Identification of Sparse Audio Tampering Using Distributed Source Coding and Compressive Sensing Techniques
© G. Valenzise et al. 2009
Received: 16 May 2008
Accepted: 20 November 2008
Published: 11 February 2009
In the past few years, a large amount of techniques have been proposed to identify whether a multimedia content has been illegally tampered or not. Nevertheless, very few efforts have been devoted to identifying which kind of attack has been carried out, especially due to the large data required for this task. We propose a novel hashing scheme which exploits the paradigms of compressive sensing and distributed source coding to generate a compact hash signature, and we apply it to the case of audio content protection. The audio content provider produces a small hash signature by computing a limited number of random projections of a perceptual, time-frequency representation of the original audio stream; the audio hash is given by the syndrome bits of an LDPC code applied to the projections. At the content user side, the hash is decoded using distributed source coding tools. If the tampering is sparsifiable or compressible in some orthonormal basis or redundant dictionary, it is possible to identify the time-frequency position of the attack, with a hash size as small as 200 bits/second; the bit saving obtained by introducing distributed source coding ranges between 20% to 70%.
With the increasing diffusion of digital multimedia contents in the last years, the possibility of tampering with multimedia contents—an ability traditionally reserved, in the case of analog signals, to few people due to the prohibitive cost of the professional equipment—has become quite a widespread practice. In addition to the ease of such manipulations, the problem of the diffusion of unauthorized copies of multimedia contents is exacerbated by security vulnerabilities and peer-to-peer sharing over the Internet, where digital contents are typically distributed and posted. This is particularly true for the case of audio files, which represent the most common example of digitally distributed multimedia contents. Some versions of the same audio piece may differ from the original because of processing, due for example to compression, resampling, or transcoding at intermediate nodes. In other cases, however, malicious attacks may occur by tampering with part of the audio stream and possibly affecting its semantic content. Examples of this second kind of attacks are the alteration of a piece of evidence in a criminal trial, or the manipulation of public opinion through the use of false wiretapping. Often, for the sake of information integrity, not only it is useful to detect whether the audio content has been modified or not, but also to identify which kind of attack has been carried out. The reasons why it is generally preferred to identify how the content has been tampered with are twofold: on one hand, given an estimate of where the signal was manipulated, one can establish whether or not the audio file is still meaningful for the final user; on the other hand, in some circumstances, it may be possible to recover the original semantics of the audio file.
In the past literature, the aim of distinguishing legitimately modified copies from manipulations of a multimedia file has been addressed with two kinds of approaches: watermarks and media hashes. Both approaches have been extensively applied to the case of image content types, while fewer systems have been proposed for the case of audio signals. Digital watermarking techniques embed information directly into the media data to ensure both data integrity and authentication. Even if digital watermarks can be categorized based on several properties, such as robustness, security, complexity, and invertibility , a common taxonomy is to distinguish between robust and fragile watermarks. It is the latter category that is particularly useful for checking the integrity of an audio file; a fragile watermark is a mark that is easily altered or destroyed when the host data is modified through some transformation, either legitimate or not. If the watermark is designed to be robust with respect to legitimate, perceptually irrelevant modifications (e.g., compression or resampling), and at the same time to be fragile with respect to perceptually and semantic significant alterations, then it is a content-fragile watermark . With this scheme, a possible tampering can be detected and localized by identifying the damage to the extracted watermark. Examples of this approach for the case of image content types are given in [2, 3]. The authors of  propose an image authentication scheme that is able to localize tampering, by embedding a watermark in the wavelet coefficients of an image. If a tampering occurs, the system provides information on specific frequencies and space regions of the image that have been modified. This allows the user to make application-dependent decisions concerning whether an image, which is JPEG compressed for instance, still has credibility. A similar idea, also working on the signal wavelet domain, has been applied to audio in , with the aim of copyright verification and tampering identification. The image watermarking system devised in  inserts a fragile watermark in the least significant bits of the image on a block-based fashion; when a portion of the image is tampered with, only the watermark in the corresponding blocks is destroyed, and the manipulation can be localized. Celik et al.  extend this method by inserting the watermark in a hierarchical way, to improve robustness against vector quantization attacks. In , image protection and tampering localization is achieved through a technique called "cocktail watermarking"; two complementary watermarks are embedded in the original image to improve the robustness of the detector response, while at the same time enabling tampering localization. The same ideas have been applied by the authors to the case of sounds , by inserting the watermark in the host audio FFT coefficients. For a more exhaustive review of audio watermarking for authentication and tampering identification see Steinebach and Dittmann .
