A secure fractional-order polar harmonic transforms based image watermarking scheme using funk singular value decomposition

Abstract

In addition to existing cryptographic systems, watermarking technologies have been developed to add extra security. Digital watermarking utilizes embedding or hiding techniques to protect multimedia files from copyright violations. Fundamental procedures of digital watermarking techniques are embedding and extraction. Singular value decomposition (SVD) based Image watermarking schemes become popular owing to its better trade-off among robustness and imperceptibility. Nevertheless, false positive problem (FPP) is a major issue of SVD-based watermarking schemes. The singular value that is a fixed value and does not contain structural information about image is the primary cause of FPP problem. Therefore, Message Digest algorithm image watermarking scheme based on Funk Singular Value Decomposition and Fractional-Order Polar Harmonic Transform (FSVD-FOPHT) is proposed in this paper to address this problem. The MD-5 algorithm is used to extract data from the host and watermark imageries and then create secret key. The FSVD-FOPHT method is utilized to hide watermark information in host image. The secret keys are extracted from hided image using inverse process of Fractional-Order Polar Harmonic Transforms with Funk Singular Value Decomposition algorithm. By using the extraction procedure, watermark image is extracted, and then reconstructs original watermarked image. During extraction procedure, the secret key is used for authentication to address FPP. Then, the proposed method is implemented in MATLAB and performance is analyzed with evaluation metrics, such as Embedding capacity, MSE, PSNR, and NC. The proposed method provide 14.6%, 17.34%, 19.53%, 21.46% and 23.89% high PSNR for cold-snow-landscape-water test image, 14.29%, 16.47%, 18.39%, 20.16% and 21.93% high PSNR for landscape-nature-sky-blue Test image, 16.85%, 19.99%, 22.70%, 27.22% and 29.16% high Embedding Capacity for cold-snow-landscape-water test image 22.83%, 24.64%, 27.92%, 29.60% and 31.77% high Embedding Capacity for landscape-nature-sky-blue Test image 35.38%, 32.63%, 30.95%, 28.61% and 26.08% low extraction time compared with existing methods SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Keywords

Fractional-order polar harmonic transforms funk singular value decomposition embedding and extraction message digest algorithm secure image watermarking

1 Introduction

Due to issues with unauthorized copying, editing, distribution, and integrity, the demand for improved data protection techniques has significantly expanded with the mass adoption of digital applications as well as developed the services of network technology. In addition to existing cryptographic systems, watermarking technologies have been developed to give more security. Digital watermarking involves embedding or hiding techniques to protect multimedia data from copyright violations [1]. The two fundamental operations in digital watermarking techniques are embedding together with extraction. The embedding operation conceals watermark details in another digital data object, like an image, while the extraction operation entails extracting the embedded data [2]. Besides, digital watermarking can be thought of as a secret information embedding technique [3]. Digital watermarking offer copyright security, and also utilized content identification with authentication, digital forensics, tamper detection, broadcast monitoring, fingerprinting, and media file archiving [4].

Two types of digital image watermarking are: visible and invisible. Visible insert logos or labels as proof of content ownership in a host image [5]. It is easily identifiable, but easily hacked and eradicated by adversaries. Invisible watermarks are typically employed on account of it hard to be observed through human visual system (HVS) [6]. The idea behind invisible watermarks is to include watermark data in unidentified areas of a host image [7]. There are three fundamental requirements that apply to all current invisible watermarking systems: robustness, imperceptibility, security [8]. The robustness assures that the extracted watermark is still identifiable even after geometrical as well as non-geometrical attacks are made in the watermarked image [9]. A watermarking method is undetectable if there is no discernible variation among the host and watermarked image [10]. As a result, it is challenging to locate the embedded watermark or recognize patterns caused by the embedding procedure [11]. A secure watermarking system is one that is protecting from threats.

Due to the fact that it influences the degree of embedding, the scale factor is crucial to image watermarking [12]. Single scaling factor (SSF) may lead to a steady embedding, but it may not accomplish the desired balance betwixt robustness and invisibility [13]. For example, a small SSF value leads to higher imperceptibility but lower robustness against typical attacks. On the contrary, a large SSF value enhances robustness when sacrificing imperceptibility [14]. Because of this, investigators have utilized multi-scaling factors (MSF) in lieu of SSF to attain needed robustness with imperceptibility [15]. Optimum MSF values are defined with the help of optimization approaches, like genetic algorithms, particle swarm optimization (PSO), multi-objective ant colony optimization (MOACO), differential evolution (DE) [16]. MSF-base optimization strategies reach all necessities from better image watermarking method [17]. Nevertheless, SVD base optimization strategies have computationally cost, also having proper MSF values limited range for all watermarking methods [18].

To overwhelm the false positive problem and other safety issues, some solutions have been presented, like hashing, encryption, digital signature, singular vector embedding, principle component embedding. By one-way hash function, the side information, UW and VW are hashed in hashing method, and saved while the process of embedding. The side information is hashed together with analyzed against the saved hash values during extraction process. If it is equal, the side information effectively authentic and complete the extraction. Else, quit the extraction. Prior to embedding, the watermark is encrypted when encryption is used. To recover the original watermark during extraction, the effective decryption must be enabled. Else, it is invalid, resulting arbitrary image will be appeared. Similar to other approaches, the scheme depends on digital signature embeds the watermark straightforwardly to SH. The host image also contains the side information’s digital signature. The extracted watermark can be authenticated using the digital signature after extraction. These models have proven a better trade-off among robustness and imperceptibility. Even though, these models still dependent upon side information as the extraction key, and are vulnerable to diverse FPP variations, because the side information has non-sensitive to minute variations, estimating the watermark is simple. Also, the additional authentication procedures are needed while the process of extraction that incurs computational overhead.

1.1 Stability of SVD singular values

The singular values of an image matrix after SVD transformation are invariant under transpose, flip, rotation, scale, and translation. After mentioned attacks, the singular values are less affected by them. They make excellent candidates for embedding the watermark image. Each level of the SVD decomposition is anticipated to inherit these qualities. It concludes that the singular values in higher level of SVD are more robust to geometric along signal processing attacks. Finally, Message Digest algorithm image watermarking scheme utilizing Funk Singular Value Decomposition and Fractional-Order Polar Harmonic Transform addresses the disadvantages of the existing techniques.

1.2 Correction of false positive detection

The majority of SVD-based watermarking methods suffer from false positive detection as a result of ambiguous circumstances where an attacker can demonstrate ownership of the watermark image. A digital signature is created and applied to be placed to the cover image alongside the watermark image to prevent this effect. An authentication scheme is developed to verify the false negative effects of the digital signature. When a watermarked image is subjected to severe attack, the digital signature is more likely to be corrupted. This demonstrates the need for an authentication mechanism to check the false negative effect of the digital signature. This is a fact that has not predicted in the prior schemes. In this work, a Message Digest algorithm image watermarking scheme utilizing Funk Singular Value Decomposition and Fractional-Order Polar Harmonic Transform mechanism, wherein if attackers pass the digital signature level, another test is enabled to prevent them from watermark extraction or verified the imprecision of digital signature extraction from severe attacks.

By using invisible image hiding procedure, the watermarked image is enduringly demolished and their original version of image cannot be reconstructed after excerpting the secret watermark image [19, 20]. So, invisible image hiding procedure cannot be used in some fields, for instance, military, medical, or law administration [21–23]. This can be motivated to do this work.

In this manuscript, Message Digest Algorithm Image Watermarking Scheme based upon Funk Singular Value Decomposition with Fractional-Order Polar Harmonic Transforms is proposed. The proposed method has 3 phases: key generation phase, embedding phase and extraction phase. In key generation phase, Message Digest algorithm-5(MD-5) is used to extract data from the host images and watermark imageries, and then produce a secret key. In embedding phase, FSVD-FOPHT method hides the watermark details in host image. In extraction phase, the secret keys are extracted from the hided image using inverse process of FSVD-FOPHT. By using extraction process, extract the watermark image, and then reconstructs the original watermarked image. During extraction process, the secret key is employed authentication to address FPP.

The main contribution of this manuscript is as follows,

In this paper, a new Funk Singular Value Decomposition image watermarking scheme in the frequency coefficient domain is proposed. The proposed technique utilizes a novel embedding method whereby the whole transformed watermark image is embedded in R_LL matrix of the decayed host image. The innovation of this proposed technique is to use Funk Singular Value Decomposition and Fractional-Order Polar Harmonic Transform for image watermarking that is embedded in the host image under secret key from host and watermark images. The host image first undergoes Fractional-Order Polar Harmonic Transforms, to make approximate image LL which is disintegrated by FSVD to acquire R_LL. The secret key is applied for authentication to deal FPP during the process of extraction. There is no requirement for optimization techniques when employing FSVD-FOPHT, which results in greater efficiency. The proposed technique attains the desired necessities of robustness, imperceptibility, high capacity, and security of image watermarking scheme. The proposed approach can help a wide range of CMSF values and is resistant to certain well-known attacks. The proposed technique has highly sensitive to the secret key.

