Application of a novel image moment computation in X-ray and MRI image watermarking

Orthogonal polar image moments which are deﬁned over a unit disk, are used often in watermarking applications. One such image moment is polar harmonic Fourier transformation (PHFT). The quality of the extracted watermark depends on the accuracy of the computed image moments. Various methods have already been used for the computation of the moments with varying degrees of accuracy. This study proposes application of a novel image moments computation method which is a synergy of analytic, and numerical techniques, to image watermarking. The computed moment results in a robust watermark extraction. The PHFT have rotation, and scaling invariance property, which allows the watermark to resist various geometric attacks. In addition to the geometric attacks the watermark has the capability to resist various signal processing attacks. The performance of the proposed watermarking scheme is performed with the methods used in recently published articles. The experimental results show that the proposed method has the better performance compared to other techniques, and has better capability to resist various attacks.


INTRODUCTION
With growing speed of internet accessibility and ever increasing internet speed, the amount of data transfer has increased [1,2]. The transfer of data involved in many fields including medical, satellite data and internet of things applications [3][4][5][6][7]. Medical images are frequently transferred, uploaded and downloaded over the internet in telemedicine. In some cases, images are compromised and we just don't know if the images are attacked. Therefore, to identify the copyright of the image, often watermarks are inserted into the medical images [8][9][10][11][12]. Over the years, many state of the art techniques have been proposed in the literature, to insert watermarks in medical images and retrieve the watermarks. Many of these algorithms are robust against various attacks including addition of noise, JPEG compression, cropping and flipping [8,[13][14][15]. However, these algorithms are not very effective against geometric attacks such as rotation and scaling.
For geometric invariant watermarking, the algorithm must use geometric invariant image descriptors. The family of orthogonal moments defined over a unit disk is used frequently in rotational and scaling invariant watermarking. The authors in This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Image Processing published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology [16] used Zernike moments (ZM) and pseudo Zernike moments (PZM) for watermarking. It was shown that the watermarking algorithm was robust against various attacks including rotation, scaling and flipping. The authors in [17] used polar harmonic transformation (PHT) for watermarking. PHT moments too are capable of attaining rotational and scaling invariance watermarking. However, the PHTs have higher capacity to embed watermark than the ZM and PZM. Radial harmonic Fourier moments (RHFMs) are also used in watermarking applications [18]. The authors in [18] shown that RHFMs provide more robust watermarking than the ZMs and PZMs. Orthogonal Fourier-Mellin moments can also used in image watermarking [19]. The authors in [20] used PHT moments for geometric invariant watermarking. The robustness of the watermark depends on the accuracy of the computed image moments. The authors in [20], have computed the PHT moments on a polar grid, and it was shown that the accuracy of the computed moments was improved significantly. It was also shown by the authors that the improved accuracy of the moments have contributed in more robust watermarking algorithm against various attacks when compared with PHT watermarking algorithm in [17]. [21] used polar harmonic Fourier moments (PHFM) for image watermarking and shown that their watermarking scheme is more robust than the ZMs, PZMs, PHTs and RHFMs.
In image moment based watermarking, the watermark bits are embedded into the image moments. Watermarked image is reconstructed using the modified moments. The watermarked bits are extracted at the destination by recomputing the image moments from the watermarked image. Therefore, the quality of the extracted watermark depends on the accuracy of the computed moments. For example, the authors in [17] and [20], both used PHT moments for watermark embedding. Since, the accuracy of the moments was better in [20], the watermarking in [20] is more robust than the watermarking in [17]. The moment computation involve integration over a unit disk. The domain of the image function is mapped to the unit disk. There are two approaches to map an image function over the unit disk. In one approach the image is completely inscribed into a square and in the other approach unit circle is inscribed into a square. In watermarking applications the later approach is adopted often as the former is not rotational invariant. In this approach, integration is performed over each pixel falling within and on the unit circle. Since image function is discrete in nature, only an approximation of the integration can be found. Therefore, the accuracy of the moment reduced to the approximation of the integration. The authors in [16] and [17], have reduced the integration to just a value at centre of the pixel lying inside the unit circle. This method is known as zeroth order approximation. This method has many shortcomings. In this method, the image function remain constant over a pixel, but the kernel function varies over a pixel. Therefore, the computed moment will not be very accurate. The pixels partially intersecting with the unit circle and their centres falling outside the unit circle are completely ignored. In addition, the pixels partially intersecting with the unit circle and their centres falling inside the unit circle are treated similar to pixels completely falling inside. Therefore, the moments computed using the zeroth order approximation are not accurate and therefore, not robust for watermarking. The authors in [20], have computed the moments in polar coordinate system and their accuracy was better than the zeroth order approximation. However, in the watermarking algorithm the image function is interpolated two times in the watermark embedding and extraction process. Therefore, there is still an scope for the further improvement in the moment computation and their applications in the watermarking. The authors in [21] have used Gaussian quadrature numerical integration method [22] as an approximation to the integration. This method improved the robustness of the watermarking techniques as their results show that the watermarking robustness is better than the ZMs, PZMs, PHTs and RHFMs. Their method uses weights computed from the Legendre polynomials roots and the weights, but the sum of weights is not always sum to two. One of the properties of the Gaussian quadrature integration is that the weights must sum to two. In addition, this method also does not take into account the whole unit disk. Therefore, there is a requirement for an algorithm which takes into account the whole unit disk and improves the accuracy of the moments and hence improves the robustness of the watermarking algorithm.
In this paper, we have utilised an integration technique proposed by the authors in another study, which uses the entire unit disk and improves accuracy of the PHFT moments. Watermark bits are embedded into this improved moments. The improved moments contribute considerably in improving the robustness of the watermarking technique. The orthogonal moments are not explored much in the medical images. In additional, the accurately computed PHFT moment is not used in watermarking. Therefore, the novelty of this work is the application of the improved accurate PHFT moments in the medical images. We have also tested the robustness of the watermarking with various attacks. We have compared the results with the ZM, PZM [16], PHT [17,20] and PHFT [21]. In the next section we will define the PHFT and the watermarking algorithm is constructed. The results are discussed in Section 3, finally we conclude with a conclusion section.