Despite their widespread diffusion as a tool for multimedia protection, watermarking schemes suffer from a series of disadvantages: (1) watermarking authentication is not backward compatible with previously encoded contents (unmarked contents cannot be authenticated later by just retrieving the corresponding hash); (2) the original content is distorted by the watermark; (3) the bit rate required to compress a multimedia content might increase due to the embedded watermark. An alternative solution for authentication and tampering identification is the use of multimedia hashes. Unlike watermarks, content hashing embeds a signature of the original content as part of the header information, or can provide a hash separately from the content upon a user's request. Multimedia hashes are inspired by cryptographic digital signatures, but instead of being sensitive to single-bit changes, they are supposed to offer proof of perceptual integrity. Despite some audio hashing systems (also named audio fingerprinting) being proposed in the past few years [10–12], most of the previous research, as for the case of watermarking, has concentrated on the case of images [13, 14]. In , the authors build audio fingerprints by collecting and quantizing a number of robust and informative features from an audio file, with the purpose of audio identification as well as fast database lookup. Haitsma and Kalker  build audio fingerprints robust to legitimate content modifications (mp3 compression, resampling, moderate time, and pitch scaling), by dividing the audio signal in highly overlapping frames of about 0.3 seconds; for each frame, they compute a frequency representation of the signal through a filter bank with logarithmic spacing among the bands, in order to resemble the human auditory system (HAS). The redundance of musical sounds is exploited by taking the differences between subbands in the same frame, and between the same subbands in adjacent time instants; the resulting vector is quantized with one bit, and similarities between each short fingerprint are computed through the Hamming distance. By concatenating all the fingerprints of each frame, a global hash is obtained, which is used next to efficiently query a song database of previously encoded fingerprints. Though in principle such an approach could be used for identifying possible localized tampering in the audio stream, the authors do not explicitly address this problem. An excellent review of algorithms and applications of audio fingerprinting is presented in .
To the best of the authors' knowledge, no audio hashing technique has been used up to now with the purpose of detecting and localizing unauthorized audio tampering. One of the main reasons of that is probably the great amount of bits of the audio hashes required for enabling the identification of the tampering, when traditional fingerprinting approaches as the ones described above are employed. In fact, in order to limit the rate overhead, the size of the hash needs to be as small as possible. At the same time, the goal of tampering localization calls for increasing the hash size, in order to capture as much as possible about the original multimedia object. Recently, Lin et al. have proposed a new hashing technique for authentication  and tampering localization  for images, which produce very short hashes by leveraging distributed source coding theory. In this system, the hash is composed of the Slepian-Wolf encoding bitstream of a number of quantized random projections of the original image; the content user (CU) computes its own random projections on the received (and possibly tampered) image, and uses them as a side information to decode the received hash. By setting some maximum predefined tampering level on the received image (e.g., a minimum tolerated PSNR between the original and the forged image is allowed), it is possible to transmit the hash without the need of a feedback channel, performing rate allocation at the encoder side (a similar bit allocation technique has been adopted by the authors also in the context of reduced-reference image quality assessment ). When decoding succeeds, it is possible to identify tampered regions of the image, at the cost of additional hash bits. This scheme has been applied also to the case of audio files ; instead of random projections of pixels, the authors compute for each signal frame a weighted spectral flatness measure, with randomly chosen weights, and encode this information to obtain the hash. Though this scheme applies well to the authentication task (which can be attained with a hash overhead less than 100 bits/second), it is not clear how to extend the application to identification of general kinds of tampering.