Remaining manuscript is arranged as: section 3 describes the proposed method, section 4 proves the results and discussion, section 5 concludes the manuscript.

2 Literature review

Among the numerous research works related to image watermarking technique, few recent research works are reviewed below,

Zainol et al. [24] have presented singular value decomposition-base image watermarking utilizing chaotic map. The major disadvantage of singular value decomposition -based watermarking techniques was FPP. The secret key was 1st extracted from host image and watermark image and create chaotic matrix and chaotic multiple scaling factors to improve sensitivity. The watermark image was transferred to chaotic matrix was directly embedded into the host imageries using the chaotic multiple scaling factors. It provides higher security with high PSNR rate.

Mellimi et al. [25], have presented a rapid and effective image watermarking method utilizing deep neural network. To validate the efficacy of the proposed technique, some attacks like compression, noise and filtering are considered. To recognize the alterations caused by these attacks on the various frequency bands, Deep Neural Network is trained. The application of higher frequency sub-band low-high/LH1/high-low 2 maximizes invisibility for watermark insertion. In contrast, diverse sub-bands act differently against different kinds of attacks. The experimental result provides lower PSNR and higher image quality.

Zhong et al. [26] have presented, an automated and robust image watermarking scheme using deep neural networks. A blind image watermarking was dependent upon deep learning neural networks. Deep neural network was used to train and extend an automated image watermarking method. Deep learning structure was structured for the tasks of image watermarking that was trained in unsupervised mode to avert human interference. The experimental result shows best trade-off among robustness and invisibility with lower Structural similarity index (SSIM) rate.

Khare and Srivastava, [27] have presented the trustworthy and safe image watermarking technique uses homomorphic transform (HT) in discrete wavelet transformation domain. Where, high low sub band was selected, and then it was converted by homomorphic transform and it was decomposed into the components of illumination along reflectance. The watermark was scrambled with Arnold transform to make stronger security towards spiteful attacks. By the way, the reflectance components have the features of prominent image. The embedding process provides improved immunity against various attacks and imperceptibility was obtained and it was varied very quickly. It provides low PSNR rate and higher embedding capacity.

Gao and Chen, [28] have presented the robust and safe image watermarking technique using Speed-Up Robust Feature and better Artificial Bee Colony algorithm in discrete wavelet transform to solve FPP in different singular value decomposition -based watermark methods. Block wise singular components were applied for principal component insertion of watermark. The presented method was used in 2 ways: (1) IABC method improved scaling factors for balancing unseen, robustness, (2) improved convergence speed. Experimental outcomes demonstrated that high security level and lower imperceptibility.

Singh, [29] have presented the Robust and invisible image watermarking method dependent on singular value decomposition, Discrete cosine transform (DCT), bi-dimensional empirical mode decomposition (BEMD) and particle swarm optimization in wavelet domain. During embedding process, discrete wavelet transform decay the cover image as sub bands. BEMD decomposition executes in the chosen band of discrete wavelet transform. For optimization, particle swarm optimization was utilized for complex searches and multiple dimensional searches. DCT and SVD were chosen band. Factors, like embedding and scaling were embedded using the security key. It attains watermarked image with better quality and low PSNR rate.

Garg and Rama Kishore, [30] have presented the secured digital image watermarking method under Discrete cosine transforms, fuzzy entropy, and image scrambling in hybrid domain. Here, the secured and robust method was presented using fuzzy entropy, discrete cosine transform, image scrambling in hybrid domain. The presented method attains satisfying outcomes by giving better Mean square error, Peak signal to noise ratio values and different kinds of attacks were performed on watermarked image to give better Normalized correlation value and lower imperceptibility and robustness.

Hatoum et al. [34] have presented a deep learning approach for image watermarking attack. In this presented the effect of Fully Convolutional Neural Network (FCNN) as a denoising attack. The encoder-decoder removes the noise while maintaining the structure of the image by the use of denoising, which is improved by the deep architecture. FCNNDA overtakes the other attacks because it abolishes the watermarks. Spread Transform Dither Modulation and Spread Spectrum were employed to embed the watermarks at the images utilizing certain scenarios. It provides lower MSE with minimum embedding capacity.

Ge et al. [35] have presented a screen-shooting resilient document image watermarking scheme using deep neural network. A screen-shooting resilient watermarking method was presented for document image utilizing deep neural network. By utilizing that method, the watermark can be still extracted from the captured imageries when the watermarked image was displayed on the screen and captured by a camera. The end-to-end neural network with encoder was to embed watermark and a decoder to extract watermark. To simulate the distortions induced by the screen-shooting process in real situations, like camera distortion, shooting distortion, light source distortion, a distortion layer between the encoder and decoder was included during the training phase. A background sensitive loss upgraded the visual quality of watermarked document images at the training procedure. It provides higher bit extraction accuracy with higher MSE.

Ge et al. [36] have presented a robust document image watermarking scheme under deep neural network. An end-to-end document image watermarking scheme was suggested with the help of deep neural network. An encoder with decoder was considered to embed and extract the watermark. A noise layer was added to simulate the attacks that were encountered in reality, like Cropout, Dropout, Gaussian blur, Gaussian noise, Resize, JPEG Compression. To limit the embedding alteration on characters, a text-sensitive loss function was created. An embedding strength adjustment was presented to develop the watermarked image quality with tiny loss of extraction accurateness with minimum embedding capacity.

3 Proposed methodology

In this manuscript, FOPHT-FSVD is proposed for Secure Image Watermarking Technique. The lack of structural information refers to the absence of clearly identifiable patterns or relationships in a dataset or system. Fractional order Singular Value Decomposition (SVD) extends the traditional SVD by introducing a fractional exponent parameter, allowing for a more flexible representation of the data. In case, where structural information is lacking, fractional order SVD becomes a useful tool. By adjusting the fractional order parameter, it enables the exploration and extraction of potential hidden structures or patterns that may not be apparent with traditional SVD. Thus, fractional order SVD can capture different levels of structure, even in the absence of explicit knowledge or well-defined patterns. The fractional order polar harmonic transform (FOPHT) in the context of image watermarking is a specific technique that incorporates fractional orders into the polar harmonic transform for watermark embedding and extraction. The polar harmonic transform (PHT) is a mathematical tool that decomposes an image into a set of polar harmonic components. It represents the image in terms of radial frequency components (r) and angular frequency components (θ). The PHT is typically defined with integer orders, but the fractional order extension allows for more flexibility in capturing subtle image features. To incorporate fractional orders into the PHT for image watermarking, the fractional orders are introduced as parameters in the transform. These fractional orders control the behaviour and characteristics of the transform, influencing the quality and robustness of the watermark embedding and extraction process. The detailed discussion regarding Message Digest Algorithm Image Watermarking Scheme depends on Funk Singular Value Decomposition with Fractional-Order Polar Harmonic Transforms is given below,

3.1 Key generation

In key generation, Message Digest algorithm-5 (MD-5) is used to extract data from host imageries and watermark imageries and then hash them to produce secret keys. Commonly, these secret keys are employed in symmetric encryption models to guarantee data secrecy. Data reliability refers to the accuracy and trustworthiness of the data, which can be assessed by considering the credibility of the data source, the methods used for data collection, consistency within the dataset, cross-referencing with other reliable sources, and performing data quality checks. On the other hand, data richness refers to the depth and breadth of information in the dataset. It can be evaluated by examining the number of variables and dimensions included the level of detail or granularity, historical coverage, diversity in the data, and the potential to incorporate supplementary data sources. Evaluating data reliability and richness ensures that the data used for analysis is dependable and provides a comprehensive understanding of the subject under study. Generally, secret keys are 128 bits longer to tolerate brute force attacks. By the use of proposed MD-5 algorithm, a secret key must increase the sensitive data with little changes in its key bits. The initial conditions together with system parameters of improved chaotic maps are produced using these crucial bits. Then, maps repeated to create matrices that used in transforming watermark. In key generation phase, using Message Digest algorithm-5 algorithm, the host image is converted into 4 sub-bands. First the approximation image and the next 3 sub-bands has other information.

