nips nips2002 nips2002-38 knowledge-graph by maker-knowledge-mining

38 nips-2002-Bayesian Estimation of Time-Frequency Coefficients for Audio Signal Enhancement

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Author: Patrick J. Wolfe, Simon J. Godsill

Abstract: The Bayesian paradigm provides a natural and effective means of exploiting prior knowledge concerning the time-frequency structure of sound signals such as speech and music—something which has often been overlooked in traditional audio signal processing approaches. Here, after constructing a Bayesian model and prior distributions capable of taking into account the time-frequency characteristics of typical audio waveforms, we apply Markov chain Monte Carlo methods in order to sample from the resultant posterior distribution of interest. We present speech enhancement results which compare favourably in objective terms with standard time-varying ﬁltering techniques (and in several cases yield superior performance, both objectively and subjectively); moreover, in contrast to such methods, our results are obtained without an assumption of prior knowledge of the noise power.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract The Bayesian paradigm provides a natural and effective means of exploiting prior knowledge concerning the time-frequency structure of sound signals such as speech and music—something which has often been overlooked in traditional audio signal processing approaches. [sent-9, score-0.926]

2 Here, after constructing a Bayesian model and prior distributions capable of taking into account the time-frequency characteristics of typical audio waveforms, we apply Markov chain Monte Carlo methods in order to sample from the resultant posterior distribution of interest. [sent-10, score-0.588]

3 As a result, so-called timefrequency representations are ubiquitous in audio signal processing. [sent-13, score-0.562]

4 The focus of this paper is on signal enhancement via a regression in which time-frequency atoms form the regressors. [sent-14, score-0.474]

5 This choice is motivated by the notion that an atomic time-frequency decomposition is the most natural way to split an audio waveform into its constituent parts—such as note attacks and steady pitches for music, voiced and unvoiced speech, and so on. [sent-15, score-0.504]

6 Moreover, these features, along with prior knowledge concerning their generative mechanisms, are most easily described jointly in time and frequency through the use of Gabor frames. [sent-16, score-0.164]

7 Then (roughly speaking) if is reasonably well-behaved and the lattice is sufﬁciently dense, the Gabor system provides a (possibly non-orthogonal, or even redundant) series expansion of any function in a Hilbert space, and is thus said to generate a frame. [sent-25, score-0.088]

8 ¡ ¡ ¥ ¨ ¡ © © ¨ ¥ § ¢£ is a dictionary of time-frequency shifted versions of More formally, a Gabor frame a single basic window function , having the additional property that there exist constants (frame bounds) such that ¡ ¡ % # ! [sent-26, score-0.309]

9 This property can be understood as an approximate Plancherel formula, guaranteeing comcan pleteness of the set of building blocks in the function space. [sent-29, score-0.059]

10 That is, any signal be represented as an absolutely convergent inﬁnite series of the , or in the ﬁnite case, a linear combination thereof. [sent-30, score-0.166]

11 Such a representation is given by the following formula: Q I UG0 ¢£ ¡ , (1) ¢£ C ¢£ ` Y X9 0 ¡ 5 ¡ 7 6 8 ¢£ V W0 ¢£ B f¦ecV d b ¢£ a ¢£ BY where is a dual frame for . [sent-31, score-0.233]

12 Dual frames exist for any frame; however, the canonical dual frame, guaranteeing minimal (two-)norm coefﬁcients in the expansion of (1), is given by , where is the frame operator, deﬁned by . [sent-32, score-0.398]

13 ¡ b ¡ ¡ 7 6 ¢£ B p ¢£ B 8 ¢£ C A9 0 ¢£ aY ¡ ¡ ¡ h V 0 iUgb The notion of a frame thus incorporates bases as well as certain redundant representations; ) with ; the union of for example, an orthonormal basis is a tight frame ( . [sent-33, score-0.581]

14 Importantly, a two orthonormal bases yields a tight frame with frame bounds key result in time-frequency theory (the Balian-Low Theorem) implies that redundancy is a necessary consequence of good time-frequency localisation. [sent-34, score-0.617]

15 1 However, even with redundancy, the frame operator may, in certain special cases, be diagonalised. [sent-35, score-0.243]

16 If, furthermore, the are normalised in such a case, then analysis and synthesis can take place using the same window and inversion of the frame operator is avoided completely. [sent-36, score-0.445]

17 2 Short-Time Spectral Attenuation The standard noise reduction method in engineering applications is actually such an expansion in disguise (see, e. [sent-43, score-0.299]

18 In this method, known as short-time spectral attenuation, a time-varying ﬁlter is applied to the frequency-domain transform of a noisy signal, using the overlap-add method of short-time Fourier analysis and synthesis. [sent-46, score-0.148]

19 The observed signal y is ﬁrst divided into overlapping segments through multiplication by a smooth, ‘sliding’ window function, which is non-zero only for a duration on the order of tens of milliseconds. [sent-47, score-0.245]

20 The coefﬁcients of this transform are attenuated to some degree in order to reduce the noise; as shown in Fig. [sent-49, score-0.061]

21 1, individual short-time intervals Y are then inverse-transformed, multiplied by a smoothing window, and added together in an appropriate manner to form a time-domain signal reconstruction x. [sent-50, score-0.203]

22 Moreover, previous approaches in this vein have relied (either explicitly or implicitly) on independence assumptions amongst the time-frequency coefﬁcients; see, e. [sent-54, score-0.077]

23 Thus, with the aim of improving upon this popular class of audio noise reduction techniques, we have used these approaches as a starting point from which to proceed with a fully Bayesian analysis. [sent-57, score-0.605]

24 As a step in this direction, we propose a Gabor regression model as follows. [sent-58, score-0.105]

25 1 Gabor Regression ¢ £¡I Let x denote a sampled audio waveform, the observation of which has been corrupted by additive white Gaussian noise of variance , yielding the simple additive model y x d. [sent-60, score-0.695]

26 We consider regression in this case using a design matrix obtained from a Gabor frame. [sent-61, score-0.105]

27 2 V 2 ¤ ¥ In our particular case, this choice of regressors is motivated by a desire for constant absolute bandwidth, as opposed to, e. [sent-62, score-0.088]

