Hybrid model of alternating least squares and root polynomial technique for color correction

Color correction is an image-altering technique that modifies image color in such a way that it matches a reference image. Many approaches have already been proposed by various researchers; however, those models have been unable to reduce color errors between two images, which results in inefficiency and poor-quality images. This research paper presents an effective and improved color correction model wherein alternate least square (ALS) and root polynomial (RP) are used together. The main objective of the proposed model is to reduce the error between a reference image and a target image to enhance the image quality and make them look realistic. To achieve this objective, the proposed model used the Amsterdam library of object images which contains a picture of single objects captured under various illumination angles and colors. The main contribution of this paper is a hybrid ALS + RP color correction technique, implemented on the dataset image that fixes its color as per the reference image and enhances its quality. The target image is then converted into three color models, i.e., LAB, LUV, and RGB into XYZ format. Finally, the color difference between a reference image and a target image is observed by calculating values for parameters like Mean, median, 95% quantile, and maximum error. The effectiveness of the suggested hybrid color correction approach is assessed and validated in MATLAB software for each color model. Through extensive experiments, it is observed that the proposed hybrid model yields the least errors for the RGB color model. This is followed up by LUV and then LAB, to prove its supremacy over other models.


Introduction
The quality of a colored image taken by a digital camera is influenced by the surface attributes of the items present in the image and the illuminating conditions and features of the camera through which the image is captured (Gasparini and Schettini 2003a). As each image sensor has characteristics that provide distinct digital images, color variations can be observed in multi-view images taken using several types of cameras. In addition to that, various external and internal factors like the direction of light, exposure time, and the white balance, etc. may give rise to color inconsistency issues in images even when captured using the same camera. This differentiation in image color degrades the potential of applications that make use of multi-view images like image stitching related to panoramic images on mobile phones (Brown and Lowe 2007a), video stitching with multiple camera-based surveillance systems (Kim et al. 2017;Xu and Mulligan 2010), and 3D reconstruction (Schonberger and Frahm 2016). There are often graphically unpleasant artifacts around the delineation of overlapped zones as a result of these color discrepancies. Therefore, many scholars have formulated (Shin et al. 2012) many color correction approaches for minimization of color inconsistency which arises in multiview images by applying the color features of an intended image to different source images. While editing pictures or videos, one of the easiest and simplest tasks for a professional image editor is to balance color patterns in images. Generally, an image is selected by the expert who they consider as a required target image, and then perform changes to further images for equating the required color pattern. This entire procedure is known as a color transfer. In this procedure, the expert artists need to thoroughly change or balance the distinct characteristics like mapping of color, exposure, white point, brightness, etc., and hence, it is not an easy task. Moreover, these changes are not independent and may impact other characteristics, like features that were formerly adjusted may get disturbed due to the tuning of another feature. Also, nonlinear processing of an image, like JPEG block edges, results in the emergence of artifacts followed by color adjustment, and hence, the development of an automated system would be preferable to perform this time taking a job. Reinhard et al. (2001) were the first to adopt an example-based color transfer system. And afterward, a lot of study work has been carried out in this regard (Nguyen et al. October 2014;Pitié et al. 2007;Pouli and Reinhard 2010). Color homography theorem is the latest development that has confirmed homography-based regulation of color approaching an interchange according to conditions like camera, illuminating, shading, etc. To estimate several color transfer outcomes, a method has been presented in this work that relies upon the color homography theorem. But it is very essential to understand the meaning of the color homography theorem.

