MUSIC STRUCTURE BOUNDARY DETECTION AND LABELLING BY A
DECONVOLUTION OF PATH-ENHANCED SELF-SIMILARITY MATRIXTian Cheng, Jordan B. L. Smith, Masataka Goto
National Institute of Advanced Industrial Science and Technology (AIST), Japan{tian.cheng, jordan.smith, m.goto}@aist.go.jp
ABSTRACT180

We propose a music structure analysis method that converts apath-enhanced self-similarity matrix (SSM) into a block-enhanced
SSM using non-negative matrix factor 2-D deconvolution (NMF2D).
With a non-negative constraint, the deconvolution intuitively corresponds to the repeated stripes in the path-enhanced SSM. Thenthe block-enhanced SSM is constructed without any clustering technique. We fuse block-enhanced SSMs obtained using different parameters, resulting in better and more robust results. Discussion
shows that the proposed method can be a potential tool for analysing
music structure at different scales.

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(a) Smoothed recurrence plot and annotated boundaries(in vertical lines)

Index Terms— Music structure analysis, NMF 2D deconvolution, path-enhanced self-similarity matrix, fusion

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1. INTRODUCTION
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Time [s]A self-similarity matrix (SSM) is an important mid-level representation for music structure analysis, which is generated by computing
frame-to-frame similarity. The repetition of segments typically leads
to diagonal stripes in the SSM [1]. The stripes can be emphasised in
the path-enhanced SSM by applying diagonal smoothing to the SSM[2] or to the recurrence plot [3]. As shown in Figure 1, the stripe
patterns clearly illustrate the repetition of segments (e.g., verse or
chorus) and the repetition within a segment (e.g., the bridge in Figure 1). The stripes of the same group can be generated by shifting a
specific structure pattern. Based on this observation, we propose to
decompose the path-enhanced SSM into structure patterns (as shownin Figure 2(a)) and shift activations corresponding to each patternusing non-negative matrix factor 2-D deconvolution (NMF2D) [4].
We then construct a new, block-enhanced SSM by computing theSSM of the normalised activations. In order to obtain a robust result,
we fuse block-enhanced SSMs obtained with different parameters.
Then we detect boundaries on the fused, block-enhanced SSM using a checkerboard kernel [5], and label the detected segments bycomparing each pair of segments.
Several previous methods decompose the SSM to analyse music structures. Two methods, [6] and [7], apply non-negative matrix
factorisation (NMF) to SSMs by assuming that columns within thesame segment types have similar distributions. This assumption apparently works better on block-enhanced SSMs. In practice, to beeffective, [6] only works on segment labelling by clustering NMF
activations, and [7] roughly estimates the main segments by performing low-rank convex-NMF. In [8], the original SSM is decomposedusing symmetric NMF by considering the symmetry of the SSM.
Then the reconstruction of the symmetric NMF is taken as the block(b) Annotations: introduction (I), bridge (B), verse (V),chorus (C), bridge 1 (B1) and ending (E)Fig. 1: An illustration of piece ‘RWC-MDB-P-2001 No.23’ in RWC
Pop Database [9].
structure of the SSM. [1] decomposes the path-enhanced SSM byeigenvalue decompositions, obtaining different eigenvectors corresponding to different stripe structures. The clustered eigenvectors
are used to compute an SSM in the form of a block structure. Then
NMF is used to decompose the block-enhanced SSM for labelling.
We consider our method to be most similar to [1]: both methods convert a path structure to a block structure. However, there
are several differences. First, we compute the block-enhanced SSM
without a clustering step because of the intuitive representation of
the NMF2D. Second, we fuse the block-enhanced SSMs obtainedusing different parameter settings by taking the sum of them for amore robust result. Third, we directly use the average value fromthe block structure to indicate the similarity of two segments. Then
segment labelling is achieved by simple thresholding, without any
further post-processing steps. Our method may also be seen as a
stripe (i.e., sequence)-oriented version of [7], which used NMF todecompose blocks (i.e., homogeneous segments) in SSMs. Focusing
on stripes vs. blocks reflects two different interpretations of musicalstructure, and it is important to be able to analyse both.
The proposed method is evaluated using the RWC Pop Database[9], and compares favourably to some released results on the same
database in the Music Information Retrieval Evaluation eXchange(MIREX) campaign.1 Results show that the proposed method isamong the top of the state-of-the-art.

