HARMONIC-PERCUSSIVE SOURCE SEPARATION USINGHARMONICITY AND SPARSITY CONSTRAINTS

Jeongsoo Park, Kyogu LeeMusic and Audio Research Group

Seoul National University, Seoul, Republic of Korea{psprink, kglee}@snu.ac.kr

ABSTRACT

In this paper, we propose a novel approach to harmonic-percussive sound separation (HPSS) using Non-negativeMatrix Factorization (NMF) with sparsity and harmonicityconstraints. Conventional HPSS methods have focused ontemporal continuity of harmonic components and spectralcontinuity of percussive components. However, it may notbe appropriate to use them to separate time-varying har-monic signals such as vocals, vibratos, and glissandos, asthey lack in temporal continuity. Based on the observa-tion that the spectral distributions of harmonic and percus-sive signals differ – i.e., harmonic components have har-monic and sparse structure while percussive componentsare broadband – we propose an algorithm that successfullyseparates the rapidly time-varying harmonic signals fromthe percussive ones by imposing different constraints onthe two groups of spectral bases. Experiments with realrecordings as well as synthesized sounds show that the pro-posed method outperforms the conventional methods.

1. INTRODUCTION

Recently, musical signal processing has received a greatdeal of attention especially with the rapid growth of digi-tal music sales. Automatic musical feature extraction andanalysis for a large amount of digital music data has beenenabled with the support of computational power. The ma-jor purposes of such tasks include extracting musical infor-mation such as melody extraction, chord estimation, onsetdetection, and tempo estimation.

Because most music signals often consist of both har-monic and percussive signals, the extraction of tonal at-tributes is often severely degraded by the presence of per-cussive interference. On the other hand, when we analyzerhythmic attributes such as tempo estimation, the harmonicsignals act as interference that may prevent accurate anal-ysis. Consequently, the separation of harmonic and per-cussive components in music signals will function as an

c� Jeongsoo Park, Kyogu Lee.

Licensed under a Creative Commons Attribution 4.0 International Li-cense (CC BY 4.0). Attribution: Jeongsoo Park, Kyogu Lee.“Harmonic-Percussive Source Separation Using Harmonicity and Spar-sity Constraints”, 16th International Society for Music Information Re-trieval Conference, 2015.

important pre-processing step that allows efficient and pre-cise analysis.

For these reasons, many researchers have focused oninvestigating HPSS using various approaches. Uhle etal. performed singular value decomposition (SVD) fol-lowed by independent component analysis (ICA) to sep-arate drum sounds from the mixture [1]. Gillet et al. pre-sented a drum-transcription algorithm based on band-wisedecomposition using sub-band analysis [2].

Other researchers have employed matrix factoriza-tion techniques such as non-negative matric factoriza-tion (NMF). Helen et al. proposed a two-stage pro-cess composed of a matrix-factorization step and a basis-classification step [3]. Kim et al. employed the matrix co-factorization technique, where spectrograms of the mix-ture sound and drum-only sound are jointly decomposed[4]. NMF with smoothness and sparseness constraintswas utilized by Canadas-Quesada et al. [5]. The algo-rithm was developed based on assumptions regarding theanisotropic characteristics of the harmonic and percussivecomponents; harmonic components have temporal con-tinuity and spectral sparsity, whereas percussive compo-nents have spectral continuity and temporal sparsity.

Most HPSS algorithms have employed the same as-sumption. Ono et al. presented a simple technique to rep-resent a mixture sound spectrogram as a sum of harmonicand percussive spectrograms based on the Euclidean dis-tance [6]. Their technique aims to minimize the tempo-ral dynamics of harmonic components and the spectral dy-namics of percussive components. They further extendedtheir work to use an alternative cost function based onthe Kullback-Leibler (KL) divergence [7]. More recently,FitzGerald presented a median filtering-based algorithm[8], where a median filter is applied to the spectrogram ina row-wise and column-wise manner for the extraction ofharmonic and percussive sounds, respectively. Gkiokas etal. also proposed a non-linear filter-based HPSS algorithm[9].

However, the assumption regarding the temporal con-tinuity, which is considered to be crucial for conven-tional harmonic-percussive studies, does not account forthe rapidly time-varying harmonic signals often present invocal sounds and musical expressions such as slides, vi-bratos, or glissandos. This is because their spectrogramsoften fluctuate over short periods of time. Thus, it may de-grade the performance of the algorithms, particularly when

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loud vocal components or such musical expressions aremixed.

