A Proper version of Synthesis-based Sparse Audio Declipper

A PR OPER VERSION OF SYNTHESIS- B ASED SP ARSE A UDIO DECLIPPER P avel Záviška ⋆ P avel Rajmic ⋆ Ond ˇ rej Mokrý ⋆ Zden ˇ ek Pr ˚ uša † ⋆ Signal Processing Laboratory , Brno University of T echnology , Brno, Czech Republ ic † Acoustics Research Institute, Austrian Academy of Science, V ienna, Austri a Email: ⋆ {xza vis01, rajmic, 170583}@vu t br .cz, † zdenek.prusa@oeaw .ac.at ABSTRA CT Methods based on spar se representation have found gr eat use in the recovery of audio signals d egraded b y clipping. The state of the ar t in declipping within the spa rsity-based ap- proach e s h as been achieved by the SP ADE algo r ithm by Ki- ti ´ c et. al. (L V A/ICA ’15). Our recent stu d y (L V A/ICA ’18) has shown that alth o ugh the o riginal S-SP ADE can b e im proved such that it converges faster than th e A-SP A DE , the restora- tion quality is significantly worse. In the present paper, we propo se a new versio n of S- SP ADE. Exp e r iments show that the novel version of S-SP ADE ou tp erforms its old version in terms of restoration quality , and that it is compar able with the A-SP ADE while being even slightly faster than A-SP ADE. Index T erms — Declipping, Sp arse, Cosparse, Synthesis, Analysis 1. INTR O DUCTION Clipping is o ne of the co mmon type s of signal degradatio n. It is usually cau sed b y an elemen t in the sign al path whose d y- namic rang e is insufficient comp ared to the dynam ics of the signal. This fact causes the peaks of the signal to be cut (sat- urated). More exactly , in th e so- c a lled hard clipping , samples of the input signa l x ∈ R N that exceed the dynam ic rang e giv en by the thr esholds [ − θ c , θ c ] ar e modified such th at the signal ou tput c an be descr ib ed b y th e form ula y [ n ] = ( x [ n ] for | x [ n ] | < θ c , θ c · sgn( x [ n ]) for | x [ n ] | ≥ θ c . (1) Due to the great number of h igher harmon ics that ap pear in the clippe d signal, the c lip ping has a negativ e e ffect o n the pe r cei ved au dio quality [1]. Therefo re it is inevitable to perfor m restoration, so-called declipp ing , i.e. a recovery of the clipp e d samp les, based on th e observed signal y . In the hard clipping case, which is the co ntext of the pa- per, the sign al samples can be di vided into three sets R , H , The authors thank S. Kiti ´ c for providing them with his implementati on of the SP ADE algorithms and for discussion. The work was supporte d by the joint project of the FWF and the Czec h Science Fo undation (GA ˇ CR): numbers I 3067-N30 and 17-33798L, respecti vely . Research described in this paper was financed by the National Sustainab ility Program under grant LO1401. Infrastructu re of the SIX Center was used. and L , which correspond to “reliable” samples and samples that ha ve been clipped to “hig h” and “lo w” clipping th resh- olds, respectively . T o select only samples from a specific set, the r estriction op er ators M R , M H , M L will be used. These operator s can be und erstood as linear projectors or id entity matrices with specific colu m ns removed th at co r respond to the thre e cases. W ith the additional infor mation that th e positiv e clipp ed samples should lie ab ov e the θ c > 0 and the negati ve clipped samples below − θ c , a set of feasible solutions Γ is defined as a (conve x) set of tim e-domain signals su c h that Γ = Γ( y ) = { ˜ x | M R ˜ x = M R y , M H ˜ x ≥ θ c , M L ˜ x ≤ − θ c } . (2) The restored signal, ˆ x , is naturally r equired to be a member of the set, i.e. ˆ x ∈ Γ . Finding ˆ x is an ill-posed