METHODOLOGY
This section has five subsections. First two subsections describe the definition of PHFT and its computations, the third section presents a image scrambling method used to provide an additional layer of security. The watermark embedding and watermark extraction procedure are described in the final two subsections.

Polar harmonic Fourier transform
Let us denote Z as set of integers and Z + denotes the set of non negative integers. The PHFT with order u ∈ Z + and repetition v ∈ Z , of a function g(r, ) is a complex function defined over a unit disk as [23]: where T u (r ) is known as the radial basis function and defined as sin (u + 1) r 2 , if u is positive and odd. cos u r 2 , if u is positive and even. ( The kernel functions H uv (r, ) are orthogonal to each other, i.e., The function g(r, ) can be reconstructed using Equation (4) [23]: Here umax is the maximum order and vmax is the maximum repetition used for the function reconstruction. To reconstruct and image f (x, y) the coordinates (x, y) are transformed to (r, ).

PHFT computation
Let us consider a square image f (i, j ) of size N × N . Let us consider a circle inscribed into the square image with the four boundaries as the tangent to circle. Let us give a coordinate system to the image by considering that the centre of the image as origin and the two axes parallel to the columns and rows. Assign the left boundary as x = −1 and right boundary as x = 1. Similarly, assign the lower boundary as y = −1 and the upper boundary as y = 1. With this configuration the radius of the inscribed circle will be 1 and width x along x-axis and y along y-axis are x = y = 2 N . Let us denote the centres of the image pixels as (x i , y j ), i, j = 0, 1, … , N . The PHFT moments for an image can be expressed by modifying Equation (1) as where (r i , j ) is the polar form of centres (x i , y j ) of the pixels. If we restrict the index i, j in Equation (6) such that the centres (x i , y j ) is within and on the unit circle, the corresponding approximation of the moments is known as zeroth order approximation. This method is used in [16] and [17]. The accuracy of the moments can be increased by refining the function H uv (r i , j ) x y as Here, (r, ) is polar form for the points lying within the pixel with centre (x i , y j ) and in unit disk domain. The domain of integration depends on the pixel location. If a pixel falls completely inside the circle, the domain of integration will be the square boundaries of the pixel. If the pixel falls completely outside the unit circle the integral in Equation (7) will be zero and contribution of the pixel in the moment will be zero. If the circle passes through a pixel, the limit of the integral will be circular boundary instead of the rectangular boundary. In earlier studies such circular boundary was not taken into account and therefore complete unit disk was not taken into consideration for the computation of moments. For example, Figure 1 shows the three different cases of the intersection of the circular boundary with pixel boundaries. In the following the difference between the proposed method and other methods. In the zeroth order approximation the first case will be neglected and approximated with zeros. The next two cases will be treated as equivalent and approximated with Equation (8) where a, b are the lower and upper boundaries along the y-axis, respectively, and c, d are the left and right boundaries along the x-axis, respectively, of the respective pixels. In the proposed method all the three cases are treated differently and the double integration is performed in the required region only, no region is omitted and no unnecessary regions are added. The first case of Figure 1 is computed by using Equation (9) ∫ ∫ H uv (r, )dxdy = ∫  Figure 1 is computed by using Equation (10) The third case is computed by using Equation (11) where a 1 is the intersection of the right boundary with the circle. The method used in [21] uses Gaussian quadrature numerical integration method [22] as an approximation to the integration. As discussed in the introduction, the weights used in this method when applied in the three cases in Figure 1 do not sum to two. Figure 1 and Equations (9)- (11) illustrated that the proposed method takes the circular boundary into consideration for pixels intersecting the unit circle. Therefore, the limits of integration of such pixels in Equation (7) is no longer constant but a variable. Double integration of such integration is computed numerically. One of the popular methods for such integration is Quadpack [24] which is a subroutine package for automatic integration. Quadpack is part of the integrate module in python library scipy. Our method for computing Equation (7) consists of two steps. In first step, which is analytical part, we have found the boundary of each pixel. In the second step, which is numerical part, these boundaries are passed into the dblquad function of the integrate module. The image is reconstructed using Equation (12) f The computations are computationally intensive. However, as noted in Equation (6), the computationally intensive part which is the computation of Equation (7), is performed only once for each pixel. Then this computed value can be applied on any image to compute the image moments. We have used an Ubuntu laptop with 8GB RAM for our computations.

