Hierarchical ownership and deterministic watermarking of digital images via polynomial interpolation

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Abstract

This paper presents a novel method for the secure management of digital images formulated within the mathematical theory of polynomial interpolation. As main innovative features, our approach is based on a hierarchical joint ownership of the image by a trusted layered authority and on a deterministic watermarking procedure, embedding a short meaningful or random signature into the image. Experimental results show that the inserted signature can almost always be fully recovered even in presence of a reasonable amount of image degradation due to image processing operators, such as filtering, geometric distortions and compression.

Introduction

In the last decade, digital watermarking techniques have raised a great deal of interest in the scientific community since the pioneering contribution by Cox et al. [8]. Indeed, the practice of imperceptible alteration of a document to embed a message into it plays a key role in the challenging field of ownership right protection. Much progress has been done in the last few years (see for instance [12]), but no general solution has been reached so far. This can be explained by several different factors, among which the heterogeneity of the requirements imposed by each application context and the clear definition and the operational mechanisms of the authority that would deal with the ownership verification process. In other words, currently proposed watermarking techniques strongly depend on the application scenario.

Let us consider the specific case, where a public or private organization needs to keep control on internal resources distributed to a number of users. In such a context, the organization can be regarded as an authority which has free access to the original data and assigns watermarked copies to users, who ignore the presence of the watermark. In this particular application scenario, two different problems need to be addressed: (i) the management of the original data by the authority (which is trustworthy as a whole but includes possibly untrusted members); and (ii) the selection of a proper watermarking technique. A well-known example is provided by the distribution of uniquely identifiable copies of a confidential British cabinet document to each minister by Margaret Thatcher in 1981. Hence, when the document was printed in the newspapers, the source of the leak could be discovered [9, p. 4]. Another realistic example for this application scenario is the case of the management of a set of confidential images involved in a legal prosecution. The control of their access is shared by the judging court (the authority) which is a hierarchical structure (president, members of the jury, etc.). The images should be made available to a group of persons (the users) involved directly or indirectly in the prosecution process such as lawyers, officers, etc. Since the users may be not trusted, the authority wants to be able to detect possible illegal leakages. To achieve this goal, the images given to the users are watermarked with a signature that uniquely identifies them. In case of violation, an authorized subgroup of the authority can identify the source of the leakage. Accordingly, the two main ingredients of this kind of copyright management scheme are: (i) a joint ownership of the original data in a group with hierarchical structure; and (ii) a watermarking procedure which can exploit the original data in the reconstruction phase (i.e., it is not necessarily blind) and whose existence is hidden to users (i.e., it is steganographic according to the terminology of [9]).

As far as the authority is concerned, we stress that we consider the possibility to cope with untrusted member and therefore we need to find a way to manage the original data in a distributed way, exploiting the hierarchical structure of the organization. The hierarchical ownership handling, has been recently addressed in the context of digital image watermarking by Guo and Georganas [14], whose work exploits a secret sharing procedure generalizing the basic scheme by Shamir [18]. A (k,n)-threshold sharing scheme allows to divide a secret into n shares and requires the knowledge of at least k out of n shares to reconstruct the original content. Each share does not carry any meaningful partial plaintext of the secret and, if the number of shares available is less than k, a potential attacker can do no better than guessing, even with infinite computing time and power. Nevertheless, the solution in [14] has the annoying drawback that the procedure of shares distribution is expensive in terms of storage and complexity, since a huge number of shares is assigned to each participant. For a critical analysis of this algorithm, we refer to [20].

As an alternative, in this paper we propose a more sophisticated approach based on Birkhoff polynomial interpolation. Its main advantage is that the secret sharing is simplified by assigning just a single real number to each member of the group (no matter how complicated the corresponding access structure). An authorized subset of the authority can access the original data only if it includes a sufficient number of members for each level in the organization. Referring to the above example, this means that to access an image used in a legal prosecution one can need one actor of the highest level (e.g., the president) and two actors of the second level (e.g., two members of the jury). An interesting property of the proposed hierarchical scheme is its flexibility. Indeed, it can be applied to manage the access to different types of data (not only images) and combined with various kinds of watermarking methods depending on the application requirements and not only with that described in the following.