In this paper, we propose a HPSS algorithm that is clas-sified as a spectrogram decomposition-based method. Weconsider the spectrum of harmonic components to have aharmonic and sparse structure in the frequency domain,whereas the spectrum of percussive components to havean unsparse structure. To realize the successful separationof harmonic/percussive sounds, we apply constraints thatimpose a particular structure of the spectral bases. Thenovelty of the proposed method resides in the harmonic-ity constraint, which is an extension of the sparsity con-straint presented in previous works [10]. The constraint isclosely related to the Dirichlet prior, which is frequentlyused in probabilistic analysis. Because the proposed al-gorithm does not assume temporal continuity for the sep-aration of harmonic signals, we can successfully separateharmonic signals from the mixture sound, even when thereare significant fluctuations over time.

The rest of this paper is organized as follows. Section2 explains in detail how the proposed method works. InSection 3, we present experimental results, and in Section4, we conclude the paper.

2. PROPOSED METHOD

In this section, we present a detailed explanation of theproposed HPSS method. The proposed algorithm uses thespectrogram-decomposition technique, NMF, with the har-monicity and sparsity constraints based on the Dirichletprior. For the efficient description of the proposed method,we first introduce the conventional NMF. Then, the algo-rithm description for the proposed method is presented. Fi-nally, the theoretical relations of the proposed method tothe Dirichlet prior are described.

2.1 Conventional NMF

Lee and Seung introduced the multiplicative update rule ofNMF for KL divergence [11]. As we iteratively update theparameters, we can represent a non-negative matrix, whichmay correspond to a magnitude spectrogram, as a multi-plication of two non-negative matrices that may containspectral bases and temporal bases. The update rule can berepresented as:

Hk,n Hk,n

P

m

n

Wm,kFm,n

.

Fm,n

o

P

m0Wm0,k

(1)

Wm,k Wm,k

P

n

n

Hk,nFm,n

.

Fm,n

o

P

n0Hk,n0

(2)

where F and F denote the M ⇥N magnitude spectrogramof an audio mixture, and its estimation, respectively, Wand H denote the M ⇥K matrix of the spectral bases andthe K ⇥N matrix of their activations.

2.2 Formulation of Harmonic-Percussive Separation

We present a modified NMF algorithm to impose the char-acteristics of harmonic/percussive sounds. The update ruleis separately represented for the harmonic source basis andpercussive source basis as follows:

Hk,n Hk,n

P

m

n

Wm,kFm,n

.

Fm,n

o

P

m0Wm0,k

(3)

Wm,k Wm,k

P

n

n

Hk,nFm,n

.

Fm,n

o

P

n0Hk,n0

(4)

wk �

1� �HH

�

wk + �HH ifft ({fft (wk)}p) , k 2 �H

(5)

wk max (wk, 0) , k 2 �H (6)

⇢

wk �

1� �HS

�

wk + �HS (wk)q, k 2 �H

wk �

1� �PS

�

wk + �PS (wk)r, k 2 �P

(7)

where �H and �P denote a set of harmonic bases and per-cussive bases, respectively, fft (·) and ifft (·) denote thefunctions of the fast Fourier transform (FFT) and the in-verse FFT (IFFT), respectively, wk denotes the kth col-umn of W, �H

H denotes the harmonicity weight parameterfor the harmonic signal, and �H

S and �PS denote the sparsity

weight parameters for harmonic and percussive signals, re-spectively. Note that Eqns (3) and (4) are identical to Eqns(1) and (2), respectively. Eqns (5)-(7) contribute to shapingthe spectral bases as desired as the iteration proceeds.

Mixing weights that have values between 0 and 1 rep-resent the importance of each constraint imposition, andindicate the degree to which we need to impose the charac-teristic. To enable the harmonic bases to have a harmonicand sparse structure while preserving the original figuresof spectral bases, �H

H and �HS are set to have small positive

numbers, as the effect of the constraint is accumulated overthe iteration.

The exponents p, q, and r have to be determined consid-ering the range of each parameter, 0 r 1 p, q. Here,p and q respectively reflect the degree of harmonicity andsparsity of the destination, and they have to be controlledconsidering the spectral characteristics of the original har-monic sources. Likewise, r reflects the degree of “unspar-sity” of the percussive sources.

Among the update equations shown above, the functionof the conventional NMF update equations in Eqns (3) and(4) is to minimize the error between F and its estimationF. On the other hand, the remainders of the equationsaim to shape the spectral bases. The sparsity constraintin Eqn (7) has been similarly adopted for the matrix de-composition [10], and it is based on the fact that the squareoperation increases the differences among the vector com-ponents. If the square root operation is used instead, as

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in the percussive case of Eqn (7), unsparsity can be im-posed to the basis. Similarly, we can extend this conceptto the harmonicity. The second term in Eqn (5) denotesthe harmonics-emphasized basis, which is due to the factthat the spectrum of the spectrum is sparse. To prevent ele-ments from being negative, the max (·, ·) operation in Eqn(6) has to be jointly involved.