The secret keys are measured related to the mean value of 3 detailed subbands then convert the mean value of watermark histogram. It is estimated using Equation (1) as follows, $N (K) = (\begin{matrix} Mean (PV (LH)) + Mean (PV (HL)) \\ + Mean (PV (HH)) + TF (W) \end{matrix}) \times 10^{14}$ (1) here LH, HL and HH represents the detailed sub bands, PV (LH), PV (HL) and PV (HH) represents the function which returns positive values, TF (W) represents transform operation that is utilized in watermark and it is expressed in Equation (2), $TF (W) = Mean (({FQ}_{lm} ∥ lm) \times 2^{14}); \forall lm \in [0, 255]$ (2)

From Equation (2), FQ_lm represents the frequency of a pixel lm in W. The purpose of TF (W) ensures that every pixels of watermark affect the corresponding secret key i.e, pixels are indirectly diffused all over secret key. Here, the factors, like 10¹⁴ is chosen to expand the values and improves the general sensitivity of model with smaller changes in watermark imageries and the host imageries. The resultant intermediary key N (K) is hashed through hash function Message Digest algorithm-5 and the attained outcome is 128- bits.

For enhancing the logistic and sine map, initial values, like (a₀, d₁) and (b₀, d₂) are used. The maps are repeated N × 256 times because the chaotic trajectory store in A and B matrix of N × 256 sizes. Intermediary matrix based, the last chaotic matrix X is measured using Equation (3), $X = \sum_{p = 1}^{N} \sum_{q = 1}^{256} (((a (p, q) + b (p, q)) mod 0.5)$ (3) where a (p, q) and b (p, q) specifies the elements of A and B matrices, p and q specifies rows and columns of X. The resultant matrix X contains the random number among the phase space [0, 0.5]. Some change in secret key leads to a totally various matrix by initial condition sensitivity of MD-5. The secret keys are themself different in every watermark and the database through the trusted 3rd party that deals with the ownership protection. At last, by using Message Digest algorithm-5 (MD-5), the secret key is generated.

3.2 Watermark embedding

In watermark embedding, Funk Singular Value Decomposition technique and Fractional-Order Polar Harmonic Transforms (FSVD-FOPHT) method is employed to hide the watermark datas in host imagery. FSVD-FOPHT is an arithmetical tool that splits any of the matrices into 3 matrices.

This image is known as matrix I, it has 8-bit numbers with various dimensions based upon image types. By using Funk Singular Value Decomposition technique and Fractional-Order Polar Harmonic Transforms for watermark embedding is embedded by not affecting the visual perception of host imageries. Figure 1 shows the proposed FOPH- FSVD- SIW scheme embedding process. The parameter update rule in image watermarking refers to how the watermarking parameters are adjusted during the procedure of embedding a watermark in a digital image. It determines how the watermark is inserted and controlled within the image. The process typically involves dividing the image into smaller blocks, also calculating the embedding strength for each block based on watermark data and block characteristics. The parameter update rule is then applied to modify the embedding parameters based on the embedding strength and block characteristics, ensuring the watermark is properly embedded and remains robust against attacks. This iterative process continues until all blocks have been processed. The specific update rule can vary depending on the watermarking algorithm used, aiming to find the right balance between watermark visibility and robustness. The process of proposed Funk Singular Value Decomposition technique and Fractional-Order Polar Harmonic Transforms (FSVD-FOPHT) is as follows,

Fig. 1

Proposed FOPH- FSVD- SIW scheme embedding process.

Initially, host image LL sub band is selected then FSVD is applied and it is expressed mathematically in Equation (4),

$FSVD (LL) = V_{LL} * R_{LL} * U_{LL}^{T}$ (4)

Then, watermark W is transformed to a new matrix W_New by using the resulting matrix X. Then, the transformation process is expressed in Equation (5), $\begin{matrix} W_{New} (a, b) = X (a, W (a, b)), \\ a = 1, 2, 3, \dots N and b = 1, 2, 3, \dots M \end{matrix}$ (5)

The values are embedded as R_LL from matrix W_New and it is expressed mathematically in Equation (6), $R_{LL}^{New} = \frac{4 δ}{W_{New}} \cdot R_{LL}$ (6) where “·” represents the dot product operation, δ represents the single and multiple scaling factor. In single scaling factor, δ represents that it is only value applied in the scheme of watermarking. When δ is smaller, the unseen watermarked and host image is higher but this scheme lacks in robustness. Next, multiple scaling factor contains multiple scaling values, which providesa best balance among imperceptibility and robust. The elements of scaling factor is appropriate for Funk Singular Value Decomposition (FSVD) technique and requires less computational effort and it is calculated using Equations (7 and 8) as follows, $δ = (a_{p, q} - a_{p, q}^{'}) = {(a_{p, q} - \sum_{q = 1}^{q} λ_{1} + λ_{2})}^{2}$ (7) $δ = [\begin{matrix} δ_{1, 1} & δ_{1, 2} & \dots & δ_{1, n} \\ δ_{2, 1} & δ_{2, 2} & \dots & δ_{2, n} \\ \dots & \dots & \dots & \dots \\ δ_{N, 1} & δ_{N, 2} & \dots & δ_{N, n} \end{matrix}]$ (8) here δ specifies one value or more values in bounded intervals of [λ₁, λ₂] and a_p,q represents the spatial feature matrix. If λ₁ = 1 and λ₂ = 5, then δ is a single scaling factor equal to 5. If λ₁ = 50 and λ₂ = 50, then δ represents the multiple scaling factor ranging from [50, 100]. Thus, multiple scaling factors are randomly generated based on bounded interval. These points are based on random numbers, which leads incompatible interval ranges if it remains unbounded. Therefore, bounded intervals are executed to regulate multiple scaling factor values. By permitting λ₁ and λ₂ bounding interval variables, the values of multiple scaling factor are more flexible and random. Then, the values of every components in the spatial feature vector is based on every temporal features and it is extracted by using Equation (9) as follows, $X_{p, q} = δ \frac{λ E_{p, q}^{2}}{λ A_{p, q}} = 2 δ (E_{p, q} B_{q, p} - λ A_{p, q})$ (9) where $λ E_{p, q}^{2}$ represents the time used to factories, λA_p,q represents the gradient descent function, B_q,p specifies loss function, E_p,q represents hyper parameter that controls regularization degrees.

The modified LL sub and is acquired via inverse Funk Singular Value Decomposition technique and is expressed in Equation (10),

${LL}_{Modified} = V_{LL} * R_{LL}^{New} * V_{LL}^{T}$ (10)

The inverse Fractional-Order Polar Harmonic Transforms (FOPHT) is applied using LL_Modified then the remainder detailed sub-bands are LH, HL and HH used to attain watermarking image W_Image. λ₁, λ₂ and $R_{LL}^{New}$ represents side information extraction to key bits as secret keys. By using Funk Singular Value Decomposition (FSVD) technique, the watermark information is hidden in another part of digital data.

3.3 Watermark extraction

In watermark extraction phase, the Funk Singular Value Decomposition technique and Fractional-Order Polar Harmonic Transforms (FSVD-FOPHT) method is used to retrieve the embedded information from the digital data. FSVD-FOPHT is signal processing technique employed in different applications, like image processing, compressing. The main features of FSVD-FOPHT are it aids floating point integers or numbers unlike classical transfers deals with values of floating points. In image processing, the floating point representation results in information loss with the help of round-off operations. Similarly, FSVD-FOPHT is appropriate for image processing is due to eight-bit integers to specify pixels. The lifting transform (FOPHT) is reversible and converts input data to integers without quantization errors. Figure 2 shows the proposed FOPH- FSVD- SIW scheme extraction process.

Fig. 2

Proposed FOPH- FSVD- SIW scheme extraction process.

