A hybrid approach for stain normalisation in digital histopathological images

Stain in-homogeneity adversely affects segmentation and quantification of tissues in histology images. Stain normalisation techniques have been used to standardise the appearance of images. However, most the available stain normalisation techniques only work on a particular kind of stain images. In addition, some of these techniques fail to utilise both the spatial and textural information in histology images, leading to image tissue distortion. In this paper, a hybrid approach has been developed, based on an octree colour quantisation algorithm combined with the Beer-Lambert law, a modified blind source separation algorithm, and a modified colour transfer approach. The hybrid method consists of two stages the stain separation stage and colour transfer stage. An octree colour quantisation algorithm combined with Beer-Lambert law, and a modified blind source separation algorithm are used during the stain separation stage to computationally estimate the amount of stain in an histology image based on its chromatic and luminous response. A modified colour transfer algorithm is used during the colour transfer stage to minimise the effect of varying staining and illumination. The hybrid method addresses the colour variation problem in both H&DAB (Haemotoxylin and Diaminobenzidine) and H&E (Haemotoxylin and Eosin) stain images. The stain normalisation method is validated against ground truth data. It is widely known that the Beer-Lambert law applies to only stains (such as haematoxylin, eosin) that absorb light. We demonstrate that the Beer-Lambert law applies is applicable to images containing a DAB stain. Better stain normalisation results are obtained in both H&E and H&DAB images.


Introduction
In histopathology, tissue samples are examined for disease diagnosis.Image staining is performed to increase the contrast of the image.In-homogeneity in histology images is still a major problem that impairs visual inspection as well as image analysis by computer systems [40,65,85,86].Figure 1 demonstrates the colour variation problem.
Stain normalisation is performed to standardise the appearance of the images so as to improve image analysis during image segmentation and quantification [14].However, serious tissue distortion may occur if stain normalisation is poorly performed leading to poor image interpretation [8,54,85,86].For example, several objects may appear to be linked as a single object after segmentation [28,69].For more accurate diagnosis, an automated stain normalisation method should preserve both tissue structure and texture in the image.Different stain normalisation approaches have been proposed.However, most of these stain normalisation techniques are based on histogram specification [26,33], grey world normalisation [88], comprehensive colour normalisation [18], and histogram equalisation [45,58].These colour normalisation techniques work on natural images rather than histology images [40,85,86].
Stain normalisation techniques employ non-negative matrix factorisation (NMF) [40,85,86], singular value decomposition (SVD) [57], principal component analysis (PCA) [44], [32], independent component analysis (ICA) [84], statistical based algorithm ICA [1,51], and machine learning [34,79,97,98].However, these methods are only applicable to certain histology images.In addition, some of these methods produce undesirable image tissue distortions and also introduce considerable visual artefacts in histology images [40,86].Further more, some of these methods require expertise and are computationally expensive.For accurate stain normalisation, a precise estimation of each applied stain in both source image and target image is necessary to modify stain colour to match that of the target image [57,85,86].For example in H&E stained image slides, purple and pink stains are estimated respectively based on the Beer-Lambert law.It is widely known that the Beer-Lambert law applies to only stains (such as haematoxylin, eosin) that absorb light [89].We demonstrate that Beer-Lambert law is applicable to DAB images.In this paper, a hybrid approach has been developed, based on an octree colour quantisation algorithm combined with Beer-Lambert law, a modified blind source separation algorithm, and a modified colour transfer approach.Stain normalisation is carried out in similar way as existing approaches [57,86].However, our method utilises blind source separation in [63] rather than NMF during stain separation.Note: we modified the blind source separation in [63] for a better stain normalisation.
The rest of the paper is structured as follows: Section 2 briefly reviews some stain normalisation algorithms, Section 3 presents the proposed methodology, Section 4 presents experimental results and the discussion of results, Section 5 concludes the paper.

