Sparse Watermark Embedding and Recovery using Compressed Sensing Framework for Audio Signals Mohamed Waleed Fakhr Electrical and Electronics Department, University of Bahrain Isa Town, Manama, Bahrain mfakhr@uob.edu.bh Abstract In this paper a new watermark embedding and recovery technique is proposed based on the compressed sensing framework. Both the watermark and the host signal are assumed to be sparse, each in its own domain. In recovery, the L1- minimization is used to recover the watermark and the host signal perfectly in clean conditions. The proposed technique is tested on MP3 audio where the effects of MP3 compression/decompression, sampling rate reduction and additive noise attacks are considered and bit error rate is compared with spread spectrum embedding. The proposed technique offers significantly better performance in all tested conditions and opens a new research approach for watermark embedding and recovery. I. INTRODUCTION One of the approaches to control Audio MP3 music files piracy is to enforce copyright detection and tracking. Digital robust watermarking embeds a private secure code within a multimedia file (e.g., MP3 music) where the watermark detection is possible only for the certified authority. The watermark should be imperceptible, secure, with high payload and robust to some attacks that do not destroy the original host signal. This is a truly challenging problem for audio signals, and in particular, for MP3 audio where coding/decoding, sampling rate conversion are available to everyone [1,2]. Traditionally, spread spectrum (SS) and quantization index modulation (QIM) approaches have been used for watermark embedding and recovery, however, both suffer from problems when it comes to compression/decompression attacks, synchronization attacks and additive noise attacks. In particular, SS suffers from host rejection problem and QIM suffers from low immunity to additive noise and both have problems with compression/decompression attacks. Recently, there has been a considerable amount of work in robust watermarking of MP3 audio to overcome some of the problems of the traditional techniques [1-6]. Some of the recent papers propose an interesting idea which is to make the watermarking moderately audible for free peeking and downloading, and that those audible effects can be removed by the correct secret key [5,6]. Most of the recent work focuses on the effects of MP3 coding/decoding and sampling rate conversion, which cause degradation to most watermark embedding techniques [1-4]. SS and QIM are still the dominant embedding approaches in all recent work. Compressed sensing is a relatively new signal processing framework in which a sparse representation of a signal can be recovered from its noisy compressed measurements using the L1-norm minimization [7-10]. More recently, it has been shown that a sparse signal and an additive sparse interference can be both recovered perfectly under some conditions [11,12]. This recent work has inspired the novel idea of a sparse watermark which is presented in this paper in the compressed sensing framework. To show the practicality of the proposed technique, it is used for MP3 music robust watermarking with various attacks, and is compared to the traditional SS approach. This paper is organized as follow. Section II discusses the related recent work in MP3 watermarking. Section III gives a brief description of the related theory of compressed sensing and the recent work in the pursuit of justice [11] and extended LASSO [12] techniques which inspired the proposed technique. Section IV describes the details of the sparse watermark embedding and recovery, and the method used for sparsifying the audio host signal. Experimental results and comparisons are shown and discussed in section V and finally section VI gives the conclusions and future work. II. AUDIO WATERMARKING RECENT WORK In a recent paper [1] authors have made a comparison between most important MP3 watermarking research based on 7 criteria, namely, 1- the methodology being used (and that includes the complexity of the embedding and detection algorithms), 2- the imperceptibility of the watermark, 3- the robustness against attacks (compression/decompression, sampling rate conversion and de-synchronization), 4- the payload, 5- the security of the watermark, 6- whether the technique is blind or not and 7- if the algorithm and watermark are known to the attacker, can he use them to embed another audio media to frame the owner. Most of the reviewed techniques suffer from degraded performance against MP3 compression, sampling conversion and synchronization attacks. Payloads from as low as 10bps to up to 230bps were reported with MP3 compression attacks with bit error rates in the ranges between 0.01% to 5%. [1]. Most of the techniques discussed rely essentially on either the concept of quantizing some parameters in a transform space or spread spectrum embedding [3] while some rely on the patchwork algorithm and histogram modification of selected coefficients [4]. It is concluded from the review papers and the recent efforts in this area that there is still room for a lot of improvement in particular in increasing the payload and in robustness against MP3 and sampling rate conversion attacks. III. COMPRESSED SENSING RELATED THEORY Compressed sensing relies on the concept of a sparse domain representation of compressible signals. In the basic and pioneering formulation by Candes and others [7-10], if a