We have recently proposed a new image hashing technique  which exploits both the distributed source coding paradigm and the recent developments in the theory of compressive sensing. The algorithm proposed in this paper extends these ideas to the scenario of audio tampering. It also shares some similarities with the works in [15, 17]; as in , the hash is generated by computing random projections starting from a perceptually significant time-frequency representation of the audio signal and storing the syndrome bits obtained by Low-Density Parity-Check Codes (LDPC) encoding the quantized coefficients. With respect to , the proposed algorithm is novel in the following aspect: by leveraging compressive sensing principles, we are able to identify tamperings that are not sparse in the time domain only, but that can be represented by a sparse set of coefficients in some orthonormal basis or redundant dictionary. Even if the spatial models introduced in  could be thought of as a representation of the tampering in some dictionary, it is apparent that the compressive sensing interpretation allows much more flexibility in the choice of the sparsifying basis, since it just uses off-the-shelf basis expansions (e.g., wavelet or DCT) which can be added to the system for free.
The rest of the paper is organized as follows: Section 2 provides the necessary background information about compressive sensing and distributed source coding; Section 3 describes the tampering model; Section 4 gives a detailed description of the system; Section 6 describes how it is possible to estimate the rate of the hash at the encoder without feedback channel or training; the tampering identification algorithm is tested against various kinds of attacks in Section 7, where also the different bit-rate requirements for the hash with or without distributed source coding are compared; finally, Section 8 draws some concluding remarks.
In this section, we review the important concepts behind compressive sensing and distributed source coding, that constitute the underlying theory of the proposed tampering identification system. In spite of the relatively large amount of literature published on these fields in the past few years, this is a very concise introduction; for a more detailed and exhaustive explanation the interested reader may refer to [19–21] for compressive sensing and to [22–24] for distributed source coding.
2.1. Compressive Sampling (CS)
where simply counts the number of nonzero elements of . This program can correctly recover a -sparse signal from random samples . Unfortunately, such a problem is NP hard, and it is also difficult to solve in practice for problems of moderate size.
Problem (4) is an instance of a second-order cone program (SOCP)  and can be solved in time. Several fast algorithms have been proposed in the literature that attempt to find a solution to (4). In this work, we adopt the SPGL1 algorithm , which is specifically designed for large-scale sparse reconstruction problems.
2.2. Distributed Source Coding (DSC)
Consider the problem of communicating a continuous random variable . Let denote another continuous random variable correlated to . In a distributed source coding setting, the problem is to decode to its quantized reconstruction given a constraint on the distortion measure when the side information is available only at the decoder. Let us denote by the rate-distortion function for the case when is also available at the encoder, and by the case when only the decoder has access to . The Wyner-Ziv theorem  states that, in general, but for Gaussian memoryless sources and mean square error (MSE) as distortion measure.
The Wyner-Ziv theorem has been applied especially in the area of video coding under the name of distributed video coding (DVC), where the source (pixel values or DCT coefficients) is quantized with levels, and the bitplanes are independently encoded, computing parity bits by means of a turbo encoder. At the decoder, parity bits are used together with the side information to "correct" into a quantized version of , performing turbo decoding, typically starting from the most significant bitplanes. To this end, the decoder needs to know the joint probability density function (pdf) . More recently, LDPC codes have been adopted instead of turbo codes [28, 29].
As we will see in Section 4, relates to the energy of the original signal, while to the energy of the tampering. Equation (8) shows that the advantage of using a DSC approach with respect to a traditional quantization and encoding becomes consistent when the signal and the side information are well correlated, that is, when the energy of the tampering is small relative to the energy of the original sound.