The extraction procedure starts off by improving $W_{New}^{'}$ from the watermarked image. $W_{New}^{'}$ is transferred as watermark W. Initially, apply one level of Fractional-Order Polar Harmonic Transforms (FOPHT) on the water marked image W_Image to attain 4 sub-band, LL^{W
_Image}, LH, HL and HH. Additionally, decompose LL^{W
_Image} using Funk Singular Value Decomposition (FSVD) technique and it is expressed mathematically in Equation (11),

$FSVD ({LL}^{W_{Im age}}) = V_{{LL}^{W_{Im age}}} * R_{{LL}^{W_{Im age}}} * U_{{LL}^{W_{Im age}}}$ (11)

$W_{New}^{'}$ is obtained using Equation (12) as follows, $W_{New}^{'} = (R_{LL}^{New} - R_{{LL}^{W_{Im age}}}) \cdot δ$ (12) where δ represents secret key and the values of λ from Equation (7), $R_{LL}^{New}$ is attained via side information. Prior transferring $W_{New}^{'}$ to watermark, W_N specifies secret key of individuals by claiming the ownership to host image that is against the secret key and it is saved using trusted 3rd party. If 2 keys match, the transferring process is permitted to continue. Otherwise, this process is halted. Therefore, the attackers with fake secret keys unable to demonstrate the ownership of host image successfully and it is exhibited in Equation (13), ${HI}_{pq} = \frac{δ}{2 π} \int_{0}^{2 π} \int_{0}^{1} {[W_{pq} (d, θ)]}^{*} f (d, θ) drdr θ$ (13)

From Equation (13), the p and q values are given as |p| = |q| = 0, 1, 2, 3, … , ∞, δ represents fractional parameter, W_pq (d, θ) denotes basis functions. This basis function satisfies the orthogonally relations in the interval [0, 1], [0, 2π] and it is given in Equation (14) as follows, $\int_{0}^{2 π} \int_{0}^{1} W_{pq} (d, θ) {[W_{nm} (d, θ)]}^{*} drdr θ = \frac{2 π}{δ} η_{pn} η_{qm}$ (14)

where η_pn and η_qm denotes the Kronecker function. Suppose, the matching process is effectual, the transformation procedure continues by generating the matrix X using Equation (3). Image reconstruction using FSVD-FOPHT moments can be achieved as illustrated in Equation (15) as follows, $\begin{matrix} K_{RC} (d, θ) = \sum_{p = - \infty}^{\infty} \sum_{q = - \infty}^{\infty} {HI}_{pq} W_{pq} (d, θ) \\ \approx \sum_{p = - p max}^{p max} \sum_{q = - q max}^{q max} {HI}_{pq} W_{pq} (d, θ) \end{matrix}$ (15)

Then, improve the watermark image W_N by computing the absolute difference in every row of 2 matrices $W_{New}^{'}$ and X, which is expressed mathematically using Equation (16) as follows, $W_{N} (p, q) = {\begin{matrix} f - 1; & if PV (W_{New}^{'} (p, q) - X (p, f)) = 0, \\ W_{N} (p - 1, q); & otherwise . \end{matrix}$ (16)

From Equation (16), p ={ 1, 2, 3, …, N } and q = f ={ 1, 2, 3, …, M }, W_N denotes extracted watermark image that leads to any distortions in the watermarked host image. For p and q value, f is repeated from 1 to M. While the condition is satisfied, f - 1 is selected and if it not satisfied, chooses the preceding value from the similar watermark. If p is not equals to one, choose the value from 0 i.e, 0 is not present in image width. If any changes in the key, the extracted watermark has every 0 (black image).

To minimize the effect of negative values and to address false positive problem appearing in watermark images are due to FSVD effect, the average function is applied on it and it is expressed in Equation (17), $W (p, q) = \frac{W (p, q - 1) + W (p, q + 1)}{2}$ (17) where p and q denotes the index values of the pixel intensity of watermark image. Finally, by using inverse process of FOPHT-FSVD method the secret keys are extracted from hided image and the original watermarked image is reconstructed. For the verification of copyright, the secret key must be registered and saved along 3rd party called certified authority (CA). Because of the adversaries are covered with its own watermark and generate its own secret keys, this circumvent attacks will occur. Any adversary claims the watermarked image ownership without the certified authority. The certified authority contributes to the connection of every image to its side data.

A single scaling parameter δ does not relevant to perturb all values v_i, because diverse spectral components express more or lesser tolerance to modification. In general, one having multi-scaling parameters δ₁, …, δ_n and utilize update rules v_i = v_i (1 + δ_ix_i). Here, δ_i implies relative measure of change one would have to make to change v_i the perceptual quality of document. A huge δ_i as one can perceptually get away with altering, v_i through a huge factor without document degradation.

The choice of diverse scaling is still a challenge. In few cases, the choice of δ_i depends upon common assumption. A special case of the generalized function (v_i = v_i + δ_ix_i) for δ_i = δv_i, this leads to the logical inference that a huge value is lesser susceptible to additive changes than a smaller value. Typically, it’s possible to underestimate how sensitive an image is to different values. Determining the distortion brought on by various attacks on the original image is one method of empirically evaluating these sensitivities. For example, one might determine the degraded image D^* from D, extract the associated values $v_{1}^{*}, \dots, v_{n}^{*}$ and consider δ_i as proportional to the deviation $| v_{i}^{*} - v_{i} |$ . To maximal robustness, one should try many formats of distortion and create δ_i proportional to $| v_{i}^{*} - v_{i} |$ average value. One may choose to take the median or maximum deviation instead of the average deviation. This empirical model can be strengthened using broad, global hypotheses on the sensitivity values. For eg one require δ_i ⩾ δ_j whenever v_i ⩾ v_j. To integrate this constraint to the empirical method is set δ_i according to $δ_{i} \approx \underset{j | v_{j} ⩽ v_{i}}{max_{︸}} | v_{j}^{*} - v_{j} |$ . Weakening the monotonicity restriction to be robust against sporadic outliers would be a more advanced strategy. All the experiments utilize a single parameter δ = 0.1. The higher energy frequency components of the original image not altered proportionally to their magnitude when JPEG-based distortions of the image are computed.

4 Results and discussion

Here, the simulation performance of Message Digest Algorithm Image Watermarking Scheme utilizing Funk Singular Value Decomposition and Fractional-Order Polar Harmonic Transforms is discussed. In every experiment, five gray scale images, such as Lena, Peppers, Baboon, Couple, Boat of size 512 as host imageries, 256_256 Cameraman image as watermark imagery. The proposed scheme is simulated with the help of MATLAB R2012b on 32-bit on processor, 4 GB random access memory. The host image is split into four sub-bands using FOPHT in the proposed scheme. Despite being computationally expensive, computing the FOPHT improves the resolution of image and accepts integers lacking round-off errors. Only after performing the FOPHT transformation on the host image, the watermark image transformed matrix is embedded straight to the approximation sub-band singular values. The performance of FOPH- FSVD for Secure Image Watermarking is tested using the performance metrics. Then, the efficacy of proposed FOPH- FSVD- SIW is analyzed to the existing models, like SVD-CMSF-SIW [24], FE-IWS-DNN [25], AR-IWS-DNN [26] BBET-SHA1-SIW [32] and LSB-DWT-SIW [33].

The experiment is activated in MATLAB R2012b with 4GB RAM, Intel ® coretrademark processor. The embedding with extraction procedures done in different standard images, namely cold-snow-landscape-water Test image and landscape-nature-sky-blue Test image. The performance is scaled to compute its strength and imperceptibility. PSNR and MSE are computed for imperceptibility and NC value proving robustness of the method. Several kinds of geometric as well as image processing attacks are performed on dissimilar images for experimental purpose and assessed its presentation by scaling NC factor. The attacks are deemed as median and average filtering, JPEG compressing under value 30, rotation, Gaussian noise, wiener filtering, cropping from center, resizing. The majority of attacks produced NC values closer to 1, demonstrating the proposed method is effective against diverse attacks. When compared to existing methods, the proposed method shows more robustness. NC values in contradiction of various attacks and the extracted watermark images are portrayed in Figs. 5 and 6.

4.1 Dataset

The performance of FOPHT-FSVD for Secure Image Watermarking is taken from the felicepollano image dataset [31]. Data is divided in training and validation with an 80/20 proportion.

Here, the proposed FOPHT- FSVD- SIW image watermarking performance is tested using 2 images is given in Fig. 3. Then, the different confidential images, such as animal-africa-wilderness-zoo, man-hand-car-black, street-animals-birds-doves, summer-dog-vibes-doge and sunset-mood-sun-afterglow are represented in Fig. 4. Table 1 shows the size of each watermark image.

Fig. 3

Testing images.

Fig. 4

Watermark image.

Table 1

Size of watermark image

Watermark images	Size
Animal-africa-wilderness-zoo	350 × 525 × 3
Man-hand-car-black	350 × 525 × 3
Street-animals-birds-doves	350 × 525 × 3
Summer-dog-vibes-doge	350 × 467 × 3
Sunset-mood-sun-afterglow	350 × 525 × 3

4.2 Performance metrics

The performance of FOPH- FSVD based image watermarking is verified using the following performance metrics.