problem sinc e there are infinitely m a ny possible solutions. A possible way to treat this problem is to exploit the fact that aud io signals are sparse with r espect to a ( tim e-)frequen cy tran sform. I n other words, the goal is to find the signal ˆ x from th e set Γ of th e highest sparsity . In th e past, se veral ap p roaches to declipp ing were intr o- duced. Focusing on th e sparsity-based meth ods, the very first method using the sparsity assumption was reported in [2]; it was based on th e gr e edy app roximation o f a signal within the reliable p a rts. I n [3], conve x optimization was used. Accord- ing to [4], adding th e structure to the coefficients may lead to the improvement in the restora tion qua lity . The auth ors o f [5] used an iter ati ve hard thresholdin g algorithm that was con- strained to solve the declipping task and in [6] reform ulated the task to the analysis app roach to the sparsity . On top of these app r oaches, non-negative matrix factor- ization has also b een recently ado p ted to audio declipping [7]. As far as the au thors know , [8 ] presented the cu rrent state- of-the- a r t, a heuristic declipp ing alg orithm for bo th th e anal- ysis and the synth esis mo dels ( the SP ADE algorithm ). Un- til rece ntly , the synthesis variant was considered sig n ificantly slower due to the difficult projection step , but [9] has shown that the o pposite is true—the acceleration makes th e synthe - sis model r equire even fewer iterations to conver ge, making it faster than the analysis variant. Unfortunate ly , the r estoration quality of the synthesis variant h as been shown to be sub - stantially worse. In this paper, the problem of the original synthesis variant is briefly explained and a n e w , more prope r, synthesis version of the algo rithm is pre sen ted. 2. SP ADE ALGORITHMS SP ADE (SParse Audio DEclipp er) [8] by Kiti ´ c et. al. is a sparsity-based heuristic declipping algorithm. It is de- riv ed u sing the Alternatin g Direction Method o f Multip lier s (ADMM), which is briefly revised first. For details and proof s, see [1 0]. 2.1. ADMM The ADMM [1 1] is a means for solving problems of the f o rm min f ( x ) + g ( A x ) , or equiv alently min x , z f ( x ) + g ( z ) s.t. A x − z = 0 , (3) where x ∈ C N , z ∈ C P and A : C N → C P is a linea r operator . ADMM is based o n minim izin g the Au gmented La- grangian , defined f or (3) as: L ρ ( x , y , z ) = f ( x ) + g ( z ) + y ⊤ ( A x − z ) + ρ 2 k A x − z k 2 2 , (4) where ρ > 0 is called the penalty parameter . T he ADMM consists of three steps: minimizatio n of (4) over x , over z , and the u pdate of the dual variable, for mally [ 11]: x ( i +1) = ar g min x L ρ  x , z ( i ) , y ( i )  , (5a) z ( i +1) = ar g min z L ρ  x ( i +1) , z , y ( i )  , (5b) y ( i +1) = y ( i ) + ρ  A x ( i +1) − z ( i +1)  . (5c) The ADMM can b e often seen in th e so-called sca led form , which we o btain by substituting a dual v ariable y with the scaled du a l variable u = y /ρ . 2.2. SP ADE The SP ADE algorithm [ 8] appro ximates the solution o f the following NP-h ard regularized in verse p roblems min x , z k z k 0 s.t. x ∈ Γ( y ) and k A x − z k 2 ≤ ǫ, (6a) min x , z k z k 0 s.t. x ∈ Γ( y ) and k x − D z k 2 ≤ ǫ, (6b) where ( 6a) and ( 6 b) repr e sent the pr o blem formulatio n for the an a lysis and the syn thesis variant, respecti vely . Here, Γ denotes the set o f feasible solution s (see Eq. (2) ), x ∈ R N stands for the unknown signal in the time d omain, and z ∈ C P contains the (also u n known) coefficients. As fo r th e linear operator s, A : R N → C P