Color homography theorem
The color homography theorem Inlayson et al. 2017;Grogan and Dahyot 2019) states that chromatic behavior around a modification during capturing of an image in various conditions like imaging device, light color, and shading is a homography apart. Using a 3 9 3 full rank matrix C, mapping of an RGB q to correspondent RGI (red-green-intensity) c is carried out as follows: Here the chromaticity coordinates, that is, r, can be represented as r = R/(R ? G ? B) and g can be represented as g = G/(R ? G ? B). The right-hand side of the above equation is elucidated as a homogeneous coordinate, and it is equal to c / rg1 ½ T which can also be represented as While the shading effect in an image is kept constant, any change in the lightening or camera can be reflected in their respective RGBs that are interrelated to a 3 9 3 linear transform M by the below-given equation: where q 0 represents the respective RGBs ratio when it is analyzed by the lightning or taken by any other capturing apparatus (Maloney 1986;Marimont and Wandell 1992;Hwang, et al. 2019). Furthermore, H=C -1 MC is responsible for mapping the colors in RGI among various illuminating conditions. The impact of multiple shading can be seen on the RGI as well in second light and can be denoted as: where / represents the unknown scaling. For an instance, let us consider that c is a homogenous coordinate and doesn't lose any generality; this means the 3rd component will be 1; in this case, where r, g represents the chromaticity coordinates, where The mapping between the two chromaticity charts is exactly a homography if the lighting color shift is linear.
To map the geometrical spatial coordinates of one picture to its identical in another picture, homography is employed (Abebe, et al. 2018;Fan et al. 2021). The 2D-2D match issue like mapping of chromaticity to chromaticity involved in color homography is also a point of concern, although, for the color transfer in other applications, there exists an appropriate alternative which is 3D color-3D color mapping homography in a way unrelated to shading. Concerning actual environmental changes like summer to autumn or lightning changes like illumination, a color transfer is elucidated as the re-rendering of an image.
In this research, an effective color correction model is presented in which one photometric view is mapped to other by mapping the color precisely without ant shading effect. We are essentially looking for the 3 * 3 map that maps the color rays in one photometric view to matching rays in another one because shading only alters the luminance, or amplitude, of the RGB vector. Many color correction models have been presented in the literature; however, the problem with current methods is that they employ only a single color correction technique in their work. Because of this, the error between two images, i.e., reference image and color-corrected images, is higher which leads to poor performance of the entire color correction model, which ultimately leads to poor visual quality as well. To solve this problem, we have proposed an effective and highly accurate color correction model in which alternating least squares and root polynomial techniques are hybridized. The proposed hybrid color correction model can solve the color transfer problem more effectively than a single technique-based color correction model. By doing so, an efficient color transfer mechanism is proposed wherein the error between two images is the least which means high visual quality.

Research motivation
According to our theory, a homography will connect the chromaticity distributions for a similar scene illuminated by two distinct lights, while the picture shading may alter. The goal of this paper is to fix colors in the target image as per the reference image in such a way that error between two images is reduced that aids in enhancing the visual quality of images. Moreover, the color correction method to be implemented must have a low computational cost and can be implemented in real-time scenarios. By applying color space homography, the final image obtained resembles very much with input image of the same scene under different lighting conditions. Several color homography models can be interpreted into a majority of color transfer effects. Color homographies may be computed quickly, unlike many color transfer procedures, which makes this conclusion useful. We have discovered that it is possible to implement the homography to the full-resolution image after running a computationally expensive technique on a thumbnail image and calculating its homography results. In the proposed work, we have solved the color transfer problem by finding the best least square transform mapping. Results demonstrate how effective our theory of color space homography is. In terms of color correction, we observe a notable increase in color accuracy when compared to the usual color correction techniques.

Major contributions of the research
The major contributions of this research are: • To read and analyze the images that are present in the given Amsterdam Library of Object Images (ALOI) dataset and select images from this dataset. • Convert these images into three different color models in XYZ format so that it is easy to identify color coordinates • To implement a hybrid ALS ? RP color correction technique for fixing the color issues in the target image. • In the next step, the color difference between a reference image and a target image is examined. • Finally, the performance of the proposed ALS ? RP model is tested and validated by comparing it with a few state of art methods in the context of various performance metrics.

