Applied Mathematical Sciences, Vol. 11, 2017, no. 54, 2681 - 2695

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2017.78271

A Robust Public Key Watermarking Scheme

Based on Improved Singular Value Decomposition

Cao Thi Luyen, Nguyen Hieu Cuong and Pham Van At

Faculty of Information Technology, University of Transport and Communication

Ha Noi, Viet Nam

Copyright © 2017 Cao Thi Luyen, Nguyen Hieu Cuong and Pham Van At. This article is

distributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Digital watermarking has been considered as an effective solution for multimedia

rightful protection and authentication. Singular value decomposition (SVD) is

used for many watermarking schemes. This method has encountered some

challenges, such as computational complexity and robustness. In this paper,

instead of using the conventional SVD, we employed a new algorithm to directly

compute the largest eigenvalue and eigenvector of segmented image blocks.

Theoretically, by using this approach, our proposed watermarking scheme has

computational complexity lower than various SVD based schemes. This

improvement is essential for watermarking systems in practice, where ones often

have to work with large-scale image datasets. Experimental results also showed

that our watermarking scheme outperforms several widely used schemes in terms

of robustness. Moreover, using the proposed algorithm, we designed a new robust

public key watermarking scheme, where watermarked images can be verified

without using the pre-defined watermark and a secret key.

Keywords: Public key watermarking, SVD, eigenvalue, eigenvector

1 Introduction

With the advent of the Internet and the wide availability of technical equipments,

most people easily copy and distribute illegally digital products. For decades,

digital watermarking is commonly used for copyright protection and

authentication of multimedia content (i.e. digital images, audio, video, etc.). With every watermarking scheme, a watermark is embedded into digital objects in order to

2682 Cao Thi Luyen et al.

present their authorship. In fragile schemes, the embedded watermarks can be

distorted when any operation is applied to the watermarked objects. Conversely,

in robust ones, the embedded watermarks should be retained, even when some

operations are deliberately applied to the watermarked objects.

Robust watermarking has been proposed as an important method for protecting

the copyright of digital images. To this end, it is required to provide trustworthy

evidence for protecting rightful ownership. A good robust watermarking

algorithm is mostly resisted due to common image manipulations, such as

Gaussian noise addition, JPEG compression, filtering, etc. Moreover, the

perceptual difference between an original image and its watermarked version

should not be distinguishable to human eyes [12].

In order to design robust watermarking schemes, many image operations and

transformations were employed. Among them, the singular value decomposition

(SVD) is widely used due to its robust properties. In the SVD based image

watermarking schemes, first each segmented image block is decomposed into

three matrices U, D and V (see 2.1). After that, watermarks can be embedded in

different positions in the matrices: (1) in D [6, 15, 17, 19, 28] (2) in the first

column U or V [4, 13, 28] or (3) in both D and U [2, 7]. In various SVD based

algorithms [2, 6, 8, 10, 13], only the first coefficient of D and the first column of

U (or V) are used. Besides, SVD can be combined with some other image

transforms, such as discrete Cosine transform (DCT) and discrete wavelet

transform (DWT) [16, 24, 25].

The most important problem of every SVD based algorithms is the step of SVD

analysis. This decomposition requires finding all eigenvalues and eigenvectors of

the analyzed matrices. Since the transform is complex and must be employed to

every block, this step is rather time consuming. Inspired by the work of [7], we

show that in watermarking schemes, one does not need to compute SVD for

image blocks in the conventional way, but only need to use the largest eigenvector

and eigenvalue of the blocks. We proposed a new algorithm to directly compute

the aforementioned values and utilize them in watermarking schemes.

Consequently, both of the procedures of embedding and extracting of

watermarking schemes are speeded up. This improvement is essential for

watermarking systems in practice, when ones often have to work with large-scale

image dataset. We also conducted experiments to evaluate the robustness the

proposed scheme and some widely used watermarking schemes. The results show

that all of the schemes are robust against common attacks and the proposed

scheme is more robust than others.