3. Tampering Model
where is the modified signal received by the user. We postulate without loss of generality that has only nonzero components (in fact, it suffices for to be sparse or compressible in some basis or frame).
and thus we are able to approximate the energy of the tampering from the projections computed at the decoder and the encoder-side projections reconstructed exploiting the hash. This fact comes out to be very useful to estimate the energy of the tampering at the CU side and will be exploited in Section 4. Furthermore in order to apply the Wyner-Ziv theorem, we need to be i.i.d. Gaussian with zero mean. This has been verified through experimental simulations on several tampering examples. Indeed, a theoretical justification can be provided by invoking the central limit theorem, since each element is the sum of random variables whose statistics are not explicitly modeled.
4. Description of the System
4.1. Generation of the Hash Signature
(1) Frame-Based Subband Log-Energy Extraction
The values provide a time-frequency perceptual map of the audio signal (see Figure 1). The log-energy values are "rasterized" as a vector , where is the total number of log-energy values extracted from the audio stream.
(2) Random Projections
A number of linear random projections is produced as . The entries of the matrix are sampled from a Gaussian distribution , using some random seed , which will be sent as part of the hash to the user.
(3) Wyner-Ziv Encoding
The random projections are quantized with a uniform scalar quantizer with step size . As mentioned in Section 1, to reduce the number of bits needed to represent the hash, we do not send directly the quantization indices. Instead, we observe that the random projections computed from the possibly tampered audio signal will be available at the decoder side. Therefore, we can perform lossy encoding with side information at the decoder, where the source to be encoded is and the "noisy" random projections play the role of the side information. The vector contains the log-energy values of the audio signal received at the decoder. With respect to the distributed source coding setting illustrated in Section 2.2, we have . Following the approach widely adopted in the literature on distributed video coding , we perform bitplane extraction on the quantization bin indices. Then each bitplane vector is LDPC coded to create the hash.
4.2. Hash Decoding and Tampering Identification
Frame-Based Subband Log-Energy Extraction
Wyner-Ziv DecodingA quantized version is obtained using the hash syndrome bits and as side information. LDPC decoding is performed starting from the most significant bitplane.
If a feedback channel is available, decoding always succeeds, unless an upper bound is imposed on the maximum number of hash bits.
Conversely, if the actual distortion between the original and the tampered signal is higher than the maximum tolerated distortion determined by the original CP, decoding might fail.
Distortion EstimationIf Wyner-Ziv decoding succeeds, an estimate of the distortion in terms of a perceptual signal-to-noise ratio is computed using the projections of the subsampled energy spectrum of the tampering. Let be the projections of the subsampled energy spectrum of the tampering; we define the perceptual signal-to-noise ratio ( ) of the received audio stream as
This definition needs some further interpretation. In fact, we compute the from the projections in place of the whole time-frequency perceptual map of both the signal and the tampering. This is justified by the energy conservation principle stated in (11) and by the fact that, at the CU side, no information about the authentic audio content is available; hence, this is an approximation of the actual , which uses the quantized projections obtained by decoding the hash signature, in the reasonable hypothesis that and .
5. Choice of the Hash Parameters
6. Rate Allocation
Let denote the bitplane index and the bitrate (in bits/symbol) needed to decode the th bitplane. As mentioned in Section 3, the probability density function of and can be well approximated to be zero mean Gaussian, respectively, with variance and . The rate estimation algorithm receives in input the source variance , the correlation noise variance , the quantization step size , and the number of bitplanes to be encoded and returns the average number of bits needed to decode each bitplane . The value of can be immediately estimated from the random projections at the time of hash generation. The value of is set to be equal to the maximum MSE distortion between the original and the tampered signal, for which tampering identification can be attempted.
where denotes the th bitplane of . In fact LDPC decoding of bitplane exploits the knowledge of the real-valued side information as well as previously decoded bitplanes . Since we use nonideal channel codes with a finite sequence length to perform source coding a rate overhead of approximately [bit/sample is added. The integral needed to compute the value of the conditional entropy in (19) is factored out in detail in our previous work .