Imagel scrambling
To provide an additional layer of security, we have scrambled the watermark before embedding. A chaotic sequence generated from three dimensional quadratic system is used for the scrambling of watermark. The system consists of three first order ordinary differential equations in variables x, y, z (shown below) with five constant parameters a, b, c, d, e [25] (Note that the parameters a, b, c, d are different from the parameters a, b, c, d used in the PHFT computation section as limits of integration). For description of the variables and parameter refer [25].
Recently [26] showed that chaotic sequence generated from such system of differential equations can be effectively used in image encryption. With initial condition x = y = z = 0.   Figure 2 shows the 256 iterations of the x and y variables. We have followed algorithm discussed in [26] for the scrambling of the watermark image (size as 16 × 16) using the sequence x and y of length 256 (=16 × 16). For complete description of the encryption and decryption algorithms refer [26]. Since the algorithm was primarily meant for gray scale image and our watermark is a binary image, we did a modification in the algorithm.
In the algorithm of [26], the y-sequence between 0 and 255. We have changed the y-sequence as a binary sequence by replacing elements of the sequence by 0 for the non-negative elements and by 255 for positive elements.

Watermarking algorithm
The watermark algorithm presented in this paper broadly function in three stages. In the first stage watermark bits are scrambled, in the second stage, the scrambled watermark bits are embedded into the moments of the image, in the third stage the modified moments are used for reconstruction of the watermarked image using Equation (12). Let us consider a watermark image of size n × n. Let us denote the n 2 bits of the scrambled watermark image as W i , i = 1, 2, … , n 2 . Since one watermark bit is embedded per image moment, a total of n 2 moments are required for embedding a n × n watermark image. The selection of proper moments are also an important task. It was shown in an earlier study by the authors that, the accuracy of the lower order moments are better than the accuracy of the higher order moments. Therefore, the moments order is restricted between 0 and 25 and repetition is restricted between −25 and (-25). Since, C uv = C u −v , that is the moment corresponding to the repetition v is complex conjugate of the moment corresponding to the repetition −v, only positive repetitions are considered as candidate for selection. Therefore, we have selected randomly n 2 moments with a fixed seed from a total of 26 × 25 = 650 moments (u >= 0, v > 0). Let us denote M i , i = 1, 2, … , n 2 , as the selected moments. Let A i represents the absolute values of the moments M i , that is, A i = |M i | and i represents the argument of the moments M i for i = 1, 2, … , n 2 . By division algorithm, there exists real numbers Q i and 0 ≤ R i < Δ, such that A i = Q i Δ + R i , where Δ is a quantisation step. The absolute values A i are modified as following equations for i = 1, 2, … , n 2 . Using the modified absolute values, the image moments M i are modified as, here j = √ −1. To obtain the watermarked image from the modified moments, Equation (12) is adjusted as, Here 'Re' stands for real part of a complex number and the index k contains the information of u, v so that H k (x i , y j ) can be computed from corresponding u and v. Let k is associated with u and v. If we make modification in the moment M k , then similar change will take place in the moment with order u and repetition −v. Sum of the corresponding two complex number in the right hand side of Equation (12) will be twice the real part any one of then, i.e., Let f (i, j ) denotes the original image and g(i, j ) represents the reconstructed image using Equation (12). Let us define the error in the reconstructed image as where i, j = 0, 1, … , N − 1. Let us denote by h(i, j ), the reconstructed image by replacing M k by M ′ k in Equation (12), that is, the reconstructed image with the modified moments. Then the watermarked image will be obtained by adding e(i, j ) and h(i, j ).

Watermark extraction
The seed is the only information required to extract the scrambled watermark. In this sense the watermarking algorithm is blind. The moments M ′ k of the watermarked image is computed and let us denote their absolute values as A ′ k , k = 1, 2, … n 2 . Then by division algorithm, there exist, Q k and 0 ≤ We can extract the scrambled watermarked bits W ′ k using the following rule.
for k = 1, 2, … , n 2 . The watermark image W k can be restored by decrypting the W ′ k using the chaotic sequence method discussed earlier in the section.