4.2.1 Embedding capacity

It is determined with the help of the Equation (18), $\begin{matrix} Embedding Capacity = \\ \frac{Number of bits embedded in one frame}{Frame Size} \end{matrix}$ (18)

4.2.2 Peak signal to noise ratio

PSNR as well as normalized correlation are the 2 major tests for measuring unseen and efficacy of watermarking method. Hence, it is expressed in Equation (19), $PSNR = 10 {log}_{10} [\frac{maax (I (i, j))^{2}}{MSE}]$ (19) here maax (I (i, j)) represents host images bigger pixel values.

4.2.3 Mean square error

It is known as host image I and watermarked image W_Image. Hence, it is calculated in Equation (20), $MSE = \frac{1}{N \times M} \sum_{i = 1}^{N} \sum_{j = 1}^{M} [I (i, j) - W_{Im age} (i, j)^{2}]$ (20) here N, M denotes count of rows including columns of imagery. Higher PSNR values are chosen to specify a slight difference in host image I and watermarked image W_Image.

4.2.4 Normalized Correlation

This is scaling of variation among the extracted watermark W_N and original watermark W. Hence, it is measured using Equation (21) $\begin{matrix} NC (W, W_{N}) \\ = \frac{\sum_{i = 1}^{N} \sum_{j = 1}^{M} [W (i, j) - μ_{1} {IW}_{N} (i, j) - μ_{2}]}{\sqrt{\sum_{i = 1}^{N} \sum_{j = 1}^{M} [W (i, j) - μ_{1}]^{2}} \sqrt{\sum_{i = 1}^{N} \sum_{j = 1}^{M} [W_{N} (i, j) - μ_{2}]^{2}}} \end{matrix}$ (21) where μ₁ and μ₂ specifies mean values of W and W_N. Original watermark and the extracted watermark image are closer resembled with each other NC ≈ 1. Particularly, the watermarks of original and extracted are same in NC = 1.

Tables 2 to 6 shows the image quality of different watermark test image using different method. Tables 6 to 11 shows the PSNR value for watermark test image using different method. Tables 12, 13 shows the Mean square error for Test image using different method. Tables 14, 15 shows the Normalized correlation value for Test image using different method. Tables 16, 17 shows the embedding capacity for Test image using different method. Table 18 shows the Extraction Time of Watermark Test Image using different method.

Table 2 shows the Image Quality for animal-africa-wilderness-zoo image using different method. Here the proposed FOPH-FSVD-SIW method provide 61.818%, 14.68%, 7.142%, 31.993% and 19.778% high image quality at noise scale 1; 38.18%, 12.02%, 64.818%, 41.903% and 22.726 % high image quality at noise scale 2; 58.06%, 42.26%, 67.692%, 48.292% and 20.107 % high image quality at noise scale 3; 11.49%, 24.5%, 69.4118%, 53.541% and 22.805% high image quality at noise scale 4; 61.66%, 8.99%, 65.517%, 64.869% and 29.363% high image quality at noise scale 5 compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 2

Image Quality for animal-Africa-wilderness-zoo image using different method

Image Quality for animal-Africa-wilderness-zooimage
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Noise Scale 1	0.54	0.4	0.55	0.67	0.77	0.89
Noise Scale 2	0.41	0.38	0.45	0.7	0.8	0.8
Noise Scale 3	0.43	0.35	0.4	0.7	0.8	0.84
Noise Scale 4	0.39	0.39	0.3	0.6	0.66	0.8
Noise Scale 5	0.42	0.43	0.3	0.46	0.8	0.82

Table 3 shows the Image Quality for man-hand-car-black image using different method. Here, the proposed FOPH-FSVD-SIW method provide 6.27%, 58.06%, 61.66%, 9.79% and 32.09% high image quality at noise scale 1; 77.77%, 71.43%, 58.06%, 43.26% and 64.33% high image quality at noise scale 2; 47.76%, 65%, 11.49%, 25.05% and 50.23% high image quality at noise scale 3; 2.89%, 6.59%, 6.27%, 59.06% and 23.5% high image quality at noise scale 4; 47.76%, 65%, 77.77%, 72.43% and 76.66% high image quality at noise scale 5 compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 3

Image Quality for man-hand-car-black image using different method

Image Quality for man-hand-car-black image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Noise Scale 1	0.68	0.7	0.8	0.86	0.79	0.87
Noise Scale 2	0.78	0.75	0.78	0.76	0.80	0.85
Noise Scale 3	0.6	0.68	0.65	0.7	0.8	0.89
Noise Scale 4	0.59	0.49	0.55	0.7	0.66	0.9
Noise Scale 5	0.49	0.65	0.48	0.56	0.66	0.88

Table 4 shows the Image Quality for street-animals-birds-doves image using different method. Here the proposed FOPH-FSVD-SIW method provide 26.92%, 15.11%, 2.89%, 7.59%, 6.43% high image quality at noise scale 1; 57.14%, 15.47%, 47.76%, 66.88%, and 24.05% high image quality at noise scale 2; 60%, 77.77%, 47.76%, 67.34%, and 24.75% high image quality at noise scale 3; 2.89%, 6.59%, 26.92%, 16.11%, and 31.26% high image quality at noise scale 4; 58.06%, 42.26%, 57.14%, 16.47%, and 26.64% high image quality at noise scale 5 compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 4

Image Quality for street-animals-birds-doves image using different method

Image Quality for street-animals-birds-doves image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Noise Scale 1	0.64	0.71	0.8	0.77	0.76	0.89
Noise Scale 2	0.72	0.77	0.8	0.76	0.66	0.85
Noise Scale 3	0.61	0.69	0.8	0.8	0.7	0.86
Noise Scale 4	0.53	0.45	0.6	0.7	0.56	0.82
Noise Scale 5	0.44	0.61	0.58	0.63	0.6	0.81

Table 5 shows the Image Quality for summer-dog-vibes-doge image using different method. Here the proposed FOPH-FSVD-SIW method provide 53.97%, 24.5%, 60%, 77.57% and 67.06% high image quality at noise scale 1; 57.14%, 26.92%, 2.89%, 7.59%, 6.43% high image quality at noise scale 2; 54.76%, 55.33%, 58.06%, 43.26% and 56.55%, high image quality at noise scale 3; 11.03%, 22.06%, 53.97%, 25.5% and 45.78% high image quality at noise scale 4; 41.03%, 32.06%, 57.14%, 27.92% and 16.11 high image quality at noise scale 5; compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 5

Image Quality for summer-dog-vibes-doge image using different method

Image Quality for summer-dog-vibes-doge image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Noise Scale 1	0.61	0.78	0.78	0.69	0.70	0.8
Noise Scale 2	0.73	0.74	0.68	0.66	0.66	0.78
Noise Scale 3	0.65	0.66	0.65	0.63	0.64	0.84
Noise Scale 4	0.59	0.42	0.53	0.54	0.56	0.8
Noise Scale 5	0.47	0.61	0.6	0.63	0.65	0.8

Table 6 shows the Image Quality for sunset-mood-sun-afterglow image using different method. Here the proposed FOPH-FSVD-SIW method provide 32.10%, 33.12%, 54.76%, 56.33% and 58.23% high image quality at noise scale 1; 32.83%, 328.02%, 11.03%, 23.06% and 14.23% high image quality at noise scale 2; 21.08%, 6.07%, 41.03%, 33.06% and 35.12% high image quality at noise scale 3; 69.67%, 53%, 32.10%, 34.12% and 31.71% high image quality at noise scale 4; 45.75%, 37.64%, 21.08%, 7.07% and 4.67% high image quality at noise scale 5 compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 6

Image Quality for sunset-mood-sun-afterglow image using different method

Image Quality for sunset-mood-sun-afterglow image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Noise Scale 1	0.54	0.52	0.7	0.66	0.9	0.84
Noise Scale 2	0.23	0.64	0.55	0.55	0.9	0.85
Noise Scale 3	0.68	0.47	0.52	0.6	0.69	0.88
Noise Scale 4	0.71	0.68	0.45	0.49	0.69	0.89
Noise Scale 5	0.64	0.66	0.3	0.35	0.56	0.85

Table 7 shows the PSNR value for image (Watermark Test Image- animal-africa-wilderness-zoo image) using different method. Here the proposed FOPH-FSVD-SIW provide 6.263%, 7.526%, 24.606%, 22.802% and 21.95% high PSNR value at cold-snow-landscape-water Test image; 8.319%, 3.525%, 23.362%, 23.143% and 18.604% high PSNR value at cold-snow-landscape-water Test image compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 7

PSNR for image (Watermark Test Image-animal-africa-wilderness-zoo image)

PSNR for image (Watermark Test Image-animal-africa-wilderness-zoo image)
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Cold-snow-landscape-water test image	45	50	55	57	58	69
Landscape-nature-sky-blue test image	48	52	56	58	59	69