7. Experimental Results
We have carried out some experiments on 32 seconds of speech audio data, sampled at 44100 Hz and 16 bits per sample. The test audio consists of a piece of a newspaper article read by a speaker; the recording is clean but for some noise added at a few time instants, including the high frequency noise of a shaken key ring, the wide-band noise of some crumpling paper, and some impulsive noise in the form of coughs of the speaker. We have set the size of the audio frame to samples (0.25 seconds), and the number of Mel frequency bands to , obtaining a total of 128 audio frames corresponding to log-energy coefficients. We have then assembled a testbed considering 3 kinds of tampering.
Time Localized Tampering (T)
We have replaced some words in the speech at different positions, for a total tampering length of 3.75 seconds (about 11.7% of the total length of the audio sequence).
Frequency Localized Tampering (F)
A low-pass phone-band filter (cut-off frequency at 3400 Hz and stop frequency at 4000 Hz) is applied to the entire original audio stream.
Time-Frequency Localized Tampering (TF)
A cough at the beginning of the stream and the noise of the key ring in the middle are canceled out using the standard noise removal tool of the "Audacity" free audio editing software . The noise removal tool implemented in this application is an adaptive filter, whose frequency response depends on the local frequency characteristics of the noise. In this case, the total time length of the attack is 4.36 seconds.
7.1. Rate-Distortion Performance of the Hash Signature
We can make two main comments on the curves in the two graphs of Figure 4. The first difference between the frequency and the time tampering is that all the rate-distortion functions in the frequency attack are shifted upwards to higher rates, and have a steeper descending slope as the distortion increases. This is due to the fact that the frequency manipulation has a higher sparsity coefficient , that is, more measurements are needed for signal reconstruction. Although in the real application no guess about the sparsity of the tampering can be made at the CP side, here we have fixed a different sparsity for the two kinds of attacks, in order to visually prove the effect of the number of measures on the hash length. Thus, even if the rate per measurement is the same in both the cases (it only depends on the signal energy, as expressed in (5) and (7)), the rate in bits per second has slopes and offsets proportional to the number of measurements . Clearly, if we did not use compressive sensing to reduce the dimensionality of the data (i.e., in our setting), the rate required for the hash would have been equivalent to using random projections with ; therefore, the rate saving due to compressive sensing is approximately equal to the ratio . The second interesting remark that emerges from Figure 4 is the different gap between the family of WZ rates (ideal, with feedback and without feedback) and the NO-WZ curves. As (8) suggests, the coding gain from NO-WZ to WZ strongly depends on the energy of the tampering, that is, to (see Table 1). In the case of time attack, we have dB, while dB, thus according to (8) the bit saving achieved with WZ is smaller in the case of the frequency attack. As can be inferred from the graphs, this gain ranges from 20% to 70%.
7.2. Choice of the Best Tampering Reconstruction
In practice, the tampering may be sparse or compressible in more than one basis: this may be the case, for instance, of piece-wise polynomials signals which are generally sparse in several wavelet expansions. When this situation occurs, multiple tampering reconstructions are possible, and at the CU side there is an ambiguity about what is the best tampering estimation. As described in Section 4.2, we are ultimately interested in finding the sparsest tampering representation. This requires in practice to evaluate the sparsity of the tampering in each basis expansion; we use for this purpose the inverse-tangent norm defined in (15). To validate the choice of this norm, we compare the optimal basis expansion predicted from the norm and the inverse tangent norm with the actual best basis in terms of reconstruction quality.
Inverse tangent norm of the tampering using a fixed bit-rate for the hash signature of 200 bps.
We presented a hash-based tampering identification system for detecting and identifying illegitimate manipulations in audio files. The algorithm works with sparse modifications, leveraging the recent compressive sensing results for reconstructing the tampering from a set of random nonadaptive measurements. Perhaps the most distinctive feature of the proposed system is its ability to reconstruct a tampering that is sparse in some orthonormal basis or frame, without knowing at the CP side the actual content alteration. In practice, such an approach is feasible only if the bit length of the hash is not too large; we have found that encoding the hash signature through a distributed source coding paradigm enables a consistent reduction of the transmitted bits, especially when the strength of the tampering is small compared to the original signal energy. The hash size may be further decreased in the future by considering weighted minimization  to reduce the number of measurements required by the algorithm.
