can be mathematically represented as

$ {t}_{\lambda }\left(x{\mathrm{,}}\; y\right)={e}^{-\beta \left(\lambda \right)d\left(x{\mathrm{,}} \; y\right)} $ (2)

where β denotes the extinction coefficient of the medium and d represents the distance from the source within that medium.

By following the formation of the residual energy ratio in (2), the distance-dependent model can be formulated as

$ {I}_{\lambda }\left(x{\mathrm{,}}\; y\right)={J}_{\lambda }\left(x{\mathrm{,}}\; y\right){N}_{\lambda }^{d\left(x{\mathrm{,}} \; y\right)}+\left(1-{N}_{\lambda }^{d\left(x{\mathrm{,}} \; y\right)}\right){A}_{\lambda } $ (3)

where water composition affects the normalized residual energy ratio N.

In the work of Land et al. [29], oceanic water is divided into three categories. In Chiang’s work [13], the following value is used to quantify the normalized residual energy ratio for clearest coastal waters:

$ {N}_{\lambda }=\left\{ \begin{array}{l}0.80 - 0.85{\mathrm{,}}\; \lambda =650 \; {\text{μ}} {\rm{m}} - 750 \; {\text{μ}} {\rm{m}} \left({\mathrm{R}}\right)\\ 0.93 - 0.97{\mathrm{,}}\; \lambda =490 \; {\text{μ}} {\rm{m}} - 550 \; {\text{μ}} {\rm{m}} \left({\mathrm{G}}\right)\\ 0.95 - 0.99{\mathrm{,}}\; \lambda =400 \; {\text{μ}} {\rm{m}} - 490 \; {\text{μ}} {\rm{m}} \left({\mathrm{B}}\right)\end{array}\right. $ (4)

where the transmission of the red wavelength decreases faster as the distance increases.

Therefore, the red channel is always described as the dark channel. According to Fu’s work [30], we can solve the background light by

$ {A}_{\lambda }=\underset{\left(x{\mathrm{,}} \; y\right)\in I\left(i{\mathrm{,}} \; j\right)}{\mathrm{max}}{\text{min}}_{\in \Omega \left(x{\mathrm{,}}\;y\right)}{I}_{\lambda }\left(i{\mathrm{,}}\;j\right) {\mathrm{.}} $ (5)

A homomorphic filter is applied for every wavelength channel. Then, we obtain three Poisson equations for each wavelength channel. Finally, enhanced images are obtained using the Jacobi method.

The underwater imaging model in (3) can be formulated as follows:

$ {I_\lambda }\left( {x{\mathrm{,}}\;y} \right) = ({J_\lambda }\left( {x{\mathrm{,}}\;y} \right) - {A_\lambda }) + {A_\lambda }N_\lambda ^{{{d}}\left( {x{\mathrm{,}}\;y} \right)} {\mathrm{.}} $ (6)

Observing (6), the distance is part of the exponent. It is very difficult to obtain a numerical solution for this nonlinear relationship. This can be done using a logarithmic operation to remove the distance d from the exponent. Therefore, put the logarithmic operation on both sides of (6) given as

$ {\text{log}}({I_\lambda }\left( {x{\mathrm{,}}\;y} \right) - {A_\lambda }) = {\text{log}}({J_\lambda }\left( {x{\mathrm{,}}\;y} \right) - {A_\lambda }) + {\text{log}}{N_\lambda }d\left( {x{\mathrm{,}}\;y} \right){\mathrm{.}} $ (7)

This relationship provides a simpler expression than (6) with the distance d.

In (7), the distance d and normalized residual energy ratio N are unknown. Although (4) gives a range for the normalized residual energy ratio N, we also need to estimate its exact value. Previous algorithms often preferred to estimate these two unknowns from degraded underwater images [31]. Unlike these algorithms, our algorithm recovers scene radiance without solving these unknowns. The adjacent pixels are always from the same object. The positions of adjacent pixels belonging to the same object are very close. Therefore, the distance between the observer and adjacent pixels is approximately equal. However, the distances of adjacent pixels are slightly different due to the variant detail. However, the pixels’ distances in the same local patch are homogeneous. The most frequently used local patches are four neighborhoods and eight neighborhoodhomogeneousof the same local patch leads to (8).