Table 8 shows the PSNR value for image (Watermark Test Image- man-hand-car-black image) using different method. Here the proposed FOPH-FSVD-SIW provide 6.710%, 6.91%, 21.944%, 20.255% and 19.438% high PSNR value at cold-snow-landscape-water Test image; 76.3%, 45.26%, 25.948%, 25.678% and 20.949% high PSNR value at cold-snow-landscape-water Test image compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 8

PSNR for image (Watermark Test Image- man-hand-car-black image)

PSNR for image (Watermark Test Image- man-hand-car-black image)
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Cold-snow-landscape- water test image	45	50	55	57	58	69
Landscape-nature-sky- blue test image	48	52	56	58	59	69

Table 9 shows the PSNR value for image (Watermark Test Image-street-animals-birds-doves image) using different method. Here the proposed FOPH-FSVD-SIW provide 74.12%, 15.22%, 18.452%, 15.937% and 13.66% high PSNR value at cold-snow-landscape-water Test image; 19.3%, 32.26%, 20.903%, 16.05% and 12.016% high PSNR value at cold-snow-landscape-water Test image compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 9

PSNR for image (Watermark Test Image- street-animals-birds-doves image)

PSNR for image (Watermark Test Image- street-animals-birds-doves image)
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Cold-snow-landscape-water test image	45	50	55	57	58	69
Landscape-nature-sky-blue test image	48	52	56	58	59	69

Table 10 shows the PSNR value for image (Watermark Test Image-summer-dog-vibes-dogeimage) using different method. Here the proposed FOPH-FSVD-SIW provide 11.31%, 22.06%, 21.847%, 19.159% and 16.771% high PSNR value at cold-snow-landscape-water Test image; 26.03%, 14.55%, 23.429%, 20.723% and 14.142% high PSNR value at cold-snow-landscape-water Test image compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 10

PSNR for image (Watermark Test Image-summer-dog-vibes-doge image)

PSNR value for image (Watermark Test Image-summer-dog-vibes-dogeimage)
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Cold-snow-landscape-water test image	45	50	55	57	58	69
Landscape-nature-sky-blue test image	48	52	56	58	59	69

Table 11 shows the PSNR value for image (Watermark Test Image- sunset-mood-sun-afterglow image) using different method. Here the proposed FOPH-FSVD-SIW provide 14.12%, 26.12%, 13.854%, 11.306% and 9.494% high PSNR value at cold-snow-landscape-water Test image; 61.22%, 11.42%, 18.449%, 16.032% and 11.999% high PSNR value at cold-snow-landscape-water Test image compared with existing methods, like SVD-CMSF- SIW, FE-IWS-DNN, AR-IWS-DNN, BBET- SHA1-SIW and LSB-DWT-SIW respectively.

Table 11

PSNR for image (Watermark Test Image- sunset-mood-sun-afterglow image)

PSNR value for image (Watermark Test Image- sunset-mood-sun-afterglow image)
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Cold-snow-landscape-water test image	45	50	55	57	58	69
Landscape-nature-sky-blue test image	48	52	56	58	59	69

Table 12 shows the Mean square error for cold-snow-landscape-water Test image using different method. Here the proposed FOPH-FSVD-SIW provide 19.3%, 32.26%, 4.041%, 5.136% and 2.969% low Mean square error at test image 1, animal-africa-wilderness-zoo; 10.13%, 21.66%, 5.126%, 6.244% and 5.126% low Mean square error at test image 2, man-hand-car-black; 39.29%, 15.59%, 6.244%, 2.958% and 6.244% low Mean square error at test image 3, street-animals-birds-doves; 9.11%, 19.56%, 2.969%, 3.502% and 1.918% low Mean square error at test image 4, summer-dog-vibes-doge; 10.12%, 29.04%, 2.757%, 2.546% and 1.71% low Mean square error at test image 5, sunset-mood-sun-afterglow compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively

Table 12

Mean square error for cold-snow-landscape-water Test image

Mean square error for cold-snow-landscape-water test image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test imagery 1	0.87	0.87	0.88	0.80	0.89	0.4
Test imagery 2	0.68	0.81	0.83	0.79	0.80	0.38
Test imagery 3	0.79	0.79	0.81	0.72	0.9	0.36
Test imagery 4	0.77	0.78	0.79	0.69	0.85	0.31
Test imagery 5	0.69	0.77	0.84	1.0	0.77	0.45

Table 13 shows the Mean square error for landscape-nature-sky-blue Test image using different method. Here the proposed FOPH-FSVD-SIW provide 18.23%, 62.89%, 69.67%, 53.54% and 25.44% low Mean square error at test image 1, animal-africa-wilderness-zoo; 9.33%, 59.71%, 45.75%, 38.64% and 24.64% low Mean square error at test image 2, man-hand-car-black; 6.42%, 14.55%, 32.83%, 38.02% and 30.12% low Mean square error at test image 3, street-animals-birds-doves; 18.01%, 11.06%, 14.68%, 7.142%, and 4.65% low Mean square error at test image 4, summer-dog-vibes-doge; 48.22%, 32.15%, 38.18%, 12.02%, and 7.59% low Mean square error at test image 5, sunset-mood-sun-afterglow compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 13

Mean square error for landscape-nature-sky-blue Test image

Mean square error for landscape-nature-sky-blue Test image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test imagery 1	0.86	0.81	0.78	0.83	0.76	0.34
Test imagery 2	0.69	0.82	0.8	0.77	0.84	0.4
Test imagery 3	0.71	0.72	0.75	0.83	0.9	0.5
Test imagery 4	0.76	0.79	0.7	0.79	0.89	0.38
Test imagery 5	0.62	0.71	0.81	0.8	0.76	0.3

Table 14 shows the Normalized correlation for cold-snow-landscape-water Test image using different method. Here the proposed FOPH-FSVD-SIW provide 59.23%, 78.09%, 22.76%, 25.98% and 31.86% high Normalized correlation at test image 1, animal-africa-wilderness-zoo; 11.93%, 56.49%, 30.56%, 28.08% and 26.87% high Normalized correlation at test image 2, man-hand-car-black; 9.44%, 11.55%, 26.98%, 26.98% and 43.87% high Normalized correlation at test image 3, street-animals-birds-doves; 63.48%, 59.15%, 16.86%, 25.86% and 34.87% high Normalized correlation at test image 4, summer-dog-vibes-doge; 39.12%, 13.02%, 32.54%, 27.97%, and 26.45% high Normalized correlation at test image 5, sunset-mood-sun-afterglow compared with existing methods, like SVD-CMSF-SIW, DFE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 14

Normalized correlation for cold-snow-landscape-water Test image

Normalized correlation for cold-snow-landscape-water Test image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test imagery 1	0.71	0.7	0.88	0.88	0.86	0.9
Test imagery 2	0.76	0.75	0.95	0.9	0.82	0.95
Test imagery 3	0.72	0.8	0.85	0.87	0.88	0.92
Test imagery 4	0.8	0.81	0.79	0.9	0.86	0.96
Test imagery 5	0.82	0.85	0.84	0.87	0.89	0.92

Table 15 shows the Normalized correlation for landscape-nature-sky-blue Test image using different method. Here the proposed FOPH-FSVD-SIW provide 23.59%, 8.77%, 32.87%, 26.43%, and 16.87% high Normalized correlation at test image 1, animal-africa-wilderness-zoo; 19.32%, 17.05%, 26.87%, 16.87%, and 21.65% high Normalized correlation at test image 2, man-hand-car-black; 23.15%, 27.05%, 25.87%, 32.54%, and 25.85% high Normalized correlation at test image 3, street-animals-birds-doves; 59.22%, 81.26%, 32.98%, 32.87%, and 25.86% high Normalized correlation at test image 4, summer-dog-vibes-doge; 8.98%, 11.13%, 34.54%, 22.58%, and 27.97% high Normalized correlation at test image 5, sunset-mood-sun-afterglow compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 15

Normalized correlation for landscape-nature-sky-blue Test image

Normalized correlation for landscape-nature-sky-blue Test image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test imagery 1	0.55	0.66	0.78	0.83	0.76	0.9
Test imagery 2	0.6	0.67	0.8	0.77	0.82	0.92
Test imagery 3	0.65	0.81	0.75	0.83	0.9	0.89
Test imagery 4	0.72	0.79	0.7	0.79	0.89	0.95
Test imagery 5	0.7	0.69	0.81	0.8	0.76	0.93