Literature survey
The authors in Finlayson et al. (2017) produced three illustrations of homography. At first, the homography problem was derived from color calibration. In the second step, homography matching was used to equate a colorful image chromaticity distribution with the dataset containing object chromaticity distributions. The mapping of color present in an image was established the obtaining a final image matching the target image. It was concluded in the study that the re-interpretation of natural image color transfer was taken as a color homography mapping. Experimental observations confirmed that enhanced object recognition based on color and precise calibration was obtained by the elucidation of the color homography problem. This further enabled the authors to propose a novel framework for the development of natural color transfer algorithms. Similarly, in Gong et al. (1608), color-homography-derived color transfer decomposition was suggested by scholars in which shading adjustment and chromaticity shift were blended to encipher the color transfer strategy. The global shading curve was formulated to present the robust shading adjustment, and with its use, the same shading homography was available anywhere. For improvement in color transfer and re-implementation in video color grading, the authors illustrated two applications. It was noted that the suggested color transfer method would be useful for the implementation of modern color transfer models. A method for color correction was proposed in Finlayson et al. (2015) that computed estimation in varying intensity and matrix of 333 transformations at the same time taken from an image of a color chart. The authors developed a simple procedure called an alternating least squares procedure to solve the color correction problem which was originally taken as the discovery of the 333 correction matrix. The experimental observations formed the basis for the validation of their suggested method. The authors in Zhou and Li (2019) presented Homography FpnNet which when applied on an image pair aligned centrally, resulted in a minute approximation error regarding other methods available. When an image pair that was not aligned centrally was employed with the suggested hierarchical Homography FpnNet approach, better outcomes were obtained in comparison to the approaches which used random sample consensus and artificially designed features. The 3-stage Homography FpnNet was also applied on wall images taken from mounting robots, and its experimental results showed a minute mean corner pixel error while an enhanced performance was showcased in homography estimation of camera images and wall map. On a GPU, a 10.8 ms average processing time was obtained using the suggested hierarchical Homography FpnNet. The processing speed taken in real time met the needs of wall-mounting robots. The scholars in Hua et al. (2019) proposed a novel strategy to approximate the homography sequence online. This homography sequence was applied to those sequences of images that were taken from robotic vehicles having embedded sensors for vision. The suggested strategy used components of a set of homographies, viz. the gyroscope measurements as well as the subsidiary special linear group structure, and to produce a temporal filter to approximate the homography, it performed point feature correspondences between images. To estimate the photo-realistic color transfer method, the scholars in Gong et al. (2019) presented a color homography method powered by 3D. The method worked as a hybrid model of mean intensity mapping and a 3D perspective transform. The suggested recoded 3D color transfer was beneficial as it was precise and not complex. It was well established through demonstrations that the proposed color transfer method would be able to provide eminent color transfer potential in terms of recoding. Moreover, several operations like colorrobust image stitching, complex color transfer acceleration, and color transfer artifact fixing could also be executed using the suggested color homography method. Moreover, as pictures could be captured using several cameras, with distinct settings of the camera, and at any time of the day, the color of the scenery showed variations in every captured image. Therefore, for the adjustment of colors in every image, the scholars in Hwang et al. (2019) used a scattered point interpolation scheme to demonstrate a new color transfer algorithm. In this study, the scholars used to move least squares method for elucidation of 3D RGB space color mapping using a nonparametric and full nonlinear color mapping. To enhance the color transfer in the suggested techniques, the transfer was integrated with probabilistic modeling in the spatial constraints and in the 3D color space with the purpose to handle misalignment, illuminations that varied spatially, and noise. The authors in Takahashi et al. (2016) took into consideration the process of color correction and hence framed an efficient pipeline of color correction for a noise-containing image. There were two major parts of the pipeline namely, denoising and color correction. The matrices of color correction were computed by spatially varying color correction (SVCC) in the color correction process for a particular local image block while reviewing the noise effect. While for the process of denoising, an efficient model was presented by the authors to denoise the color-corrected image containing spatially distinct levels of noise. Through analysis and observations, the better performance of the suggested model was confirmed when compared with available color correction methods for a noise-contained image. The scholars in Tao, et al. (2021) adopted a 3-step strategy for efficient color correction procedures applied on below water aquatic images. In the foremost stage, for the selection of the leading and deteriorated input channel(s) or media, a new method was presented. In the second stage, the deteriorated channels were compensated locally with the help of a parameter adaptive-adjusted color compensation algorithm. In the third stage, the resultant was assessed when the obtained color-compensated image was fed with the widely used histogram equalization and the conventional gray-world assumption. Through subjective and objective evaluations, the enhanced performance of the suggested strategy was observed. The authors in Diaz et al. (2021) presented a blended approach including color homography for automatic color correction of images captured in dim light and K-means clustering for color extraction from the image. The suggested approach worked with the purpose of camera calibration elucidation about color and minimization in datasets to be used.
A new approach for color correction was introduced by scholars in Li et al. (2022) to significantly reduce the discrepancies in colors of large-scale multi-view images. The approach used the graph partition method for grouping images, followed by color correction for intragroup and intergroup, respectively. Initially, with the use of the local homography principle, the dependent similar-matched regions across the sparse points were computed for every image pair based on color correspondences. Later, the partitioning of large-scale images was completed to attain several groups of images. The correction parameters of every single attained group were computed for the deletion of color discrepancies in the images of the group. At last, an intergroup correction was applied to delete the group color discrepancies with the purpose to make the groups free of inconsistencies. Several datasets were applied with the suggested approach which determined the better performance as compared to other available approaches. Also, for stereoscopic camera models, the scholars in Failed (2011) used the feature detection method and stereo matching based on RANSAC for the automated discovery of stereo pair correspondences. Next, the color differences that existed between the stereo pair were minimized using a color transform matrix. Various aspects like color transform approaches, matching processes, and feature detectors were accountable for the accuracy rate of color correction. For approximation and compensation of the color homography in a numerical format to rebuild the actual color homography, the opposite correspondences taken from various down-scaled images were extracted with the use of a method adopted by scholars in Seo et al. (2013). This numerical approach ensured adjustment in the tone of color occurred between the targeted image and referenced image if unevenness was figured out. To do so, the on-site knowledge of overlapped regions was used. Through analysis and experiential observations, it was found that the suggested approach worked fivefold quicker in 8 Mp images and threefold quicker in 3.2 Mp images as compared to the traditional methods. Hence, it was confirmed that the suggested approach proved beneficial to effectively approximate a color homography. The authors in Yamakabe et al. (2020) worked on the tunable color correction (TCC) method which was a color correction approach. The method allowed the color correction matrix tuning that existed between the PCC model and the LCC model. To figure out the efficient trade-off balance of the TCC method, a mean squared PCC model for calculating error was formulated. The efficient trade-off balance was confirmed for noise-containing images through experiential observations along with that its performance surpassed the PCC and LCC efficiency. In addition to that, TCC was summarized for multispectral cases. For determining the efficacy of the suggested method, they performed color correction on an RGB-near-infrared sensor.