To improve the security of watermarking systems, most algorithms have

employed a secret key. Thus, for verification, ones need to know both the

pre-embedded watermarks and the secret key. That limits the applicability in

practice, where the watermarks and secret keys are not easily distributed to legal

recipients. To overcome this drawback, various public key watermarking schemes

were proposed [3, 11, 18]. The schemes borrowed some ideas from public key

cryptography, where the watermark was encrypted by the private key of the legal owner. The corresponding public key was known by everyone and in the verification

A robust public key watermarking scheme 2683

phase, it will be used to decrypt embedded watermarks. Although public key

watermarking schemes have solved the problems of distribution and verification,

robustness is also a crucial requirement of public key watermarking [1, 19, 20]. In

this work, in order to improve the robustness of public key watermarking schemes,

we employ again the proposed algorithm to extract robust features from

segmented image blocks. As a result, our scheme achieved lower computational

complexity and higher degree of robustness.

The rest of this paper is organized as follows. Section 2 reviews the essential

background of SVD and some related work. In Section 3, we describe the

proposed the algorithm and design a robust public key watermarking scheme.

Sections 4 reports and discusses experimental results. Lastly, the paper is

concluded in Section 5.

2 Related word

2.1 Singular value decomposition

SVD is not only be used in linear algebra, but also in many different applications,

such as in image processing. The main benefits of SVD in image processing are that

it provides a method to decompose a large matrix to smaller and more manageable

matrices and the singular values (SVs) of an analyzed matrices have a good

stability. Thus, when a manipulation is applied to an image, its SVs do not

significantly changed. Concretely, an m×n image block (can be considered as a

matrix) is decomposed into three matrices U, D and V [9]:

𝐴 = 𝑈 × 𝐷 × 𝑉𝑇 = 𝑈1𝐷(1,1)𝑉1𝑇 + 𝑈2𝐷(2,2)𝑉2

𝑇 + ⋯ + 𝑈𝑠𝐷(𝑠, 𝑠)𝑉𝑠𝑇, (1)

where s = min(m,n), 𝑈 ∈ 𝑅𝑚×𝑚 , 𝑉 ∈ 𝑅𝑛×𝑛 are normalized orthogonal

matrices and 𝐷 ∈ 𝑅𝑚×𝑛 is a diagonal matrix. The main diagonal of D consists of

the singular values of A, where D(1,1) ≥ D(2,2) ≥ ... ≥ D(s,s) ≥ 0 (s is called the

rank of the matrix D).

2.2 Watermarking schemes based on SVD

2.2.1 The scheme of Chung et al.

In Chung et al. [13], to embed a watermark 𝑊 = {𝑤1, 𝑤2, … , 𝑤𝑠} into an image I,

first the image is divided into non-overlapped blocks. Next, in following steps,

each image block 𝐼𝑖, (𝑖 = 1, 2, … , 𝑠) can be used to insert one bit 𝑤𝑖:

Step 1: Applying SVD to every Ii:

𝐼𝑖 = 𝑈𝑖 × 𝐷𝑖 × 𝑉𝑖𝑇

Step 2: Inserting a bit wi into Ui by adjusting 𝑈𝑖(2,1) and 𝑈𝑖(3,1) following

the rule:

|𝑈𝑖(3,1)| − |𝑈𝑖(2,1)| ≥ 𝜃 if 𝑤𝑖 is 0

|𝑈𝑖(3,1)| − |𝑈𝑖(2,1)| < 𝜃 if 𝑤𝑖 is 1

2684 Cao Thi Luyen et al.

where θ is a pre-defined thereshold.

After the embedding process, 𝑈𝑖 becomes 𝑈𝑖′, which includes 𝑤𝑖.

Step 3: Computing 𝐼𝑖′

𝐼𝑖′ = 𝑈′𝑖 × 𝑉𝑖 × 𝑉𝑖

𝑇 In the processes of transmission and distribution, the watermarked image 𝐼′ can

be affected by unintentional manipulations or deliberated attacks. Subsequently,

the receiver usually obtained the attacked version 𝐼∗, which is not quite the same

as 𝐼′. Therefore, the watermark 𝑊∗ (extracted from 𝐼∗) should be compared to

the original watermark W in order to evaluate the robustness of the scheme.

In the reverse direction, the extracted watermark is recovered in following steps:

Step 1: Applying SVD to each non-overlapped segmented block 𝐼𝑖∗:

𝐼𝑖∗ = 𝑈𝑖

∗ × 𝐷𝑖∗ × 𝑉𝑖

∗𝑇

Step 2: Computing a bit 𝑤𝑖∗ from 𝑈𝑖

∗:

𝑤𝑖∗= {

0 If |𝑈𝑖(3,1)| − |𝑈𝑖(2,1)| ≥ θ1 Otherwise

Step 3: Evaluating the accuracy of the obtained watermark:

𝐸𝑅𝑅 =1

𝑡∑ 𝑤𝑖 𝑋𝑂𝑅 𝑤𝑖

∗𝑡𝑖=1 (2)

For verification, we employ a pre-define threshold τ (τ[0,1]) to conclude the

extraction results: ERR < τ means that W was legally embedded in 𝐼∗.