EXPERIMENTAL RESULTS AND DISCUSSION
We have selected an X-ray image and two MRI images both of size 256 × 256 ( Figure 3). In addition, the performance of the proposed solution is also assessed by using two publicly available datasets. First database is, chest X-ray image database of Mendeley Data, V2, [27] available publicly. The database contains two types of labeled X-ray images, normal and Pneumonia. The database is primarily used for classification problem in machine learning. However, normal X-ray images of first category (total 69) are used for the validation of the watermarking technique discussed in this paper. The second database is the chest X-ray image database of ChestX-ray8 [28] available publicly. This database contains 108948 frontal-view X-ray images of 32717 unique patients with the eight different disease labels. This database, too, is primarily used for classification problem in machine learning. However, first 50 X-ray images from the fourth category are used for the validation of the watermarking technique discussed in this paper. The watermark image is a binary image of size 16 × 16 ( Figure 3). The High values of the quantisation step Δ gives lower PSNR but high robustness against various attacks. The low values of the quantisation step gives high PSNR but less robustness against various attacks. Therefore, there is a trade-off between PSNR and robustness. We have selected experimentally the value of the quantisation step in such a way that the PSNR is above 40 and robustness is also not compromised. This way, the quantisation step Δ is set at 0.2. The maximum order of the PHFM is restricted to 25 as in [21]. The quality of the watermarked image is measured in terms of peak signal-to-noise ratio (PSNR). The quality of the extracted watermark is measured in term of bit error ratio (BER) defined as where b is the number of pixels in which the extracted and the original watermark do not match. Since, b and N 2 are positive numbers and b ≤ N 2 , BER is always between 0 and 1. If the extracted and original watermark are similar, then b will be smaller and BER will be close to 0. Similarly, if the two watermark image are very different from each other, b will be higher and BER will be close to 1. The quality of the watermarked image is quantified using the peak signal-to-noise ratio (PSNR), which is defined as in Equation (19).
where MSE is the mean squared error between the original image and the watermarked image.
We have compared the watermark algorithm with ZM, PZM in [16], PHT in [17,20] and PHFT in [21]. The robustness of the watermarking algorithm is compared using various attacks. Complete analysis is performed over an ubuntu machine with 8 GB RAM using Python 3.

Experimental results on X-ray and MRI images rotational invariance
The ZM, PZM, PCET, PCT, PST and PHFT are theoretically invariant to rotation. When watermark bits are embedded into an image and the moments are computed accurately, the watermark will be extracted from the watermarked correctly. Due to rotational invariant property of the moments, when the watermarked image is rotated the moments will be similar to the original moments and watermark will be extracted without error. Figure 4 shows the watermarked images rotated counter clockwise with angles at 10 • , 20 • and 30 • , 40 • , 50 • , 60 • , 70 • , 80 • and 90 • along with corresponding extracted watermark images. The visual results indicate the good robustness of the propose method. Table 1 shows the comparison of watermarking algorithms, when watermarked TABLE 1 The first column represents the algorithms used for the comparison of the watermarking algorithms on the chest X-ray. The second column represents PSNR between the base image and the watermarked embedded image. The third column represents the BER of the extracted watermark. The next nine columns represent the of the BER of the extracted watermark when watermarked image is rotated counter clockwise at angles at   image (chest X-ray) is rotated counter clockwise at angles at It is observed that the PSNR of the ZM and PZM is above 50. However, their property to resist rotation is poor as they have higher BER close to 0.5 at each of the angles. The PCET, PCT and PST of [17] have smaller PSNR compared to ZM and PZM but their BER is much smaller that ZM and PZM. The BER of the PCET [20] has small BER across all angles but the PSNR is only 38.76.
Since PSNR of PCET in [20] is smaller than 40, analysis involving PCET was not done. The BER of PCT [20] is comparable with our method, but PSNR is less than our method. The PSNR of PST [20] is comparable with PSNR of our method, their BER is higher than ours, however. The BER of PHFT in [21] is only smaller at 50 • and 60 • . Table 2 shows the comparison of watermarking algorithms, when watermarked image (teeth X-ray) is rotated counter clockwise at angles 0 The results are similar to X-ray images. It is observed that the PSNR of the ZM and PZM is above 50 as in the case of chest Xray. However, their robustness to resist rotation is poor as they have higher BER close to 0.5 at each of the angles. The PCET, PCT and PST of [17] have smaller PSNR compared to ZM and PZM but their BER is much smaller than ZM and PZM. The BER of PCT [20] is smaller than the proposed method only at 50 • and 60 • , and its PSNR is less than the proposed method. The PSNR of PST [20] is also less than the PSNR of the proposed method, their BER, too, is higher than ours, however. The PSNR of PHFT in [21] is little smaller than ours and the BER is also larger at 30 • and 50 • .
For chest MRI image, at 0 • -20 • and 70 • -90 • , the BER of our algorithm is at least not higher than other methods (see Table 3). At 40 • and 50 • , the BER of only PCT [17] is smaller, at 30 • , and 60 • the BER of only PCT [20] is smaller than the proposed method. The robustness of the proposed method at MRI image is comparable with PHFT in [21] and inferior only to PCT in [20] which has smaller PSNR than the proposed method.