This work was supported in part by JST ACCEL Grant Number JPMJAC1602, Japan.

978-1-5386-4658-8/18/$31.00 ©2018 IEEE

1 http://www.music-ir.org/mirex/wiki/MIREX_HOME

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ICASSP 2018

(10)2. METHOD
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To generate a path-enhanced SSM, we adopt the method of computing recurrence plots in [3].2 Firstly, we compute Harmonic Pitch
Class Profile (HPCP) features [10] with 12 pitch classes, a window
length of 743 ms and a hop size of 372 ms. Then we concatenate
HPCP features of M adjacent frames to construct a new feature. We
zero-pad the beginning and end of the HPCP features to keep theindices of new features the same as those of the original HPCP features. The concatenated features are denoted by X = [x1 , ..., xN ],where xi , for i = 1, ..., N , is a column vector with the dimension of
12 × M , and N is the total number of feature frames.
Next, we compute the distance (the Euclidean norm) for eachpair of frames. The recurrence plot R is obtained as follows. For
each frame xi , i = 1, ..., N , we search for its K nearest neighbours
in xj , j = 1, ..., N . If xi is a neighbour of xj , and xj is also aneighbour of xi , we set Ri,j = 1. K is set according to the frame
length N by K = kN , where k ∈ (0, 1].
We convolve the recurrence plot with a diagonal matrix covering
6 s to smooth the stripes and to eliminate short lines. The diagonalvalue of the matrix is a Gaussian function with a standard deviationof 3 s. A smoothed recurrence plot is shown in Figure 1(a).

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(c) Normalised activations NH

In an NMF2D model, a pattern is convolved in both the x-axis andy-axis [4]. As shown in Figure 1, the columns within a repeated segment can be generated from a single column’s pattern shifted along
both the x- and y-axes. So we apply NMF2D to the smoothed recurrence plot to separate frames of different groups. The smoothedrecurrence plot R is modelled as:

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Time [s]Time [s]2.2. Deconvolution of the path-enhanced SSM

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2.1. Generating the path-enhanced SSM(d) Block-enhanced SSM

Fig. 2: The process of computing the block-enhanced SSM.
part; 2 patterns (W2 and W6 ) correspond to the verse part; 3 patterns(W3 , W4 and W8 ) correspond to the chorus part; and patterns W5
and W1 correspond to the introduction and the ending parts, respectively. Bridge 1 is not represented by any pattern.
Despite that, the active frames of patterns with the same label
are more correlated than those of other patterns. Hence these frames
have more similar distributions in NH . Based on this observation,
we directly compute the SSM of NH to form a block-enhanced
SSM, as shown in Figure 2(d).

(1)Φ

Each column Wd (1 ≤ d ≤ D) in Figure 2(a) represents a structure
pattern, and its shifting activation Hd ∈ RΦm ×N is shown in thecorresponding sub-figure in Figure 2(b), where Φm is the range ofvertical shifting. We assume that the number of labels in a song is notlarger than 10 (D = 10), and the length of a segment is not longer
than 30 seconds (Φm equal to 30 seconds). Because the patterns arevectors, there is no need for horizontal shifting in this approach; thehorizontal shift variable τ is thus set to 0. We use the implementationof [4]3 with a sparsity weight λ = 2 to enforce sparsity on H. For
parameters not mentioned, we kept the default values.4
For the active frames in Hd , their structures in R are shiftedfrom the same structure pattern Wd . Thus we sum Hd to indicatetheP frames corresponding to the structure pattern Wd : H1 d (n) =Φ Hd (Φ, n). Then we normalise H1 d to a maximum of 1 andsmooth it using a 6-second window. The normalised activation NH
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2.4. A simple fusion
Because of the random initialisation of the NMF2D, the deconvolution is different every time. In order to obtain more robust results,
we run NMF2D 8 times with different parameters. When generating the recurrence plot in Section 2.1, we group M ∈ {5, 6, 7, 8}
frames and search for the top 2 or 3 percent nearest neighbours(k ∈ {0.02, 0.03}). To fuse the block-enhanced SSMs obtained
from the 8 parameter sets [M, k], we simply take their sum [11].
The preliminary test shows that the fusion step improves the resultsby over 6 percentage points on boundary detection in comparison to
the result of each parameter set. Besides making the results morerobust, the fused SSM keeps boundaries obtained in different blockenhanced SSMs. If a boundary is detected in more SSMs, it is easier
to detect in the fused SSM, and more likely to be a true boundary.