2.2.2 The scheme of Lai

The main steps of [6] are similar to [13], but there are several differences:

1) A watermarking bit is not embedded into every block, but only in the blocks

satisfy some criteria of visual entropy,

2) [6] employs DCT to the selected blocks before using SVD, and

3) Watermarks are embedded into 𝑈𝑖(3,1) and 𝑈𝑖(4,1) instead of using

𝑈𝑖(3,1) and 𝑈𝑖(4,1) as in [13].

2.2.3 Comments on the SVD based schemes

In numerous SVD based schemes [5, 6, 13, 17, 19, 23], each segmented image

block Ii is usually divided into matrices 𝑈𝑖 , 𝑉𝑖 and 𝐷𝑖. Although each bit wi can

be embedded into 𝑈𝑖(3,1) and 𝑈𝑖(4,1) [6,13] or only in 𝐷𝑖(1,1) [5,17,19,23],

in the processes of embedding and extraction, we have to analyze all of the

matrices. Next, in watermarking schemes, we show that, it does not need to

analyze all matrices, but only need to compute the first column of 𝑈𝑖 and 𝑉𝑖, or

the first coefficient of 𝐷𝑖 . Obviously, we have:

𝐼𝑖 = 𝑋𝑖 + 𝐶𝑖 where

Xi= Di(1,1)Ui(1)Vi(1)T

A robust public key watermarking scheme 2685

Ci= ∑ Di(j,j)Ui(j)Vi(j)Tsj=2

s = min {m, n}

After embedding a watermarking bit into Ui(1), Vi(1) or Di(1,1), then 𝑋𝑖

become 𝑋𝑖′ , while 𝐶𝑖 is not changed. Therefore:

𝐼𝑖′ = 𝑋𝑖′ + 𝐶𝑖 where

𝐶𝑖 = 𝐼𝑖 − 𝑋𝑖

Consequently, in order to insert one bit into 𝐼𝑖, we only need to compute

𝑋𝑖. That means we only determine 𝐷𝑖(1,1), 𝑈𝑖(1) and 𝑉𝑖(1), instead of finding

𝐷𝑖(𝑗, 𝑗), 𝑈𝑖(𝑗) and 𝑉𝑖(𝑗), where 𝑖 = 1 … 𝑠, 𝑗 = 1. . . 𝑛 .

Based on this idea, next section describes the new watermarking scheme,

which achieves low computational complexity. For hiding a watermarking bit in a

segmented image block, we employ a pair {𝑈𝑖(1,1), 𝑈𝑖(2,1)} instead of using

{𝑈𝑖(2,1), 𝑈𝑖(3,1)} in order to increase the robustness of watermarking schemes.

2.3 Using SVD for feature extraction

In order to design robust public key watermarking schemes, extracting robust

features from images is crucial. In embedding process, the features are utilized to

create watermarks. There are two main directions to generate watermarks in

public key watermarking schemes:

1) using standard random distribution [18],

2) using robust image features [26, 27].

In this paper, the second approach is considered. Next, we review some SVD

based algorithms for extracting robust features from images and employ them to

watermarking schemes.

2.3.1 The algorithm of Zhou and Jin

In the algorithm of Zhou and Jin [27], the features B are extracted from an image I

by following steps:

Step 1: Using the L-level DWT transform (in the paper, the authors use 2-level

DWT) to I in order to obtain low frequency part, which is denoted by 𝐹𝐿.

Step 2: Dividing 𝐹𝐿 into non-overlapped blocks, denoted 𝐹𝐿𝑖, 𝑖 = 1. . 𝑘.

Step 3: Applying SVD to every block 𝐹𝐿𝑖:

𝐹𝐿𝑖= 𝑈𝑖𝐷𝑖𝑉𝑖

𝑇 , 𝑖 = 1. . 𝑘

Step 4: Determining the special features B as following rule:

If 𝐷2𝑖−1(1,1) > 𝐷2𝑖(1,1) then B(i) = 1 else B(i) = 0, where 𝑖 = 1. . ⌊𝑘

2⌋.