$ d\left( {x{\mathrm{,}}\;y} \right) = 1/4\left[ {d\left( {x + 1{\mathrm{,}}\;y} \right) + d\left( {x - 1{\mathrm{,}}\;y} \right) + d\left( {x{\mathrm{,}}\;y + 1} \right) + d\left( {x{\mathrm{,}}\;y - 1} \right)} \right]{\mathrm{.}} $ (8)

Let left-hand side equal to J_λ(x, y), and the first item of the right-hand side is equal to I_λ(x, y). The same relation for its neighbor is utilized in this work. Then, the relation for every pixel is combined with the weight and finally, the following Poisson is obtained:

$ {\tilde J_\lambda }\left( {x{\mathrm{,}}\;y} \right) = {\tilde I_\lambda }\left( {x{\mathrm{,}}\;y} \right) + {\text{ln}}({N_\lambda }\left( {x{\mathrm{,}}\;y} \right) + c)d\left( {x{\mathrm{,}}\;y} \right) {\mathrm{.}} $ (9)

An equal relationship with its neighbor is utilized in this work. Then, the relation for every pixel is combined with the weight, and finally, the following Poisson is calculated:

$ {\nabla ^2}{\tilde J_\lambda }\left( {x{\mathrm{,}}\;y} \right) = {\nabla ^2}{\tilde I_\lambda }\left( {x{\mathrm{,}}\;y} \right) $ (10)

where $ \nabla $ is the Laplace operator.

This is a linear system of equations about J, which can be solved using the Jacobi iterative approach. Equation (11) provides the intermediate outcomes of the Jacobi iterative method as follows:

$ {\tilde J^n}_\lambda (x{\mathrm{,}}\;y) = \frac{1}{4}\left( {{{\tilde J}_\lambda }^{n - 1}\left( {x - 1{\mathrm{,}}\;y} \right) + {{\tilde J}_\lambda }^{n - 1}\left( {x + 1{\mathrm{,}}\;y} \right) + {{\tilde J}_\lambda }^{n - 1}\left( {x{\mathrm{,}}\;y - 1} \right) + {{\tilde J}_\lambda }^{n - 1}\left( {x{\mathrm{,}}\;y + 1} \right) + {\nabla ^2}I\left( {x{\mathrm{,}}\;y} \right)} \right) $ (11)

where n represents the number of iterations.

By repeating (11), we can obtain the numerical solution of (9). The solution is the estimation of the underwater scene radiance. The initial values of the iteration are set in keeping with the degraded underwater image.

The digital camera is unable to maintain the appearance of a scene with a large dynamic range due to sensor hardware restrictions. These issues were intended to be resolved by using the advanced multi-Retinex method. Although Retinex improves visibility and color consistency, it occasionally has lower global contrast, the halo effect, and color distortion.

Most underwater images appear green or blue because green light has the longest wavelength at which it may pass through water. To address the color cast, a color-correcting technique mostly based on statistical methods is used. S is the given undersea image in this case. The implied rectangle error and the inferred pricing are calculated using the formulas for the RGB channels of S. The maximum and minimum for each channel can be found by

$ S_{{\text{max}}}^c = S_{{\text{mean}}}^c + \mu S_{{\text{Var}}}^c $ (12)

$ S_{{\text{min}}}^c = S_{{\text{mean}}}^c - \mu S_{{\text{Var}}}^c $ (13)

where c$\in $ {R, G, B); S_min, S_max, S_mean, and S_var denote the minimum, maximum, mean, and variance of the c channel’s underwater captured image. The parameter µ controls the image dynamic range. If the parameter value is too small, it will cause the result image to be over-enhanced. If the parameter value is too high, the result image will be under-enhanced. We normalized the S_min and S_max values to achieve enhanced result. The color-corrected image is ultimately yielded by

$ {S_{\mathrm{CR}}^c} = ({S^c} - S_{\mathrm{min}}^c)/({S_{\mathrm{max}}^c} - {S_{\mathrm{min}}^c})255{\mathrm{.}} $ (14)

The underexposure problem can be solved with the Retinex approach since the underwater environment is similar to the shift in illumination. The human visual system can adjust to illuminations that vary in both brightness and color, as demonstrated by the Retinex theory [28,29]. An advanced multi-scale Retinex (MSR) algorithm [30] was used in the proposed study to separate the illumination and reflectance from the luminance layer. This was done to deal with the problem of not enough light.

