AN EFFICIENT ALGORITHM FOR INFORMED EMBEDDING OF DIRTY-PAPER TRELLIS CODES FOR WATERMARKING Lin Lin, Gwena¨ el Do¨ err, Ingemar J. Cox, Matt L. Miller ABSTRACT Dirty paper trellis codes are a form of watermarking with side information. These codes have the advantage of being invariant to valumetric scaling of the cover Work. However, the original proposal requires a computational expensive sec- ond stage, informed embedding, to embed the chosen code into the cover Work. In this paper, we present a computa- tional efficient algorithm for informed embedding. This is ac- complished by recognizing that all possible code words are uniformly distributed on the surface of a high n-dimensional sphere. Each codeword is then contained within an (n - 1)- dimensional region which defines an n-dimensional cone with the centre of the sphere. This approximates the detection re- gion. This is equivalent to the detection region for normal- ized correlation detection, for which there are known analytic methods to embed a watermark in a cover Work. We use a previously described technique for embedding with a constant robustness. However, rather than moving the cover Work to the closest Euclidean point on the defined surface, we find the point on the surface which has the smallest perceptual distor- tion. Experimental results on 2000 images demonstrate a 600- fold computational improvement together with an improved quality of embedding. 1. INTRODUCTION In 1999, several researchers [1, 2, 3] contemporaneously recognized that watermarking with blind detection can be mod- eled as communication with side-information at the transmit- ter [4]. This realization has led to the design of algorithms for informed coding and informed embedding. In informed coding, there is a one-to-many mapping be- tween a message and its associated codewords. The code or pattern that is used to represent the watermark message is de- pendent on the cover Work. The reader is directed to [5] for a detailed discussion of these concepts. Informed coding is based on the work of Costa [6]. Chen [7] first realized the importance of Costa’s work to watermarking and Moulin and O’Sullivan [8] have since extended Costa’s analysis to noise models more realistic in the context of watermarking. Costa’s result suggests that the channel capacity of a wa- termarking system should be independent of the cover Work. Lin Lin and Ingemar J. Cox are with the Department of Electronic and Electrical Engineering, University College London. Gwena¨ el Do¨ err is with the Eur´ ecom Institute, Sophia-Antipolis, France. Matt L. Miller is with the NEC Laboratories Princeton, NJ, 08540 USA. This is highly unexpected, since previously watermaking sys- tems were modeled as communication systems that operated in very low signal-to-noise regimes due to the strong interfer- ence from the cover Work. This limited the number of bits that could be reliably embedded in a cover Work. Costa’s result therefore offers the promise of significantly improving watermarking systems. Costa’s result relies on a very large random codebook that is impractical. In order to permit efficient search for the best dirty-paper 1 codeword three main approaches have been pro- posed based on structured codebooks. These are syndrome codes, lattice codes and trellis codes [2, 3, 9]. Lattice codes, more often referred to as quantization in- dex modulation (QIM), have received most attention due to (i) easy implementation, (ii) low computational cost and (iii) high data payloads. Quantization index modulation has been criti- cized for being very sensitive to valumetric scaling, i.e. mul- tiplicative changes to the amplitude of the cover Work. For example, changes to the volume of an audio signal can lead to complete loss of the watermark message. Recently, however, considerable progress has been made [10, 11, 12, 13] towards resolving this issue. Trellis codes are an alternative to lattice codes and were originally proposed [9, 14] to address the issue of valumetric scaling, e.g. changes in image brightness. The codes of a trel- lis lie on the surface of a high dimensional sphere. For a given message, the trellis structure permits an efficient identification of the most appropriate code to embed in a given cover Work. However, while selection of the best dirty paper code is effi- cient, subsequent informed embedding of the code word re- quires a computational expensive iterative procedure.Abrardo and Barni [15] proposed using orthogonal dirty paper codes that can be embedded computationally efficiently. However, we believe that dirty paper trellis codes have the potential for higher data payloads. In this paper, we described a computationally efficient method for informed embedding of dirty paper trellis codes. The key insights are (i) that all possible codewords are uniformly dis- tributed on the surface of an n-dimensional sphere, (ii) the Voronoi region around each codeword can be approximated by an (n - 1)-dimensional sphere, (iii) the centre of the sphere and the surface of the (n - 1)-dimensional sphere define an n- dimensional detection region that is a n-dimensional cone and (iv) this n-dimensional cone is equivalent to the detection re- 1 The term derives from that fact that Costa’s original paper was entitled “Writing on Dirty Paper”.