2.3.2 The algorithm of Ye

According to Ye [26], the particular features B are extracted by the following

steps:

Step 1: Dividing the host image I into non-overlapped blocks denoted by 𝐼𝑖 , 𝑖 =1. . 𝑘.

Step 2: Applying SVD to every block 𝐼𝑖:

𝐼𝑖 = 𝑈𝑖𝐷𝑖𝑉𝑖𝑇 , 𝑖 = 1. . 𝑘

2686 Cao Thi Luyen et al.

Step 3: Employing DCT for 𝐷𝑖 and using its DC coefficient, denoted 𝐷𝐶𝑖.

Step 4: Determining the special features B as the following rule:

If 𝐷𝐶2𝑖−1(1,1) > 𝐷𝐶2𝑖(1,1) then B(i) = 1 else B(i) = 0, where 𝑖 = 1. . ⌊𝑘

2⌋.

2.3.3 Comments on extracting features

In the mentioned algorithms [26,27], specific features were created by using SVD

and the inequalities 𝐷2𝑖−1(1,1) > 𝐷2𝑖(1,1) or 𝐷𝐶2𝑖−1(1,1) > 𝐷𝐶2𝑖(1,1). When

an image has been attacked, these relationships can be ruined and the extracted

features from attacked images become different from the features of the original

image. Therefore, if we choose a threshold T > 0, the features satisfy the

condition |𝐷1(1,1) − 𝐷2(1,1)| < 𝑇.

3 The proposed algorithm

3.1 Determining Di (1,1), Ui (1) and Ui (1)

In this section, we revise the algorithm for computing 𝐷𝑖(1,1), 𝑈𝑖(1) and 𝑉𝑖(1)

directly without using the conventional SVD. To achieve that goal, we employ the

method suggested in [14] to find the largest eigenvalue and eigenvector of a

non-negative matrix. The algorithm is described as below:

For this algorithm, the input is a matrix Ii and the output are the first coefficient of

Di and the first columns of 𝑈𝑖 and 𝑉𝑖. To this end, we set:

Ai = Ii IiT, Ai ≥ 0, 𝐴𝑖 ∈ 𝑅𝑚×𝑚

Because Ui and Vi are normalized orthogonal matrices, we have:

𝐴𝑖 = 𝑈𝑖𝐷𝑖𝑉𝑖𝑇(𝑈𝑖𝐷𝑖𝑉𝑖

𝑇)𝑇=UiDi2Ui

T ,

𝐴𝑖𝑈𝑖 = 𝑈𝑖𝐷𝑖2 and 𝐴𝑖𝑈𝑖(1) = 𝐷𝑖

2(1,1)𝑈𝑖(1).

Following the definition of eigenvalues and eigenvectors, 𝐷𝑖2(1,1) and 𝑈𝑖(1)

will be the largest eigenvalue and eigenvector of the matrix 𝐴𝑖 , respectively.

Similarly, if we set: Bi = IiTIi, Bi ≥ 0, 𝐵𝑖 ∈ 𝑅𝑛×𝑛 then Di

2(1,1) becomes the largest

eigenvalue and 𝑉𝑖(1) is the corresponding eigenvector of the non-negative matrix

𝐵𝑖.

As a result, the values of 𝐷𝑖2(1,1) , 𝑈𝑖(1) and 𝑉𝑖(1) can be computed by

evaluating the maximum eigenvalue and corresponding eigenvector of the

non-negative matrix 𝐴𝑖. In other words, by directly applying this technique to the

proposed algorithm, we can bypass the conventional SVD. Therefore, the

computational complexity of feature extracting and watermarking schemes will be

decreased.

3.2 The robust features extracting

To reduce the computational complexity as well as further improve the robustness

of extracted features, we employ a threshold T along with the proposed algorithm

(Section 3.1). The extracting algorithm includes following steps:

Step 1: Dividing an original image I into non-overlapped m×n blocks 𝐼𝑖 , 𝑖 =1. . 𝑘.

A robust public key watermarking scheme 2687

Step 2: Applying the mentioned algorithm (in Section 3.1) to each block 𝐼𝑖 in

order to compute 𝐷𝑖(1,1).

Step 3: Determining robust features of the cover image by the formulas as below:

If |𝐷2𝑖−1(1,1) − 𝐷2𝑖(1,1)| > 𝑇 then B(i) = 1 else B(i) = 0, where 𝑖 = 1. . ⌊𝑘

2⌋.

