Speech Enhancement Using the Minimum-Probability-of-Error Criterion

Transcription

1 Interspeech 8 - September 8, Hyderbd Speech Enhncement Using the Minimum-Probbility-of-Error Criterion Jishnu Sdsivn, Subhdip Mukherjee, nd Chndr Sekhr Seelmntul Deprtment of Electricl Communiction Engineering, Deprtment of Electricl Engineering, Indin Institute of Science, Bnglore 5, Indi {sdsivn, subhdipm, chndrsekhr}@iisc.c.in Abstrct We propose novel speech denoising frmework by minimizing the probbility of error (PE, which mesures the devition probbility of the estimte from its true vlue. To develop the minimum PE (MPE criterion, one requires the knowledge of the noise probbility density function (p.d.f., which my not be vilble in prmetric form in speech denoising pplictions. Therefore, we dopt two pproches for modeling the noise p.d.f.: (i Gussin modeling bsed on dptive vrince estimtion; nd (ii Gussin mixture model (GMM in view of its pproximtion cpbilities. We consider discrete cosine trnsform (DCT domin shrinkge, where the optimum shrinkge prmeter is obtined by minimizing n estimte of the PE. A performnce ssessment for rel-world noise types shows tht for input signl-to-noise rtios (SNR greter thn 5, the proposed MPE-bsed point-wise shrinkge estimtors outperform three benchmrk techniques in terms of segmentl SNR nd short-time objective intelligibility (STOI scores. Index Terms: Minimum probbility of error, Speech denoising, Gussin mixture model, point-wise shrinkge estimtor.. Introduction Ambient coustic noise introduces unwnted disturbnces in speech signls leding to degrdtion in speech qulity, thereby ffecting the downstrem processing in speech communiction systems nd limiting the bility of listeners to understnd nd concentrte. Therefore, it is impertive to suppress noise nd enhnce the qulity nd intelligibility of speech. A typicl pproch to speech denoising is to minimize n pproprite distortion mesure, lso referred to s the risk function in sttistics literture, to obtin n estimte of the clen signl. However, direct minimiztion of risk requires the knowledge of the underlying clen signl or its sttistics, which is difficult to obtin in prctice. Hence, one needs to rely on the estimte of the clen signl sttistics. Genertive processes of speech signls exhibit wide vribilities bsed on speker, phonemes nd their durtions, lnguge, etc., which render the speech signl non-sttionry stochstic process. Therefore, estimting the clen speech prior is difficult, since it necessittes intricte stochstic modeling nd requires rigorous trining phse. Speech enhncement techniques cn be brodly ctegorized into (i spectrl subtrction techniques [ ], which involve the subtrction of the noise spectrum from the spectrum of noisy speech; (ii Wiener filtering [ ], which relies on the estimtes of the power-spectr of clen speech nd noise; (iii subspce techniques [7], wherein one utilizes the properties of the signl nd noise subspces; nd (iv sttisticl model-bsed pproches, which re setup in Byesin frmework nd rely on n estimte of the clen signl prior [8 ]. Recently, Xu et l. demonstrted the use of deep neurl networks for lerning the nonliner mp from noisy speech to clen speech [7, 8]. In this pper, emphsis is plced on developing non- Byesin technique for speech denoising. Our formultion relies only on the noise sttistics in its entirety, unlike the mensqured error (MSE formultions, in which the first- nd second-order sttistics suffice [9]. A sttisticl model is not ssumed on the clen speech signl. The key devition with respect to the stte-of-the-rt lies in the choice of the distortion mesure. We do not employ the stndrd MSE metric or perceptul distortion metric. Insted, we consider novel criterion for denoising, nmely the probbility of error (PE, which mesures the probbility of devition between the ground-truth signl nd its estimte. This criterion requires one to know or t lest estimte the noise p.d.f., nd plces no sttisticl ssumptions on the clen signl. The PE criterion is mesured