Related work
This section briefly reviews existing stain normalisation algorithms addressing the stain in-homogeneity problem in histology images.The commonly used stain normalisation methods include NMF, SVD, PCA, ICA.All of these methods are based on matrix factorisation(also known as matrix decomposition).Matrix decomposition methods reduce a complex matrix into constituent parts so as to make it easier to calculate.Although these methods are based on matrix factorisation, they use various kinds of assumptions translated into different constraints and/or objective functions during stain normalisation [87].Forexample, NMF aims at considering the physical constraint that each stain has a non-negative response, SVD aims at evaluating the decomposition that best approximates a data matrix, given some rank restriction [68], PCA aim at accounting for as much of the data variability while enforcing orthogonality between their components, and ICA is based on the assumption that each stain, stains the tissue independently from all the other stains [66,87].In imaging, besides stain normalisation, matrix factorisation has been used to recover human pose [29], to extract muscle synergy [17], to segment objects [61,81], to classify images [101], to monitor human motion [90], to reduce image dimension [4,80], to extract features [83], and to retrieve images from the large database of images [96].This is because of its effectiveness and efficiency.The four stain normalisation methods are described below.

Non-negative Matrix Factorisation (NMF)
NMF was proposed by Paatero and Tapper in 1994 [62].Non-negative matrix factorization (NMF) factors a data matrix into low-rank latent factor matrices with non-negativity constraints.Given a data matrix X and a target rank R, NMF seeks a factorization model [19].
Compared to other matrix factorisation methods, NMF is capable of learning parts of objects [19,46].In addition, NMF is able to identify the ground-truth-generative factors W and H up to certain trivial ambiguities, which results into a strong interpretability.NMF can be combined with other methods such as non negative least squares to normalise stain in FAST images as the FAST images consist of more objects than it had spectral dimensions [11].The effectiveness of NMF in analyzing real-life non-negative data has sparked a substantial amount of research in image analysis.In image analysis, NMF and related techniques have been applied to different kinds of imaging(such as remote sensed images, medical images(such as histology images, CT images, MRI images), and natural images) during feature extraction [47,49,53,55,92], image classification [5,24,76,99], image ranking [13,82], image reconstruction [7,15,16], and image enhancement [20,40,41,41,42,73,78,[85][86][87]100].However, NMF, is less utilised during image enhancement of medical images, in particular histology imaging, which is affected by colour variation problem.This study focus on stain normalisation methods.
During stain normalisation, NMF-based algorithms attempt to factor the optical density (OD) matrix into W and H with the constraint that all elements must be non-negative, since stains cannot absorb a negative amount of light.It also forces all columns of the stain vectors to be unit length to keep it from having an infinite number of solutions that differ only by a constant.These methods work well on multispectral images, however they tend to inconsistently converge when only three spectral components are available [57].The standard NMF is subject to local minimisation that is initialisation-dependent and thus requires a good approximation of the colour vectors to initialise the factorization process [87].In addition, standard NMF is not capable of preserving structural and textual information during stain normalisation.Further more, NMF is not suitable for estimating the stain vector extraction [87,93].
Following NMF, various analysis have been studied.For example, to help circumvent problems that can be found with NMF, Xu et al. [93] proposed the use SNMF in place of NMF to extract the colour vectors.In SNMF, regularisation and sparsity terms are introduced in the NMF objective function.However, Xu et al. [93] does not provide information about the initialisation they use for the SNMF algorithm.In 2017,Van et al. [87] tested SNMF using the parameter in [52].According to Van et al. [87], SNMF requires good initialisation to be efficient, which is not always an easy task.Vahadane et al. [86] developed a stain normalisation approach based on SNMF to preserve the structural information of the source image after color normalization.In this approach, sparsity constraints are added on to stain channels during stain separation.Parameters for stain normalization are calculated based on the pseudo maximum of the separated stains.Sparsity constraints are able to enhance the recognition ability of the model for separating independent stains.However, the weighting of stains is processed independently after stain separation, which limits the capacity for color normalization.Furthermore, the overall intensity and proportion of the separated stains are not considered in the model of stain separation.It risks a bias in stain separation, for which most of the pixels would be normalized to share the appearance of a single stain [102].
Amit et al. [78], developed a stain normalisation method based on graph regularised sparse NMF to preserve the textural information in the image.Like the method in [86], this method also add sparsity constraints on to stain channels during stain separation.This method also does not consider intensity and proportion of the separated stains in the model of stain separation.In [102], an adaptive color deconvolution (ACD) model for stain separation and color normalization of histological images is presented.In this method, the stain color appearance matrix is utilised as variables, the ACD model is solved through an inte-grated optimization.The normalization is achieved by a unified transformation for the pixels in the image.Unlike in [78,86], This method considers both intensity and proportion of the stains.Both intensity and proportion are embedded in the ACD model to effectively reduce the failure of stain separation and also to avoid color artifacts in the normalized images.
Recently Li et al. [50] proposed a majorisation-minimisation principle to develop a convergence-guaranteed algorithm NMF.However, the method was applied during communication analysis of signals.