Three components make up Retinex’s foundation. There is no color in the world in which we live; instead, shade is the product of the interaction of light and matter. The second is that RGB can be used to create any coloration. According to the 1/3 rule, RGB colors reflect the arbitrary unit position. A well-known method for improving images, the Retinex algorithm is predicated on the idea of color constancy. It separated the image into two components, radiation and reflection, each expressed as

$ I\left( {x{\mathrm{,}}\;y} \right) = L\left( {x{\mathrm{,}}\;y} \right)R\left( {x{\mathrm{,}}\;y} \right) $ (15)

where I(x, y) denotes the original image, L(x, y) stands for the lighting component, noise, haze, and other environmental interference that must be eliminated from the image, and R(x, y) represents the reflection component including the image details of the target item.

Since the human eye sensitivity has the logarithm property, we can write (15) as

$ \text{log }R\left(x{\mathrm{,}}\;y\right)=\mathrm{log} L\left(x{\mathrm{,}}\;y\right)+\mathrm{log} I\left(x{\mathrm{,}}\;y\right) {\mathrm{.}} $ (16)

In (16), L(x, y) is written as

$ L\left(x{\mathrm{,}}\;y\right)=F\left(x{\mathrm{,}}\;y\right)I\left(x{\mathrm{,}}\;y\right) {\mathrm{.}} $ (17)

We can express the Gaussian function F(x, y) as

$ F\left(x\mathrm{,}\; y\right)=Ke^{\left[\frac{-\left(x^2+y^2\right)}{\sigma^2}\right]}\mathrm{.} $ (18)

If σ represents the standard deviation, the improved image’s hue may become warped. The normalization coefficient, K, must meet the following requirement:

$ \iint F\left(x{\mathrm{,}}\;y\right)\text{d}x\text{d}y=1{\mathrm{.}} $ (19)

When the Gaussian core function simply looks at the geometric distance between adjacent pixels and ignores the brightness difference between nearby pixels, the edge fuzzy phenomenon happens. A kind of Gaussian filtering known as “bilateral filtering” involves adding a parameter to the filter coefficient that indicates the brightness difference between adjacent pixels. Bilateral filtering eliminates the edge fuzzy problem associated with Gaussian filtering and improves edge preservation. The formula for bilateral filtering is as follows:

$ F\left( {x{\mathrm{,}}\;y} \right) = \frac{{ {\displaystyle\sum \limits_{m = i - p}^{i + p} } \displaystyle\sum \limits_{n = j - p}^{j + p} d\left( {m{\mathrm{,}}\;n} \right)\chi \psi }}{{ {\displaystyle\sum \limits_{m = i - p}^{i + p} }\; {\displaystyle\sum \limits_{n = j - p}^{j + p} } \chi \psi }} $ (20)

where $d\left( {m{\mathrm{,}}\;n} \right)$ and $F\left( {x{\mathrm{,}}\;y} \right)$ are the input and output images, respectively; $p$ represents the filtering window size; $\chi $ and $\psi $ are geometric distances. We can now express the brightness difference between adjacent pixels as follows:

$ \chi = {e^{ - {\sigma _d}\left( {{{\left( {i - m} \right)}^2} + {{\left( {j - n} \right)}^2}} \right)}} $ (21)

$ \psi = {e^{ - {\sigma _l}{{\left( {d\left( {i{\mathrm{,}}\;j} \right) - d\left( {m{\mathrm{,}}\;n} \right)} \right)}^2}}}$ (22)

where d is the geometric distance parameter and l is the brightness difference parameter. On the basis of traditional bilateral filtering, Gaussian-weighted bilateral filtering is an enhanced algorithm. With the following equation, it enhances the parameter expressing brightness difference.