Experimental results (in Section 4) show that the proposed algorithm is more

robust than some other related schemes [26, 27].

3.3 The watermark embedding procedure

The input of the embedding algorithm is an image I, a watermark 𝑊 =(𝑤1, 𝑤2, … , 𝑤𝑡), and pre-defined thresholds (𝜃, 𝜏). Its output is the watermarked

image 𝐼′.

The embedding algorithm consists of following steps:

Step 1: Dividing the original image I into non-overlapped m×n blocks 𝐼𝑖.

Step 2: Using the proposed algorithm to directly compute the first column of 𝑈𝑖.

We obtained 𝑈𝑖(1) as the eigenvector corresponding to the largest eigenvalue of

the non-negative matrix 𝐼𝑖 × 𝐼𝑖𝑇.

Step 3: Embedding 𝑤𝑖 into the the image block 𝐼𝑖 using the pair

{𝑈𝑖(1,1), 𝑈𝑖(2,1)}.

In order to obtain 𝑈𝑖′, we adjust 𝑈𝑖(1,1) and 𝑈𝑖(2,1) according to 𝑤𝑖 such that

they satify the following conditions:

|𝑈𝑖(2,1)| − |𝑈𝑖(1,1)| ≥ θ if 𝑤𝑖 is 0

|𝑈𝑖(2,1)| − |𝑈𝑖(1,1)| < θ if 𝑤𝑖 is 1 Step 4: Figuring out

𝐼𝑖′ = 𝐷𝑖(1,1) × 𝑈𝑖

′(1) × 𝑉𝑖𝑇(1) + 𝐼𝑖 − 𝐷𝑖(1,1) × 𝑈𝑖(1) × 𝑉𝑖

𝑇(1)

As a result, 𝐼𝑖′ can be computed without using conventional SVD analysis. The

correctness of the calculation of 𝐼𝑖′ is described as below.

According to the formula (1) in [9]:

𝐼𝑖 = 𝐷𝑖(1,1) × 𝑈𝑖(1) × 𝑉𝑖𝑇(1) + ∑ 𝐷𝑖(k,k) × 𝑈i(k) × 𝑉𝑖(𝑘),𝑠

𝑘=2

where 𝑠 = 𝑚𝑖𝑛 (𝑚, 𝑛).

Therefore:

∑ 𝐷𝑖(k,k) × 𝑈i(k) × 𝑉𝑖(𝑘)𝑠𝑘=2 = 𝐼𝑖 − 𝐷𝑖(1,1) × 𝑈𝑖(1) × 𝑉𝑖

𝑇(1) (3)

Besides, we have:

𝐼𝑖′ = 𝐷𝑖

′(1,1) × 𝑈𝑖′(1) × 𝑉𝑖

′𝑇(1) + ∑ 𝐷𝑖

′(k,k) × 𝑈𝑖′(k) × 𝑉𝑖

′(𝑘)𝑠𝑘=2 (4)

Since the values of 𝐷𝑖(k,k), 𝑈i(k) and 𝑉𝑖(𝑘) are retained after embedding, we

have:

𝐷𝑖′(𝑘,k) = 𝐷𝑖(𝑘,k), 𝑈𝑖

′(k) = 𝑈i(k) and 𝑉𝑖′(𝑘) = 𝑉𝑖(𝑘), 𝑘 ≥ 2.

Thus,

𝐼𝑖′ = 𝐷𝑖(1,1) × 𝑈𝑖

′(1) × 𝑉𝑖𝑇(1) + ∑ 𝐷𝑖(k,k) × 𝑈i(k) × 𝑉𝑖(𝑘)𝑠

𝑘=2 (5)

Substitute (3) for (5), we obtain the next equation:

Ii′ = 𝐷𝑖(1,1) × 𝑈𝑖

′(1) × 𝑉𝑖𝑇(1) + 𝐼𝑖 − 𝐷𝑖(1,1) × 𝑈𝑖(1) × 𝑉𝑖

𝑇(1)

2688 Cao Thi Luyen et al.

3.4 The watermark extracting procedure

The extracting procedures of the proposed algorithm and the algorithm of of [13]

are alike, except the first and the second steps. In the first step, we solve the

largest eigenvalue and the correspond eigenvector of the non-negative matrix as

below:

𝐴𝑖∗ = 𝐼𝑖

∗ × 𝐼𝑖∗𝑇

Instead of applying SVD on each blocks 𝐼𝑖

∗ to determine 𝐷𝑖∗(1,1) and 𝑈𝑖

∗(1). In

the second step, 𝑤1∗ is extracted by following way:

If |𝑈𝑖∗(2,1)| − |𝑈𝑖

∗(1,1)| ≥ θ if 𝑤𝑖 is 0 then 𝑤1∗ = 0, otherwise w1

∗ = 1.