Singular Value Decomposition (SVD)
The singular value decomposition of a matrix A is an attractive algebraic transform that factorises a matrix A into the product of three matrices(U,D,V): Let A be a m × n matrix.The Singular Value Decomposition of A is as follows: Where U and V are orthogonal matrices with singular values.V T is the conjugate transpose of V. D is a diagonal matrix with singular elements [38].The process of SVD depends on the task at hand.
During stain normalisation, SVD is used for automated stain matrix estimation to address colour variation problems [57].The SVD method works by calculating the plane from the two vectors corresponding to the two largest singular values of the SVD decomposition of the OD(optical density) transformed pixels and then later projecting OD transformed pixels onto the plane [1,57,84].SVD easily adapts to variations in the local statistics of an image [3,25,37,56,95].SVD is also stable and robust to noise [2,6,36,38,43,43,60,71,91,94].However, although SVD can effectively determine independent components, it changes the colour distribution of both source and reference images, which is not necessary [78].Compared to NMF, SVD is fast.However, it tends to produce unrealistic colours in the normalised source image [85,86].

Principal component analysis
Principal component analysis was first introduced by Karl Pearson in 1901 [64], and developed independently by Hotelling in 1933 [31].Principal component analysis uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables known as principal components.
During stain normalisation, in PCA [39], a plane is constructed that optimally approximates variability between image pixels.The plane is spanned by the two first principal component vectors of the image pixel colour data.The deconvolution error is minimised, by making sure that the optimal colour deconvolution vectors of the two stains in the original image are part of the plane.Although PCA can identify an appropriate plane for describing the clouds of bi-coloured pixels for H&E staining, it is not good at representing variability in stain colours as it assumes fixed stain vectors for H and E images [1,66,87].In comparison to PCA and NMF algorithms, ICA gives a superior projection of the data with respect to each stain [84].

Independent Component Analysis (ICA)
Independent Component Analysis is blind source separation method that obtains non-Gaussian signals from a Gaussian source.Independent Component Analysis is quite similar to PCA [84].The only difference lies in the way PCA and ICA deal with data.PCA dispense the data in a way that best separates variance while ICA dispense the data along the axes in such a way that each independent component directly corresponds to a given stain [84].ICA may not achieve an orthogonal distribution if the two stains in the image are of the same color.Although Trahearn et al. [84] introduced a correction step to estimate the true stain vector from within the ICA space, the correction step involves two steps that include (a) the true stain vectors are a minor adjustment from the unit vectors in ICA space, (b) Most of the data in ICA space is close to one of the true stain vectors.However ICA is applicable to only images with more than two stains.Rabinovic et al. [66] compared two stain deconvolution approaches, NMF and ICA.They showed that while NMF performed better however neither of the methods was sufficient to fully deconvolve the images [1], [12].The study was performed on hyper-spectral images, rather than light microscopy images.Hyper-spectral imaging are different from microscopy image in such a way that they have higher number of input channels as compared to the three-channel RGB microscopy images, which may limit the comparison that can be made between methods for the two modalities.

Statistical blind source separation algorithm ICA
Alsubaie N et al. [1] chose to recover the original source signals blindly by using the statistical blind source separation algorithm ICA.In this method, independent component analysis is used in the wavelet domain where each colour channel of the input image is decomposed into a series of narrow sub-band images using a decimated wavelet transform.Then later each sub-band is statistically analysed in order to find the least Gaussian sub-bands.Finally, ICA is applied to the selected sub bands to estimate the stain matrix.The method cannot estimate weaker stains such as eosin.

Methods and material
This section presents the proposed methodology for automated stain normalisation of H&E and H&DAB stain histology images.The proposed methodology (Fig. 2) involves two stages: (a) stain separation, and (b) colour transfer.

Proposed automated stain normalisation
The hybrid method consist of two stages: the stain separation stage and colour transfer stage.An octree colour quantisation algorithm combined with the Beer-Lambert law, and a modified blind source separation algorithm are used during stain separation stage to computationally estimate the amount of stain in an histology image based on its chromatic and luminous response.A modified colour transfer algorithm is used during the colour transfer stage to minimise the effect of varying staining and illumination.