in the short-time discrete-cosine trnsform (DCT domin. We rely on the prsimony of representtion nd energy compction of clen speech in the DCT bsis. Soon et l. showed tht the DCT is superior to the discrete Fourier trnsform (DFT for speech denoising []. The noise, however, is non-sprse in the DCT bsis. This representtion therefore justifies the use of point-wise shrinkge estimtor for denoising. Since the orcle PE requires one to know the ground-truth, we pproximte it by surrogte function tht depends solely on the noisy observtions, leding to prcticlly relizble estimte. Since denoising entils reduction in noise vrince in ech spectrl bnd, the PE is minimized with respect to the shrinkge prmeter over the intervl [, ], which is gret convenience s fr s optimiztion is concerned. We develop two different vrints of the PE risk, both point-wise, but one is n instntneous estimtor wheres the other incorportes temporl smoothing. Since the key objective is to combt relworld noise types (street, trin, nd F noise, which re nonsttionry nd whose distributions re not vilble priori, we dopt two p.d.f. models, one bsed on the Gussin nd the other employing GMM. Experimentl results re presented on the rel-world noise types for vrious input signl-to-noise rtios (SNRs nd compred with the stte of the rt.. MPE for Speech Denoising Consider the dditive observtion model x n = s n + w n, n =,,, N, ( where s n denotes the clen signl nd x n is the observtion corrupted by noise smples w n, which re independent nd identiclly distributed (i.i.d. with zero men nd vrince σ. Shorttime discrete cosine trnsform (DCT domin processing is considered for denoising within the proposed MPE formlism. The short-time DCT representtion of ( tkes the form X k,i = S k,i +W k,i, k =,, K, nd i =,, M, ( where k nd i denote the DCT coefficient nd the speech frme indices, respectively. For estimting S k,i, we develop point-.7/interspeech.8-9

2 wise estimtor Ŝk,i = k,i X k,i, where the shrinkge fctor k,i [, ] is selected optimlly bsed on the MPE criterion... MPE criteri for point-wise shrinkge Since the estimtor is point-wise, we drop the indices k nd i to mintin brevity of nottion, nd define the PE s ( R = P Ŝ S > ɛ, ( where ɛ > is predefined tolernce prmeter. Substituting Ŝ = X = (S + W, the risk in ( evlutes to R (, S = P ( (S + W S > ɛ ( ( ɛ ( S ɛ + ( S = F + F, ( where F ( is the cumultive distribution function (c.d.f. of the noise in the DCT domin. Since R depends on the groundtruth S, it is imprcticl to optimize it directly over, s the estimtor would be unrelizble. Therefore, we minimize n estimte of R, which is obtined by replcing S in ( with its noisy counterprt X. Such n estimte R tkes the form R(, X = F ( ɛ ( X + F ( ɛ + ( X nd correspondingly, the optiml shrinkge is obtined s opt = rg min R, by performing grid-serch over [, ] with grid-spcing of.. We consider two types of shrinkge estimtors. The first one, referred to s MPE-, pplies different shrinkge fctors to ech spectrl coefficient {X k,i } in the i th frme. The optiml shrinkge is selected coefficient-wise s opt k,i = rg min R(, X k,i. In the second vrint, which we refer to s MPE-, single shrinkge fctor is pplied to group of coefficients bunched long i, resulting in n estimtor of the form Ŝ k,i = opt k,i X k,i, where opt k,i = rg min +τ t= τ R(, X k,i t. (5 The prmeter τ determines the extent of temporl verging... Approximting unknown noise distributions In rel-world speech denoising scenrios, the noise distribution is often not known priori in prmetric form. In such scenrios, one hs to model the noise p.d.f. ppropritely. We consider two pproches for noise modeling: In the first one, we use Gussin, whose vrince is estimted dptively from the noisy speech signl, wheres in the second pproch, we employ GMM-bsed model. The effectiveness of the models will be vlidted experimentlly.... Gussin model nd dptive vrince estimtion This pproch relies on the ssumption tht the time-domin noise smples within frme re i.i.d. rndom vribles. Since