2.3. Generating the block-enhanced SSM
The structure patterns obtained via NMF2D are not guaranteed tocorrespond one to one to the structural groups. It is clear in Figure 2(c) that 3 patterns (W7 , W9 and W10 ) correspond to the bridge

2.5. Boundary detection and labelling
After obtaining the fused block-enhanced SSM, we normalise it to a
maximum of 1, denoted by MB . We apply a checkerboard kernel [5]to MB to generate a novelty curve. All peaks in the novelty curve
are detected as boundaries. We add two other boundaries at the firstand last frames. The fused SSM and detected boundaries are shownin Figure 3(a) and (b) respectively.

2 Using SSM in [3] outperforms SSM in [2] in preliminary experiments.
3 http://www2.imm.dtu.dk/pubdb/views/publication_
details.php?id=4499
4 It takes around 9 seconds to converge for a song of 200 seconds on a 2.8
GHz Intel Core i7 computer.

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We test the proposed method on the RWC Pop Database [9].
Two different annotations of this database are available, indicatedby ‘RWC-POP-A’5 and ‘RWC-POP-B’6 respectively. We use bothannotations to evaluate the proposed method for boundary detection
and labelling.

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3.2. Results

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Among the results obtained by using checkerboards of different
lengths, the best boundary detection result is achieved by the 2 × 14second (28-second) checkerboard.7 As shown in Table 1(a), for the
RWC-POP-A annotations the proposed method achieves F-measureof 73.6% and 63.7% for boundary detection and labelling, respectively. For comparison, we select the 4 results published in MIREX
from 2012 to 2017 with a boundary detection F-measure above
67% or a labelling F-measure above 60% (this ignores 33 competitors with lower scores). The SMGA method [3] achieved thebest F-measure on both boundary detection (78.5%) and labelling(68%); see Table 1(a)). The proposed method has the second best Fmeasures, outperforming the rest of the methods on both tasks. Boththe proposed and SMGA methods are based on the same recurrence
plot. In Section 4, we discuss these two methods in detail.
Using the RWC-POP-B annotations (see Table 1(b)), SMGA [3]still has the best boundary detection F-measure of 79.7%, followed
by GS1 and GS3 with F-measures of 79.3% and 72.9%, respectively.
Both GS methods [13] train a Convolutional Neural Network (CNN)
on the combined self-similarity lag matrix (of two different lags).
The trained model works as a binary classifier to detect boundaries
in a moving window. The proposed method achieves an F-measureof 72.1%, better than the other two methods (NB2 [14] and FK2[15]) by 2.1 and 6.3 points, respectively.
In the RWC-POP-B annotations, the silent parts at the beginningand end of the music pieces are annotated as separate segments. Thismeans that 4 correct boundaries can be detected only by detecting thesilent parts. The results show that the improvement of 6.4 percentage
points from GS3 to GS1 is achieved by identifying the silent segments. We do not think these boundaries are as important as others.
In addition, the results may be biased, considering the high ratio of
the number of silence-related boundaries to all boundaries. Thus, we
also report the boundary detection results when ignoring all boundaries in the first 5 seconds and last 5 seconds [16] for the proposed
and GS methods. The proposed method achieves 70.6% and 71.7%
on F-measure with two annotations, respectively.8 The F-measuresof the GS method [13] are 70.9% and 75.2%, respectively, with nodifference between GS1 and GS3.
The labelling F-measure of the proposed method is as high as
73.5% in Table 1(b). However, MIREX does not yet publish resultswith the RWC-POP-B labels, which were updated in 2014, so wecannot directly compare our work. Our results can serve as a baseline
for future research.

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(c) Similarity of each pair of detected segments

Fig. 3: Boundary detection and labelling based on the blockenhanced SSM.

Having detected the boundaries, we compute the similarity ofeach pair of segments. The similarity is indicated by the averagevalue of the block-enhanced SSM MB . Given L detected boundaries, we have L-1 segments. The similarity between the ith and j thsegments is computed as follows:
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Where, (Ii , Ij ) indicates the frame indices in the ith and j th segments, with lengths of Ni and Nj , respectively. The similarity ofeach segment to itself is forced to be Si,i = 1. The similarity of thedetected segments is shown in Figure 3(c). If the similarity value islarger than a threshold (0.5 in this paper), the pair of segments aredetected as the same group. We enforce transitivity generously: if
A and B belong to the same group, and A and C belong to the samegroup, then all A, B and C belong to the same group.