To perform recoloring of transparent images along with maintenance of color relationships and comprehensive details of the captured image, the authors in Huang et al. (2018) presented a novel strategy. In this approach, the problem of recoloring was considered a location transformation issue and was solved in two major stages, namely automatic palette extraction and homography estimation. For this aim, the influential colors of the captured image were extracted using the proposed H-means approach which relied upon clustering and histogram statistics. Next, the mapping of source colors with the wanted colors of the CIE-LAB was carried out with another homography estimation method presented by the authors. In addition to that, the authors introduced a non-linear optimization strategy for enhancing the outcomes produced in the final step. Hence, greater accuracy and fidelity of the source image were affirmed by the suggested model. Several practicals were conducted which showcased that the methodology produced an enhanced visual outcome, specifically in the areas covered with shadow. Also, the efficacy of the approach was validated by the source images possessing the real characteristics which were produced by a ray tracer. In (Zhong and Quan 2021), the scholars focused on CP discovery which might be present between right and left images and hence proposed two approaches, namely homography transform-based stereo matching (HTSM) and stereo rectification-based stereo matching (SRSM). There were many applications of HTSM despite HTSM applicability to the planar objects. With the help of this suggested method, the process of coarse searching in stereo matching was simplified and the effect of change in shape which was arisen due to the difference of perspectives (DOP) was reduced. Through simulated and practical experiments, comparisons were evaluated among HTSM, conventional stereo matching (CSM), and SRSM algorithms. These experiments showcased the supremacy of HTSM efficacy and accuracy rate among all three algorithms in the case of the planar surface of an object, and CSM efficacy was comparatively lower than SRSM. The authors in Molina-Cabello et al. (2020) attempted to build a novel method to calculate homography estimation. Convolutional neural network (CNN) was used to approximate homography, and input pair of images in a set of different versions were given. These different versions were resultant of color level disturbance that occurred in the images provided. Next, a differentiated estimation of color homography was produced by every single image pair, and to attain dynamic and ultimate estimation, the produced estimations were integrated collectively. Several experiments were planned and applied for verification of the performance of the method. Through experimental observations, the suggested method was validated both in terms of quality and quantity. The proposed method confirmed its efficiency in comparison to the baseline method which utilized the outcomes of homography estimation deep network when applied to the actual pair of input images. To calculate homography estimation in which exclusively single-point correspondence was needed, the scholars in Diaz-Ramirez et al. (2022) developed an iterative approach. The suggested approach used particle swarm optimization to calculate the search problem for homography estimation of parameters, by maximizing a match score between a projective transformed fragment of the input image using the estimated homography and a matched filter constructed from the reference image, while minimizing the reprojection error. The results showed that regarding available approaches, the suggested method confirmed précised homography estimation from a single-point correspondence in which four points were needed at the minimum. Various synthetic, as well as experimental projective transformed, images were processed with the purpose to validate the efficiency of the suggested approach. The authors in Rodríguez (2022) offered RANSAC (Random Sample Consensus) algorithm variations in which a-contrary method and affine approximations were used for enhancement of homography estimation that was calculated in between an image pair. In the contrarious method, the estimation soundness was defined as well as it permitted adaptive thresholds of inlier/outlier differences. The results showcased an enhanced performance of the suggested technique which outperformed many available techniques, specifically regarding the distinct selection of datasets of images and image descriptors. Moreover, the system also approved the enhancement in success possibility while image pairs were figured out from complex matching datasets. Additionally, the multi-image matching issue was framed as stitching by the authors in Brown and Lowe (2007b), and hence, to figure out image matching between available images, similar local characteristics were utilized. Due to this reason, factors like input image scale, orientation, illumination, and order were not able to influence the outcomes of the suggested method. The methodology was even unresponsive regarding images containing noises as well as which did not belong to the panorama. Further, in the case of disordered databases of images, it was found capable of identifying various panoramas. Moreover, this study work was an extension of the last study work which was related to the BL03 area; here in the latest study, the new concepts added were automatic straightening steps and gain compensation.
To identify and delete a color cast which was an imposed leading color from a captured image with no requirement of previous information about semantics, the scholars in Gasparini and Schettini (2003b) introduced a new robust and dependable approach. Firstly, the classification of the input image was performed with the help of a multi-step technique which classified the image based on cast into intrinsic, ambiguous, evident, and no cast (images presenting a cast due to a predominant color that must be preserved). The removal of the cast was carried out with the help of an enhanced white balancing method on the identification of an ambiguous cast or an evident cast. This step which involved the deletion of the cast was employed for ambiguous or evident casts. To validate the suggested approach, a 650 images database was utilized which showcased promising results. The applications containing multiple cameras expected all cameras to have one major color reception, although, because of differences in radiometric features of various cameras and continuously varying lighting conditions, there existed chrominance and luminance differentiations of considerable degree in various views taken by the camera. And these differences posed major implications for the methods which are based on color correspondence. The authors in Wang et al. (2009) attempted a dynamic color correction method to meet the aforementioned challenge. In this method, color differences were adjusted considering each region despite managing an image by taking the entire image or applying a color calibration object. Hence, the local image parts on which the global correction methods proved insufficient by providing inconsistent correction outcomes could be avoided by the suggested method. For validating the efficacy of the proposed model, several experiments were conducted which showcased an enhanced performance. The approach applied to several applications while the authors originally developed this approach for stereo vision. The scholars in Jeong et al. (2021) presented a new color correction technique which was mainly designed for the enhancement of color consistency in multi-view images. The suggested approach used a 3D point set registration method with the purpose to distort the non-rigid color allocation of an image to form a reference image. The authors extracted a negligible count of illustrative color points which would be utilized in the 3D registration process. This was done for ensuring a reduction in computational complexity involved in 3D registration. Through comprehensive experiments, the efficiency and robustness of the suggested approach improvement in color consistency were validated. Also, the approach proved computationally efficient using representative color point set approximation. Also, the colors of an image could be transformed for attaining certain purposes, the entire process was called Color Correction. Generally, the characteristics which was taken into consideration while conducting color constancy and white balancing were similar to color correction.