Table 16 shows the embedding capacity for cold-snow-landscape-water Test image using different method. Here the proposed FOPH-FSVD-SIW provide 29.55%, 55.63%, 22.54%, 26.87%, and 32.98% high embedding capacity at test image 1, animal-africa-wilderness-zoo; 11.33%, 18.05%, 25.87%, 26.54%, and 32.87% high embedding capacity at test image 2, man-hand-car-black; 7.14%, 11.25%, 42.98%, 29.08% and 27.09% high embedding capacity at test image 3, street-animals-birds-doves; 32.51%, 5.27%, 32.06%, 20.94% and 32.04% high embedding capacity at test image 4, summer-dog-vibes-doge; 15.02%, 18.55%, 27.03%, 28.94% and 39.04% high embedding capacity at test image 5, sunset-mood-sun-afterglow compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 16

Embedding capacity for cold-snow-landscape-water Test image

Embedding capacity for cold-snow-landscape-water Test image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test image 1	48	50	53	56	55	59
Test image 2	40	48	43	46	45	65
Test image 3	39	44	38	59	41	63
Test image 4	35	47	44	47	45	65
Test image 5	44	41	40	523	55	61

Table 17 shows the embedding capacity for landscape-nature-sky-blue Test image using different method. Here the proposed FOPH-FSVD-SIW provide 55.25%, 36.55%, 38.94%, 28.84% and 29.04% high embedding capacity at test image 1, animal-africa-wilderness-zoo; 8.23%, 19.15%, 33.76%, 19.54%, and 26.09% high embedding capacity at test image 2, man-hand-car-black; 33.41%, 15.39%, 30.97%, 27.07%, and 25.86% high embedding capacity at test image 3, street-animals-birds-doves; 22.59%, 21.81%, 29.05%, 31.54%, and 32.97% high embedding capacity at test image 4, summer-dog-vibes-doge; 9.88%, 13.11%, 18.97%, 32.84%, and 34.79% high embedding capacity at test image 5, sunset-mood-sun-afterglow compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 17

Embedding capacity for landscape-nature-sky-blue Test image

Embedding capacity for landscape-nature-sky-blue Test image
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test image 1	30	50	52	41	59	62
Test image 2	32	48	57	43	31	65
Test image 3	38	44	50	31	41	58
Test image 4	34	47	42	31	51	55
Test image 5	40	33	52	42	39	57

Table 18 shows the Extraction time analysis test image using different method. Here the proposed FOPH-FSVD-SIW provide 55.29%, 63.03%, 91.667%, 75.65% and 93.59% low extracting time at test image 1, animal-africa-wilderness-zoo; 13.13%, 15.85%, 90.588%, 86.206% and 88.571% low extracting time at test image 2, man-hand-car-black; 51.31%, 7.57%, 82.29%, 78.205% and 67.307% low extracting time at test image 3, street-animals-birds-doves; 10.12%, 18.57%, 82.456%, 87.951% and 89.362% low extracting time at test image 4, summer-dog-vibes-doge; 13.19%, 24.09%, 95.294%, 93.548% and 95.294% low extracting time at test image 5, sunset-mood-sun-afterglow compared with existing methods, like SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Table 18

Extraction time of watermark test image using different method

Extraction Time of Watermark Test Image using different method
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test image 1	0.5	0.51	0.6	0.22	0.77	0.05
Test image 2	0.4	0.58	0.8	0.56	0.8	0.06
Test image 3	0.6	0.61	0.9	0.77	0.57	0.01
Test image 4	0.2	0.3	0.8	0.7	0.9	0.03
Test image 5	0.7	0.5	0.55	0.9	0.86	0.07

Another significant problem with watermarking system performance is time complexity. Time complexity of proposed method for computing all the maximum order P_max is $O (N^{2} P_{max}^{2})$ . Here, comparing the speed of proposed and other methods under average time taken for watermark embedding. Time taken for embedding overall count of bits in maximum order of p_max ⩽ 36 moments is in a gray scale host image of 128 × 128 pixels size. The weighted averages for the proposed and existing models for computations are 0.167 secs, 1.615 secs, 2.567secs, 1.963secs, 3.647secs, 1.743secs signifying substantial enhancement in time complexity of proposed method. Table 19 shows the comparison of embedding speed.

Table 19

Comparison of embedding speed

Comparison of embedding speed
	BBET-SHA1-SIW	LSB-DWT-SIW	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	FOPH-FSVD-SIW (Proposed)
Test imagery 1	1.578	3.65	0.378	0.236	0.675	0.145
Test imagery 2	0.501	0.89	1.78	0.75	0.523	0.109
Test imagery 3	1.031	0.76	0.45	0.13	1.639	0.123
Test imagery 4	0.678	0.87	0.96	0.63	2.694	0.189
Test imagery 5	2.063	1.32	1.63	0.76	0.471	0.154

4.3 Imperceptibility and robustness analysis

Table 20 shows that the Imperceptibility Results. Table 20 tabulates the PSNR findings for 5 host images using MSF ranges. The existing schemes reach higher PSNR values but they subpar under robustness against geometrical including non-geometrical attacks. This is depicted in Table 20, wherein small including large scaling factors can attain higher PSNR. Table 21 shows that the Robustness Results. Table 21 represents the NC outcomes of proposed approach using MSF ranges. Here, five MSF intervals are employed under various intervals. Whether the MSF values are small or large, the proposed approach successfully extracts the watermark, with NC outcomes of approximate 1.

Table 20
Imperceptibility results

Test image [1 - 10²] [1 - 10³] [1 - 10⁴] [1 - 10⁵] [1 - 10⁶]

Cold-snow-landscape-water test image 52.53 52.52 52.84 52.66 52.03

Landscape-nature-sky-blue test image 52.34 52.83 52.76 52.96 52.37

Test image	[1 - 10²]	[1 - 10³]	[1 - 10⁴]	[1 - 10⁵]	[1 - 10⁶]
Cold-snow-landscape-water test image	52.53	52.52	52.84	52.66	52.03
Landscape-nature-sky-blue test image	52.34	52.83	52.76	52.96	52.37

Table 21

Robustness results

Test image	[1 - 10²]	[1 - 10³]	[1 - 10⁴]	[1 - 10⁵]	[1 - 10⁶]
Cold-snow-landscape-water test image	0.99345	0.99234	0.99543	0.99246	0.99345
Landscape-nature-sky-blue test image	0.99543	0.99532	0.99453	0.99245	0.99643

Tables 22 and 23 display the NC values of proposed technique for cold-snow-landscape-water test image and landscape-nature-sky-blue test image using MSF ranges. On the watermarked image, various geometric and non-geometrical modifications are used, and subsequently the watermarks are removed. The proposed approach extracts the watermark successfully in all cases without MSF range. The proposed technique is excellent in terms of resilience. The proposed approach reaches great imperceptibility and robustness when the MSF range is small. When the MSF range is huge, the proposed technique has higher imperceptibility with robustness against a number of well-known attacks. To attain superior outcomes in PSNR and NC, the proposed technique create MSF values this is effectiveness than optimization algorithms. Figures 5 and 6 depicts watermarked and extracted watermark images.

Fig. 5

Extracted watermark image and its Normalized cross correlation values for various attacks in cold-snow-landscape-water Test image.

Fig. 6

Extracted watermark image and its Normalized cross correlation values for various attacks in landscape-nature-sky-blue Test image.