4. DISCUSSION
When we inspected errors made by our algorithm, we found it wasusually not due to poor modelling of the stripes, but a mismatch be-

3. EXPERIMENT
3.1. Evaluation metrics and database

5 https://staff.aist.go.jp/m.goto/RWC-MDB/
AIST-Annotation/
6 http://musicdata.gforge.inria.fr. We use the reduced
version of the annotation.
7 Testing a reasonable range of checkerboard sizes (between 22 s to 30 s),
the variance of the F-measures is less than 1 percentage.
8 The smoothness in Section 2.2 reduces false alarms but blurs the boundaries. The FB s (within ±0.5 s) of the proposed method are less than 25%.

Two subtasks of music structure analysis are evaluated here: boundary detection and labelling. A boundary is considered to be correct
if it lies within ±3 s from a ground truth boundary [12]. We reportboundary detection results of F-measure (FB ), precision (PB ) andrecall (RB ). Labelling is evaluated in 0.1 s frames, with pairwise
F-measure (FL ), precision (PL ) and recall (RL ).

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Fig. 4: Two examples: Left, ‘ABCAB’, right, ‘aaBCaaB’.
tween our hypothesis and the song being analysed: the stripes thatexist are found reliably, but they can be insufficient to model thestructure. Methods based on path-enhanced SSMs (including [1, 3]
and the proposed method) are sensitive to repetitions between segments and within segments. We illustrate this with two examples.
In Figure 4(a), we show two typical path-enhanced SSMs corresponding to segment sequences labelled ‘ABCAB’, but in the rightexample, segment ‘A’ consists of repeating sub-segments ‘aa’. The
simulated results of SMGA [3] and the proposed method are shownin Figure 4(b) and (c). We can see that in the first case ‘ABCAB’,
both methods miss the boundaries between A and B; if both A and
B are through-composed, this boundary is, in a sense, undiscoverable, and might only be found when considering musical novelty or
homogeneity, or segment size uniformity.
In the second case ‘aaBCaaB’, both methods find the boundaries
between A and B. SMGA also detects the boundary internal to ‘aa’,
and this boundary is more noticeable than the one between A andB, whereas our method recognizes both as belonging to the same
“repetition type”. So, despite the gap in performance, we still recognize the special ability of our NMF2D approach to resolve intrasegment repetitions as being a kind of “homogeneously repetitive”
state, which we hope to exploit better in future work.
Of course, there may be disagreement on the ground truth
whether ‘aa’ should be labelled as one segment or two segments. So
beyond this evaluation, we would like to provide meaningful representations for analysing music structure at different scales, such asin [17], [18] and the SALAMI database9 . NMF2D shows promise asa multi-scale music structure analyser. In this paper, we find framesof the same group based on the summed shift activations. Then
within the group, the repetition is clearly indicated by the diagonal
stripes in the shift activations as shown in Figure 4(c) and also in thereal example in Figure2(b).
Among the MIREX algorithms omitted from Table 1(b) are versions of [6] and [7], two NMF approaches directed at blocks instead of stripes. By targeting sequences instead of homogeneity, we
improve F-measure from below 67 to 72.1, closer to the state-ofthe-art performance of SMGA (F-measure = 79.7). We also found,like [11], that a simple late fusion approach led to increased robustness: both precision and recall increased in a balanced way to improve F-measure by 7 points. Based on these observations, we speculate that by fusing non-negative factorisation models of stripes andblocks, performance may improve even more.

5. CONCLUSION AND FUTURE WORK
In order to analyse music structure, we propose to deconvolve a pathenhanced SSM using NMF2D and generate a block-enhanced SSM
based on the summed activations. We fuse block-enhanced SSMs
obtained with different parameters. The analysis of the decomposition results shows that the summed activations intuitively separate
frames belonging to different labels. The fusion step makes the results less sensitive to the parameters of computing the path-enhanced
SSM, and also improves the performance.
The proposed method sums the shift activations to indicate theframes of the same pattern. However, the stripes in the original shift
activations can be potentially used to detect the repeated segments ofthe same label, providing an analysis at a different scale. Exploiting
this remains a task for future work. We are also interested in theperformance of the proposed method on other databases, especially
the SALAMI database with different annotation scales.

9 http://ddmal.music.mcgill.ca/research/salami.

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