Research findings
After analyzing the above literature section, it is observed that in the last few decades several techniques have been proposed by various experts for correcting the color in two images captured under different angles. Although the techniques are working smoothly to some extent, we observed that there is a scope for improvement in these methods. It was observed that the majority of the researchers are using only one-color correction technique in their work, due to which the error between two images, i.e., reference image and color-corrected images came out to be higher. The increased error attained in images leads to poor performance of the entire color correction model, which ultimately leads to poor visual quality as well. Moreover, it was also analyzed that among all the color correction techniques, alternate least square and root polynomial techniques generate the best results. Keeping these facts in mind, an improved and effective color correction model will be proposed that enhances the image quality by reducing the errors between a reference image and a target image (color-corrected image) while matching color coordinates.

Present work
Color correction can be defined as the technical process wherein the color issues of images are fixed to make the picture look realistic. As discussed in previous sections the conventional color correction model undergoes many limitations that disrupt its smooth functioning and increase error between two images. Therefore, a new and effective color correction model is proposed in this research article in which two color correction techniques are hybridized for getting enhanced results. The primary goal of the proposed work is to reduce errors between a reference image and a color-corrected image so that its visual quality is enhanced. To combat this task, two best-performing color correction techniques, i.e., alternate least square (ALS) and root polynomial (RP), are clubbed together in the proposed work to develop a hybrid color correction model.
Several steps are followed up by the proposed hybrid color correction model to achieve the desired objective: data collection, converting images into xyz format for different color models, implementing hybrid ALS ? RP color correction model on each color model individually, calculating color difference for different color models, and finally analyzing the performance of proposed hybrid color correction model in each color model. One of the major reasons for incorporating ALS and RP together in the proposed work is to neutralize the drawbacks of each other. This will enhance the efficacy of the results. To fix color issues in images, the proposed hybrid model uses the Amsterdam Library of Object Images (ALOI) dataset, which is available publicly on the internet for scientific purposes. After this, one image is selected as a reference image that is converted into LAB, LUV, and RGB color models in XYZ format. After this, another image of the same object is selected upon which a hybrid ALS ? RP color correction technique is implemented. This image is then converted into three color models of LAB, LUV, and RGB in XYZ format for comparison purposes. Finally, the color difference between the reference image and the target image is obtained and performance is calculated on this basis. Figure 1 illustrates the flow chart of the proposed hybrid ALS ? RP color correction model. Each step is described briefly in the following subsections of the paper.

