The blind source separation (BSS) problem entails separating a collection of signals, each comprised of a superposition of some unknown sources, into their constituent components. A canonical example of the BSS task arises in the so-called cocktail party problem
, and a number of methods have been proposed to address this problem. Perhaps the most well-known among these is independent component analysis (ICA)2, 3, 4], or, when appropriate for the underlying model, non-negative matrix factorization (NNMF) .
Here we focus on a slightly different, and often more challenging setting – the so-called single channel source separation problem – where only a single mixture of the source signals is observed. Single channel source separation problems require the use of some additional a priori knowledge about the sources and their structure in order to perform separation[6, 7, 8, 9]. Here, we assume that the local structure of one of the source signals is approximately
known (in a manner described in more detail below), and our aim is to separate this partially known source from an unknown “background” source. Our task is motivated by an audio processing application in law enforcement scenarios where electroshock devices are used. A key forensic task in these scenarios is to determine, from audio data recorded by the device itself, the resistive load encountered by the device (corresponding to qualitatively “low” and “high” resistance loads). The approach proposed here can aims to separate the audio corresponding to a nominally periodic and approximately known (up to the resistive load ambiguities) discharge from otherwise unknown, but often highly structured, background audio. The separated audio signal can subsequently be used to classify the state of the resistive load (we consider only the separation task here).
Our separation approach is based on local sparse approximations of the mixture data. A novel feature of our proposed method is in our representation of the unknown background source – we describe a technique for learning (from the data itself) a model that sparsely represents the unknown background source, using tools from the dictionary learning literature (see, eg., [10, 11, 12]). The next section describes the problem we consider here more formally, and discusses the nature of our contributions in the context of existing works in sparse representation, dictionary learning, and low-rank modeling.
2 Background and Problem Formulation
Our effort here is motivated by a single-channel semi-blind audio source separation problem, in which the goal is to separate a nominally periodic and approximately known signal from unknown but structured background interference, given only a superposition of the two sources. Let represent our observed data, and suppose that may be decomposed as a sum of two sources – one of which () exhibits local structure that is partially or approximately known, and the other () is unknown. In our motivating audio application for example, is comprised of samples of an underlying continuous time waveform, and we consider to be samples of a source that is a nominally regular repetition of one of a small number of prototype signals. One example scenario where this model is applicable is the case where is, up to some unknown offset jitter, periodic. Our aim is to separate the sources and from observations of , which may be noisy or otherwise corrupted.
Our proposed approach is based on the principle of local sparse approximations. In order to state our overall problem in generality, we describe an equivalent model for our data that facilitates the local analysis inherent to our approach. Let us suppose that is an integer that divides evenly, such that , an integer. Then may be represented equivalently as a matrix :
where is a matrix whose columns are non-overlapping length- segments of , and similarly for . The goal of our effort is, in essence, to separate into its constituent matrices and .
As alluded above, our separation approach entails leveraging local structure in each of the components of . Our main contribution comes in the form of a procedure that, given our “partial” information about the columns of , enables us to learn in an online fashion and from the data itself a dictionary such that columns of are accurately expressed as linear combinations of (a small number of) columns of . In a broader sense, our work is related to some classical approximation approaches as well as several recent works on matrix decomposition. We briefly describe these background and related efforts here, in an effort to put our main contribution in context.
2.1 Prior Art
2.1.1 Low Rank and Robust Low Rank Approximation
Consider the model (1) and suppose that the columns of can each be represented as a linear combination of some linearly independent vectors, implying that is a matrix of rank . Now, different separation techniques may be employed depending on our assumptions of . Perhaps the simplest case is where is random noise (e.g., having entries that are iid zero-mean Gaussian); in this case, the problem amounts to a denoising problem, which can be solved using ideas from low-rank matrix approximation. In particular, it is well-known that the approximation obtained via the truncated (to rank
) singular value decomposition (SVD) ofis a solution of the optimization
where is the function that returns the rank of .