$ \psi = {e^{ - {\sigma _l}{{\left( {\left| {d\left( {i{\mathrm{,}}\;j} \right) - d\left( {m{\mathrm{,}}\;n} \right)} \right| - \xi } \right)}^2}}} $ (23)

where the selection of parameter ξ will affect the image enhancement. This shows that as the value decreases, the halo phenomenon becomes less visible. When the value of 20 is reached, the halo effect is fully erased, and we are left with enhanced visuals with crisp contours. The MSR algorithm selects the small scale, middle scale, and large scale, so it is better than the simple Retinex algorithm.

$ {I}\left({x}{{{\mathrm{,}}}}\;{y}\right)=\sum _{{i}=1}^{{K}}{{W}}_{{i}}{{R}}_{{i}}\left({x}{{{\mathrm{,}}}}\;{y}\right) $ (24)

where $ \displaystyle\sum \limits_{i = 1}^K {W_i}{R_i} = 1$ and K is the scale number. If K=1, MSR returns to the Retinex algorithm. The advanced MSR algorithm is based on an arbitrary Gaussian distribution function. R_i(x, y) is the reflectance result of single scale Retinex (SSR) with the i-th color channel and the n-th scale. The color restoration factor W_i (weighted sum of the outputs of several SSR) is expressed as

$ {W_i}\left( {x{\mathrm{,}}\;y} \right) = {\mathrm{log}} \left[ {\alpha {f_i}\left( {x{\mathrm{,}}\;y} \right)} \right] - {\mathrm{log}} \left[ {{f_i}\left( {x{\mathrm{,}}\;y} \right)} \right] {\mathrm{.}}$ (25)

To normalize I(x, y) into the 8-bit range (0–255), the mean value μ_i and the mean-square standard deviation σ_i of I(x, y) are calculated. The coefficient α is a dynamic coefficient, generally [0.5, 2]. The value of the smaller dynamic coefficient provides bigger image contrast enhancement. The normalized result of I(x, y) defined using the values of b = 256, Max_i = µ_i + ασ_i, Min_i = µ_i − ασ_i, and ∆_i= Max_i − Min_i as follows:

$ {I}^{\prime }\left(x{\mathrm{,}}\;y\right)=b\left(I\left(x{\mathrm{,}}\;y\right)- {\text{Min}}_{i}\right)/{\varDelta }_{i} {\mathrm{.}} $ (26)

The guided image filter (GIF) is used to prevent gradient reversal artifacts. The GIF filtering process begins with the image $G$, which could be the input image $I$ or a different image. Suppose ${W_k}$ is the kernel window centered at pixel $k$ and ${\left( I \right)_P}$. The GIF process computes the intensity value at pixel $p$ and pixel $k$ of the input and guided images, denoted as $G_p$:

$ \text{GIF}{\left(I\right)}_{P}=\frac{1}{\displaystyle \sum _{qϵ{W}_{k}}{W}_{{\text{GIF}}_{p{\mathrm{,}\;}q}}\left(G\right)}\displaystyle \sum _{qϵ{W}_{k}}{W}_{{\text{GIF}}_{p{\mathrm{,}\;}q}}\left(G\right) {I}_{q}{\mathrm{.}} $ (27)

From (27), the kernel weights function ${W_{{\text{GI}}{{\text{F}}_{p{\mathrm{,}}q}}}}$can be written as

$ {W}_{{\text{GIF}}_{p{\mathrm{,}}\;q}}\left(G\right)=\frac{1}{{\left|w\right|}^{2}}\displaystyle \sum _{k\left(p{\mathrm{,}}\;q\right)ϵ{w}_{k}}\left(1+\frac{\left({G}_{p}-{\mu }_{k}\right)\left({G}_{q}-{\mu }_{k}\right)}{{\sigma }_{k}^{2}+\epsilon }\right) {\mathrm{.}} $ (28)

where ${\mu _k}$ and $\sigma _k^2$ are the mean and variance of the guided image G in the local window w_k, and |w| is the number of pixels in this window. When both G_p and G_q are simultaneously on the same side of an edge, the weight assigned to pixel q is the maximum. Pixel q will receive a small weight when G_p and G_q are on opposing sides. GIF can also be minimized as

