Analysis of digital watermarks subjected to optimum linear filtering and additive noise

Using a theoretical approach based on random processes, signal processing, and information theory. we study the performance of digital watermarks subj ected to an attack consisting of linear shift-invariant filtering and addit ive colored Gaussian noise. Watermarking is viewed as communication over a hostile channel, where the attack takes place. The attacker attempts to min imize the channel capacity under a constraint on the attack distortion (dis tortion of the attacked signal), and the owner attempts to maximize the cap acity under a constraint on the embedding distortion (distortion of the wat ermarked signal). The distortion measure is frequency-weighted mean-squared error (MSE). In a conventional additive-noise channel, communication is mo st difficult when the noise is white and Gaussian, so we first investigate an effective white-noise attack based on this principle, We then consider t he problem of resisting this attack and show that capacity is maximized whe n a power-spectrum condition (PSC) is fulfilled. The PSC states that the po wer spectrum of the watermark should be directly proportional to that of th e original signal. However, unlike a conventional channel, the hostile atta ck channel adapts to the watermark, not vice versa. Hence, the effective wh ite-noise attack is suboptimal, We derive the optimum attack. which minimiz es the channel capacity for a given attack distortion. The attack can be ro ughly characterized by a rule-of-thumb: At low attack distortions, it adds noise, and at high attack distortions. it discards frequency components. Ag ainst the optimum attack, the PSC does not maximize capacity at all attack distortions. Also, there is no unique watermark power spectrum that maximiz es capacity over the entire range of attack distortions. To find the waterm ark power spectrum that maximizes capacity against the optimum attack, we a pply iterative numerical methods, which alternately adjust the watermark po wer spectrum and re-optimize the parameters of the optimum attack. Experime nts using ordinary MSE distortion lead to a rule-of-thumb: White watermarks perform nearly optimally at low attack distortions, while PSC-compliant wa termarks perform nearly optimally at high attack distortions. The effect of interference from the original signal in suboptimal blind watermarking sch emes is also considered; experiments examine its influence on the optimized watermark power spectra and the potential increase in capacity when it can be partially suppressed. Additional experiments demonstrate the importance of memory. and compare the optimum attack with suboptimal attack models. F inally, the rule-of-thumb for the defense is extended to the case of freque ncy-weighted MSE as a distortion measure. (C) 2001 Elsevier Science B.V. Al l rights reserved.