28 Moreover, audio signal enhancement results with wavelets have been for the most part disappointing (witness the dearth of literature in this area), whereas standard engineering practice has evolved to use time-varying ﬁltering—which is inherently Gabor analysis. [sent-66, score-0.729]

29 , [5]), as a ﬁnal consideration it is interesting to note that Gabor’s original formulations [6]–[7] were motivated by psychoacoustic as well as information theoretic considerations. [sent-69, score-0.041]

30 ¦ © § ¨¦ 2 mod , under the assumption (without loss of generTechnically, we consider the ring ality) that the vector of sampled observations y has been extended to length in a proper way at its boundary before being periodically extended on . [sent-70, score-0.035]

31 Thus, one has the model d £ y ¥WI ¥ V Gc y £ ¡ y ¤¢¢ UI where the columns of G form the Gabor synthesis atoms, and elements of c represent the respective synthesis coefﬁcients. [sent-73, score-0.09]

32 To complete this model we assume an independent, identically distributed Gaussian noise vector, conditionally Gaussian coefﬁcients, and inverted-Gamma conjugate priors: ( 1 )' 0 ( & ©¦ %2# $ 2 ¤ c ¤ ¨ ¦ ¤ 2 D% ©§2 ' ¤ diag ¦ 2 D% ¨ "¦ 2 ! [sent-74, score-0.182]

33 We note that it is possible to obtain vague priors through the choice of these hyperparameters; alternatively, one may wish to incorporate genuine prior knowledge about audio signal behaviour through them. [sent-76, score-0.712]

34 2, we consider the case in which frequency-dependent coefﬁcient priors are speciﬁed in order to exploit the time-frequency structure of natural sound signals. [sent-78, score-0.134]

35 4 2 3 5 2 The choice of an inverted-Gamma prior for is justiﬁed by its ﬂexibility; for instance, in many audio enhancement applications one may be able to obtain a good estimate of the noise variance, which may in turn be reﬂected in the choice of hyperparameters and . [sent-79, score-0.745]

36 However, in order to demonstrate the performance of our model in the ‘worst-case’ scenario of little prior information, we assume here a diffuse prior for . [sent-80, score-0.235]

37 3 Implementation As a means of obtaining samples from the posterior distribution and hence the corresponding point estimates, we propose to sample from the posterior using Markov chain Monte Carlo (MCMC) methods [8]. [sent-82, score-0.092]

38 By design, all model parameters may be easily sampled from their respective full conditional distributions, thus allowing the straightforward employment of a Gibbs sampler [9]. [sent-83, score-0.21]

39 In all of the experiments described herein, a tight, normalised Hanning window was employed as the Gabor window function, and a regular time-frequency lattice was constructed to yield a redundancy of two (corresponding to the common practice of a 50% window overlap in the overlap-add method. [sent-84, score-0.464]

40 ) The arithmetic mean of the signal reconstructions from 1000 iterations (following 1000 iterations of ‘burn-in’, by which time the sampler appeared to have reached a stationary regime in each case) was taken to be the ﬁnal result. [sent-85, score-0.341]

41 As a further note, colour plots and representative audio examples may be found at the URL speciﬁed on the title page of this paper. [sent-86, score-0.517]

42 It can also be expected that, where possible, convergence may be speeded by starting the sampler in regions of likely high posterior probability, via use of a preliminary noise reduction method to obtain a robust coefﬁcient initialisation. [sent-89, score-0.43]

43 1 Coefﬁcient Shrinkage in the Overcomplete Case To test the noise reduction capabilities of the Gabor regression model, a speech signal of the short utterance ‘sound check’, sampled at 11. [sent-91, score-0.665]

44 025 kHz, was artiﬁcially degraded with white Gaussian noise to yield signal-to-noise ratios (SNR) between 0 and 20 dB. [sent-92, score-0.298]

45 At each SNR, ten runs of the sampler, at different random initialisations and using different pseudo-random number sequences, were performed as speciﬁed above. [sent-93, score-0.147]

46 2, along with estimates of the noise variance averaged over each of the ten runs. [sent-95, score-0.288]

47 25 True noise variance Estimated noise variance −3 10 15 log(σ2) Output SNR (dB) 20 −2 10 − − Wiener filter rule − Gabor regression − . [sent-96, score-0.571]

48 Individual realisations corresponding to the ten sampler runs are so closely spaced as to be indistinguishable. [sent-100, score-0.263]

49 20 10 0 5 10 Input SNR (dB) 15 20 (b) True and estimated noise variances (each averaged over ten runs of the sampler) Figure 2: Noise reduction results for the Gabor regression experiment of 3. [sent-101, score-0.402]

50 1 5 As it is able to outperform many of the short-time methods over a wide range of SNR (despite its relative disadvantage of not being given the estimated noise variance), and is also able to accurately estimate the noise variance over this range, the results of Fig. [sent-102, score-0.382]

51 2 would seem to indicate the appropriateness of the Gabor regression scheme for audio signal enhancement. [sent-103, score-0.667]

52 However, listening tests reveal that the algorithm, while improving upon the shortcomings of standard approaches discussed in 1. [sent-104, score-0.039]

53 The EMSR, on the other hand, is known for its more colourless residual noise (although as can be seen from Fig. [sent-106, score-0.225]

54 2 Coefﬁcient Shrinkage Using Wilson Bases In the case of a real signal, it is still possible to obtain good time-frequency localisation without incurring the penalty of redundancy through the use of Wilson bases (also known in the engineering literature as lapped transforms; see, e. [sent-109, score-0.188]

55 4 2 To test the effects of such a frequency-dependent prior in the context of a Wilson regression model (in comparison with the diffuse priors employed in 3. [sent-113, score-0.372]

56 1), the speech signal of the previous example was degraded with white Gaussian noise of variance , to yield an SNR of 10 dB. [sent-114, score-0.698]

57 Once again, posterior mean estimates over the last 1000 iterations of a 2000-iteration Gibbs sampler run were taken as the ﬁnal result. [sent-115, score-0.221]