$ \text{GIF}{\left(I\right)}_{P}=\displaystyle \sum _{qϵ{\omega }_{k}}{W}_{{\text{GIF}}_{p{\mathrm{,}}\;q}}\left(G\right) {I}_{q} {\mathrm{.}} $ (29)

The parameter $ \varepsilon $ modifies the amount of GIF smoothing. With a higher value of $ \varepsilon $, the filtered image will be smoother. Using (1), we may reconstruct the scene radiance by computing the transmission depth map with $ \varepsilon = 0.2 \times 0.2$. Because we can limit the transmission t(x) to a lower bound t, very dense haze regions retain a tiny amount of haze. We can express J(x), the final scene’s radiance, as

$ J\left( x \right) = \frac{{{I_c}\left( x \right) - {A_c}}}{{{\text{max}}\left( {t\left( x \right){\mathrm{,}}\;{t_0}} \right)}} + {A_c}. $ (30)

Because the transmission is not always constant in a patch, the restored image may contain block artifacts due to rough transmission. He et al. [11] proposed guided filtering to replace the soft matting to reduce the complexity and greatly improve the computational efficiency. The restored image demonstrates the effectiveness of the proposed model. To avoid producing too much noise, we restrict the value of the transmission t(x) to the range between 0.1 and 0.9. RUDIE achieves a promising visual outcome by transmitting t(x) to a lower bound t, while very dense haze regions retain a tiny amount of haze. Fig. 2 shows the proposed model’s final results of dehazing images.

We compare the proposed model RUDIE with conventional underwater image models [12,22–24] using standard image performance metrics ENTROPY, perception-based contrast quality index (PCQI), underwater color image quality evaluation (UCIQE), and peak signal-to-noise ratio (PNSR). These metrics are essential to evaluating the degraded image and are standard metrics for evaluating image improvement. Fig. 2 demonstrates that the original aquatic snap input image exhibits significant color distortion, resulting in an overall green color tone despite the statue’s sanguine hue and the diver’s black attire.

The images used in the proposed model are 640×480 pixels in size. The visualization shows that the proposed model works efficiently for aquatic images. We conducted quantitative analyses in addition to the visual examination of the images outlined in Fig. 2. Tables 1, 2, 3, and 4 illustrate the experimental results conducted for Fig. 2. The results demonstrate that the RUDIE method effectively reduces haziness and improves the visibility of deep ocean photos.

ENTROPY is a standard image evaluation metric to measure randomness or uncertainty in the image, mathematically formulated as

$ {\text{ENTROPY}} = - \mathop \sum \limits_{i = 1}^n {p_i}\left( {{\text{lo}}{{\text{g}}_2}{p_i}} \right) $ (31)

where p_i represents the probability that two adjacent pixels differ by i.

Table 1 shows the results of ENTROPY’s quantitative performance for Fig. 2. The ENTROPY scores demonstrate that the proposed method can effectively increase the information about the contrast and visibility of underwater images.

The proposed RUDIE method has the best average ENTROPY score compared to the benchmark methods listed in Table 1. The proposed method can restore the balanced chroma, contrast, and saturation of improved underwater images, as well as achieving higher average ENTROPY than the standard models [12,22–24].

We chose the UCIQE metric as the alternate measure. It assesses and corrects the non-uniform color cast, poor discrepancy, and blurring that characterize aquatic photos. UCIQE can be mathematically defined as

$ {\text{UCIQE}} = {c_1}{\sigma _c} + {c_2}{\text{co}}{{\text{n}}_l} + {c_3}{\mu _s} $ (32)

where $ {\sigma _c} $ is the hue standard deviation; $ {\text{co}}{{\text{n}}_l} $ is the contract of luminance; $ {\mu _s} $ is the average of saturation; $ {c_1} $, $ {c_2} $, and $ {c_3} $ are weighted portions. $ {c_1} $ = 0.4680, $ {c_2} $ = 0.2745, and $ {c_3} $ = 0.2576 are the portions utilized in this paper.

From Table 2, we can see that the proposed model gives better UCIQE scores than previous models [12,22–24], and fashion generates a better UCIQE average score than existing models.