It is well-known that certain (non-Gaussian) forms of interference
may cause the accuracy of estimators of the low-rank component obtained via truncated SVD to degrade significantly. This is the case, for example, whenis comprised of sparse large (in amplitude) impulsive noise. In these cases, the low-rank approximation problem can be modified to its robust counterpart, which goes by the name of robust PCA in the literature [13, 14]. The robust PCA approach aims to simultaneously estimate both the low-rank and the sparse , by solving the convex optimization
where is a regularization parameter. Here denotes the nuclear norm of , which is the sum of the singular values of . The nuclear norm is a convex relaxation of the non-convex rank function . Further, is the sum of the absolute entries of – essentially the norm of a vectorized version of , which is a convex relaxation of the non-convex quasinorm that counts the number of nonzeros of .
Here, of course, we explicitly assume that is more highly structured, making the separation problem more well-suited to a new suite of techniques that explicitly exploit such structure.
2.1.2 Low Rank Plus Sparse in a Known Dictionary
A useful extension of the robust PCA approach arises in the case where is not itself sparse, but possesses a sparse representation in some known dictionary or basis. One example is the case where the background source is locally smooth, implying it can be sparsely represented using a few low-frequency discrete cosine transform or Fourier basis elements. Formally, suppose that for some known matrix , we have that , where the columns of are sparse. The components of can be estimated by solving the following optimization 
Note that an estimate of may be obtained directly as . This approach assumes (implicitly) a priori knowledge of a dictionary that sparsely represents the background signal, which may be a restrictive assumption in practice.
2.1.3 Morphological Component Analysis
A more general model arises when is not low-rank, but instead, its columns are also sparsely represented in a known dictionary. Suppose that and are sparsely represented in some known dictionaries and , such that and , and that the columns of and are sparse. Such models were employed in recent work on Morphological Component Analysis (MCA) [16, 17, 18], which aimed to separate a signal into its component sources based on structural differences codified in the columns of the known dictionaries. The MCA decomposition can be accomplished by solving the following optimization
for some , where the estimates of and are formed as and , respectively. When and are each comprised of a single column, this optimization is equivalent to the so-called Basis Pursuit (or more specifically, Basis Pursuit Denoising) technique , which formed a foundation of much of the recent work in sparse approximation. Note that, as with the previously mentioned approach, this approach also assumes a priori knowledge of a dictionary that sparsely represents the background.
2.2 Our Contribution: “Semi-blind” Morphological Component Analysis
Our focus here is similar to the MCA approach above, but we assume only one of the dictionaries, say , is known. In this case, the MCA approach transforms into a semi-blind separation problem where we try to also learn a dictionary to represent the unknown signal. Our main contribution comes in the form of a “Semi-Blind” MCA procedure, designed to solve the following modified form of the MCA decomposition
and this problem forms the basis of the remainder of this paper. Specifically, in Section 3 we propose a procedure, based on alternating minimization, for obtaining local solutions to optimizations of the form (6). In Section 4 we examine the performance of our proposed approach in an application motivated by an audio source separation problem in audio forensics. Finally, we discuss conclusions and possible extensions in Section 5.
3 Semi-blind MCA
As described above, our model assumes that the data matrix can be expressed as the superposition of two component matrices, and . Further, we assume that each of the component matrices possesses a sparse representation in some dictionary, such that and , where is known a priori. Our essential aim, then, is to identify an estimate of the coefficient matrix and estimates and of the matrices and . Our estimates of the separated components are then given by , and .
We propose an approach to solve (6) that is based on alternating minimization, and is summarized here as Algorithm 1. Let be user specified regularization parameters. Our initial estimate of coefficients , corresponding to the coefficients of in the known dictionary , is obtained via
which is a simple LASSO-type problem. We then proceed in an iterative fashion, as outlined in the following subsections, for a few iterations or until some appropriate convergence criteria is satisfied. It should be noted that the lack of joint convexity makes the SBMCA algorithm sensitive to initialization. Therefore, any suitable initialization using sparse approximation techniques, depending upon the problem setting, can be employed. This is well illustrated in Section 4, where we consider an audio forensics application.
3.1 Dictionary learning stage
Given the estimate , we can essentially “subtract” the current estimate of from , and apply a dictionary learning step to identify estimates of the unknown dictionary and the corresponding coefficients . In other words, we solve
Now, given the estimate , we update our current estimate of the overall dictionary . We then update the overall coefficient matrix by solving another sparse approximation problem, as described next.