K-sparse domain vector x with dimension (N 1) is sampled randomly with a Gaussian orthogonalized M N matrix Φ producing a measurement vector y (M 1) where M < N (under-determined system), then, given the measurement vector and knowing the sampling matrix Φ =Φ + (1) An exact recovery of the sparse vector x is possible through L1-minimization using the basis pursuit denoising algorithm: minimize subject to Φ ε (2) Where e is a small additive noise and ε is its variance. Under the condition that Φ has to satisfy the restricted isometry property (RIP) which is met for Gaussian random orthonormal matrices [7,8], recovery is possible under the following condition: > (3) Where K is the number of non-zero elements in x. More recently, there have been two extensions to this framework. The first by Laska et. al. [11] in what was named the pursuit of justice algorithm. If the measurement vector is corrupted by noise that is sparse in some domain, (1) becomes: =Φ + Ω + (4) Where Ω is some full or partial transform matrix with orthonormal columns and dimensions (M L) where and is a sparse noise vector with possibly large amplitude non-zero components of length L and sparsity k. The authors have shown that both sparse vectors (x and ) can be perfectly recovered if: > ( + ) + (5) + Where k is the sparsity of and e is a small additive noise. Another interesting work by Nguyen et. al. [12] named the extended LASSO assumed that the sparse vector is additive directly with the measurement vector (thus of same dimension M). In that sense, it can be considered an extension of the pursuit of justice with the Ω is the identity matrix of size M M. In both cases, the recovery algorithm assumes a new sparse vector U=[x ] of size (N+L) 1 and a new matrix ψ=[φ Ω] with size M (N+L). And the basis pursuit becomes: Minimize Subject to: ε (6) And hence, when the sparse vector U is estimated, its first N elements are those of x and the remaining L are those of. Many fast algorithms have been implemented for the basis pursuit and the LASSO algorithms, of those, the L1-magic library [13] was used for all the L1-minimization algorithms in the proposed technique. IV. SPARSE WATERMARKING FORMULATION A. Sparse Watermark The watermark is a sparse vector of length L and with only one non-zero component which is ±1 based on the required watermark value. For embedding in a measurement vector of length M, the watermark signal W is generated: =Ω Where Ω is a random orthogonalized Gaussian matrix of size (M L). B. Sparse Host Signal Audio (music and speech) signals are highly compressible signals for which sparse domains exist. In this paper, the discrete Hadamard (DHT) transform is used to find a sparse representation for the host audio signal. The audio signal is divided into frames of length M each. The DHT is applied to each frame to produce the DHT-domain coefficients. All the coefficients with magnitude less than a fraction of the largest one are put to zero (sparsifying process). A similar process was used by Shiekh and Baraniuk [14] and by others to sparsify image data. The threshold used in this paper has been chosen experimentally so that the sparsity K is always less than 50% and that the sparsified audio signal quality degradation is imperceptible. Let the audio frame in the sparse domain be x which is now K-sparse (where K changes per frame), then, going back to the time-domain we take the inverse Hadamard transform. In this paper, the Hadamard matrix is of size M M and is normalized by the square root of M so the Hadamard matrix and its inverse are the same, and is called H. The sparsified host signal in time domain is thus: ( )= (7) C. Watermark Embedding The watermarked host signal is given by: = +.Ω (8) Where α is an embedding strength scalar factor which is made proportional to the frame energy and it is simply absorbed in the random matrix. If the location of the non-zero element is fixed, then one bit is embedded at each M-length frame since the sparse watermark contains one non-zero value. If the position of the non-zero element is allowed to change, then more bits are encoded in each frame. In future work, more than one non-zero elements will be used. D. Host and Watermark Recovery One major advantage of the proposed approach is that one can recover the host signal almost perfectly, an advantage which does not exist in most watermarking methods in the literature and may have some important practical applications. One of these applications is to use a perceptible watermark where the noisy signal is free for previewing and downloading, but the clean signal can only be obtained by the L1- minimization recovery, which requires the secret random matrix Ω. Three different, almost equivalent, methods are used here to recover the sparse watermark and the clean host signal:
1) Direct Pursuit of Justice: Having the measurement vector y, we apply the basis pursuit algorithm as in (6): Minimize Subject to: ε Where in this case = [x ] and =[H Ω] Where the watermark sparse sequence which contains only one bit is and once the sparse vector x is recovered, the clean host signal in time domain is obtained by X(t)=H x. 2) Multiplying by the Inverse of Ω: (Ω ) In this case, we have a new measurement vector y1=ω with a new dimension L and is given by: 1=Ω + (9) We use =[Ω H ] where is the identity matrix of size L L and we get the sparse vector = [x ]. 3) Multiplying by the annihilator of Ω: (Ω ) The annihilator of Ω is a matrix Ω of dimension (M-L) M. In this case, the new measurement vector y2=ω with a new dimension (M-L) and is given by: 2=Ω (10) We use =[Ω H] and in this case, we only get the sparse vector x, and by getting x we do the subtraction which produces 3=Ω (11) Then we can get either multiplying by Ω or by applying the basis pursuit to get the sparse vector. The watermark value detection looks at the sign of the recovered watermark value. A simple voting is used between the three methods described to make the final decision. E. Averaging to Enhance Robustness To overcome the effects of additive noise and compression/decompression attacks, the averaging option was adopted. The watermark information estimated from D frames are averaged and the watermark bit value is re-estimated based on the averaged information. For method-1, the averaging is done over y vector for D frames and the basis pursuit is applied to recover the watermark. For method-2, the averaging is done over y1 for D frames and for method-3 is done over y3. The value D corresponds to a redundancy coding of the watermark bits, in the sense that the same bit is repeated D times over D successive frames, as a mean of channel coding to enhance the robustness. This of course results in a decreased payload as will be discussed in the experimental results section. F. Spread Spectrum(SS) as a Baseline for Comparison Spread spectrum based watermarking is still a popular and reliable embedding technique for its robustness to additive noise. It is taken as a baseline for comparison in this paper. The embedding equation is given by: = +μ. (12) Where is the sparsified signal in time domain, μ is an embedding factor which is taken proportional to the frame energy, is a diagonal matrix with pseudo random ±1 binary values for spreading, and is an all +1 or -1 vector of length M representing the watermark bit value for this frame. The embedding factor is selected so that SS embedding would have the same SNR as the proposed compressed sensing (CS) embedding. At recovery, the received watermarked measurement is multiplied by the diagonal matrix and the resulting sequence is summed and the sign is detected. G. Related Work In 2007 Shiekh and Baraniuk (SB technique for brevity) [14] proposed a transform domain watermarking model based on compressed sensing as follows = + is the transform domain watermarked signal where f is the spread spectrum watermark sequence, A is an m n random matrix where (m>n) and e is the sparse transform domain vector for the host signal. The annihilator of A, ( ) is multiplied by the transform domain vector to give a new vector =. for which the L1-minimization is performed. Once the sparse transform domain signal e is detected, it is subtracted from and the result is multiplied by the inverse of A to get the watermark f. One major difference between their technique and the one proposed here is that in this paper here the watermark itself is a sparse vector allowing for simultaneous recovery of the watermark and the host signal, and potentially better robustness against attacks. More importantly, the proposed sparse watermark allows to encode more than one bit in each frame by changing its position and inserting more than one non-zero element, since the L1-minimization can detect the positions and signs of the non-zero elements. The (SB) technique is implemented (with m=128 and n=32) and compared to the proposed technique as shown in section V. V. EXPERIMENTAL RESULTS A. Experimental Setup To prove the practicality of the proposed compressed sensing based technique, it was applied on a 128kbps MP3 music file of duration 40 seconds containing slow rock music with vocals. The MP3 music file is divided into frames of length M=128 samples each, the watermark embedding is done for each frame and the results show the average performance over the 13,700 frames used. If additive noise is to be added it is done before the MP3 attack. With no noise, the watermarked file is converted back to MP3 with same bit rate and with 64kbps bit rate to measure the effects of reduced bit rates. The watermark recovery performance is measured at 3 different points. First, after the watermark embedding and before writing back the MP3 file to measure the effects of additive noise. Second, after writing the MP3 file to measure the effects of MP3 coding/decoding and bit rate conversion, and finally, after applying the averaging process. The watermark embedding for CS and SS in all experiments produced an SNR of 27dB which gives minor effects to the host signal. The lower dimension of the CS embedding matrix Ω is taken L=2 with sparsity k=1 In CS embedding two approaches are used in watermark embedding and detection; CS1 where the non-zero element position is fixed and known to the receiver, and only the sign is estimated, and CS2 where