Scaling invariance
The moments discussed in the paper are also scaling invariant. Therefore, if computed correctly, the extracted watermark will  be similar to origin watermark. Figure 5 shows the watermarked images scaled with a factor of 0.5, 0.75, 1.5, 2, respectively, and corresponding extracted watermark images. The visual results show that the proposed method is robust against various scaling attacks. Table 4 shows the BER of the extracted watermark when watermarked image is scaled with a factor 0.5 (second column), 0.75 (third column), 1.5 (fourth column) and 2 (fifth column). It is clear from the table that the proposed method along with PCT of [20] and PHFT of [21] are completely scaling invariant. The ZM and PZM are do not show scaling invariant property. In [17], the BER of PCET is higher than PCT and PST. Therefore, PCT and PST have better resistance to rotational attack than the PCET. However, in the [20], the scaling invariance property is followed better in the PCET than PST. The scaling resistance shown by PCT is better in [20] than that shown in [17]. In [17], zeroth order approximation was used for the moment computation while in [20] polar interpolation method was used for the moment computations. Therefore, an approach for computing the moments plays an important role. Table 5 shows the BER of the extracted watermark when teeth X-ray watermarked image is scaled with a factor 0.5 (second column), 0.75 (third column), 1.5 (fourth column) and 2 (fifth column). Similar to chest X-ray, the proposed method along with PCT of [20] and PHFT of [21] are completely scaling invariant. The ZM and PZM do not show the scaling invariant property. In [17], the BER of PCET is higher than the PCT and PST. Therefore, PCT and PST have better resistance to rotational attack than the PCET. The scaling resistance shown by PST in [20] is better than the PST in [17].
The results of scaling attack on the MRI image is similar to the X-ray images. Therefore, the corresponding table is not shown here. The BER of the proposed method along with PCT of [20] and PHFT of [21] is zero. Therefore, the proposed method is completely scaling invariant, i.e., the extracted watermark will be same as original after the scaling attacks on the MRI images.

Addition of noise
The success of a watermarking algorithm is also depend on its ability to resist addition of noise including salt and pepper, speckle, Poisson and Gaussian noises. Figure 6 shows the watermarked images with addition of four types of noises, salt & pepper, Speckle, Gaussian and Poisson noise (from top left in clockwise) along with corresponding extracted watermark images.
The visual results of the figure show that the proposed method has good robustness against addition of noise. Table 6 shows the comparison of BER extracted from watermarked image when it is attacked with addition of noise. The table shows the BER for four attacks. Equal amount of salt and pepper with total of 0.01 is taken for the addition of salt and pepper noise.  As clear from the table no method has smaller BER than the proposed method. Therefore, the proposed method performed best when salt and pepper noise is added to the watermarked image. When the amount of salt and pepper in the watermarked image is reduced to 0.003, the BER of the proposed method becomes zero. Similar to salt and pepper noise, the BER of the proposed method when speckle noise (uniform distribution between -0.2-0.2) is added, is better than all methods. For uniform distribution between -0.1-0.1, the BER of the proposed method becomes zero. Therefore, we conclude by considering the PSNR values that the proposed method performs best when speckle noise is added to the watermarked image. When Poisson noise is added to the watermarked image, the BER of the retrieved watermark is zero. The BER of PCT in [20] and PHFT in [21] is also zero. Other methods has positive BER. Therefore, in the case of Poisson noise the proposed method is not inferior to any method. In Gaussian noise ( = 0.05), the BER of the proposed method is smallest. Therefore, the proposed method has best resistance to Gaussian noise attack compared to other methods discussed in this article. In addition, when = 0.03 was used in the Gaussian noise, the BER was 0. Table 7 shows the comparison of BER extracted from the teeth X-ray watermarked image, when it is attacked with addition of noise. Similar to chest X-ray, equal amount of salt and pepper with total of 0.01 is taken for the addition of salt and pepper noise. In the teeth X-ray image, the BER of the proposed method along with the PHFT in [21] is least. Therefore, the two methods performed best when salt and pepper noise is added to the teeth X-ray watermarked image. When the amount of salt and pepper in the watermarked image is reduced to 0.003, the BER of the proposed method becomes zero, as was the case with chest X-ray. Unlike the chest X-ray, the BER of the proposed method when speckle noise (uniform distribution between -0.2-0.2) is added is slightly larger than the PHFT [21] which has smaller PSNR than the proposed method. For uniform distribution between -0.1-0.1, the BER of the proposed method becomes zero. When Poisson noise is added to the watermarked image, the BER of the retrieved watermark is zero. The BER of PCT in [20] and PHFT in [21] is also zero. Other methods have positive BER, as noted earlier in the case of chest X-ray. Therefore, in the case of Poisson noise the proposed method is not inferior to any method. In Gaussian noise ( = 0.05), the BER of the proposed method is smallest. Therefore, the proposed method has best resistance to Gaussian noise attack compared to other methods discussed in this article. In addition, when = 0.03 was used in the Gaussian noise, the BER was 0.
The BER of the proposed method is least when salt and pepper noise, Poisson noise and Gaussian noise are added to the MRI watermarked image. When Speckle noise is added only PCT [20] has lower BER than the proposed method. Therefore, the proposed method is more robust than other methods when salt and pepper noise, Poisson noise and Gaussian noise are added to the MRI watermarked image.