This work has been partially sponsored by the EU under Visnet II Network of Excellence.
- Steinebach M, Dittmann J: Watermarking-based digital audio data authentication. EURASIP Journal on Applied Signal Processing 2003,2003(10):1001-1015. 10.1155/S1110865703304081View ArticleGoogle Scholar
- Fridrich J: Image watermarking for tamper detection. Proceedings of IEEE International Conference on Image Processing (ICIP '98), October 1998, Chicago, Ill, USA 2: 404-408.Google Scholar
- Eggers JJ, Girod B: Blind watermarking applied to image authentication. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '01), May 2001, Salt Lake, Utah, USA 3: 1977-1980.Google Scholar
- Kundur D, Hatzinakos D: Digital watermarking for telltale tamper proofing and authentication. Proceedings of the IEEE 1999,87(7):1167-1180. 10.1109/5.771070View ArticleGoogle Scholar
- Tu R, Zhao J: A novel semi-fragile audio watermarking scheme. Proceedings of the 2nd IEEE Internatioal Workshop on Haptic, Audio and Visual Environments and Their Applications (HAVE '03), September 2003, Ottawa, Canada 89-94.Google Scholar
- Wong PW: A public key watermark for image verification and authentication. Proceedings of IEEE International Conference on Image Processing (ICIP '98), October 1998, Chicago, Ill, USA 1: 455-459.Google Scholar
- Celik MU, Sharma G, Saber E, Tekalp AM: Hierarchical watermarking for secure image authentication with localization. IEEE Transactions on Image Processing 2002,11(6):585-595. 10.1109/TIP.2002.1014990View ArticleGoogle Scholar
- Lu C-S, Huang S-K, Sze C-J, Liao H-YM: Cocktail watermarking for digital image protection. IEEE Transactions on Multimedia 2000,2(4):209-224. 10.1109/6046.890056View ArticleGoogle Scholar
- Lu C-S, Liao H-YM, Chen L-H: Multipurpose audio watermarking. Proceedings of the 15th International Conference on Pattern Recognition (ICPR '00), September 2000, Barcelona, Spain 3: 282-285.View ArticleGoogle Scholar
- Mıhçak MK, Venkatesan R: A perceptual audio hashing algorithm: a tool for robust audio identification and information hiding. Proceedings of the 4th International Workshop on Information Hiding (IH '01), April 2001, Pittsburgh, Pa, USA 2137: 51-65.View ArticleGoogle Scholar
- Haitsma J, Kalker T: A highly robust audio fingerprinting system with an efficient search strategy. Journal of New Music Research 2003,32(2):211-221. 10.1076/jnmr.220.127.116.1146View ArticleGoogle Scholar
- Cano P, Batlle E, Kalker T, Haitsma J: A review of algorithms for audio fingerprinting. Proceedings of IEEE Workshop on Multimedia Signal Processing (MMSP '02), December 2002, St. Thomas, Virgin Islands, USA 169-173.Google Scholar
- Roy S, Sun Q: Robust hash for detecting and localizing image tampering. Proceedings of the 14th IEEE International Conference on Image Processing (ICIP '07), October 2007, San Antonio, Tex, USA 6: 117-120.Google Scholar
- Lin Y-C, Varodayan D, Girod B: Image authentication based on distributed source coding. Proceedings of IEEE International Conference on Image Processing (ICIP '07), September-October 2007, San Antonio, Tex, USA 3: 5-8.Google Scholar
- Lin Y-C, Varodayan D, Girod B: Spatial models for localization of image tampering using distributed source codes. Proceedings of the International Picture Coding Symposium (PCS '07), November 2007, Lisbon, PortugalGoogle Scholar
- Chono K, Lin Y-C, Varodayan D, Miyamoto Y, Girod B: Reduced-reference image quality estimation using distributed source coding. Proceedings of IEEE International Conference on Multimedia and Expo (ICME '08), June 2008, Hannover, GermanyGoogle Scholar