58 Figure 3 shows samples of the noise variance parameter in this case. [sent-116, score-0.233]

59 While both the diffuse and frequency-dependent ¤ ¥d % r £u ¡r ¢ 5 −4 2 x 10 Noise variance, identical prior case Noise variance, true value Frequency−dependent prior case 1. [sent-117, score-0.235]

60 2 1 1 500 1000 Iteration 1500 2000 Figure 3: Noise variance samples for the two Wilson regression schemes of 3. [sent-121, score-0.233]

61 2 5 prior schemes yield an estimate close to the true noise variance, and indeed give similar SNR gains of 3. [sent-122, score-0.352]

62 85 dB, respectively, the corresponding restorations differ greatly in their perceptual quality. [sent-124, score-0.059]

63 Figure 4 shows spectrograms of the clean and noisy test signal, as well as the resultant restorations; whereas Fig. [sent-125, score-0.162]

64 5 shows waveform and spectrogram plots of the corresponding residuals (for greater clarity, colour plots are provided on-line). [sent-126, score-0.282]

65 4 and 5 that the residual noise in the case of the frequencydependent priors appears less coloured, and in fact this restoration suffers much less from the so-called ‘musical noise’ artefact common to audio signal enhancement methods. [sent-128, score-1.067]

66 It is well-known that a ‘whiter-sounding’ residual is perceptually preferable; in fact, some noise reduction methods have attempted this explicitly [10]. [sent-129, score-0.326]

67 4 Discussion Here we have presented a model for regression of audio signals, using elements of a Gabor frame as a design matrix. [sent-130, score-0.694]

68 , [11]–[12]); in fact, the priors discussed in [13] constitute special cases of those employed in the Gabor regression model. [sent-133, score-0.211]

69 4 Figure 4: Spectrograms for the two Wilson regression schemes of 3. [sent-151, score-0.149]

70 frequency-dependent priors (grey scale is proportional to log-amplitude) 5 ‘dictionary’ of regressors, in order to obtain the most parsimonious representation possible. [sent-153, score-0.076]

71 3 Multi-resolution wavelet-like schemes are one of many possibilities; for an example application in this vein we refer the reader to [14]. [sent-154, score-0.121]

72 Speech enhancement using a minimum mean-square error short-time spectral amplitude estimator. [sent-182, score-0.213]

73 3 It remains an open question as to whether the resultant variable selection problem would be amenable to approaches other than MCMC—for instance, a perfect sampling scheme. [sent-186, score-0.072]

74 1 Sample Number Figure 5: Waveform and spectrogram plots of the Wilson regression residuals [5] Wolfe, P. [sent-206, score-0.232]

75 A new method for speech denoising and robust speech recognition using probabilistic models for clean speech and for noise. [sent-290, score-0.517]