3.5 The proposed public key watermarking scheme

To improve the usability of watermarking in practice, various watermarking

schemes borrowed some ideas from public key cryptography [11]. In this section,

we utilize the proposed algorithm and public key in order to design a robust public

key watermarking scheme.

3.5.1 The embedding watermark algorithm

The input of the algorithm is an original image I. This private key belongs to the

legal owner of I. The output is the watermarked image 𝐼′.

The embedding process is performed as below:

Step 1: Dividing the original image 𝐼 into two separated part 𝐼1 and 𝐼2.

Step 2: Creating the watermark W by using the proposed algorithm (Section 3.2) to

extract robust features (denoted by W) from 𝐼2. Assuming that, the length of W is k

(bits).

Step 3: Embedding the watermark W into the part 𝐼1following the proposed

algorithm. Consequently, we get 𝐼1′ .

Step 4: Combining 𝐼1′ and 𝐼2 to receive the watermarked image 𝐼′.

3.5.2 The extracting and detecting algorithms

The input of the algorithm is an attacked image 𝐼∗ and a pre-defined

threshold 𝜏 (𝜏[0,1]) . The output is the conclusion that the image I∗ legally

belongs to the owners of 𝐼′.

The process of determining the copyright of 𝐼∗ is conducted in the following

steps:

Step 1: Dividing the attacked image 𝐼∗ into two separated part 𝐼1∗ and 𝐼2

∗ follows

the same way in Step 1 of the embedding watermarking process.

Step 2: Recovering 𝐵1∗ from 𝐼1

∗ by applying the proposed extracting procedure (in

Section 3.4).

Step 3: Extracting the features of 𝐼2∗ by employing the extracting algorithm

(presented in Section 3.2) to obtain 𝐵2∗.

Step 4: Inspired by the formula to calculate ERR (formula (2), Section 2.2.1), we

will verify the copyright of attacked images by the following sub-steps:

- Computing 𝐶 = 𝑆𝑈𝑀(𝐵1∗ 𝑋𝑂𝑅 𝐵2

∗)/𝑘 (6)

A robust public key watermarking scheme 2689

- If 𝐶 < τ then we can conclude that the image 𝐼∗ was copyrighted.

4 Experimental Results

We use some standard images with size of 512×512 (Figure 1) to evaluate the

robustness of our feature extraction algorithm and related work (Section 4.1) and

the robust watermarking schemes (Section 4.2). The experimental images are

downloaded from the standard image database USC-SIPI.

(A)

(B)

(C)

(D)

Figure 1. Images used in experiments.

To this end, first we applied attacks to watermarked images. Different operations

of Gaussian noise (15%), JPEG compression (50%), Gaussian blurring (2%),

median filtering (with the window size of 2×2) and resizing (up to 20% and down

to the original size of the host image) and were independently employed to

watermarked images. In the following sections, we used the original images and

attacked images to evaluate the robustness of feature extraction algorithm and

watermarking schemes.

4.1 Robustness of feature extraction algorithms

In order to measure the robustness of the proposed feature extraction algorithm,

we obtain the features B and 𝐵∗ by applying the analyzed algorithms to the

extracted features from the original images (Figure 1, A, B, C or D) and its

corresponding attacked versions.

We considered the algorithms of [26, 27] and our proposed algorithm. We used

the threshold T = 100 in the proposed algorithm.

2690 Cao Thi Luyen et al.

Table 1. Robust feature extraction against “adding Gaussian noise”.

Algorithms A B C D

Ye 0.2852 0.1978 0.2998 0.2944

Zhou & Jin 0.2402 0.1309 0.1553 0.2231

Proposed 0.0313 0.0610 0.0713 0.0449

Table 2. Robust feature extraction against “JPEG compression”.