Image filtering
The robustness of our watermarking technique is also tested using three image filters, median filter, average filter and the Gaussian filter. The size of the window for all the filters is fixed at 5 × 5. Table 8 shows the BER of the watermark, when the watermarked image is filtered using three filters. The ZM and PZM do not show robustness against the three filtering attacks. When median and Gaussian filtering are applied on the watermarked image, the BER of the proposed method is only second to the PST in [20]. However, as noted earlier, the PST is not robust against the addition of noise, image scaling and image rotation. When the window size was reduced to 3 × 3 the BER for our algorithm was 0 for all the three filters, indicating a better against filtering attacks. Figure 7 shows the watermarked image attacked with the three types of filters of the size 3 × 3 and the corresponding extracted watermark image with BER 0. The size of the window for all the filters for teeth X-ray is also fixed at 5 × 5. Table 9 shows the BER of the watermark, when the watermarked image is filtered using the three filters. Similar to other attacks, the ZM and PZM do not show robustness against the three filtering attacks. When median filtering is applied on the teeth X-ray watermarked image, the BER of the proposed method is least, indicating the best robustness of the proposed method along with PHFT in [21]. In average filtering, the BER in the proposed method is only larger than PST in [17]. However, as noted earlier, the PST is not robust against other attacks. In Gaussian filtering, the BER in the proposed method is only larger than PCT in [20]. When the window size was reduced to 3 × 3 the BER for our algorithm was 0, as in the case of chest X-ray, for all the three filters, indicating a better against filtering attacks for the teeth X-ray as well. When median filtering is applied on the MRI watermarked image, the BER of the proposed method is larger than the BER of PCT and PST in [17]. However, as noted in this article, they are not robust against other attacks. We observed that the PHFT is not robust against average filtering as the BER of PHFT used in our and in [21] are better only than ZM and PZMs. However, for Gaussian filtering, the proposed method is more robust (BER = 0.0156) than the other methods in MRI images.

JPEG compression
JPEG compression is also a common form of attacks on images. The robustness of proposed algorithm is tested with JPEG compression attack with quality 90, 75, 50 and 25. The BER of the watermark is 0 for the first three qualities and very small, 0.0078 for quality 25. Therefore, the proposed algorithm is very robust against JPEG compression attack. Figure 8 shows  Table 10 shows the comparison with other algorithms when watermarked image is attacked with JPEG compression. When the quality factor is kept at 90, 75 or 50, the BER of PCT in [20] is also zero. However, with the quality factor 25, the proposed algorithm has smaller BER than the PCT in [20]. Only the BER of PHFT in [21] is comparable with the proposed algorithm at all four quality index. However, as noted already, the proposed Watermarked images with three types of filtering attacks, median, average and Gaussian filter (from left) and corresponding extracted watermark  algorithm has better robustness [21] when watermarked image is attacked with rotation, filtering and noise attacks. Similar to chest X-ray, the BER of the watermark applied over the teeth X-ray is also 0 for the first three quality factors and very small (0.0039) with quality factor 25. Therefore, the proposed algorithm is robust against the JPEG compression attack. Table 11 shows the comparison with other algorithms when watermarked image is attacked with JPEG compression. Similar to chest X-ray, when the quality factor is kept at 90, 75 or 50 for the teeth X-ray, the BER of PCT in [20] is also zero. However, with quality factor 25, the proposed algorithm has better BER than PCT. Only the BER of PHFT in [21] is comparable with the proposed algorithm at all the four quality index. However, when the quality for JPEG is further reduced to 25, the BER in the proposed algorithm is 0.0312 while the BER in PHFT of [21] is 0.0430. Therefore, the proposed method is most robust against the JPEG compression attack as compared to other methods for X-ray images. For MRI image, the BER for the quality factors 90, 75 and 50 is zero. At quality factor 25,

Image flipping
Robustness with image flipping attacks (vertical, horizontal and both side flipping) is also tested in this paper. It is found that the proposed method has zero BER for all the three image flipping. Therefore, the method is robust against these attacks. Table 12 shows BER of various methods. The BER of PCT and PHFT are zero. However, the BER of other method is positive, indicating these methods are not robust against image flipping attacks. The visual results shown in Figure 9 also indicate the robustness of the proposed method against three types of image flipping attacks. Similar to chest X-ray, in the case of teeth X-ray image, it is found that the proposed method has zero BER for all three image flipping attacks. Therefore, the method is robust against these attacks. Table 13 shows BER of various methods. The BER of PCT and PHFT are zero. However, the BER of other method is positive, indicating these methods are not robust against image flipping attacks in the case of teeth X-ray image as well.
Similar to the X-ray images, in the case of chest MRI image, it is found that the proposed method has zero BER for all three  image flipping attacks. Therefore, the method is robust against these attacks for the MRI image as well.

Motion blur attack
The proposed method is also robust against the motion blur attack as BER for both the vertical and horizontal motion blur is zero for kernel size 3. For the kernel size 5 (Table 14), only BER for PCT in [20] is smaller for vertical motion and only BER of the PST in [20] is smaller than the proposed method. But, other algorithms except the PHFT in [21] have larger value of BER. The first row in Figure 10 shows the watermarked image attacked with vertical and horizontal motion blur and their corresponding extracted watermarks also. The visual results show the robustness of the proposed method against the motion blur attacks of both types. In the case of teeth X-ray image also, the proposed method is robust against the motion blur attack as BER for both the vertical and horizontal motion blur is zero for the kernel size 3. For the kernel size 5 (Table 14), the BER for the proposed method along with that of PHFT in [21] is smaller than the other methods which indicates the better robustness against motion blur attack. Similar to the X-ray images, in the case of chest MRI image, it is found that the proposed method has zero BER for both the two motion blur attacks for the kernel size 3. Therefore, the method is robust against blur attacks for the MRI image as well.