the receiver has to estimate the position and the sign of the non-zero element. The first experiment is the additive noise attack effects, where the SB technique is also included. The second experiment is the MP3 compression/decompression, with and without rate conversion. B. Additive Noise Attack Additive noise was added with increasing levels starting from the clean condition (27dB) until the SNR reached (8dB) and the noise was quite annoying. Note that starting from a SNR below 20dB the quality is not acceptable. Table I shows the %success rate of watermark bits recovery for the proposed CS1 technique with and without averaging, for the SS technique, with and without averaging, for the SB technique and for the CS2 technique (random positions are taken and the detector estimated the position and sign of the non-zero element). The averaging factor D used is 4 frames corresponding to a payload rate of 86bps. The first 3 rows show the performance of the proposed technique versus SS and SB techniques for acceptable noise levels, where the proposed technique gives perfect watermark detection. The SB technique gives comparable results with the non-averaged CS techniques, and the averaging makes the bit error rate degrades gracefully with noise. It is noted that the CS2 technique gives comparable results with CS1, in particular for lower noise levels. This is a promising result since CS2 encodes both the sign and the position of the sparse watermark. TABLE I ADDITIVE NOISE EFFECTS ON WATERMARK RECOVERY SNR (db) CS1 CS1 SS SS- SB CS2 CS2 27 100 100 76.5 93.8 99.1 100 100 23.5 100 100 76.5 93.3 98.8 99.9 100 20 100 100 76.2 92.6 97.7 96.9 100 17.7 98 100 76.1 92.5 96.5 93.2 100 15.3 96 100 75.5 91.8 94.3 88.7 100 13.2 92 99 75.2 91 92.4 86.2 98.8 12.4 89 98.8 74.7 90.8 88.8 82 97.3 10.3 86 98.4 76 90.7 87.8 80 95 9.8 85 97 75.8 90.5 85.8 78.6 94.2 8.2 80 96.9 74 90 83.7 75.7 93.4 C. MP3 Compression/Decompression Attack With no additive noise, the MP3 compression/decompression attack is used to test the proposed technique. Table II shows the %success rate of watermark bits recovery for the proposed CS technique (CS1 for the fixed position of non-zero element and CS2 for the random position), and for the SS technique, with and without averaging. The SB technique was tested and it gives 92.5% success rate. The first column shows the number of averaged frames D. The first 4 rows show the performance with the same MP3 bit rate (128kbps), while the remaining 5 rows show the performance with the reduced MP3 bit rate of 64kbps. To maintain the same number of samples in the detection process, interpolation with a factor of 2 was used on the whole 64kbps MP3 signal. The performance shows a significant advantage of the CS technique over the SS, with and without averaging. Again, CS2 gives comparable performance with CS1, and both reach very small bit error rates with the 64kbps MP3, however, at small payload of 11bps. This is compared to zero bit error rates for 128kbps MP3 with a payload of 43bps. TABLE II MP3 COMPRESSION/DECOMPRESSION EFFECTS ON WATERMARK RECOVERY D MP3 bitrate (kbps) Payload rate (bps) CS1 CS2 SS 1 128 345bps 93 95 71 4 128 86bps 98.5 100 90 8 128 43bps 100 100 94 12 128 29bps 100 100 100 16 128 22bps 100 100 100 1 64 345bps 70 69 57 4 64 86bps 77 80 65 8 64 43bps 91 87 72 12 64 29bps 93 93.5 78 16 64 22bps 94 95 86 32 64 11bps 97 98 91 D. L1 Recovery of the Host Signal One of the major differences between the compressed sensing framework and the other embedding approaches is that in CS we can recover the host signal and separate it blindly from the watermark. The recovered host signal SNR is tested in 3 situations, in the clean condition case, after the additive noise and after the MP3 codec. In the clean condition, an SNR of more than 80dB is obtained from the 27dB watermarked signal. This is quite remarkable and it shows that the original host signal can be almost perfectly recovered from the watermarked one. In the additive noise case, an average gain of +3dB is obtained over the noisy watermarked signal. In the MP3 attack, an average gain of +2dB is obtained over the received signal. Such gains can be enhanced with a careful adjustment of the parameters of the basis pursuit denoising procedure for the additive noise case, and by characterizing the distortion caused by the MP3 codec to mitigate during the L1-minimization step by adjusting the matrices used. E. Effect of the L Dimension One of the major