is the analysis (thus P ≥ N ) and D : C P → R N is the synthe sis, while it hold s D = A ∗ . For compu tatio nal reason s, we restrict ourselves only to the Parse val tigh t frames [12], i.e. DD ∗ = A ∗ A = Id , with unitary op erators as their spe c ia l cases. The prob le m s (6) ca n be recast as the sum of two indicator function s: arg min x , z ,k ι Γ ( x ) + ι ℓ 0 ≤ k ( z ) s. t. ( k A x − z k 2 ≤ ǫ, k x − D z k 2 ≤ ǫ, (7) where ι Γ ( x ) makes the restored signal lie in Γ and ι ℓ 0 ≤ k ( z ) is a shortha n d no ta tio n f or ι { ˜ z | k ˜ z k 0 ≤ k } ( z ) , which enf orces the sparsity of the coe fficients. The signal is cut into ov erlappin g blocks and wind o wed prior to p rocessing. Th erefore, in (7), y should be u nderstood as one of the signal chunks. The overall resulting signal is made up by the overlap-add procedur e. As the transfor ma- tions, [8] uses an overcomplete DFT and IDFT , r especti vely . 2.3. A- SP ADE T o solve the analysis variant of ( 7), the A u gmented La- grangian is formed acco rding to (4) and the three ADMM steps acco rding to (5) are constru cted: x ( i +1) = arg min x k A x − z ( i ) + u ( i ) k 2 2 s.t. x ∈ Γ , (8a) z ( i +1) = arg min z k A x ( i +1) − z + u ( i ) k 2 2 s.t. k z k 0 ≤ k , (8b) u ( i +1) = u ( i ) + A x ( i +1) − z ( i +1) . (8c) The r eport [10] shows in detail that the subprob lem (8a) is, in fact, a pr ojection of ( A ∗ ( z ( i ) + u ( i ) )) onto Γ , efficiently implemented as an elem entwise m apping in the time dom ain [8, 9]. Further more, the solutio n of (8 b ) is ob tained by app ly- ing the hard - thresholding oper ator H k to ( A x ( i +1) + u ( i ) ) , setting all but k its largest elemen ts to zero , taking into ac- count the complex con jugate co e fficients. The A- SP ADE al- gorithm is finally o btained by ad ding the spar sity relaxation step to the above steps ( 8), in which the spa r sity of the re p re- sentation is allowed to increase d uring iterations. See Alg. 1. 2.4. S-SP ADE orig inal and S- SP ADE new In the syn th esis variant, th e situation is different. Alg. 2 presents the S-SP ADE a lg orithm from [8]. Here, the two minimization step s are bo th carried over z . Although this approa c h is based on the ADMM, it is explained in [10] that this algorithm solves a problem that is different from (6b). Therefo re the original S-SP ADE is no t really a synthesis counterp art of the A-SP ADE. The repor t [10] shows that on ly with unitary op e r ators ( A = D − 1 ) do all the three problems coincide. Next, we show how th e sy nthesis v ariant o f the SP ADE algorithm is d eri ved suc h that it ind eed solves (6b). First of all, the prob lem (7) is altered as arg min x , z ,k ι Γ ( x ) + ι ℓ 0 ≤ k ( z ) s. t. D z − x = 0 . (9) Algorithm 1: A-SP ADE fr o m [ 8] Require: A, y , M R , M H , M L , s, r, ǫ 1 ˆ x (0) = y , u (0) = 0 , i = 0 , k = s 2 ¯ z ( i +1) = H k  A ˆ x ( i ) + u ( i )  3 ˆ x ( i +1) = arg min x k A x − ¯ z ( i +1) + u ( i ) k 2 2 s.t. x ∈ Γ 4 if k A ˆ x ( i +1) − ¯ z ( i +1) k 2 ≤ ǫ th en 5 terminate 6 else 7 u ( i +1) = u ( i ) + A ˆ x ( i +1) − ¯ z ( i +1) 8 i ← i + 1 9 if i mod r = 0 then 10 k ← k + s 11 end 12 go to 2 13 end 14 return ˆ x = ˆ x ( i +1) Algorithm 2: S-SP ADE from [8] Require: D, y , M R , M H , M L , s, r, ǫ 1 ˆ z (0) = D ∗ y , u (0) = 0 , i = 0 , k = s 2 ¯ z ( i +1) = H k  ˆ z ( i ) + u ( i )  3 ˆ z ( i +1) = arg min z k z − ¯ z ( i +1) + u ( i ) k 2 2 s.t. D z ∈ Γ 4 if k ˆ z ( i +1) − ¯ z ( i +1) k 2 ≤ ǫ th en 5 terminate 6 else 7 u ( i +1) = u ( i ) + ˆ z ( i +1) − ¯ z ( i +1) 8 i ← i + 1 9 if i mo d r = 0 then 10 k ← k + s 11 end 12 go to 2 13 end 14 return ˆ x = D ˆ z ( i +1) Algorithm 3: S-SP ADE pro posed Require: D, y , M R , M H , M L , s, r, ǫ 1 ˆ x (0) = y , u (0) = 0 , i = 0 , k = s 2 ¯ z ( i +1) = H k  D ∗ ( ˆ x ( i ) − u ( i ) )  3 ˆ x ( i +1) = arg min x k D ¯ z ( i +1) − x + u ( i ) k 2 2 s.t. x ∈ Γ 4 if k D ¯ z ( i +1) − ˆ x ( i +1) k 2 ≤ ǫ th en 5 terminate 6 else 7 u ( i +1) = u ( i ) + D ¯ z ( i +1) − ˆ x ( i +1) 8 i ← i + 1 9 if i mod r = 0 then 10 k ← k + s 11 end 12 go to 2 13 end 14 return ˆ x = ˆ x ( i +1) Next, the Augm e nted Lagrangian is formed, L ρ ( x , y , z ) = ι ℓ 0 ≤ k ( z ) + ι Γ ( x ) + y ⊤ ( D z − x ) + ρ 2 k D z − x k 2 2 . (10) Using the scaled for m, ( 10) appears as L ρ ( x , z , u ) = ι ℓ 0 ≤ k ( z ) + ι Γ ( x ) + ρ 2 k D z − x + u k 2 2 − ρ 2 k u k 2 2 , (11) leading to the following ADMM steps: z ( i +1) = ar g min z k D z − x ( i ) + u ( i ) k 2 2 s.t. k z k 0 ≤ k , ( 12a) x ( i +1) = ar g min x k D z ( i +1) − x + u ( i ) k 2 2 s.t. x ∈ Γ , (12b) u ( i +1) = u ( i ) + D z ( i +1) − x ( i +1) . (12c) As in Sec. 2.2, add ing the spar sity relaxa tio n step an d a termination cr iterion lead s to the fin a l shape of the p ro- posed S-SP ADE algorithm —see Alg. 3. Unlike the or iginal variant of S-SP ADE in [8], where th e projection in the frequen c y domain w as re quired and a special projection lem ma had to be used [ 9], the projection step (12b) in the pro posed S-SP ADE algor ithm is a simp le elem entwise mapping as is the case of the ana lysis variant. The solution o f the minimizatio n step (1 2a) is obtained in Alg. 3 by app lying th e har d -thresholding H k [10]. Note that due to the no n-orthogo nality of D , such a vector is only an approx imate solutio n to (12a) (in contradiction to A-SP ADE where H k solves (8b) exactly , cf . Alg. 1). The compu tational cost of the SP ADE alg orithms is d om- inated b y the signal transfo rmations (i.e. the synthesis and analysis). All the three algo rithms req uire precisely o ne syn- thesis and one analysis per iteration, and therefor e, in theory , the com putational complexity of the algorithms is the same. 3. EXPERIMENT S AND RESUL TS The following experime n ts were designed to compare the pro- posed variant o f S-SP ADE (deno ted S-SP ADE DP with th e in- dex for “done prop e r ly”) with the original A-SP ADE and with the origin al S-SP ADE from [8] (S-SP ADE O ) in ter m s of the quality of restoration and the spee d of convergence. Experime nts were p erformed on five div erse au d io files with a 16 k Hz sampling rate. In th e prep rocessing step, the signals wer e peak-n ormalized and th en artificially clipped us- ing mu ltiple clip ping thresholds θ c ∈ { 0 . 1 , . . . , 0 . 9 } . All au dio sam ples were proc e ssed frame-wise, usin g the 1024- sample-long Hann wind o w with 75 % overlap. T he al- gorithms we r e implemen ted in MA TLAB 2 0 17a using the L TF A T toolb ox [13] for signal sy n thesis and analy sis. As the signal transform ation, the oversamp led DFT is used . The relaxation param e ters o f all the algo rithms wer e set to r = 1 , s = 1 an d ǫ = 0 . 