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Abstract: Monaural speech separation has been studied in previous systems that incorporate auditory scene analysis principles. A major problem for these systems is their inability to deal with speech in the highfrequency range. Psychoacoustic evidence suggests that different perceptual mechanisms are involved in handling resolved and unresolved harmonics. Motivated by this, we propose a model for monaural separation that deals with low-frequency and highfrequency signals differently. For resolved harmonics, our model generates segments based on temporal continuity and cross-channel correlation, and groups them according to periodicity. For unresolved harmonics, the model generates segments based on amplitude modulation (AM) in addition to temporal continuity and groups them according to AM repetition rates derived from sinusoidal modeling. Underlying the separation process is a pitch contour obtained according to psychoacoustic constraints. 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Monaural separation has also been studied using phasebased decomposition [3] and statistical learning [17], but with only limited evaluation. While speech enhancement remains a challenge, the auditory system shows a remarkable capacity for monaural speech separation. According to Bregman [1], the auditory system separates the acoustic signal into streams, corresponding to different sources, based on auditory scene analysis (ASA) principles. Research in ASA has inspired considerable work to build computational auditory scene analysis (CASA) systems for sound separation [19] [4] [7] [18]. Such systems generally approach speech separation in two main stages: segmentation (analysis) and grouping (synthesis). In segmentation, the acoustic input is decomposed into sensory segments, each of which is likely to originate from a single source. In grouping, those segments that likely come from the same source are grouped together, based mostly on periodicity. In a recent CASA model by Wang and Brown [18], segments are formed on the basis of similarity between adjacent filter responses (cross-channel correlation) and temporal continuity, while grouping among segments is performed according to the global pitch extracted within each time frame. In most situations, the model is able to remove intrusions and recover low-frequency (below 1 kHz) energy of target speech. However, this model cannot handle high-frequency (above 1 kHz) signals well, and it loses much of target speech in the high-frequency range. In fact, the inability to deal with speech in the high-frequency range is a common problem for CASA systems. We study monaural speech separation with particular emphasis on the high-frequency problem in CASA. For voiced speech, we note that the auditory system can resolve the first few harmonics in the low-frequency range [16]. It has been suggested that different perceptual mechanisms are used to handle resolved and unresolved harmonics [2]. Consequently, our model employs different methods to segregate resolved and unresolved harmonics of target speech. More specifically, our model generates segments for resolved harmonics based on temporal continuity and cross-channel correlation, and these segments are grouped according to common periodicity. For unresolved harmonics, it is well known that the corresponding filter responses are strongly amplitude-modulated and the response envelopes fluctuate at the fundamental frequency (F0) of target speech [8]. Therefore, our model generates segments for unresolved harmonics based on common AM in addition to temporal continuity. The segments are grouped according to AM repetition rates. We calculate AM repetition rates via sinusoidal modeling, which is guided by target pitch estimated according to characteristics of natural speech. Section 2 describes the overall system. In section 3, systematic results and a comparison with the Wang-Brown system are given. Section 4 concludes the paper. 2 M od el d escri p t i on Our model is a multistage system, as shown in Fig. 1. Description for each stage is given below. 2.1 I n i t i a l p r oc e s s i n g First, an acoustic input is analyzed by a standard cochlear filtering model with a bank of 128 gammatone filters [15] and subsequent hair cell transduction [12]. This peripheral processing is done in time frames of 20 ms long with 10 ms overlap between consecutive frames. As a result, the input signal is decomposed into a group of timefrequency (T-F) units. Each T-F unit contains the response from a certain channel at a certain frame. The envelope of the response is obtained by a lowpass filter with Segregated Speech Mixture Peripheral and Initial Pitch mid-level segregation tracking processing Unit Final Resynthesis labeling segregation Figure 1. Schematic diagram of the proposed multistage system. passband [0, 1 kHz] and a Kaiser window of 18.25 ms. Mid-level processing is performed by computing a correlogram (autocorrelation function) of the individual responses and their envelopes. These autocorrelation functions reveal response periodicities as well as AM repetition rates. The global pitch is obtained from the summary correlogram. For clean speech, the autocorrelations generally have peaks consistent with the pitch and their summation shows a dominant peak corresponding to the pitch period. With acoustic interference, a global pitch may not be an accurate description of the target pitch, but it is reasonably close. Because a harmonic extends for a period of time and its frequency changes smoothly, target speech likely activates contiguous T-F units. This is an instance of the temporal continuity principle. In addition, since the passbands of adjacent channels overlap, a resolved harmonic usually activates adjacent channels, which leads to high crosschannel correlations. Hence, in initial segregation, the model first forms segments by merging T-F units based on temporal continuity and cross-channel correlation. Then the segments are grouped into a foreground stream and a background stream by comparing the periodicities of unit responses with global pitch. A similar process is described in [18]. Fig. 2(a) and Fig. 2(b) illustrate the segments and the foreground stream. The input is a mixture of a voiced utterance and a cocktail party noise (see Sect. 3). Since the intrusion is not strongly structured, most segments correspond to target speech. In addition, most segments are in the low-frequency range. The initial foreground stream successfully groups most of the major segments. 