Stain separation
Stain separation is performed to computationally estimate the amount of stain in an histology image based on its chromatic and luminous response [21,22,66,70].Octree colour quantisation algorithm is combined with Beer-Lambert law,and a modified blind source separation algorithm are used during stain separation stage.Note: We modified the blind source separation [63].As compared to blind source separation method in [63], our blind Colour quantisation is the process of choosing M colours from I to construct a quantised colour space, Ī , where Ī = Ī j , j = {1, 2, ...., M} [77].Colour quantisation is introduced due its ability to enhance the image [8,27,35,59,77].Quantisation algorithms can be categorised into two classes: uniform quantisation and non-uniform quantisation algorithm.The difference between the two lies in the way they handle colour space.For example, in uniform colour quantisation, the colour space is broken into equal sized regions where the number of regions is less than or equal to M. Non-uniform colour quantisation breaks the colour space depending on distribution of colours in the image.In terms of performance, nonuniform algorithms outperforms uniform quantisation algorithms.Non-uniform algorithms are easy to implement and offer consistent results [48,77].We choose Octree algorithm that is highly non-uniform to extract the classes [72].The colour classes are then labelled and the labelled image is converted into optical density.The octree quantisation algorithm quantises the colour following the steps below: given an image I, every colour in the image is stored in an octree of depth 8 (every leaf at depth 8 represents a distinct colour).A limit of M (in this case M = 256) leaves are placed on the tree.The colours are filtered based on the limit of M leaves set.For example colour in the tree can result in two outcomes: (a) if there are less than M leaves the colour is filtered down the tree until either it reaches some leaf node that has an associated representative colour or it reaches the leaf node representing its unique colour, and (b) If there are greater than M leaves in the tree some set of leaves in the tree must be merged (their representative colours averaged) together and a new representative colour stored in their parent.
Leaves are selected and merged following two criteria: (a) Reducible nodes that have the largest depth in the tree should be chosen first.They represent colours that lie closest together, (b) If there is more than one group of leaves at the maximum depth the algorithm either merges leaves that represent the fewest number of pixels to keep the error small or reduce leaves that represent the most pixels.In this case large areas will be uniformly filled in a slightly wrong colour while maintaining detailed shading.
Once the entire image has been processed, the colour in the image is quantised by simply filtering each colour down the tree until a leaf is hit.(ii) Convert the colour quantised image to optical density space where I 0 is the illuminating light associated with the quantised RGB Image I.
(b) Obtain stain vector The stain vector is obtained following the steps below (i) Pre-whiten data following the steps below [31].
Compute the correlation matrix using (4):- where R s is the received sigma.R s is the conjugate transpose of R s .L s is the length of the sigma.Obtain Eigen values U and Eigen vectors V using Schroedinger Potential in [9,10], rather than the common Eigen values and vector.Note: As compared to blind source separation method in [63], our blind source separation method uses spatial-spectral Schroedinger-Eigen vectors and Eigen values during prewhitening process to determine the set of stain vectors and stain matrix (3 X 3).
Perform Eigen values decomposition using shur decomposition method in (5).Shur decomposition method is expressed as [23,30,63,75]: where Q is a unitary matrix,  (6) Given that ideally , the non-zero part of the matrix LD 1 2 equals H up to a unitary ambiguity matrix: where U is a unitary matrix.L is constructed as a matrix containing p large norm.Column in LD 1 2 .Then, the received signal is pre-filtered using: where # denote matrix pseudo-inverse.(ii) Perform multi-kurtosis maximisation Multi user kurtosis (MUK) can be defined by M q×p the following set of q × p complex matrices: We perform MUK following the steps below:-Compute the gradient of F (G) using the formula in ( 9) with respect to W. Then later obtain an adaptive stochastic-gradient algorithm. Let And by invoking C2 with the assumption that symmetrical inputs ( E( a 2 ( K) = 0) ) , the gradient of F ( G) with respect to W is given by: Update W (k) in the direction of the instantaneous gradient using (10).
Satisfy the orthogonality constraint at the next iteration of the algorithm by utilising the unitary constraint using Gram-Schmidt orthogonalisation in (11).(11) Undo the remaining unitary mixture which makes the equaliser W of size p×p, hence saving computational cost.Then, to satisfy the constraint in (11), it suffices to satisfy W (k + 1) ∈ M p×p , i.e.
Since there is no guarantee that W (k + 1) will satisfy the constraint in ( 12), we have to transform W (k + 1) into a unitary By treating W (k + 1) as a p × p matrix satisfying ( 12) that is as close as possible to W (k + 1) in the Euclidean sense using the iterative procedure that satisfies the following criterion successfully for j = 1, ...., p: where X 2 = X H X is squared Euclidean norm of vector X, σ ij is kronecker delta, W j is obtained from To solve (14), we first construct the Lagrangian of ∇( W j ) using ( 16) Where λ j , l j , and V l j are real scalar parameters.Setting the gradient of L ∇ ( W j , λ j , μ, v) with respect to W j to zero ∂L ∇ ∂W j = 0, we obtain the following equations for each j: Where B lj = 1 2 ( μ l j + iV lj ).From (17), we obtain (18) According to ( 18) Colour deconvolution scheme is used to transform quantised RGB channels of a histology image into a density map for each stain.
(i) Determine the stain concentration of individual stains: where Z is the optical density map obtained in (3), W is the stain matrix obtained in (18).The optical density maps of each stain are obtained by taking the columns of stain matrix W and taking rows of the stain concentration matrix using the equation below:

Colour transfer
Colour transfer is performed to minimise the effect of varying staining and illumination [74].We modified the colour transfer method in [57,78,86] to achieve better stain normalisation results.We also introduce a weighting factor in the colour to easily adjust the contrast and brightness of the image.Our colour transfer involves 6 steps as follows: (a) Let W be the stain matrix and C be the coefficient matrix from ( 20) (b) Adjust the dynamic range of C Source so as it matches that of C target to obtain a normalized source stains C snorm , by Calculating the 95 percentile of each row of the C source and C target as C s−extreme (i) and C t−extreme (i) to obtain a scale matrix.The dynamic range is estimated with a pseudo maximum (95%) as it is roburst.We used (22) instead of (23).
Instead of where i = 1, ....., r which is the row number in a coefficient matrix C. (c) Obtain the scale row of C source using (24) instead of (25).
Instead of (d) Colour transfer in optical density space.
(e) Apply a weighting factor to a colour transferred image (27) where T ≥ 1, T can be adjusted to improve the brightness and contrast of the normalised image.The default values of T for both H&DAB and H&E stain images is 1. (f) Convert the normalised image from optical density space to RGB space where

Experiments
We tested the proposed method using 48 H&DAB image slides with dimensions of 1920 × 1080 pixels from the mouse cortex, and 100 H&E stain images with dimensions of 1539 × 1376 pixels provided by Dr Abhishek Vahadane.We also used 22 H&E histology images with the dimension of 2000 × 2000 pixels of different datasets (i.e.breast dataset, colon dataset and lung dataset) by Alsubaie N et al. [1] under the support information section.H&E images came with ground truth.Please refer to [1] for more information about the ground truth.H&DAB images were stained on different dates.H&E images are from different scanners (Hamamatsu scanner and Aperio scanner).To increase reliability, more than one sample of slides were extracted from each mouse.48 H&DAB image slides were extracted from different mice.As compared to other colour transfer methods, a weighting factor is introduced in the colour transfer stage.The weighting factor improves the brightness and contrast of the image as shown in Fig. 3 4

.1 Validation and evaluation of stain normalisation
We evaluated results from stain separation stage and the colour transfer stage both qualitatively and quantitatively.The accuracy of stain separation is evaluated using both estimated stain matrix and density maps.This is because stain matrix represents the principal colour of each stain and each vector in the stain matrix represents RGB values within an optical density space for each stain colour [1,86].An accurate stain matrix leads to better stain estimation.In this paper, ground truth data is used during the evaluation of the estimated stain matrix.During this stage, the root smean square error between the estimated stain vector and the ground truth is used.Bar charts are used during the comparison of our method with existing methods.The density maps represent the amount of stain absorbed at each pixel.
A similarity quality metric between the results obtained using the proposed method and the ground truth is used to evaluate the accuracy of the proposed method.Box plots and a bar graph are used to compare stain separation obtained using our methods with that of other methods (Fig. 4).We used the median for easy comparison.The quantitative results shown in Figs. 5 and 4, show the superior performance of the proposed method over others.The accuracy of colour transfer is evaluated both qualitatively as shown in Figs.6f, 7f.Our stain normalisation method produces better results compared to other methods.