Table 22

Robustness outcomes for the cold-snow-landscape-water test image

Attacks	[1 - 10²]	[1 - 10³]	[1 - 10⁴]	[1 - 10⁵]	[1 - 10⁶]	FOPH-FSVD-SIW (Proposed)
Cropping (Center, 20)	0.99262	0.99364	0.99345	0.99234	0.99323	0.99131
Cutting (10 rows)	0.99335	0.99453	0.99577	0.99546	0.99544	0.99044
Shareing (1,0.2)	0.99565	0.99344	0.99345	0.99435	0.99435	0.99240
Translating (20,20)	0.99215	0.99454	0.99764	0.99654	0.99232	0.99153
Shiftting (30)	0.99437	0.99543	0.99704	0.99522	0.99563	0.99443
Rotating (110)	0.99434	0.99766	0.99843	0.99564	0.99323	0.99133
Scaling (0.25,4)	0.99234	0.98230	0.99432	0.99643	0.99562	0.99124
Median Filter (3,3)	0.98343	0.99764	0.99564	0.99876	0.99764	0.99264
Gamma Correction (0.8)	0.99345	0.99536	0.99642	0.99734	0.99544	0.99533
Wiener Filter (3.3)	0.99565	0.99845	0.99363	0.99847	0.99443	0.99433
Histogram Equalization	0.99564	0.99876	0.99654	0.99211	0.99763	0.99156
Salt Peppers Noise (0.3)	0.98695	0.99846	0.993494	0.99743	0.99348	0.99224
Speckle Noise (0.3)	0.99375	0.99342	0.99756	0.99754	0.99544	0.99533
Gaussian Filter (0.3)	0.99324	0.99574	0.99323	0.99824	0.99245	0.99288
JPEG Compression (30)	0.99567	0.99322	0.99578	0.99324	0.9978	0.99109

Table 23

Robustness outcomes for the landscape-nature-sky-blue test image

Attacks	[1 - 10²]	[1 - 10³]	[1 - 10⁴]	[1 - 10⁵]	[1 - 10⁶]	FOPH-FSVD-SIW (Proposed)
Cropping (Center, 20)	0.99443	0.99976	0.99224	0.99976	0.99345	0.99221
Cutting (10 rows)	0.99453	0.99573	0.99865	0.99435	0.99764	0.99124
Shareing (1,0.2)	0.99356	0.99567	0.99547	0.99356	0.99356	0.99170
Translating (20,20)	0.99554	0.99745	0.99245	0.99754	0.99343	0.98456
Shiftting (30)	0.99765	0.99356	0.99643	0.99654	0.99357	0.99103
Rotating (110)	0.99433	0.99563	0.99635	0.99234	0.99654	0.97133
Scaling (0.25,4)	0.99637	0.98233	0.99122	0.99567	0.99543	0.99220
Median Filter (3,3)	0.98234	0.99754	0.99854	0.99356	0.99534	0.95264
Gamma Correction (0.8)	0.99823	0.99764	0.99753	0.99754	0.99554	0.97533
Wiener Filter (3.3)	0.99655	0.99534	0.99765	0.99432	0.99423	0.99055
Histogram Equalization	0.99564	0.99876	0.99654	0.99211	0.99763	0.98106
Salt Peppers Noise (0.3)	0.98695	0.99846	0.993494	0.99743	0.99348	0.99120
Speckle Noise (0.3)	0.99375	0.99342	0.99756	0.99754	0.99544	0.99407
Gaussian Filter (0.3)	0.99324	0.99574	0.99323	0.99824	0.99245	0.97307
JPEG Compression (30)	0.99567	0.99322	0.99578	0.99324	0.9978	0.99016

PSNR imperceptibility values are evaluated for some host images. When assessed to other existing schemes, the proposed method achieves better invisibility for all images. NC robustness comparison of proposed with existing SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW schemes under certain well-known attacks are tabulated in Table 24. According to the outcomes, the proposed method surpasses the competition and can correctly recover the watermark without distorting it. This is a result of the second extraction step’s ability to eliminate distortions and recover the right pixels.

Table 24

Robustness comparison under various attacks

Attacks	SVD-CMSF-SIW	FE-IWS-DNN	AR-IWS-DNN	BBET-SHA1-SIW	LSB-DWT-SIW	FOPH-FSVD-SIW (Proposed)
Histogram Equalization	0.9945	0.9934	0.9975	0.9934	0.9953	0.99253
Gaussian Filter	0.9923	0.9945	0.9943	0.9953	0.9956	0.98236
JPEG Compression	0.9955	0.9962	0.9962	0.9953	0.9985	0.99342
Gamma Correction	0.9965	0.9954	0.9978	0.9956	0.9976	0.99146
Median Filter	0.9969	0.9974	0.9945	0.9923	0.9965	0.99233
Cropping	0.9963	0.9545	0.5734	0.9836	0.9946	0.99422
Shiftting	0.9935	0.9843	0.5473	0.9954	0.9846	0.99134

4.4 FPP analysis

The proposed technique secures copyright ownership of digital imageries successfully, as it addresses false positive problems by activating secret key match to authorize owner legality. The side information, like λ₁, λ₂ and $R_{LL}^{new}$ , secret key Keybits embed as well as extract the watermark. Most of the existing watermarking techniques image solely uses side information as the secret key produced from embedding process, which is highly vulnerable to attack and changed. When making false claims of ownership of the host image, adversaries may take gain of the fact that the side information is not sensitive to even minor modifications. From the host as well as watermark imageries, a secret key is extracted. If varies in any one of them leads to vary in the secret key. On the basis of 3 FPP scenarios, the proposed technique robustness is analysed.

5 Conclusions

In this manuscript, Message Digest Algorithm Image Watermarking Scheme based upon FOPHT-FSVD is successfully implemented for Secure Image Watermarking (FOPHT-FSVD-IWT). Here, the performance metrics, like Embedding capacity, PSNR, MSE and Normalized correlation (NC)are analyzed.Then, the proposed FOPHT-FSVD-IWT method attains lower MSE for cold-snow-landscape-water test image, 4.12%, 23.14%, 3.79%, 2.44% and 3.88% low MSE for landscape-nature-sky-blue Test image, 5.31%, 4.69%, 4.547%, 3.894% and 4.479% high Normalized cross correlation for cold-snow-landscape-water test image, 3.28%, 5.19%, 2.56%, 3.38% and 3.15% high Normalized cross correlation for landscape-nature-sky-blue Test image compared to existing methods, SVD-CMSF-SIW, FE-IWS-DNN, AR-IWS-DNN, BBET-SHA1-SIW and LSB-DWT-SIW respectively.

Uncertainties can be classified as internal or external, parametric or non-parametric, constant, characteristic, or random. Internal uncertainties arise within a system, while external uncertainties come from outside influences. Parametric uncertainties involve uncertain parameter values, while non-parametric uncertainties are more complex and difficult to characterize. Constant uncertainties remain fixed over time, while characteristic uncertainties are specific to a particular system. Random uncertainties are unpredictable and result from stochastic processes. Determining the structures and amounts of uncertainties in real-time applications is challenging due to limited data, evolving conditions, and complex interactions. It often requires a combination of statistical analysis, data-driven approaches, sensitivity analysis, and expert knowledge.

Implementing a proposed algorithm in real-time applications can face challenges. Strict timing requirements must be met, and computational complexity should be manageable within the available time. Limited resources like processing power or memory need to be considered, and the algorithm should be robust to handle variable sensor inputs. Safety, reliability, and energy efficiency are crucial, and the algorithm must adapt to complex real-world environments. Fault tolerance and error handling are important, as well as addressing interference with other systems. By

Considering these factors, the algorithm can be better suited for real-time applications. Implementing a proposed algorithm in real-time applications presents several challenges that must be overcome. The algorithm must meet strict timing requirements and manage its computational complexity within the available time constraints. Limited resources, such as processing power and memory, need to be considered and optimized. It is crucial for the algorithm to be robust and handle variable sensor inputs effectively. Safety, reliability, and energy efficiency are essential factors that should be prioritized. The algorithm must also adapt to complex real-world environments, accounting for dynamic changes. Fault tolerance and error handling mechanisms need to be in place, and interference with other systems should be addressed. By considering these factors and implementing optimization techniques, resource management strategies, robustness measures, adaptability mechanisms, safety and reliability features, and interference mitigation methods, the algorithm can be better suited for real-time applications.

Even though, it is quite challenging to confirm the validity of the host data. An important tool for ensuring this authenticity is digital image watermarking. So in Future work, the secured image watermarking is dedicated to develop efficient methods with Novel deep learning optimized utilizing Novel Optimization algorithm for guaranteeing a feasible trade-off amongst basic design necessities, such as imperceptibility, proficient, security that can be accomplished with this novel image encryption approach.

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A secure fractional-order polar harmonic transforms based image watermarking scheme using funk singular value decomposition

Abstract

Keywords

1 Introduction

1.1 Stability of SVD singular values

1.2 Correction of false positive detection

2 Literature review

3 Proposed methodology

3.1 Key generation

4.1 Dataset

4.2.1 Embedding capacity

Table 20 Imperceptibility results Test image [1 - 102] [1 - 103] [1 - 104] [1 - 105] [1 - 106] Cold-snow-landscape-water test image 52.53 52.52 52.84 52.66 52.03 Landscape-nature-sky-blue test image 52.34 52.83 52.76 52.96 52.37

5 Conclusions

References

Table 20
Imperceptibility results

Test image [1 - 10²] [1 - 10³] [1 - 10⁴] [1 - 10⁵] [1 - 10⁶]

Cold-snow-landscape-water test image 52.53 52.52 52.84 52.66 52.03

Landscape-nature-sky-blue test image 52.34 52.83 52.76 52.96 52.37