2.2 P i t c h tr a c k i n g In the presence of acoustic interference, the global pitch estimated in mid-level processing is generally not an accurate description of target pitch. To obtain accurate pitch information, target pitch is first estimated from the foreground stream. At each frame, the autocorrelation functions of T-F units in the foreground stream are summated. The pitch period is the lag corresponding to the maximum of the summation in the plausible pitch range: [2 ms, 12.5 ms]. Then we employ the following two constraints to check its reliability. First, an accurate pitch period at a frame should be consistent with the periodicity of the T-F units at this frame in the foreground stream. At frame j, let τ ( j) represent the estimated pitch period, and A(i, j,τ ) the autocorrelation function of uij, the unit in channel i. uij agrees with τ ( j) if A(i , j , τ ( j )) / A(i, j ,τ m ) > θ d (1) (a) (b) Frequency (Hz) 5000 5000 2335 2335 1028 1028 387 387 80 0 0.5 1 Time (Sec) 1.5 80 0 0.5 1 Time (Sec) 1.5 Figure 2. Results of initial segregation for a speech and cocktail-party mixture. (a) Segments formed. Each segment corresponds to a contiguous black region. (b) Foreground stream. Here, θd = 0.95, the same threshold used in [18], and τ m is the lag corresponding to the maximum of A(i, j,τ ) within [2 ms, 12.5 ms]. τ ( j) is considered reliable if more than half of the units in the foreground stream at frame j agree with it. Second, pitch periods in natural speech vary smoothly in time [11]. We stipulate the difference between reliable pitch periods at consecutive frames be smaller than 20% of the pitch period, justified from pitch statistics. Unreliable pitch periods are replaced by new values extrapolated from reliable pitch points using temporal continuity. As an example, suppose at two consecutive frames j and j+1 that τ ( j) is reliable while τ ( j+1) is not. All the channels corresponding to the T-F units agreeing with τ ( j) are selected. τ ( j+1) is then obtained from the summation of the autocorrelations for the units at frame j+1 in those selected channels. Then the re-estimated pitch is further verified with the second constraint. For more details, see [9]. Fig. 3 illustrates the estimated pitch periods from the speech and cocktail-party mixture, which match the pitch periods obtained from clean speech very well. 2.3 U n i t l a be l i n g With estimated pitch periods, (1) provides a criterion to label T-F units according to whether target speech dominates the unit responses or not. This criterion compares an estimated pitch period with the periodicity of the unit response. It is referred as the periodicity criterion. It works well for resolved harmonics, and is used to label the units of the segments generated in initial segregation. However, the periodicity criterion is not suitable for units responding to multiple harmonics because unit responses are amplitude-modulated. As shown in Fig. 4, for a filter response that is strongly amplitude-modulated (Fig. 4(a)), the target pitch corresponds to a local maximum, indicated by the vertical line, in the autocorrelation instead of the global maximum (Fig. 4(b)). Observe that for a filter responding to multiple harmonics of a harmonic source, the response envelope fluctuates at the rate of F0 [8]. Hence, we propose a new criterion for labeling the T-F units corresponding to unresolved harmonics by comparing AM repetition rates with estimated pitch. This criterion is referred as the AM criterion. To obtain an AM repetition rate, the entire response of a gammatone filter is half-wave rectified and then band-pass filtered to remove the DC component and other possible 14 Pitch Period (ms) 12 (a) 10 180 185 190 195 200 Time (ms) 2 4 6 8 Lag (ms) 205 210 8 6 4 0 (b) 0.5 1 Time (Sec) Figure 3. Estimated target pitch for the speech and cocktail-party mixture, marked by “x”. The solid line indicates the pitch contour obtained from clean speech. 0 10 12 Figure 4. AM effects. (a) Response of a filter with center frequency 2.6 kHz. (b) Corresponding autocorrelation. The vertical line marks the position corresponding to the pitch period of target speech. harmonics except for the F0 component. The rectified and filtered signal is then normalized by its envelope to remove the intensity fluctuations of the original signal, where the envelope is obtained via the Hilbert Transform. Because the pitch of natural speech does not change noticeably within a single frame, we model the corresponding normalized signal within a T-F unit by a single sinusoid to obtain the AM repetition rate. Specifically, f ,φ f ij , φ ij = arg min M ˆ [r (i, jT − k ) − sin(2π k f / f S + φ )]2 , for f ∈[80 Hz, 500 Hz], (2) k =1 ˆ where a square error measure is used. r (i , t ) is the normalized filter response, fS is the sampling frequency, M spans a frame, and T= 10 ms is the progressing period from one frame to the next. In the above equation, fij gives the AM repetition rate for unit uij. Note that in the discrete case, a single sinusoid with a sufficiently high frequency can always match these samples perfectly. However, we are interested in finding a frequency within the plausible pitch range. Hence, the solution does not reduce to a degenerate case. With appropriately chosen initial values, this optimization problem can be solved effectively using iterative gradient descent (see [9]). The AM criterion is used to label T-F units that do not belong to any segments generated in initial segregation; such segments, as discussed earlier, tend to miss unresolved harmonics. Specifically, unit uij is labeled as target speech if the final square error is less than half of the total energy of the corresponding signal and the AM repetition rate is close to the estimated target pitch: | f ijτ ( j ) − 1 | < θ f . (3) Psychoacoustic evidence suggests that to separate sounds with overlapping spectra requires 6-12% difference in F0 [6]. Accordingly, we choose θf to be 0.12. 2.4 F i n a l s e gr e g a t i on a n d r e s y n t he s i s For adjacent channels responding to unresolved harmonics, although their responses may be quite different, they exhibit similar AM patterns and their response envelopes are highly correlated. Therefore, for T-F units labeled as target speech, segments are generated based on cross-channel envelope correlation in addition to temporal continuity. The spectra of target speech and intrusion often overlap and, as a result, some segments generated in initial segregation contain both units where target speech dominates and those where intrusion dominates. Given unit labels generated in the last stage, we further divide the segments in the foreground stream, SF, so that all the units in a segment have the same label. Then the streams are adjusted as follows. First, since segments for speech usually are at least 50 ms long, segments with the target label are retained in SF only if they are no shorter than 50 ms. Second, segments with the intrusion label are added to the background stream, SB, if they are no shorter than 50 ms. The remaining segments are removed from SF, becoming undecided. Finally, other units are grouped into the two streams by temporal and spectral continuity. First, SB expands iteratively to include undecided segments in its neighborhood. Then, all the remaining undecided segments are added back to SF. For individual units that do not belong to either stream, they are grouped into SF iteratively if the units are labeled as target speech as well as in the neighborhood of SF. The resulting SF is the final segregated stream of target speech. Fig. 5(a) shows the new segments generated in this process for the speech and cocktailparty mixture. Fig. 5(b) illustrates the segregated stream from the same mixture. Fig. 5(c) shows all the units where target speech is stronger than intrusion. The foreground stream generated by our algorithm contains most of the units where target speech is stronger. In addition, only a small number of units where intrusion is stronger are incorrectly grouped into it. A speech waveform is resynthesized from the final foreground stream. Here, the foreground stream works as a binary mask. It is used to retain the acoustic energy from the mixture that corresponds to 1’s and reject the mixture energy corresponding to 0’s. For more details, see [19]. 