the DCT coefficients re liner combintions of i.i.d. rndom vribles, considering the frme length to be sufficiently lrge, we invoke the centrl limit theorem, which ssures tht ech DCT coefficient W k,i is pproximtely Gussin distributed., A stochstic model bsed voice-ctivity detector (VAD [] is employed to estimte the vrince of W k,i. Going by the recommendtion in [], we use the following recursion to estimte the noise vrince dptively: ˆσ k,i = { ηˆσ k,i + ( η Xk,i, if i th frme is noise-only, ˆσ k,i, otherwise, where η =.98. Essentilly, the noise vrince is updted if the VAD identifies tht the frme under considertion corresponds to noise lone. In the sequel, the point-wise shrinkge estimtors MPE- nd MPE- for the Gussin noise model re referred to s MPE--G nd MPE--G, respectively.... Noise modeling using GMM The motivtion for using GMM stems from the fct tht it cn pproximte ny p.d.f. with finite number of discontinuities sufficiently ccurtely []. The L-component GMM p.d.f. with prmeters {α m, θ m, σ m} L m= is given by f(w = L m= α m (W θm exp (, ( σ m π σm nd the corresponding PE risk turns out to be L ( ɛ ( X θm R = α m [Q + σ m m= ( ] ɛ + ( X + θm Q, (7 σ m ( where Q(u = exp π t dt. The number of GMM u components M is selected following the Byesin informtion criterion (BIC []. For ech subbnd, the prmeters of the GMM re estimted using the expecttion-mximiztion (EM lgorithm [] bsed on trining dt corresponding exclusively to noise. The GMM-bsed p.d.f. modeling, when used in conjunction with the MPE- nd MPE- estimtors, re referred to s MPE--GMM nd MPE--GMM, respectively. The noise smples during trining nd testing re tken to be different.. Simultion Results Clen speech recordings from the Noizeus dtbse (8 khz smpling frequency [] re used in our experiments. The noise smples re tken from both Noizeus (trin nd street noises nd Noisex-9 (for F noise; downsmpled to 8 khz dtbses [5]. We consider frme-by-frme processing, with Hmming window, frme length of ms, nd n overlp of 75% between consecutive frmes. The vlue of τ in MPE- in (5 is set to, nd we choose ɛ = σ, where σ is the noise stndrd devition. We perform comprtive ssessment of the MPE-bsed techniques with three benchmrking lgorithms under different noise conditions. The lgorithms chosen for the comprison re: (i Wiener filter technique, which uses decisiondirected pproch for priori SNR estimtion (WFIL []; (ii log-spectrl mplitude estimtor (LSA, which minimizes the men-squred error (MSE of the logrithm of clen speech spectrl mplitude [9]; nd (iii Byesin non-negtive mtrix Exmple speech files re vilble t wixsite.com/mpe-se.

3 SSNR GAIN ( SSNR GAIN ( SSNR GAIN ( 8 BNMF LSA WFIL MPE G MPE G MPE GMM MPE GMM ( SSNR (F noise (b PESQ (F noise (c STOI (F noise (d SSNR (Trin noise (e PESQ (Trin noise (f STOI (Trin noise (g SSNR (Street noise (h PESQ (Street noise (i STOI (Street noise 5 5 Figure : Performnce comprison of vrious lgorithms for different noise types in terms of SSNR, PESQ, nd STOI scores fctoriztion method (BNMF, wherein one optimizes the MSE of the clen speech spectrl mplitude with the help of dictionry trined offline on clen speech [5]. Mtlb implementtions of WFIL nd LSA re vilble in [7]. The implementtions use the VAD proposed in []. For MPE--G/MPE-- G, we use the sme VAD. For the GMM pproch, VAD is not needed, since it is pre-trined model. For the BNMF implementtion, we use the Mtlb code provided online by the uthors [5]. The choice of WFIL nd LSA for performnce benchmrking is motivted by the extensive comprison reported in [7], which estblished conclusively tht these result in higher speech qulity nd intelligibility thn the competing techniques. The BNMF technique hs been shown to be the best mong NMF bsed speech denoising pproches. Three objective scores re computed for performnce evlution: (i Segmentl signl-to-noise-rtio (SSNR, clculted by verging the SNRs over short speech segments; (ii Perceptul evlution of speech qulity (PESQ [8], which is widely used to mesure the perceptul