Dataset information
To examine the performance of the proposed model, we first need to collect the necessary information from publicly available datasets. Here, the Amsterdam Library of Object Images, also referred to as ALOI, is taken into consideration in which images of various objects are captured under various conditions. ALOI is one of the most used datasets that contain colored images of around 1 k small objects for scientific purposes. The pictures of objects are clicked with different viewing angles, lighting colors, and angles to capture the sensory diversity in object recordings and wide-baseline stereo pictures. Each object was photographed more than 100 times, resulting in a total of 110,250 photographs for the database. Since these images are captured under different illumination colors and angles, their color distribution is not the same. The images are selected from the available dataset upon which further advanced techniques are applied to get the desired results.

Conversion of images into different color models
In this step of the proposed hybrid color correction model, a single image is selected from many images present in the database, and further processing is performed on this image. The selected image is then converted into three different color models LAB, LUV, and RGB. This means that for one single image there are three color variants, i.e., LAB, LUV, and RGB, respectively. Also, the images are converted into the XYZ format so that color coordinates in each image can be identified easily. It must be noted here that images obtained in this step act as reference images with which the comparison for color errors needs to be done.

Color correction using a hybrid ALS 1 RP model
Once a reference image is obtained for three color models in XYZ format, it's time to implement the proposed hybrid ALS ? RP color correction method in the proposed work. For this, another image of the same object is selected from the dataset upon which the hybrid ALS ? RP color correction technique is implemented. The proposed hybrid ALS ? RP technique fixes the color in the target image by calculating the root polynomial matrix, exponential coefficient a, and PRP using Pa. After that, shading regularization is calculated in images by using the equation given in Eq. (6).
where G 1 represents the DCT basis matrix, w 1 represents the weight matrix, and I D represents the shading field image, and D is the diagonal matrix. After every iteration, the value of P, M, and R is updated in the proposed hybrid model, by using the equations given in 7, 8, and 9.
where P and R represent the color chart images. The process keeps on repeating till a fixed number of iterations and in the end; the color-corrected image is obtained. The pseudo-code for hybrid ALS ? RP color correction implemented in the proposed work is given in Algorithm 1. Once the color issues are fixed in the targeted image, the image is converted into three color models of LAB, LUV, and RGB in XYZ format. This step is necessary to compare the quality of color-corrected images using Hybrid ALS ? RP with the reference images obtained in the very beginning for three color models with XYZ format.