3 Evalu at i on an d comp ari son Our model is evaluated with a corpus of 100 mixtures composed of 10 voiced utterances mixed with 10 intrusions collected by Cooke [4]. The intrusions have a considerable variety. Specifically, they are: N0 - 1 kHz pure tone, N1 - white noise, N2 - noise bursts, N3 - “cocktail party” noise, N4 - rock music, N5 - siren, N6 - trill telephone, N7 - female speech, N8 - male speech, and N9 - female speech. Given our decomposition of an input signal into T-F units, we suggest the use of an ideal binary mask as the ground truth for target speech. The ideal binary mask is constructed as follows: a T-F unit is assigned one if the target energy in the corresponding unit is greater than the intrusion energy and zero otherwise. Theoretically speaking, an ideal binary mask gives a performance ceiling for all binary masks. Figure 5(c) illustrates the ideal mask for the speech and cocktail-party mixture. Ideal masks also suit well the situations where more than one target need to be segregated or the target changes dynamically. The use of ideal masks is supported by the auditory masking phenomenon: within a critical band, a weaker signal is masked by a stronger one [13]. In addition, an ideal mask gives excellent resynthesis for a variety of sounds and is similar to a prior mask used in a recent ASR study that yields excellent recognition performance [5]. The speech waveform resynthesized from the final foreground stream is used for evaluation, and it is denoted by S(t). The speech waveform resynthesized from the ideal binary mask is denoted by I(t). Furthermore, let e1(t) denote the signal present in I(t) but missing from S(t), and e2(t) the signal present in S(t) but missing from I(t). Then, the relative energy loss, REL, and the relative noise residue, RNR, are calculated as follows: R EL = e12 (t ) t I 2 (t ) , S 2 (t ) . (4b) ¡ ¡ R NR = (4a) t 2 e 2 (t ) t t (a) (b) (c) Frequency (Hz) 5000 2355 1054 387 80 0 0.5 1 Time (Sec) 0 0.5 1 Time (Sec) 0 0.5 1 Time (Sec) Figure 5. Results of final segregation for the speech and cocktail-party mixture. (a) New segments formed in the final segregation. (b) Final foreground stream. (c) Units where target speech is stronger than the intrusion. Table 1: REL and RNR Proposed model Wang-Brown model REL (%) RNR (%) N0 2.12 0.02 N1 4.66 3.55 N2 1.38 1.30 N3 3.83 2.72 N4 4.00 2.27 N5 2.83 0.10 N6 1.61 0.30 N7 3.21 2.18 N8 1.82 1.48 N9 8.57 19.33 3.32 Average 3.40 REL (%) RNR (%) 6.99 0 28.96 1.61 5.77 0.71 21.92 1.92 10.22 1.41 7.47 0 5.99 0.48 8.61 4.23 7.27 0.48 15.81 33.03 11.91 4.39 15 SNR (dB) Intrusion 20 10 5 0 −5 N0 N1 N2 N3 N4 N5 N6 N7 N8 N9 Intrusion Type Figure 6. SNR results for segregated speech. White bars show the results from the proposed model, gray bars those from the Wang-Brown system, and black bars those of the mixtures. The results from our model are shown in Table 1. Each value represents the average of one intrusion with 10 voiced utterances. A further average across all intrusions is also shown in the table. On average, our system retains 96.60% of target speech energy, and the relative residual noise is kept at 3.32%. As a comparison, Table 1 also shows the results from the Wang-Brown model [18], whose performance is representative of current CASA systems. As shown in the table, our model reduces REL significantly. In addition, REL and RNR are balanced in our system. Finally, to compare waveforms directly we measure a form of signal-to-noise ratio (SNR) in decibels using the resynthesized signal from the ideal binary mask as ground truth: ( I (t ) − S (t )) 2 ] . I 2 (t ) SNR = 10 log10 [ t (5) t The SNR for each intrusion averaged across 10 target utterances is shown in Fig. 6, together with the results from the Wang-Brown system and the SNR of the original mixtures. Our model achieves an average SNR gain of around 12 dB and 5 dB improvement over the Wang-Brown model. 4 Di scu ssi on The main feature of our model lies in using different mechanisms to deal with resolved and unresolved harmonics. As a result, our model is able to recover target speech and reduce noise interference in the high-frequency range where harmonics of target speech are unresolved. The proposed system considers the pitch contour of the target source only. However, it is possible to track the pitch contour of the intrusion if it has a harmonic structure. With two pitch contours, one could label a T-F unit more accurately by comparing whether its periodicity is more consistent with one or the other. Such a method is expected to lead to better performance for the two-speaker situation, e.g. N7 through N9. As indicated in Fig. 6, the performance gain of our system for such intrusions is relatively limited. Our model is limited to separation of voiced speech. In our view, unvoiced speech poses the biggest challenge for monaural speech separation. Other grouping cues, such as onset, offset, and timbre, have been demonstrated to be effective for human ASA [1], and may play a role in grouping unvoiced speech. In addition, one should consider the acoustic and phonetic characteristics of individual unvoiced consonants. We plan to investigate these issues in future study. A c k n ow l e d g me n t s We thank G. J. Brown and M. Wu for helpful comments. Preliminary versions of this work were presented in 2001 IEEE WASPAA and 2002 IEEE ICASSP. This research was supported in part by an NSF grant (IIS-0081058) and an AFOSR grant (F4962001-1-0027). References [1] A. S. Bregman, Auditory scene analysis, Cambridge MA: MIT Press, 1990. [2] R. P. Carlyon and T. M. Shackleton, “Comparing the fundamental frequencies of resolved and unresolved harmonics: evidence for two pitch mechanisms?” J. Acoust. Soc. Am., Vol. 95, pp. 3541-3554, 1994. [3] G. Cauwenberghs, “Monaural separation of independent acoustical components,” In Proc. of IEEE Symp. Circuit & Systems, 1999. [4] M. Cooke, Modeling auditory processing and organization, Cambridge U.K.: Cambridge University Press, 1993. [5] M. Cooke, P. Green, L. Josifovski, and A. Vizinho, “Robust automatic speech recognition with missing and unreliable acoustic data,” Speech Comm., Vol. 34, pp. 267-285, 2001. [6] C. J. Darwin and R. P. Carlyon, “Auditory grouping,” in Hearing, B. C. J. Moore, Ed., San Diego CA: Academic Press, 1995. [7] D. P. W. Ellis, Prediction-driven computational auditory scene analysis, Ph.D. Dissertation, MIT Department of Electrical Engineering and Computer Science, 1996. [8] H. Helmholtz, On the sensations of tone, Braunschweig: Vieweg & Son, 1863. (A. J. Ellis, English Trans., Dover, 1954.) [9] G. Hu and D. L. Wang, “Monaural speech segregation based on pitch tracking and amplitude modulation,” Technical Report TR6, Ohio State University Department of Computer and Information Science, 2002. (available at www.cis.ohio-state.edu/~hu) [10] A. Hyvärinen, J. Karhunen, and E. Oja, Independent component analysis, New York: Wiley, 2001. [11] W. J. M. Levelt, Speaking: From intention to articulation, Cambridge MA: MIT Press, 1989. [12] R. Meddis, “Simulation of auditory-neural transduction: further studies,” J. Acoust. Soc. Am., Vol. 83, pp. 1056-1063, 1988. [13] B. C. J. Moore, An Introduction to the psychology of hearing, 4th Ed., San Diego CA: Academic Press, 1997. [14] D. O’Shaughnessy, Speech communications: human and machine, 2nd Ed., New York: IEEE Press, 2000. [15] R. D. Patterson, I. Nimmo-Smith, J. Holdsworth, and P. Rice, “An efficient auditory filterbank based on the gammatone function,” APU Report 2341, MRC, Applied Psychology Unit, Cambridge U.K., 1988. [16] R. Plomp and A. M. Mimpen, “The ear as a frequency analyzer II,” J. Acoust. Soc. Am., Vol. 43, pp. 764-767, 1968. [17] S. Roweis, “One microphone source separation,” In Advances in Neural Information Processing Systems 13 (NIPS’00), 2001. [18] D. L. Wang and G. J. Brown, “Separation of speech from interfering sounds based on oscillatory correlation,” IEEE Trans. Neural Networks, Vol. 10, pp. 684-697, 1999. [19] M. Weintraub, A theory and computational model of auditory monaural sound separation, Ph.D. Dissertation, Stanford University Department of Electrical Engineering, 1985.