3.2 Sparse approximation stage
Given our current estimate of the overall dictionary, we update the corresponding coefficient matrices by solving the following LASSO-like problem:
Now, we extract the submatrix from , and repeat the overall processing (beginning with the dictionary learning step). These steps are iterated until some appropriate convergence criteria is satisfied.
4 Evaluation: An Application in Audio Forensics
We demonstrate the performance of our approach on a stylized version of the audio separation task described in the introduction, which is motivated by forensic examination of audio obtained during law enforcement events where electroshock devices are utilized. For the sake of this example, we suppose that the electroshock devices discharge approximately times per second, and the waveforms generated by the device during discharge take one of two different forms depending on the level of resistive load encountered by the device. The collected audio corresponds to the nominally periodic discharge of the device, superimposed with background noise (eg., speech). Our aim is to separate this superposition into its components.
Figure 1 shows a segment of the signals used in the simulation. We simulate the form of the approximately periodic signals (), shown in Figure 1 (a), using two distinct exponentially decaying sinusoids, to emulate a series RLC circuits with different parameters, to model the loaded and open circuit states. Specifically, we generate two distinct waveforms, which correspond to the two states (high and low resistive load), and form the overall signal by concatenating randomly-selected versions of these prototype signals, each of which is subject to a few samples of timing offset in order to model the non-idealities of the actual electroshock device. A speech signal111Speech Samples obtained from VoxForge Speech Corpus: www.voxforge.org/home shown in Figure 1 (b), was used to model background noise that may be present during the altercation. We simulate the overall raw audio data as a linear combination of , and zero-mean random Gaussian noise (Figure 1 (c) depicts the ideal case ).
The data matrix is formed from the signal as discussed in Section 2 using non-overlapping segments with samples each, and we form the dictionary by incorporating certain circular shifts of the nominal prototype pulses from which was generated. We then employ the semi-blind MCA approach (discussed in Section 3) to separate the background audio from the approximately known periodic portion.
We compare the performance of our approach with two versions of MCA, one using the DCT basis and the other using the identity basis to form the dictionary . We use the estimated , obtained via MCA-DCT procedure to initialize our approach, as follow: we apply one step of orthogonal matching pursuit (OMP)  on the estimate of obtained via MCA-DCT to form the initial (one component per column) estimate for the SBMCA algorithm.
Table 1 lists the best achievable reconstruction SNRs (in dB) of each method. We note that our interest here is in comparing the best performances achieved by MCA and our proposed method, so we clairvoyantly tune the value(s) of the regularization parameter to give the lowest error for each task. (In general, a different regularization parameter may have been utilized to obtain the reconstruction SNRs of each signal component, even for the same method and same noise level – in other words, the SNRs listed may not be jointly achievable from a single implementation of any of the stated procedures).
A second, perhaps more interesting, performance comparison is shown Figure 2, which depicts the histogram of normalized errors-per-block, measured using the vector -norm, for each method111Panels (a), (e) and (i) represent the histogram of normalized error-per-block for and (b), (f) and (j) represent the histogram of normalized error-per-block for via SBMCA, MCA-DCT and MCA-Identity respectively, with standard deviation of gaussian noise
via SBMCA, MCA-DCT and MCA-Identity respectively, with standard deviation of gaussian noise. Panels (c), (g) and (k) represent the histogram of normalized error-per-block for and (d), (h) and (l) represent the histogram of normalized error-per-block for via SBMCA, MCA-DCT and MCA-Identity respectively, with standard deviation of gaussian noise .. We observe from the distribution of -errors across blocks, that the SBMCA procedure (Figure 2 (a-d)) results in larger number of blocks with lower errors as compared to the MCA-DCT (Figure 2 (e-h)) and MCA-Identity (Figure 2 (i-l)). This feature is of primary importance in the audio forensics application where classifying each period of the nominally periodic signal , as one of the two prototype signals, is of interest.
We proposed a semi-blind source separation technique based on local sparse approximations. Our approach exploits partial prior knowledge of one of the sources, in the form of a dictionary which sparsely represents local segments of one of the sources. A key feature of our approach is the online learning of a dictionary (from the mixed source data itself) for representing the unknown background source. We posed the problem as an optimization task, proposed a solution approach based on alternating minimization, and verified its effectiveness via simulation in a stylized audio forensics application. Possible extensions to other applications (eg., image and video processing) are left to future efforts.