advantages of the sparse signal recovery is that the L1 minimization can recover the sparse elements without knowing their positions. In that sense, the position of the non-zero element in the sparse watermark vector may be used as added information so that the watermark is conveying, potentially, much more than one bit. In other words, for a sparse vector of length L and with only one non-zero element which takes ±1, one can encode ( 2L) bits. Practically, each different codeword of the 2L codewords would correspond to a stored vector in a codebook at the receiver. Tables I and II demonstrate this property where the columns for CS2 with L=2 and k=1 show the results when the single non-zero element position was changed randomly. The detection algorithm was asked to find the largest magnitude in the estimated sparse vector and take its sign. The results show that this technique performed almost as well as knowing the
position of the element (in the CS1) where the non-zero element was always at the same position. This issue and the idea of having more than one non-zero element will be investigated further in a future paper. VI. CONCLUSIONS AND FUTURE WORK A new technique for watermarking sparse watermarking based on the compressed sensing (CS) framework is proposed. The sparse watermark vector is expanded by a random matrix and added to the host signal, which is also sparse in some transform domain. The sparse watermarking technique relies on the compressed sensing framework and the L1- minimization for watermark and host signals recovery. The proposed technique is applied on audio watermarking problem for MP3 music. MP3 compression/decompression, rate conversion and additive noise attacks are used to test the practicality of the proposed technique, in comparison to spread spectrum watermarking, with significant advantage for the proposed technique. Also, the paper demonstrates the advantage of using a sparse watermark where the sparse watermark vector can encode more bits by changing the position of the non-zero element in the sparse vector, an advantage over existing techniques. More work needs to be done in the following issues: 1- Enhancing the sparsifying process of the host audio signal to improve the SNR. One can think of using an optimal adaptive transform per MP3 song. 2- Characterizing the MP3 distortion effects so that the L1- minimization can recover the host signal with better SNR. 3- Investigating the issue of using more non-zero elements in the watermark sparse vector and how to use this increased payload in embedding more sophisticated watermarks. REFERENCES [1] T. K. Tewari, V. Saxena, J. P. Gupta, Audio Watermarking: Current State of Art and Future Objectives. International Journal of Digital Content Technology and Applications, Vol. 5, No. 7, pp. 306-313 July 2011. [2] R. M. Noriega, M. Nakano, B. Kurkoski and K. Yamaguchi, High Payload Audio Watermarking: toward Channel Characterization of MP3 Compression. Journal of Information Hiding and Multimedia Signal Processing, Vol. 2, No. 2, pp. 91-107 April 2011. [3] Vivekananda Bhat K Indranil Sengupta Abhijit Das, An Audio Watermarking Scheme using Singular Value Decomposition and Dither-Modulation Quantization. Multimedia Tools and Applications Journal, Vol. 52, No. 2-3, pp. 369-383, 2011. [4] S. Xiang Histogram based Audio Watermarking Against Time Scale Modification and Cropping Attacks, IEEE Trans. On Multimedia, Vol. 9, Issue 7, pp. 1357-1372, November 2007. [5] M. K. Dutta, P. Gupta, V. K. Pathak, A Perceptible Watermarking Algorithm for Audio Signals. Multimedia Tools and Applications, Online, pp. 1-23 Feb. 2012 [6] K. Datta, I. S. Gupta, Partial Encryption and Watermarking Scheme for Audio Files with Controlled Degradation of Quality. Multimedia Tools and Applications, Online, January 2012. [7] Emmanuel Candès and Terence Tao, Near optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. on Information Theory, 52(12), pp. 5406-5425, December 2006 [8] Emmanuel Candès and Terence Tao, Decoding by linear programming. IEEE Trans. on Information Theory, 51(12), pp. 4203-4215, December 2005 [9] Emmanuel Candès and Paige Randall, Highly robust error correction by convex programming. IEEE Transactions on Information Theory (2006) Vol. 54, Issue: 7. [10] Mark Davenport, Marco Duarte, Yonina Eldar, and Gitta Kutyniok, Introduction to compressed sensing, (Chapter in Compressed Sensing: Theory and Applications, Cambridge University Press, 2012) [11] Jason Laska, Mark Davenport, and Richard Baraniuk, Exact signal recovery from sparsely corrupted measurements through the pursuit of justice. (Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, California, November 2009) [12] N. H. Nguyen, N. M. Nasrabadi, T. D. Tran, Robust Lasso with Missing and Grossly Corrupted Observations. NIPS 2011. [13] L1-magic Library Website: http://users.ece.gatech.edu/~justin/l1magic/ [14] Mona Sheikh and Richard Baraniuk, Blind error-free detection of transform-domain watermarks. (IEEE Int. Conf. on Image Processing (ICIP), San Antonio, Texas, September 2007)