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Another advantage is that this representation allows using a detailed speech model on the same footing with the ﬁlter model. This is because a speech model is deﬁned on the time scale of a single frame, whereas the original ﬁlter hm , in contrast with Hm [k], is typically as long as 10 or more frames. As a ﬁnal point on notation, we deﬁne a Gaussian distribution over a complex number Z ν by p(Z) = N (Z | µ, ν) = π exp(−ν | Z − µ |2 ) . Notice that this is a joint distribution over the real and imaginary parts of Z. The mean is µ = X and the precision (inverse variance) ν satisﬁes ν −1 = | X |2 − | µ |2 . 3 A Model for Speech Signals We assume independent sources, and model the distribution of source j by a mixture model over its subband signals Xjn , N/2−1 p(Xjn | Sjn = s) N (Xjn [k] | 0, Ajs [k]) = p(Sjn = s) = πjs k=1 p(X, S) p(Xjn | Sjn )p(Sjn ) , = (3) jn where the components are labeled by Sjn . Component s of source j is a zero mean Gaussian with precision Ajs . The mixing proportions of source j are πjs . The DAG representing this model is shown in Fig. 2. A similar model was used in [10] for one microphone speech enhancement for recognition (see also [11]). Here are several things to note about this model. (1) Each component has a characteristic spectrum, which may describe a particular part of a speech phoneme. This is because the precision corresponds to the inverse spectrum: the mean energy (w.r.t. the above distribution) of source j at frequency k, conditioned on label s, is | Xjn |2 = A−1 . (2) js A zero mean model is appropriate given the physics of the problem, since the mean of a sound pressure waveform is zero. (3) k runs from 1 to N/2 − 1, since for k > N/2, Xjn [k] = Xjn [N − k] ; the subbands k = 0, N/2 are real and are omitted from the model, a common practice in speech recognition engines. 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In a second mode, source models are trained ofﬂine using available data on each source in the problem. A third mode correspond to separation of sources known to be speech but whose speakers are unknown. In this case, all sources have the same model, which is trained ofﬂine on a large dataset of speech signals, including 150 male and female speakers reading sentences from the Wall Street Journal (see [10] for details). This is the case presented in this paper. The training algorithm used was standard EM (omitted) using 256 clusters, initialized by vector quantization. 4 Separation of Non-Reverberant Mixtures We now present a source separation algorithm for the case of non-reverberant (or instantaneous) mixing. Whereas many algorithms exist for this case, our contribution here is an algorithm that is signiﬁcantly more robust to noise. Its robustness results, as indicated in the introduction, from three factors: (1) explicitly modeling the noise in the problem, (2) using a strong source model, in particular modeling the temporal statistics (over N time points) of the sources, rather than one time point statistics, and (3) extracting each source signal from data by a Bayes optimal estimator obtained from p(X | Y ). A more minor point is handling the case of less sources than sensors in a principled way. The mixing situation is described by yin = j hij xjn + uin , where xjn is source signal j at time point n, yin is sensor signal i, hij is the instantaneous mixing matrix, and uin is the noise corrupting sensor i’s signal. The corresponding subband signals satisfy Yin [k] = j hij Xjn [k] + Uin [k] . To turn the last equation into a probabilistic graphical model, we assume that noise i has precision (inverse spectrum) Bi [k], and that noises at different sensors are independent (the latter assumption is often inaccurate but can be easily relaxed). This yields p(Yin | X) N (Yin [k] | = p(Y | X) p(Yin | X) , = hij Xjn [k], Bi [k]) j k (4) in which together with the speech model (3) forms a complete model p(Y, X, S) for this problem. The DAG representing this model for the case K = L = 2 is shown in Fig. 3. Notice that this model generalizes [4] to the subband domain. s1n−2 s1n−1 s1 n s2n−2 s2n−1 s2 n x1n−2 x1n−1 x1 n x2n−2 x2n−1 x2 n y1n−2 y1n−1 y1n y2n−2 y2n−1 y2 n Figure 3: Graphical model for noisy, non-reverberant 2 × 2 mixing, showing a 3 frame-long sequence. All nodes Yin and Xjn represent complex N/2 − 1-dimensional vectors (see Fig. 2). While Y1n and Y2n have the same parents, X1n and X2n , the arcs from the parents to Y2n are omitted for clarity. The model parameters θ = {hij , Bi [k], Ajs [k], πjs } are estimated from data by an EM algorithm. However, as the number of speech components M or the number of sources K increases, the E-step becomes computationally intractable, as it requires summing over all O(M K ) conﬁgurations of (S1n , ..., SKn ) at each frame. We approximate the E-step using a variational technique: focusing on the posterior distribution p(X, S | Y ), we compute an optimal tractable approximation q(X, S | Y ) ≈ p(X, S | Y ), which we use to compute the sufﬁcient statistics (SS). We choose q(Xjn | Sjn , Y )q(Sjn | Y ) , q(X, S | Y ) = (5) jn where the hidden variables are factorized over the sources, and also over the frames (the latter factorization is exact in this model, but is an approximation for reverberant mixing). This posterior maintains the dependence of X on S, and thus the correlations between different subbands Xjn [k]. Notice also that this posterior implies a multimodal q(Xjn ) (i.e., a mixture distribution), which is more accurate than unimodal posteriors often employed in variational approximations (e.g., [12]), but is also harder to compute. A slightly more general form which allows inter-frame correlations by employing q(S | Y ) = jn q(Sjn | Sj,n−1 , Y ) may also be used, without increasing complexity. By optimizing in the usual way (see [12,13]) a lower bound on the likelihood w.r.t. q, we obtain q(Xjn [k] | Sjn = s, Y )q(Sjn = s | Y ) , q(Xjn , Sjn = s | Y ) = (6) k where q(Xjn [k] | Sjn = s, Y ) = N (Xjn [k] | ρjns [k], νjs [k]) and q(Sjn = s | Y ) = γjns . Both the factorization over k of q(Xjn | Sjn ) and its Gaussian functional form fall out from the optimization under the structural restriction (5) and need not be speciﬁed in advance. The variational parameters {ρjns [k], νjs [k], γjns }, which depend on the data Y , constitute the SS and are computed in the E-step. The DAG representing this posterior is shown in Fig. 4. s1n−2 s1n−1 s1 n s2n−2 