1 . The restoration quality was e valuated using ∆ SDR, which expresses the signal-to-distor tio n impr ovemen t in dB, defined as ∆SDR = SDR( x , ˆ x ) − SDR( x , y ) , where y represen ts the clipped signal, x is th e o riginal undistorted signal and ˆ x denotes the restored sign al. The SDR itself is com puted as: SDR( u , v ) = 10 lo g k u k 2 2 k u − v k 2 2 [dB] . (13) The a dv antage o f using ∆ SDR is that it does no t depe n d on wh ether the SDR is computed on the whole signal or on the clipp e d samples only (assum ing that the restored signal matches the clipped signal on re liab le samples). Fig. 1 presents the overall ∆ SDR results of all SP ADE algorithm s depen ding on th e clip ping threshold θ c . When n o redund ancy (orthono rmal case) is used, all three algo r ithms perfor m equally , wh ich r esults in the b lack line. W ith hig her 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 8 10 12 14 16 18 clipping threshold θ c ∆ SDR [dB] SP ADE, red = 1 A-SP ADE, red = 2 S-SP ADE_O, red = 2 S-SP ADE_DP , red = 2 A-SP ADE, red = 4 S-SP ADE_O, red = 4 S-SP ADE_DP , red = 4 Fig. 1 . A verage perf ormance in terms o f ∆ SDR fo r all thr e e algorithm s. Notation “ red” denotes redun dancy of th e DFT . redund ancies, b oth A- SP ADE an d S-SP ADE DP significantly outperf orm S-SP ADE O , esp ecially f or lower thresholds θ c . Apart fro m overall ∆ SDR ev aluation, a mor e detailed compariso n can be seen on scatter plots in Figs. 2 and 3, where S-SP ADE DP is compar ed with S-SP ADE O and A- SP ADE, r especti vely . Each m a rk in the scatter plot corr e- sponds to the SDR value obtained from a par ticular 2048- sample-long b lock. For clarity , only results fo r clipping thresholds from 0.1 to 0.5 are displayed. Fig. 2 displays the linear regression line; clearly , a major ity of the m arks are p laced below the blue identity line, m eaning that in most of th e time chun ks, th e S-SP ADE DP perfor med bet- ter than S-SP A D E O . Results fro m the second scatter plo t in Fig. 3 prove an on-par restoration qu a lity of A-SP ADE and S-SP ADE DP , where S-SP ADE DP perfor med somewhat b etter for low SDR and vice versa. The last e xperim ent compar es the SP ADE alg orithms in terms o f the speed of co n vergence. For this pur pose, th e 10 15 20 25 30 35 40 45 50 55 60 65 70 10 20 30 40 50 60 70 SDR[dB] S-SP ADE_DP SDR[dB] S-SP ADE_O θ c = 0.1 θ c = 0.2 θ c = 0.3 θ c = 0.4 θ c = 0.5 Fig. 2 . Scatter plot of SDR values f or S-SP ADE O and S-SP ADE DP , co mputed locally on blo cks 2048 sample s long . The blue line is the identity line and th e red lin e represents lin- ear regression. The results sho wn are for the sig n al of a c o ustic guitar with th e twice oversampled DFT ( red = 2). 10 15 20 25 30 35 40 45 50 55 60 65 70 10 20 30 40 50 60 70 SDR[dB] S-SP ADE_DP SDR[dB] A-SP ADE θ c = 0.1 θ c = 0.2 θ c = 0.3 θ c = 0.4 θ c = 0.5 Fig. 3 . Scatter plot of SDR values for A-SP ADE and S-SP ADE DP , co mputed locally on blocks 20 4 8 sam ples long. number of iterations was fixed for each pro cessed bloc k and the ∆ SDR was c o mputed fro m the who le restored signal. More precisely , the nu mber o f iter ations varied from 10 to 200 (the termination criterion based o n ǫ thus does not com e into play). The results are pr esented in Fig. 4 an d they in- dicate that fo r redunda n t op erators, S-SP ADE DP conv erges faster than A-SP ADE. S-SP ADE O gains the SDR qu ickly but for a high er number of iteration s i t is not able to achiev e a suf- ficient ∆ SDR. The source cod es an d soun d signals a re available at www .utko.feec. vutbr .cz/ rajmic/software/SP ADE-DR.zip. 20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 7 8 9 10 11 number of iterations ∆ SDR [dB] SP ADE, red = 1 A-SP ADE, red = 2 S-SP ADE_O, red = 2 S-SP ADE_DP , red = 2 A-SP ADE, red = 4 S-SP ADE_O, red = 4 S-SP ADE_DP , red = 4 Fig. 4 . A verage ∆ SDR versus the nu mber of iterations. 4. CONCLUSION A novel algorithm for audio declipping based on the sparse synthesis model was introduc e d. 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