Calculating color difference or error
Once the color-corrected image using the hybrid ALS ? RP model is converted into LAB, LUV, and RGB color models, its coloring intensity is compared with the reference image of the LAB, LUV, and RGB color model to check whether an error has been reduced or not. The LUV reference image is compared with the target color-corrected LUV image, and the LAB and RGB reference image is compared with the LAB and RGB of the target image, respectively. Moreover, the values are obtained in terms of mean, median, 95% quantile, and max error. The value of these errors should be as minimum as possible for making the proposed hybrid ALS ? RP model efficient.

Performance evaluation
Finally, in the last step of the proposed hybrid color correction method, the efficacy of the proposed approach is examined and validated by putting it in comparison with a few conventional models under various performance dependency factors. The results are obtained for three color models individually and are discussed briefly in the next section of this article.

Results and discussions
In this section, the efficacy and robustness of the proposed hybrid color correction methods are analyzed and discussed. The experiment results are obtained and validated by putting it in comparison with conventional color correction models in MATLAB software. Also, the simulated outcomes are determined for three color models, i.e., LAB, LUV, and RGB separately, in terms of mean, median, 95% quantile, and max error to check which model generates the least errors. The definition and mathematical annotation of these parameters are given in the next section of this paper. Moreover, the following subsection illustrates the comparison results for three color models in terms of the abovementioned parameters.

Performance metrics
• Mean It can be simply defined as the sum of all observations divided by the total number of observations at a particular instance. Mathematically, it can be written as: where P x represents the sum of observations and N represents total observations.
• Median Median of a number can be found by ordering all points and choosing the one in between. Mathematically, it can be defined as: • Max Error The absolute value of the largest discrepancy between a predicted variable's value and its actual value is known as the maximum error. • 95% quantile Error It is a percentile score that indicated that your usage falls within that range 95% of the time and that it surpasses that range 5% of the time.

Results for LAB color model
The efficacy of the proposed hybrid color correction model is first analyzed and compared with traditional models for the LAB color model in terms of their mean values. The comparison graph obtained for the same is depicted in Fig. 2 Similarly, the performance of the proposed hybrid color correction model is examined and compared with conventional LS, RP, ALS, and homography techniques in terms of their median values. Figure 3 represents the comparison graph for the Median obtained at the end of simulations. The results showcased that the median value in the proposed hybrid model was mounted at just 1.448, whereas it was 5.67 in LS, 4.67 in RP, 3.27 in ALS, and 2.59 in homography models. Among traditional models, the homography technique is showing least median value, but still, it was 1.142 higher when compared with a proposed hybrid model.
Likewise, to prove the dominance of the proposed hybrid color correction approach over other approaches, we compared its performance in terms of 95% quantile error.
The comparison graph obtained for the same is shown in Fig. 4. After examining the graph, it is observed that traditional RP yields the highest 95% quantile error with a value of 14.6, followed up by conventional LS with 12.27, followed up by homography and ALS color models with 9.2 and 8.24 95% value. These values are significantly higher and degrade the performance of color models. However, in the proposed hybrid color correction model, the value of 95% quantile was mounted at only 4.2794 which is far lower than conventional models.
In addition to this, the effectiveness of the proposed hybrid color correction model is analyzed and put in comparison with conventional models in terms of their max error attained. The comparison graph obtained for the same is given in Fig. 5. From the above-given graph, it is observed that the value maximum error was generated by the traditional RP model with a value of 16.97, whereas it came out to be 13.83, 10.28, and 9.02 in LS, Homography and ALS color correction models. Nevertheless, the value of max-error generated by the proposed hybrid color correction model is mounted at just 4.4643 which shows its    Table 1.