- Varodayan D, Lin Y-C, Girod B: Audio authentication based on distributed source coding. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '08), March-April 2008, Las Vegas, Nev, USA 225-228.Google Scholar
- Tagliasacchi M, Valenzise G, Tubaro S: Localization of sparse image tampering via random projections. Proceedings of the 15th IEEE International Conference on Image Processing (ICIP '08), October 2008, San Diego, Calif, USA 2092-2095.Google Scholar
- Candès EJ: Compressive sampling. Proceedings of the International Congress of Mathematicians (ICM '06), August 2006, Madrid, SpainGoogle Scholar
- Baraniuk RG: Compressive sensing. IEEE Signal Processing Magazine 2007,24(4):118-121.View ArticleGoogle Scholar
- Candès EJ, Wakin MB: An introduction to compressive sampling: a sensing/sampling paradigm that goes against the common knowledge in data acquisition. IEEE Signal Processing Magazine 2008,25(2):21-30.View ArticleGoogle Scholar
- Slepian D, Wolf JK: Noiseless coding of correlated information sources. IEEE Transactions on Information Theory 1973,19(4):471-480. 10.1109/TIT.1973.1055037View ArticleMathSciNetMATHGoogle Scholar
- Wyner A, Ziv J: The rate-distortion function for source coding with side information at the decoder. IEEE Transactions on Information Theory 1976,22(1):1-10. 10.1109/TIT.1976.1055508View ArticleMathSciNetMATHGoogle Scholar
- Girod B, Aaron AM, Rane S, Rebollo-Monedero D: Distributed video coding. Proceedings of the IEEE 2005,93(1):71-83.View ArticleGoogle Scholar
- Goyal VK, Fletcher AK, Rangan S: Compressive sampling and lossy compression: do random measurements provide an efficient method of representing sparse signals? IEEE Signal Processing Magazine 2008,25(2):48-56.View ArticleGoogle Scholar
- Boyd S, Vandenberghe L: Convex Optimization. Cambridge University Press, Cambridge, UK; 2004.View ArticleMATHGoogle Scholar
- van den Berg E, Friedlander MP: In pursuit of a root. Department of Computer Science, University of British Columbia, Vancouver, Canada; June 2007.http://www.optimization-online.org/DB_FILE/2007/06/1708.pdf preprint ,Google Scholar
- Varodayan D, Aaron A, Girod B: Rate-adaptive codes for distributed source coding. Signal Processing 2006,86(11):3123-3130. 10.1016/j.sigpro.2006.03.012View ArticleMATHGoogle Scholar
- Artigas X, Ascenso J, Dalai M, Klomp S, Kubasov D, Ouaret M: The DISCOVER codec: architecture, techniques and evaluation. Proceedings of the International Picture Coding Symposium (PCS '07), November 2007, Lisbon, Portugal 6: 14496-14410.Google Scholar
- Cover TM, Thomas JA: Elements of Information Theory. John Wiley & Sons, New York, NY, USA; 1991.View ArticleMATHGoogle Scholar
- Rabiner L, Juang BH: Fundamentals of Speech Recognition. Prentice-Hall, Upper Saddle River, NJ, USA; 1993.Google Scholar
- Candès EJ, Wakin MB, Boyd SP: Enhancing sparsity by reweighted minimization. Journal of Fourier Analysis and Applications 2008,14(5-6):877-905. 10.1007/s00041-008-9045-xView ArticleMathSciNetMATHGoogle Scholar
- Bernardini R, Naccari M, Rinaldo R, Tagliasacchi M, Tubaro S, Zontone P: Rate allocation for robust video streaming based on distributed video coding. Signal Processing: Image Communication 2008,23(5):391-403. 10.1016/j.image.2008.04.004Google Scholar
- Audacity http://audacity.sourceforge.net
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