s2n−1 s2 n x1n−2 x1n−1 x1 n x2n−2 x2n−1 x2 n {y im } Figure 4: Graphical model describing the variational posterior distribution applied to the model of Fig. 3. In the non-reverberant case, the components of this posterior at time frame n are conditioned only on the data Yin at that frame; in the reverberant case, the components at frame n are conditioned on the data Yim at all frames m. For clarity and space reasons, this distinction is not made in the ﬁgure. After learning, the sources are extracted from data by a variational approximation of the minimum mean squared error estimator, ˆ Xjn [k] = E(Xjn [k] | Y ) = dX q(X | Y )Xjn [k] , (7) i.e., the posterior mean, where q(X | Y ) = S q(X, S | Y ). The time domain waveform xjm is then obtained by appropriately patching together the subband signals. ˆ M-step. The update rule for the mixing matrix hij is obtained by solving the linear equation Bi [k]ηij,0 [k] = hij j k Bi [k]λj j,0 [k] . (8) k The update rule for the noise precisions Bi [k] is omitted. The quantities ηij,m [k] and λj j,m [k] are computed from the SS; see [13] for details. E-step. The posterior means of the sources (7) are obtained by solving   ˆ Xjn [k] = νjn [k]−1 ˆ i Bi [k]hij Yin [k] − j =j ˆ hij Xj n [k] (9) ˆ for Xjn [k], which is a K ×K linear system for each frequency k and frame n. The equations for the SS are given in [13], which also describes experimental results. 5 Separation of Reverberant Mixtures In this section we extend the algorithm to the case of reverberant mixing. In that case, due to signal propagation in the medium, each sensor signal at time frame n depends on the source signals not just at the same time but also at previous times. To describe this mathematically, the mixing matrix hij must become a matrix of ﬁlters hij,m , and yin = hij,m xj,n−m + uin . jm It may seem straightforward to extend the algorithm derived above to the present case. However, this appearance is misleading, because we have a time scale problem. Whereas are speech model p(X, S) is frame based, the ﬁlters hij,m are generally longer than the frame length N , typically 10 frames long and sometime longer. It is unclear how one can work with both Xjn and hij,m on the same footing (and, it is easy to see that straightforward windowed FFT cannot solve this problem). This is where the idea of subband ﬁltering becomes very useful. Using (2) we have Yin [k] = Hij,m [k]Xj,n−m [k] + Uin [k], which yields the probabilistic model jm p(Yin | X) N (Yin [k] | = Hij,m [k]Xj,n−m [k], Bi [k]) . (10) jm k Hence, both X and Y are now frame based. Combining this equation with the speech model (3), we now have a complete model p(Y, X, S) for the reverberant mixing problem. The DAG describing this model is shown in Fig. 5. s1n−2 s1n−1 s1 n s2n−2 s2n−1 s2 n x1n−2 x1n−1 x1 n x2n−2 x2n−1 x2 n y1n−2 y1n−1 y1n y2n−2 y2n−1 y2 n Figure 5: Graphical model for noisy, reverberant 2 × 2 mixing, showing a 3 frame-long sequence. Here we assume 2 frame-long ﬁlters, i.e., m = 0, 1 in Eq. (10), where the solid arcs from X to Y correspond to m = 0 (as in Fig. 3) and the dashed arcs to m = 1. While Y1n and Y2n have the same parents, X1n and X2n , the arcs from the parents to Y2n are omitted for clarity. The model parameters θ = {Hij,m [k], Bi [k], Ajs [k], πjs } are estimated from data by a variational EM algorithm, whose derivation generally follows the one outlined in the previous section. Notice that the exact E-step here is even more intractable, due to the history dependence introduced by the ﬁlters. M-step. The update rule for Hij,m is obtained by solving the Toeplitz system Hij ,m [k]λj j,m−m [k] = ηij,m [k] (11) j m where the quantities λj j,m [k], ηij,m [k] are computed from the SS (see [12]). The update rule for the Bi [k] is omitted. E-step. The posterior means of the sources (7) are obtained by solving  ˆ Xjn [k] = νjn [k]−1 ˆ im Bi [k]Hij,m−n [k] Yim [k] − Hij j m =jm ,m−m ˆ [k]Xj m  [k] (12) ˆ for Xjn [k]. Assuming P frames long ﬁlters Hij,m , m = 0 : P − 1, this is a KP × KP linear system for each frequency k. The equations for the SS are given in [13], which also describes experimental results. 6 Extensions An alternative technique we have been pursuing for approximating EM in our models is Sequential Rao-Blackwellized Monte Carlo. There, we sample state sequences S from the posterior p(S | Y ) and, for a given sequence, perform exact inference on the source signals X conditioned on that sequence (observe that given S, the posterior p(X | S, Y ) is Gaussian and can be computed exactly). In addition, we are extending our speech model to include features such as pitch [7] in order to improve separation performance, especially in cases with less sensors than sources [7–9]. Yet another extension is applying model selection techniques to infer the number of sources from data in a dynamic manner. Acknowledgments I thank Te-Won Lee for extremely valuable discussions. References [1] A.J. Bell, T.J. Sejnowski (1995). An information maximisation approach to blind separation and blind deconvolution. Neural Computation 7, 1129-1159. [2] B.A. Pearlmutter, L.C. Parra (1997). Maximum likelihood blind source separation: A contextsensitive generalization of ICA. Proc. NIPS-96. [3] A. Cichocki, S.-I. Amari (2002). Adaptive Blind Signal and Image Processing. Wiley. [4] H. Attias (1999). Independent Factor Analysis. Neural Computation 11, 803-851. [5] T.-W. Lee et al. (2001) (Ed.). Proc. ICA 2001. [6] S. Griebel, M. Brandstein (2001). Microphone array speech dereverberation using coarse channel modeling. Proc. ICASSP 2001. [7] J. Hershey, M. Casey (2002). Audiovisual source separation via hidden Markov models. Proc. NIPS 2001. [8] S. Roweis (2001). One Microphone Source Separation. Proc. NIPS-00, 793-799. [9] G.-J. Jang, T.-W. Lee, Y.-H. Oh (2003). A probabilistic approach to single channel blind signal separation. Proc. NIPS 2002. [10] H. Attias, L. Deng, A. Acero, J.C. Platt (2001). A new method for speech denoising using probabilistic models for clean speech and for noise. Proc. Eurospeech 2001. [11] Ephraim, Y. (1992). Statistical model based speech enhancement systems. Proc. IEEE 80(10), 1526-1555. [12] M.I. Jordan, Z. Ghahramani, T.S. Jaakkola, L.K. Saul (1999). An introduction to variational methods in graphical models. Machine Learning 37, 183-233. [13] H. Attias (2003). New EM algorithms for source separation and deconvolution with a microphone array. Proc. ICASSP 2003.

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