Results obtained for LUV color model
Just like the efficacy of the proposed hybrid approach is analyzed on the LAB color model under various performance dependency factors, its efficacy is also analyzed for the LUV color model in terms of mean, median, 95% quantile, and max error. Figure 6 represents the comparison graph that is obtained after comparing the performance of conventional and proposed hybrid color correction models in terms of their Mean value. The graph revealed that the proposed hybrid model has a mean value of just 1.8949, whereas it is accounted at 7.02 in LS, 6.69 in RP, 4.17 in ALS, and 3.88 in homography approaches. These values are enough to demonstrate and serve as proof of the proposed hybrid color correction model's efficacy.
Similarly, the performance of the proposed model is analyzed and compared with traditional models in the context of their median values and is shown in Fig. 7. The efficacy of the system is inversely proportional to the median, which means the lower the median value, the higher is the system efficacy and vice versa. After analyzing the graph, it is observed that the proposed model has the lowest median value of just 1.8152, while it was highest in the traditional LS model with 6.63 (Fig. 8). The specific value for each parameter is mentioned in Table 2.

Results obtained for RGB color model
Finally, the efficacy and effectiveness of the proposed hybrid color correction approach are analyzed for the RGB color model also under various metrics. The comparison graph for the mean and median in the proposed hybrid color correction approach for the RGB color model is shown in Fig. 9a, b, respectively. As shown in the graphs,   (a) (b) Fig. 9 Comparison graph for mean in a and median in b for RGB color model Figure 10 represents the comparison graph for a 95% error in (a) and max error obtained in the proposed hybrid color correction approach in (b). After analyzing the graph, it is clear that among all the models least errors are generated by the proposed hybrid model. The 95% quantile error rate was only 0.058949, whereas its max error rate was 0.09305, respectively. When these error rates were compared with the standard model, we observed that 95% quantile error is decreased by 0.131751, 0.188, 0.0811, and 0.08 when compared with LS, RP, ALS, and homography models, while the max error rate was decreased by up to 0.208, 0.212, 0.16, and 0.151 when compared with standard LS, RP, ALS, and homography color correction approaches. Table 3 represents the exact values obtained for each factor on the RGB color model.
After carefully examining the above-given graphs and tables, it can be concluded that the proposed hybrid color correction model is showing the best results with the RGB color model with the least errors, followed up by the LUV color model and lastly LAB color model. This means that colors are transferred effectively and efficiently in the RGB color model and hence have better visualization effects.

Conclusion
In this article, we provide the unexpected finding that colors are closely related by a homography when viewing conditions (such as light hue, shading, and camera) vary.
Color homographies are used for color transfer and color correction which means mapping RAW RGBs to display counterparts. Here, an improved and effective color correction model is proposed in which two techniques, i.e., alternate least square (ALS) and root polynomial techniques, are used together for solving the color transfer problem. The main reason for hybridizing these two-color correction techniques together is because they provide good color correction results individually; therefore, hybridizing these techniques in the proposed work enhances the efficiency of the system. The efficacy and efficiency of the proposed hybrid approach are examined in MATLAB software for three different color models, i.e., LAB, LUV, and RGB. The simulated outcomes were attained and compared with conventional models in terms of their mean, median, 95% quantile error, and maximum errors. The results obtained showcased the supremacy of the proposed approach for the LAB color model whose mean, median, 95% error, and max error were only 1. 78, 1.44, 4.27, and 4.46, respectively. When these parametric values were compared with the conventional homography model, we observed a decrease in the mean by around 1.61, median by 1.142, 95% by 4.92, and max error by 5.81, respectively. Likewise, the mean, median, 95%, and max errors were analyzed for the LUV color model whose values were reduced by around 1.98, 1.16, 6.26, and 6.17 when compared with the standard homography color correction model. On the other hand, the value for mean, median, 95%, and max was also analyzed for the RGB (a) (b) Fig. 10 Comparison graph for 95% and max errors in RGB color model color model that was mounted at just 0.021, 0.015, 0.058, and 0.093 in the proposed hybrid color correction model. After further analyzing these statistics closely, it can be concluded that errors are least observed by the proposed hybrid approach in the RGB color model, which means color issues are fixed more appropriately and efficiently. Compared to the state of the art, our homography-based color-correcting method offers increased color fidelity. In future research, the visual quality of images can further be enhanced by using nature-inspired optimization algorithms in color homography techniques. Also, the optimization algorithms can be used along with enhanced hybrid models to solve the problem of color transfer in images; hence, more research needs to be done in these phases.
Funding This research received no external funding.
Data availability The data may be available from the authors upon a reasonable request.

Declarations
Conflicts of interest The authors declare that there is no conflict of interest regarding the publication of this manuscript.