Discussion
The qualitative and the overall quantitative results of stain separation in Figs. 8, 5 and 4, show that the proposed method outperforms all the other methods.The similarity metric in    method by 2.2% for E in lung dataset.This is due to usage of low weighting factor.Figure 4 shows that proposed method outperforms Alsubaie's [1] for estimated stain matrix in both the breast dataset and lung dataset.The proposed method also outperforms Macenko's method [57] for estimated stain matrix in the colon dataset but achieves the same result for estimated matrix as Macenko's method [57] in the lung dataset.Macenko's method [57] achieves better results for estimated stain matrix in breast datasets as compared to all methods.The colour transfer stage is the final stage of stain normalisation process.The qualitative results of the color transfer stage show that the proposed method can be applied to both H&E stain images and H&DAB images (Figs. 6 and 7).Khan's NMF method [40] and Macenko's SVD method [57] produces unrealistic stains when applied to H&DAB images.Khan's NMF method [40] and Macenko's SVD method [57] also introduce artefacts when applied to H&DAB images.Reinhard's method produces slightly better results when applied on H&DAB but still not better than the proposed method.Compared to all methods, our method   results into increase in brightness of the image (Fig. 3).Further work is needed to evaluate proposed method at different weighting factors.

Conclusion
Stain normalisation is an essential step to remove inherent in-homogeneity in histopathological images for diagnosing disease and its progression.A potential drawback of such a technique is that tissue structures and texture in the original image could be distorted after stain normalisation.In this paper, a hybrid approach for stain normalisation has been developed, based on an octree colour quantisation algorithm combined with Beer-Lambert law, a modified blind source separation algorithm, and a modified colour transfer approach.The hybrid method consists of two stages: the stain separation stage and colour transfer stage.An octree colour quantisation algorithm combined with Beer-Lambert law, and a modified blind source separation algorithm are used during stain separation stage to computationally estimate the amount of stain in an histology image based on its chromatic and luminous response.A modified colour transfer algorithm is used during the colour transfer stage to minimise the effect of varying staining and illumination.The method is capable of addressing colour variation in both H&DAB (Haemotoxylin and Diaminobenzidine) and H&E (Haemotoxylin and Eosin) stain image.Unlike other colour transfer, our normalisation algorithm introduces weighting factor which allows one to adjust the brightness and contrast of the normalised image.Both qualitative and quantitative validation demonstrate the superior performance of the proposed stain separation and colour normalisation technique.

Fig. 1 a
Fig. 1 a and b are the same H&E histology images captured using an Aperio scanner and Hamamatsu scanner respectively.c and d are different H&DAB histology images from the same batch stained at different times

Fig. 3
Fig. 3 H&E and H&DAB normalised histology images at different weighting factors.First row are H&E stain images normalised at weighting factor 1, 2 and 3 respectively, and second row are H&DAB stain images normalised at weighting factors 1, 2, 3 respectively

Fig. 4 Fig. 5
Fig. 4 Root mean square error between the estimated stain matrices and the ground truth data for all methods on all datasets.(1) breast dataset, (2) Colon dataset, and (3) Lungs dataset

Fig. 9
Fig. 9 Different image slides normalized at different factors.First row are original image slides from cortex, second row are images normalized at factor 4, and third row are images normalized at factor 5

Fig. 10
Fig. 10 Different image slides normalized at different factors.First row are original image slides from cortex, second row are images normalized at factor 2, and third row are images normalized at factor 3

Fig. 12
Fig. 12 System interface with normalised H&E source image is the diagonal matrix with real entries.Denoising the estimate ÃYY of A Y Y = E(Y Y − 1) is performed by estimating the noise variance.The noise variance is estimated by obtaining the average of the smallest Eigen values.

Table 1
[1]M between the stain matrices and the ground truth.The last two show the median of SSIM index for all methods Fig.5and Table1above shows that the proposed method outperforms Alsubaie et al.[1]by 4-13% for H for breast dataset, colon dataset, and the lung dataset.However Alsubaie's method[1]outperforms the proposed method by 0.1% for E in breast datasets.The proposed method still out performs Alsubaie's method[1]by 3-7% for E in colon and [57] dataset.In comparison to Macenko's method[57], the proposed method outperforms Macenko's method[57]by 2-4% and 4-5% for H&E in all three datasets (breast and colon dataset).Like Alsubaie's method [1], Macenko's method[57]also outperforms the proposed method in estimating Eosin stain.Macenko's method[57]outperforms the proposed