Concerning this last aspect, since the organization needs to keep control on copies distributed to a certain number of users, it is natural to apply a fingerprinting method by assigning to each user a unique watermark which identifies the legal recipient of the copy. Furthermore, because of the application considered, we require an exact reconstruction of the signature (watermark) assigned to each user and we assume that the users ignore the presence of the watermark (steganographic watermarking), hence no malicious attacks (e.g., collusion attack) need to be taken into account. Nevertheless, a user may need to perform some simple processing to use the data (e.g., to compress and store the data in a database or to resize it for page formatting and printing). Since the authority needs to trace the source of illegally redistributed copies, such non-malicious image processing operations should be considered in the testing of the watermarking process robustness. In order to fulfill the first requirement, instead of considering a random sequence as watermark and just a correlation measure for its detection as in [8], here a meaningful signature has to be embedded and the watermark detection should lead to a perfect recovery of the inserted signature. Such kind of watermarking schemes belong to the category of readable watermarks according to the terminology of [4] (see for instance [17], [13], just to quote a couple of recent contributions). In the specific context of image forensic, a version of the secret message has to be extracted from the stego message [6], but the problem of accepting digital image watermarking as a legal evidence of ownership is still widely open [1], [3]. As a possible solution to this problem we propose the use of a deterministic signature to be perfectly reconstructed.

Specifically, the signature written in English alphabet is first translated into a sequence of integers by means of a look-up table. Such a sequence of integers is used to set the coefficients of a trigonometric polynomial, from which a predefined number of samples is extracted evaluated at equally spaced points. Finally, the values of the samples are embedded into the lowest frequency coefficients of the original image transformed into the DCT domain (excluding the DC component as in [8]). The watermark extraction process is based on solving a system of linear equations defined by the recovered samples. It is worth mentioning that, in [7], [16], an analogous sinusoidal pattern has already been successfully exploited to embed a pseudo-random sequence. In these works, however, the detection of the watermark was just limited to a correlation measurement. In our watermarking scheme, characterized by a full reconstruction of the watermark, the choice of a trigonometric rather than an algebraic polynomial is motivated by the fact that standard polynomial interpolation is ill-conditioned, while the use of trigonometric functions allows to keep the condition number of the corresponding linear system close to the optimal value 1. In order to obtain a reliable deterministic polynomial reconstruction, we need to face the problem of image degradations due to the application of such standard image processing operators. Despite the preservation of the global quality of the image, the degradation may drastically corrupt some entries of the DCT image where the watermark is inserted. We overcome this issue by a suitable selection of the DCT samples conveying the watermark. In order to assess the effectiveness of the proposed watermarking approach, based on a hierarchical authority of ownership verification, we make use of a very large set of test images of different typologies. Experimental results show that our method exhibits a satisfactory effectiveness: the signature is reconstructed with 100% of accuracy for a wide range of image degradation operators. Furthermore, high performance is obtained in terms of false detection even in critical situations involving both users identified with very close signatures and strongly corrupted images.

The structure of the paper is as follows: in Section 2, we present a hierarchical secret sharing scheme for the joint ownership of the original image; in Section 3, we describe the generation, the embedding and the reconstruction phases of the watermarking scheme; in Section 4, we report experimental results; and in Section 5, we draw some concluding remarks.

Section snippets

Previous work

The main feature of a non-blind watermarking scheme is that the original image is needed in the reconstruction phase. As a consequence, an authority group A managing this process has to memorize the cover image, preferably storing it in a distributed (e.g., hierarchical) way for security reasons. As mentioned in the previous section, in order to do that it is natural to apply a secret sharing procedure.

In this context, the basic secret sharing scheme proposed by Shamir [18] relies on standard

The watermarking scheme

The aim of an authority A hierarchically organized into several levels is to distribute a given image I among a set of users u1,,un, keeping some control on the use of the image by each of them. In particular, for any copy of I, that may undergo some image processing operations, any subset VA in the given access structure Γ should be able to identify without ambiguity the user from whom this copy comes from. As summarized in Fig. 1, a secure management of I can be provided by the following

Experimental results

In this experimental phase, we implemented our watermarking approach setting k=16, l=8, α=0.1 and tested it on a set of 70 images of different nature to deduce meaningful conclusions. In general, the watermark inserted in the image is imperceptible since on average PSNR=43dB (see Figs. 4 and 5). The attacks we considered to verify the method robustness are the following standard image degradation operations: additive white Gaussian noise with different power values; additive uniform noise with

Conclusions

In this paper, we have proposed a novel image watermarking technique which allows a trusted authority to recover the ownership from any reasonably distorted copy of an image distributed to several users. In order to do so, we embed into the image the signature of the corresponding user in a redundant way, exploiting a suitable trigonometric polynomial. The watermark detection is performed by the authority, which is considered as a hierarchical group managing the original image with a

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