-  C. Jutten and J. Herault, “Blind Separation of Sources, Part I: An Adaptive Algorithm Based on Neuromimetic Architecture,” Signal Processing, vol. 24, no. 1, pp. 1–10, July 1991.
-  H. Hotelling, “Analysis of a Complex of Statistical Variables into Principal Components,” Journal of Educational Psychology, vol. 24, pp. 417–441, 1933.
-  C. Eckart and G. Young, “The Approximation of One Matrix by Another of Lower Rank,” Psychometrika, vol. 1, pp. 211–218, 1936.
-  I.T. Jolliffe, Principal Component Analysis, Springer Verlag, 1986.
-  A. Cichocki, R. Zdunek, and S. Amari, “New Algorithms for Non-Negative Matrix Factorization in Applications to Blind Source Separation,” Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), vol. 5, pp. V, May 2006.
G. J. Jang and T. W. Lee,
“A Maximum Likelihood Approach to Single-Channel
Journal of Machine Learning Research, vol. 4, pp. 1365–1392, Dec. 2003.
-  T. P. Jung, S. Makeig, C. Humphries, T. W. Lee, M. J. McKeown, V. Iragui, and T. J. Sejnowski, “Removing Electroencephalographic Artifacts by Blind Source Separation,” Psychophysiology, vol. 37, pp. 163–178, 2000.
-  M. N. Schmidt and R. K. Olsson, “Single-channel Speech Separation using Sparse Non-negative Matrix Factorization,” International Conference on Spoken Language Processing (INTERSPEECH, 2006.
-  M.E. Davies and C.J. James, “Source Separation using Single Channel ICA,” Signal Processing, vol. 87, no. 8, pp. 1819 – 1832, 2007.
-  B. A. Olshausen and D. J. Field, “Sparse Coding with an Overcomplete Basis Set: A Strategy Employed by V1?,” Vision Research, vol. 37, no. 23, pp. 3311–3325, 1997.
-  M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: Design of Dictionaries for Sparse Representation,” In Proceedings of SPARS’05, pp. 9–12, 2005.
-  J. Mairal, F. Bach, J. Ponce, and G. Sapiro, “Online Learning for Matrix Factorization and Sparse Coding,” Journal of Machine Learning Research, vol. 11, pp. 19–60, 2010.
-  E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust Principal Component Analysis?,” Journal of the ACM, vol. 58, no. 3, pp. 11, 2011.
-  V. Chandrasekaran, S. Sanghavi, P. A. Parrilo, and A. S. Willsky, “Rank-Sparsity Incoherence for Matrix Decomposition.,” Society for Industrial and Applied Mathematics (SIAM) Journal on Optimization, vol. 21, no. 2, pp. 572–596, 2011.
-  M. Mardani, G. Mateos, and G. B. Giannakis, “Recovery of Low-Rank Plus Compressed Sparse Matrices with Application to Unveiling Traffic Anomalies,” IEEE Transactions on Information Theory, 2012, Online: arXiv:1204.6537v1 [cs.IT].
-  J. L. Starck, M. Elad, and D. L. Donoho, “Image Decomposition via the Combination of Sparse Representations and a Variational Approach,” IEEE Transactions on Image Processing, vol. 14, no. 10, pp. 1570–1582, 2005.
-  D. L. Donoho and G. Kutyniok, “Microlocal Analysis of the Geometric Separation Problem,” CoRR, vol. abs/1004.3006, 2010.
-  J. Bobin, J. L. Starck, J. Fadili, Y. Moudden, and D. L. Donoho, “Morphological Component Analysis: An Adaptive Thresholding Strategy,” IEEE Transactions on Image Processing, vol. 16, no. 11, pp. 2675–2681, 2007.
-  S. Chen and D. Donoho, “Basis Pursuit,” in Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers. IEEE, 1994, vol. 1, pp. 41–44.
-  Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal Matching Pursuit: Recursive Function Approximation with Applications to Wavelet Decomposition,” in Proceedings of the 27th Annual Asilomar Conference on Signals, Systems and Computers, 1993, pp. 40–44.