Algorithms A B C D

Ye 0.0752 0.0449 0.0630 0.1001

Zhou & Jin 0.0557 0.0269 0.0293 0.0669

Proposed 0.0044 0.0049 0.0093 0.0059

Table 3. Robust feature extraction against “Gaussian blurring”.

Algorithms A B C D

Ye 0.1050 0.0859 0.1060 0.2739

Zhou & Jin 0.0532 0.0410 0.0361 0.0737

Proposed 0.0161 0.0376 0.0425 0.0562

Table 4. Robust feature extraction against “median filtering”.

Algorithms A B C D

Ye 0.1401 0.1401 0.0854 0.1563

Zhou & Jin 0.0879 0.0879 0.0283 0.1230

Proposed 0.0078 0.0078 0.0303 0.0811

Table 5. Robust feature extraction against “resizing”.

Algorithms A B C D

Ye 0.0630 0.0400 0.0498 0.0728

Zhou & Jin 0.0278 0.0181 0.0205 0.0298

Proposed 0.0083 0.0034 0.0190 0.0151

Next, in each case we employed the formula (6) in the previous section to

compute the difference C between B and 𝐵∗. The experimental results are shown

in Table 1 – Table 5, where the lower the difference, the higher the robustness.

The results show that, the proposed algorithm is robust.

4.2 Robustness of watermarking schemes

As the extracting and detecting processes of watermarking algorithm described in

Section 3.5.2, in the watermark validation process, comparing B1∗ and B2

∗ shows

the robustness of the watermarking algorithm. We apply the proposed algorithm

and other algorithms of [6, 13] on the attacked version of the images in Figure 1 (A

– D). The experimental outcomes (Table 6 – Table 10) exhibit the robustness of

A robust public key watermarking scheme 2691

[13] and the proposed algorithms is essentially the same and they are much more

robust than [6]. However, due to the application of the improved SVD algorithm

(presented in Section 3), our algorithms have much lower computational

complexity.

Table 6. The robustness of watermarking algorithms against “Gaussian noise”.

Algorithms A B C D

Chung 0.2609 0.2271 0.1925 0.1481

Lai 0.2937 0.2583 0.2289 0.1920

Proposed 0.2629 0.2281 0.1948 0.1526

Table 7. The robustness of watermarking algorithms against “JPEG compression”.

Algorithms A B C D

Chung 0.2662 0.0015 0.0621 0.0071

Lai 0.2789 0.2660 0.2184 0.2284

Proposed 0.1592 0.0020 0.0708 0.0075

Table 8. The robustness of watermarking algorithms against “Gaussian blurring”.

Algorithms A B C D

Chung 0.3123 0.2787 0.3127 0.2626

Lai 0.3995 0.4133 0.4175 0.4260

Proposed 0.3058 0.2665 0.3146 0.2609

Table 9. The robustness of watermarking algorithms against “filtering”.

Algorithms A B C D

Chung 0.1007 0.0638 0.0907 0.0760

Lai 0.2896 0.2879 0.2990 0.2969

Proposed 0.0994 0.0903 0.0898 0.0782

Table 10. The robustness of watermarking algorithms against “resizing”.

Algorithms A B C D

Chung 0.0297 0.0046 0.0259 0.0228

Lai 0.2917 0.3234 0.3181 0.3208

Proposed 0.0289 0.0047 0.0237 0.0244

This is very important when we often have to work with a large-scale image dataset

in practice. In addition, in order to increase the applicability in practice, our

algorithms use the public key model, so the watermark and secret key do not need

to be distributed to the recipients.

2692 Cao Thi Luyen et al.

5 Conclusion

In this paper, a fast and robust public key image watermarking scheme based on

improved SVD was presented. To achieve this goal, we employed a method to

directly compute the largest eigenvector and eigenvalue of a matrix, instead of

using the conventional SVD. Since the computational complexity of

watermarking schemes was decreased, the processes of watermarking embedding

and extracting were speed up. This is crucial for working with large-scale image

dataset in real-life applications. The robustness of feature extraction and

watermarking schemes can be further improved by using a threshold.

Experimental results showed that our watermarking scheme is more robust against

various attacks in comparison with several widely used SVD based schemes.

Besides, based on the mentioned algorithms, we suggested a new robust public

key watermarking scheme, where watermarked images can be verified without

using the pre-defined watermark and secret key. As a result, with the proposed

improvements, our scheme can be applied in practice.

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Received: September 12, 2017; Published: October 29, 2017