PCQI used an adaptable representation of the patch structure original shape to figure out how changes in discrepancy affect the quality of an image. PCQI is mathematically formulated as

$ {\text{PCQI}}\left( {x{\mathrm{,}}\;y} \right) = \frac{1}{M}\mathop \sum \limits_{j = 1}^M {q_i}\left( {x{\mathrm{,}}\;y} \right){q_c}\left( {x{\mathrm{,}}\;y} \right){q_s}\left( {x{\mathrm{,}}\;y} \right) $ (33)

where x refers to the original image, y is the test image, and M stands for the total number of patches. Based on Ref. [12], any image patch can be considered having three conceptually independent factors $ \displaystyle\sum \limits_{j = 1}^M {q_i}\left( {x{\mathrm{,}}\;y} \right){q_c}\left( {x{\mathrm{,}}\;y} \right){q_s}\left( {x{\mathrm{,}}\;y} \right)$ representing the mean intensity, signal strength, and signal structure. PCQI measures an image’s perceptual quality, especially in terms of contrast. A high PCQI value indicates a strong similarity between the perceived contrast of the test image and the reference image. This means the enrichment as likely improved the image without introducing undesirable artifacts or discrepancies. Conversely, a lower PCQI value suggests a greater discrepancy between the test and reference images. Depending on the desired outcome of PCQI could be considered “better” in terms of achieving a specific visual effect or highlighting contrast differences.

The proposed system generates the stylish average quantitative scores for the PCQI evaluation scores compared to the methodologies of Refs. [12,22–24], as illustrated in Table 3.

Tables 2 and 3 present the quantitative results of UCIQE and PCQI, respectively, for Fig. 2. This shows that the proposed approach can recover the enhanced aquatic print’s well-balanced hue, achromatism, and discrepancy.

PSNR is a standard quality metric used to analyze the quality of the restored image when affected by noise and blur. High PSNR values indicate that the image quality is high. The mean square error (MSE) values determine the image’s PSNR value. For result analysis, the PSNR values of two images are computed by considering MSE between the pixel intensities and taking the maximum possible intensity.

$ {\text{PSNR}} = 10{\text{lo}}{{\text{g}}_{10}}\frac{{{\mathrm{max}} {L^2}}}{{{\text{MSE}}}} {\mathrm{.}} $ (34)

A statistical metric called MSE is employed to assess the quality of an improved image. There is no error when MSE of the suggested model is equal to zero. The more inaccuracy there is in a given value, the higher the MSE value.

$ {\text{MSE}} = \frac{1}{{MN}}\mathop \sum \limits_{i = 1}^M \mathop \sum \limits_{j = 1}^N {(x(i{\mathrm{,}}\;j) - y(i{\mathrm{,}}\;j))^2} $ (35)

where M and N denote the number of horizontal and vertical pixels, respectively, x(i, j) represents the filtered image of coordinators i and j, and y(i, j) is the noise.

The performance of RUDIE compared with traditional methods in Refs. [12,22–24] in terms of the PSNR metric is shown in Table 4 for Fig. 2. The first input image has many color distortions, resulting in an overall green color tone despite the statue’s reddish hue and the divers’ black uniforms. The enrichment method produces the lowest MSE and greatest PSNR values while effectively increasing the visibility of underwater images. As a result, the suggested model outperforms the traditional techniques in terms of PSNR in Fig. 2.

Biological activity causes absorption and scattering at the water’s surface, affecting the visibility of underwater images. This research article proposes the RUDIE approach that uses an advanced multi-Retinex method to enhance underwater images. The technique is based on the fusion concept and requires no additional information beyond the original image. The proposed approach enhances deep sea images by improving quality and quantitative metrics compared with existing approaches. Furthermore, it improves the quality of the digital image and reduces noise. A comparison study of aquatic digital image quality assessment metrics showed that the proposed method did a much better job of post-processing images than benchmark models. The proposed research article focused on improving underwater images’ quality, turbidity, and color by reducing noise. In our future work, we plan to implement a novel image dehazing technique to restore digital images with high visual quality. Furthermore, in the future, we will address how the proposed framework will enhance the quality of various real-time images, including satellite and medical images.

The authors declare no conflicts of interest.