speech qulity in nrrowbnd telephone networks, speech codecs, nd denoised speech; nd (iii Short-time objective intelligibility score (STOI, which hs been shown to be highly correlted with the intelligibility of the denoised speech [9]. The scores re verged over different speech files corresponding to independent nd rndomly selected noise reliztions for ech input SNR. Figure shows the performnce comprison of the techniques for F, trin, nd street noise. We observe tht for ll the noise types under considertion, MPE--GMM nd MPE- -GMM exhibit higher SSNR gin compred with the competing lgorithms (cf. Figures (, (d, nd (g. Further, in the cse of F noise, nd for other noise types with input SNR greter thn 5, MPE--G exhibits better denoising performnce in terms of SSNR. Among the proposed MPE estimtors, the SSNR gin obtined using MPE--G turns out to be the lest. In terms of PESQ scores (cf. Figures (b, (e, nd (h, we observe tht, for ll the noise types considered, MPE--GMM leds to the best performnce. For F noise, MPE--G nd MPE--G lso result in firly high PESQ scores.

4 ( Clen speech (b Input SNR =, Trin noise (c BNMF (f MPE--G (d WFIL (f MPE--G (g MPE--GMM (h MPE--GMM Figure : Spectrogrms of the denoised speech obtined using different lgorithms. For input SNR exceeding 5, MPE--G, MPE--GMM, nd MPE--GMM exhibit denoising performnce superior to their competitors in terms of STOI (cf. Figures (c, (f, nd (i. To summrize, MPE--GMM exhibits better performnce compred with ll the other techniques. In the cse of street nd trin noise, GMM-bsed MPE estimtors show superior denoising performnce thn their Gussin counterprts. In the cse of F noise, the Gussin model led to better denoising. This is probbly becuse the F noise is reltively sttionry compred with street nd trin noise, nd the dptive vrince estimtion using VAD is resonbly ccurte. To demonstrte the time-frequency structure, distribution of residul noise, nd speech distortion, we show the spectrogrms of the denoised, noisy, nd clen speech signls in Figure corresponding to the trin noise. We observe tht WFIL hs higher residul noise thn ll the other lgorithms. BNMF suppresses noise, especilly in the silence regions, but it introduces speech distortions in some regions (cf. Figure (c, high frequency region (.5 to.5 khz just fter s, highlighted using red rectngle. MPE--GMM nd MPE--GMM yield superior noise suppression nd less speech distortion. In the cse of MPE--G/MPE--GMM, smll mount of musicl noise is present, which is suppressed to some extent in MPE--G/MPE- -GMM, since by construction, MPE- incorportes temporl smoothing while computing the point-wise shrinkge estimtor.. Conclusions We proposed novel criterion for speech denoising bsed on the probbility of error. Our formlism does not plce ny sttisticl ssumptions on the clen speech signl. Notwithstnding its simplicity, the performnce of the proposed denoiser turned out to be competitive with the stte-of-the-rt techniques under rel-world noise conditions. Further, n implicit ssumption of the proposed frmework is tht the clen signl dmits prsimonious representtion in chosen bsis, which is true of the speech signl in the DCT domin, nd tht the noise does not, which mkes the point-wise shrinkge nturl choice for denoising. The proposed frmework relies on modeling the noise p.d.f., for which we develop Gussin nd GMM-bsed pproximtions. The stndrd devition of the Gussin model for noise is updted recursively using VAD, wheres the prmeters for the GMM re pre-trined. Updting the GMM prmeters dptively might led to n improvement in the denoising performnce under rel-world noise conditions. Two versions of point-wise shrinkge were considered, one instntneous nd the other involving certin degree of temporl smoothing, with the ltter leding to superior performnce. All the sme, excessive smoothing might deteriorte the performnce nd the optiml degree of smoothing to be incorported in the MPE frmework must be scertined.

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