Gamma modification attack
The BER of the proposed method in case of Gamma correction attack is similar to PCT and PHFT. However, the robustness of the proposed algorithm is better than other algorithms as shown in Table 15. The Gamma correction attack with = 0.9 and = 1.1, resulted in BER= 0.0234 and 0.0312, respectively, indicating good robustness for Gamma correction attack. The visual results in the second row of Figure 10 supports the robustness of the proposed method against the gamma corrections. Similar to the chest X-ray image, the BER of the proposed method in case of Gamma correction attack is similar to PCT over the teeth X-ray image. When compared with PHFT in [21], the BER is slightly higher in the proposed method at = 0.9 but lesser at = 1.1. However, the robustness of the proposed algorithm is better than the other algorithms as shown in Table 15. For MRI image, the BER for both the values of gamma is 0.0156, which is very small. So, the BER is smaller than or equal to all other methods except the PCT in [20].
In this paragraph, the summary of the experimental results with X-ray medical images and the comparison with other algorithm is presented. The proposed algorithm is robust against various attacks as discussed above since the BER is very small. For some attacks, even the BER is zero. When compares with ZM and PZM in [16], it was observed that their methods are not robust against almost all the attacks for both the X-ray images. However, their PSNR is very high (more than 50). In both the X-ray images, the PSNR of PCT and PST in [17] is higher but PSNR of PCET is lower than the proposed method. When compared with PCET, the robustness of the proposed method is better in all the eight types of attacks discussed above except the average image filtering attack that too only over the chest X-ray image. The robustness of the proposed method is also high than PCT in all the attacks discussed above for both images. The PST is more robust than the proposed algorithm when image is rotated with 40 • and 50 • in the chest X-ray only. At all other angles, the proposed method is better. When image is scaled, the proposed method is again more robust than PST in [17]. When noise is added (salt and pepper, speckle and Poisson) in the chest X-ray watermarked image, the robustness of the proposed method is better than PST. PST is better only at addition of Gaussian noise. Again with respect to JPEG compression, flipping, motion blur and Gamma corrections, the proposed method is more robust than PST in [17]. However, the proposed method outperformed the PST in [17] at all attacks in the teeth X-ray image. When comparing with PCT in [20], it is noted that, at image scaling, image flipping and motion blur, BER is similar to the proposed method. The PCT is better than the proposed algorithm at Gamma corrections with Γ = 1.1 (at Γ = 0.9 for teeth X-ray) and image rotation with angles 60 • and 70 • (At 70 • the proposed method is better for teeth X-ray) only. The proposed method completely outperform the PCT at addition of noise, image filtering (except at Gaussian filtering), JPEG compression with lower quality. Similar to PST in [17], the PST in [20] is more robust than the proposed algorithm when image is rotated with 40 • and 50 • for chest X-ray. At all other angles the proposed method performs better. In addition, for teeth X-ray the proposed method completely outperformed the PST. At all other attacks, the proposed method is more robust than the PST. PHFT moments are used in the proposed method as well as in [21]. However, the computational approach used in this paper makes the computed moments more accurate which results in more robust watermarking scheme. In image scaling, JPEG compression, image flipping, motion blur attacks, the BER of the proposed method is same as PHFT in [21] for both the X-ray images. However, the proposed method has better PSNR and it outperformed the PHFT in [21] at image rotation, Gaussian noise, median and average filtering (only in chest X-ray). Therefore, overall the proposed method is more robust than the other method at X-ray images. Similar analysis for the MRI image shows that the proposed method is robust for the MRI too.

Comparison with the chest X-ray images at Mendeley database
The performance of the watermarking technique with improved moments is also analysed over the chest X-ray images database of Mendeley Data, V2, [27] available publicly. The database contains two types of labeled X-ray images, normal and Pneumonia. The database is primarily used for classification problem in machine learning. However, normal X-ray images of first category (total 69) are used for the validation of the watermarking technique discussed in this paper. The second row in Table 16 shows the average PSNR among 69 images from the database and their corresponding watermarked images for the methods discussed in this paper. The PSNR of the proposed method along with PHFT, PCT and PST of [20], PCET of [17] are comparable. The PSNR of the PCT and PST in [17] and ZM and PZM in [16] have higher PSNR. However, extracted watermark in not reversible. In addition, they have poorer robustness when compared with the first three methods in Table 16. The PST in [20], too, in not reversible. Therefore, the proposed method is better as far as reversibility is concerned. The PCT in [20] has very small BER, that is, it is not completely reversible. The BER of the proposed method is smaller than the PCT when the watermarked images are attacked with scaling, Poisson noise, median filtering, JPEG compression at all the qualities, flipping and motion blur. Therefore, using the fact that the watermark extraction in the proposed method is reversible and the watermarked extraction in the PCT is not reversible, we can safely state that the proposed method is better at extracting watermark and it is more robust than the PCT in many kind of known attacks (Table 16). Other than the proposed method, only PHFT in [21] provides completely reversible extraction of the watermark. However, the robustness of the proposed method is better than the PHFT at rotational attacks at all the angles within 0 • − 90 • . In addition, the proposed method show better robustness than the PHFT at scaling, speckle noise and motion blur. At all other attacks shown in Table 16 the BER of two methods are comparable. Therefore, it can be concluded that the proposed method is reversible and better robust compared with the methods discussed in this paper against many attacks.