The harmonic and percussive sounds are reconstructedusing the corresponding bases as follows:

F(Harmonic) =X

k2�H

wkhk (8)

F(Percussive) =X

k2�P

wkhk (9)

where hk denotes the kth row of H.

2.3 Relation to Dirichlet Prior

The proposed update equations can be intuitively compre-hended. However, the equations are based on a firm theo-retical background, not heuristically induced. In this sub-section, we employ Dirichlet prior from the probabilitytheory, and investigate its relations to the proposed method.

Priors were primarily adopted for the Bayesian proba-bility theory, including the probabilistic latent componentanalysis (PLCA) or probabilistic latent semantic analysis(PLSA). Such spectrogram decomposition techniques of-ten regard spectrogram components as histogram elementsof multinomial distributions. Because the Dirichlet distri-bution is a conjugate prior of a multinomial distribution,it can be adopted as a prior knowledge of a multinomialdistribution. By adopting the prior, we can modify ourgoal to be the maximizing posterior from the maximizinglikelihood. For this reason, the Dirichlet prior has beenadopted for the matrix factorization in the previous works[10], [12]. Our method employs one of the extensions ofthe Dirichlet prior for harmonicity imposition.

Because PLCA is a special case of NMF, where itscost function is KL divergence [13], we can generalize theDirichlet prior of the PLCA [12] by applying it to the NMFalgorithm as follows:

Hk,n (1� �1)

Hk,nP

m

n

Wm,kFm,n

.

Fm,n

o

P

m0Wm0,k

+�1Ak,n

(10)

Wm,k (1� �2)

Wm,kP

n

n

Hk,nFm,n

.

Fm,n

o

P

n0Hk,n0

+�2Bm,k

(11)where A and B denote the matrices of hyper parameterswith respect to H and W, respectively, and �1 and �2 de-note the mixing weights. In our research, we focus only onthe spectral bases, and thus Eqn (10) is discarded. As canbe observed, the proposed update equations, Eqns (3)-(7),

have the same form as Eqn (11), and the way in which weshape the spectral bases depends on the form of B matrix.

Frequency-domain sparsity imposition can be easilyachieved by setting the hyper parameter B as [10]

bk = (wk)u (12)

where bk denotes the kth column of B, and u denotes anexponent that controls the degree of sparsity of bk.

On the other hand, harmonicity imposition can beachieved when the hyper parameter is represented as

bk = ifft ({fft (wk)}v) (13)

where v denotes the exponent that controls the degree ofharmonicity of bk. This is because a periodic signal canbe represented as a sum of sinusoids, and the spectrumof the periodic signal is sparse. Conversely, if a spec-trum is sparse, we can assume that the original signal hasa strongly periodic characteristic. Thus, we aim to makethe spectrum of the spectrum to be sparse in order to shapea signal such that it has a harmonic structure. Note that inorder to prevent destructive interference caused by phasedistortion, we have to manipulate only the magnitudeswithin the IFFT function, preserving the original phasesof fft (wk).

3. PERFORMANCE EVALUATION

3.1 Sample Problem

In this section, we apply the proposed method and the con-ventional methods to simple sample examples, which issuitable for showing the novelty and validity of the pro-posed method. Spectrograms of synthesized sounds thatconsist of horizontal and vertical lines are presented inFigure 1(a) and Figure 2(a). Figure 1(a) models the casewhere a pitched harmonic sound is sustained for a certainperiod. The sounds of harmonic instruments such as gui-tars, pianos, flutes, and violins fall within this scenario.On the other hand, Figure 2(a) illustrates the case wherea harmonic signal alters its frequency over time. In thiscase, vibratos, glissandos, and vocal signals correspond tothe harmonic components. We compare the performanceof the proposed method to the separation results obtainedusing three conventional methods: Ono et al.’s Euclideandistance-based method [6], Ono et al.’s KL divergence-based method [7], and FitzGerald’s method [8].

As shown in Figure 1(b), both the conventional methodsand the proposed method are able to successfully separatethe sounds. This is because the horizontal lines in this ex-ample have horizontally continuous characteristics, whichare assumed by the conventional methods to be present.However, when the harmonic sound vibrates and the hor-izontal lines fluctuate, as shown in Figure 2(a), conven-tional methods cannot distinguish the horizontal lines fromvertical lines. As we can see in Figure 2(b), the estimatedpercussive components of conventional methods containharmonic partials, and only the proposed method can suc-cessfully separate them. Thus, we can claim that the pro-

150 Proceedings of the 16th ISMIR Conference, Malaga, Spain, October 26-30, 2015

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(b) Separation results of Ono’s Euclidean distance-based method,Ono’s KL divergence-based method, FitzGerald’s method, and theproposed method (from top to bottom)

Figure 1. Sample example of separating horizontal linesand vertical lines.

posed method is not affected by variations in the pitch be-cause it relies on the harmonic structure of the vertical axis,and not the degree of horizontal transition.

3.2 Qualitative Analysis

We evaluated the performance of the proposed method us-ing a real recording example. Figure 3 shows a log-scaleplot of the spectrogram of an excerpt from “Billie Jean,”by Michael Jackson. The signal was sampled at 22,050 Hz,and the frame size and overlap size were set to 1,024 and512, respectively. We can observe from the spectrogram

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Figure 2. Sample example of separating fluctuating hori-zontal lines and vertical lines.

that the excerpt contains both harmonic and percussivecomponents. The harmonic components can be seen ashorizontally connected lines, whereas the percussive com-ponents are seen as vertical lines as in the sample exam-ples.

Figure 4(a) and (b) show the separation results of theharmonic sound (up) and percussive sound (down), whichwere obtained using Ono et al.’s Euclidean distance-basedmethod and KL divergence-based method, respectively.Here, we set the parameters to the values recommended inthe references. We observe that the estimated percussive

Proceedings of the 16th ISMIR Conference, Malaga, Spain, October 26-30, 2015 151

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components still contain harmonic components that maycorrespond to the vocal components. This is because Onoet al.’s algorithms aim to minimize the temporal transitionof the harmonic spectrogram. However, vocal componentsin the original spectrogram do not match well with the un-derlying assumption.

Figure 4(c) shows the result of FitzGerald’s methodwith a median filter length of 17 and when the exponentfor the Wiener filter-based soft mask is two, as recom-mended by FitzGerald [8]. We also observe that the sep-arated percussive components still contain harmonic com-ponents, as in the previous case. This is because of the useof a one-dimensional median filter, which assumes that theharmonic components are sustained for several periods.

Figure 4(d) shows the performance of the proposedmethod. We observe that the harmonic and percussivecomponents are clearly separated, and the percussive com-ponents do not have any vocal components in these results.This is because unlike conventional methods, the proposedalgorithm does not rely on the horizontal continuity prin-ciple. Rather, the proposed algorithm tries to account forthe harmonic components using the harmonic and sparsespectral bases.

3.3 Quantitative Analysis

We performed a quantitative analysis to verify the va-lidity of the proposed algorithm. First, we compileda dataset that consists of 10 audio samples, which isa subset of the MASS database [14], but two sets ofdata, namely tamy-que pena tanto faz 6-19 and tamy-que pena tanto faz 46-57, were excluded in this exper-iment because they lack percussive signals. Then, weobtained a spectrogram for each audio sample with theframe size and hop size set to 2,048 samples and 1,024samples, respectively. Note that the sampling rate ofthe songs in the MASS dataset is 44,100 Hz. Fi-nally, we measured the signal-to-distortion ratio (SDR),signal-to-interference ratio (SIR), and signal-to-artifactratio (SAR) using the BSS EVAL toolbox (http://bass-db.gforce.inria.fr /bss eval/) supported by [15]. Table 1shows the parameter values of the proposed method used inthis experiment. The parameters of the conventional meth-ods are set to the recommended values, as in the previousexperiment.

The evaluation results are summarized in Figure 5. Wecan see that the proposed method guarantees a better av-

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152 Proceedings of the 16th ISMIR Conference, Malaga, Spain, October 26-30, 2015

erage SDR result compared to conventional methods, eventhough the proposed method has a lower SIR performancethan Ono et al.’s Euclidean distance-based method. This isbecause the proposed method far outperforms other meth-ods with respect to the SAR, which has a trade-off relationwith the SIR [16].

Parameter Valuep 1.1q 1.1r 0.5

�HH 0.001

�HS 0.001

�PS 0.1

Number of bases (H,P) (300,200)

Table 1. Experimental parameters.

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4. CONCLUSION

In this paper, we proposed a novel HPSS algorithm basedon NMF with harmonicity and sparsity constraints. Con-ventional methods assumed that the harmonic componentswere represented as horizontal lines with temporal conti-nuity. However, such an assumption could not be appliedto the vocal components or various musical expressionsof harmonic instruments. To overcome this problem, wepresented a harmonicity constraint, which is a generalizedDirichlet prior. By letting the spectrum of the spectrum beharmonic and sparse, we could refine the harmonic compo-nents and eliminate inharmonic components. The experi-mental results showed the validity of the proposed methodby comparing it with conventional methods.

5. ACKNOWLEDGEMENTS

This research was partly supported by the MSIP(Ministryof Science, ICT and Future Planning), Korea, under theITRC(Information Technology Research Center) supportprogram (IITP-2015-H8501-15-1016) supervised by theIITP(Institute for Information & communications Technol-ogy Promotion). Also, this research was supported in partby the A3 Foresight Program through the National Re-search Foundation of Korea (NRF) funded by the Ministryof Education, Science and Technology.

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