Comparison with the chest X-ray database
The performance of the watermarking technique with improved moments is also analysed over the chest X-ray images database of ChestX-ray8 [28] available publicly. This database contains 108,948 frontal-view X-ray images of 32717 unique patients with the eight different disease labels. The database is primarily used for classification problem in machine learning. However, first 50 X-ray images from the fourth category are used for the validation of the watermarking technique discussed in this paper. The second row in Table 17 shows the average PSNR between the 50 images from the database and their corresponding watermarked images for the methods discussed in this paper. The PSNR of the proposed method along with PHFT, PCT and PST of [20], PCET of [17] are comparable. The PSNR of the PCT and PST in [17] and ZM and PZM in [16] have higher PSNR. However, extracted watermark in not reversible as BER is high. In addition, they have poorer robustness when compared with the first three methods in Table 17. The PST in [20], too, in not reversible. Therefore, the proposed method is better as far as reversibility is concerned, as the BER in the proposed method is close to zero. The PCT in [20] has very small BER (larger than the proposed method), that is, it is not reversible. The BER of the proposed method is smaller than the PCT when the watermarked images are attacked with scaling, Speckle noise, Poisson noise, median filtering, Gaussian filtering, JPEG compression with higher quality, flipping, motion blur and Gamma corrections. Therefore, using the fact that the watermark extraction in the proposed method is close to reversibility and the watermark extraction in the PCT is not so FIGURE 11 Scatter plot between BER of plane watermark and encryption watermark close to reversibility, it can be stated that the proposed method is better at extracting watermark and it is more robust than the PCT in many kind of known attacks (Table 17). Other than the proposed method, only PHFT in [21] has comparable BER. However, as shown with the Mendeley Database, the robustness of the proposed method is better than the PHFT at rotational attacks at all the angles within 0 • − 90 • . In addition, the proposed method show better robustness than the PHFT at scaling, salt and pepper noise, speckle noise, median filtering and Gamma corrections. At all other attacks shown in Table 17 the BER of two methods are comparable. Therefore, it can be concluded that the proposed method is close to being reversible and better robust compared with the methods discussed in this paper against many attacks.

Experimental results on encryption with chaotic sequence
The analysis in the last section was presented with plane watermark image. The Zms and PZm in [16] and PHTs in [17] were not presented with encryption. Therefore, to be at same platform, the encryption part was removed from the comparison in the last subsection. In addition, the main focus of the proposed algorithm was increasing the accuracy of the computed moments, so that they follow the geometric invariance property, which allow the watermark to be more robust. As discussed already, for more security, an additional layer of encryption with chaotic sequence is included. Figure 11 shows the scatter plot between the BER of plane watermark and the BER of encryption watermark extracted from all the three images chest X-ray, teeth X-ray and chest MRI together. When the BER of plane watermark is zero, the BER of encrypted watermark is also zero. Therefore, if a watermark without encryption has zero BER than the BER of encrypted watermark will also be zero. If the BER of the plane watermark is less than 0.05 then the BER of encrypted watermark is less than 0.1. Since, BER of most of the attacks on plane watermarked image is less than 0.05, the BER of the attacks will not go beyond 0.1. Therefore, it can be concluded that addition of a protective layer in the form of chaotic sequence will increase the security of the embedded watermark which will be robust against many kind of attacks.

CONCLUSION
Image moments are few of the image descriptors which are rotation and scaling invariant. Image moments can be used to produce robust watermarking algorithms. The PHFT moments are used for the watermarking algorithms. In addition to PHFT moments, ZM, PZM, PCET, PCT and PST are also used for the image watermarking. These methods are applied over ordinary images and shown to be robust against many attacks. However, their robustness can be increased by increasing the accuracy of the moment computations. In this paper, a blend of analytic and numerical method is used to increase the accuracy of the PHFT moments. The accurate moments are then used for the image watermarking. The moments are applied for watermarking of the X-ray and MRI images. The algorithm using the improved moments is not restricted to medical images. It can be applied to a wide range of images. However, in this paper, the robustness of the algorithm using the improved moments are analysed using medical images only. It is found that the proposed method is more robust than some of the existing methods based on moments. In most of the attacks, it is also found that the proposed method outperformed other methods. In few attacks, there exist some methods which have BER lower than the proposed methods. Particularly, the PCT in [20] and PHFT in [21] came very close to the robustness of the proposed algorithm. However, only lower order moments are used in this paper yet the results against various attacks are better. In another work, the authors have shown that the difference between moments computed using the proposed method have more accuracy compared to other methods at higher moments. Therefore, if more number of bits are to be inserted into the images, higher moments will be used. Which will result in even higher difference in the BER between the proposed method and other methods. This will explored in more details in further study.