Spectrum histograms and fluctuation patterns are representations of audio fragments. By comparing these representations, we can determine the similarity between the corresponding fragments. Traditionally, this is done using the Euclidian distance. We propose fuzzy similarity measures as an alternative. First we introduce some well-known fuzzy similarity measures, together with certain properties that can be desirable or useful in practice. In particular we present several forms of restrictability, which allow to reduce the computation time in practical applications. Next, we show that fuzzy similarity measures can be used to compare spectrum histograms and fluctuation patterns. Finally, we describe some experimental observations for this fuzzy approach of constructing audio similarity measures

A Rauber

H Müller

K Bosteels

English

Bosteels, Klaas

Kerre, Etienne

Ghent University Academic Bibliography

Fuzzy audio similarity measures based on spectrum histograms and fluctuation patterns

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Fuzzy Audio Similarity Measures Based on Spectrum Histograms and Fluctuation Patterns

Fuzzy Audio Similarity MeasuresBased on Spectrum Histograms and Fluctuation PatternsKlaas Bosteels, Etienne E. KerreFuzziness and Uncertainty Modelling Research GroupDepartment of Applied Mathematics and Computer ScienceGhent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium{Klaas.Bosteels, Etienne.Kerre}@UGent.beAbstractSpectrum histograms and fluctuation patterns are rep-resentations of audio fragments. By comparing these rep-resentations, we can determine the similarity between thecorresponding fragments. Traditionally, this is done usingthe Euclidian distance. We propose fuzzy similarity mea-sures as an alternative. First we introduce some well-knownfuzzy similarity measures, together with certain propertiesthat can be desirable or useful in practice. In particular, wepresent several forms of restrictability, which allow to re-duce the computation time in practical applications. Next,we show that fuzzy similarity measures can be used to com-pare spectrum histograms and fluctuation patterns. Finally,we describe some experimental observations for this fuzzyapproach of constructing audio similarity measures.1. IntroductionRecent portable audio players can store about 20 000songs, and online music stores currently offer millions oftracks. This abundance of music drastically increases theneed for applications that automatically analyse, retrieveor organize audio files. Measures that are able to ex-press the similarity between two given audio fragments,are a fundamental component in many of these applications[1, 2, 6, 7, 8]. Such measures are usually constructed by fol-lowing a feature-based approach. The audio fragments arerepresented by real-valued feature vectors, and the similar-ity is calculated by comparing these vectors. We considertwo types of feature vectors: spectrum histograms and fluc-tuation patterns. So far, the Euclidian distance has alwaysbeen used to compare feature vectors of these types. In thispaper, however, we identify the feature vectors with fuzzysets, such that fuzzy similarity measures can determine thesimilarity between them.2. Preliminaries2.1. Spectrum histogramsFor a given audio segment, the Fourier transform canbe used to calculate the amplitude that corresponds witheach frequency. By dividing an audio fragment in shortsubsequent segments and applying the Fourier transform toeach of these segments, we get the amplitude for each time-frequency pair. Such a representation of an audio fragmentis called a spectrogram. The individual frequencies of aspectrogram are usually consolidated into frequency bandsto reduce the computation time.The amplitudes of the spectrogram can be converted toloudness values. Starting from these values, we can calcu-late the spectrum histogram (SH) [6, 7] by counting howmany times certain loudness levels are reached or exceededin each frequency band. In this way, we get a simple sum-marization of the spectral shape of the audio fragment. Thissummarization is, to some extent, related to the perceivedtimbre of the audio fragment.Our implementation of SHs was written in Matlab, usingthe MA toolbox [5]. Each SH is a matrix with 30 rows (fre-quency bands) and 60 columns (loudness levels). The barkscale is used for the frequencies, and the sone scale for theloudness values. We use these scales because they are psy-choacoustical, i.e., they correspond with human perception.Figure 1(a) shows an example of a SH.2.2. Fluctuation patternsBy applying the Fourier transform to the subsequentloudness values in each frequency band of the spectrogram,we get the amplitudes that correspond with the loudnessmodulation frequencies for each frequency band. A fluc-tuation pattern (FP) [6, 8] consists of weighted versions ofthese coefficients, based on the psychoacoustic model ofloudness levelfrequency band10 20 30 40 50 60102030(a)modulation frequencyfrequency band  10 20 30 40 50 60102030(b)Figure 1. The SH (a) and FP (b) for a fragmentof “Temps pour nous” by Axelle Red. Whitedepicts zero, and black represents the maxi-mum value.fluctuation strength. Hence, fluctuation patterns describethe loudness fluctuations for each frequency band. Thisimplies that they are, to some extent, related to the per-ceived rhythm. Although they are relatively simple, FPscan be very useful in practice. For instance, they were oneof the representations used for the audio similarity measurethat won the MIREX’06 (Music Information Retrieval eX-change) audio-based music similarity and retrieval task.1For implementing FPs, we again used the MA toolbox.The FPs are, like the SHs, 30 by 60 matrices in which therows correspond with frequency bands. In this case, how-ever, the columns represent modulation frequencies (rang-ing from 0 to 10 Hz). Figure 1(b) shows an example.2.3. Fuzzy similarity measuresA fuzzy set A in a universe X is a X → [0, 1] mappingthat associates with each element x from the universe X adegree of membership A(x). We use the notation F(X) forthe class of fuzzy sets in X . A fuzzy similarity measure Mis a fuzzy set in F(X) × F(X) that expresses the similar-ity between each pair of fuzzy sets in X , i.e., the degree ofmembership M(A,B) of (A,B) ∈ F(X)×F(X) denotesthe similarity between A and B. If A and B are completelysimilar then M(A,B) = 1, otherwise M(A,B) < 1.1http://www.music-ir.org/mirex2006We consider 12 well-known fuzzy similarity measures [3].The first ones are based on the fact that fuzzy sets in a fi-nite universe X can be represented as vectors in [0, 1]|X|.The other measures are fuzzy generalizations of cardinality-based similarity measures for crisp sets.We can determine how similar two vectors are by calcu-lating the Minkowski distance between them: the smallerthe distance, the greater the similarity. This observationleads to the following family of fuzzy similarity measures:M1,n(A,B) = 1−(1|X|∑x∈X|A(x)−B(x)|n) 1nfor all A,B ∈ F(X), with n ∈ N \ {0}. An alternative ap-proach is to use the cosine between the vectors as similaritymeasure:M2(A,B) =∑x∈X (A(x) ·B(x))√(∑x∈X A(x)2) (∑x∈X B(x)2)for all A,B ∈ F(X).For two crisp sets A and B in a finite universe X , i.e.,A,B ∈ P(X), we can use the following measures to deter-mine their similarity:M3(A,B) =|A ∩B||A|M4(A,B) =|A ∩B||B|M5(A,B) =|A ∩B||A ∪B|M6(A,B) =|A ∩B|√|A| · |B|M7(A,B) =2 |A ∩B||A|+ |B|M8(A,B) =|A ∩B|min(|A|, |B|)M9(A,B) =|A ∩B|max(|A|, |B|)M10(A,B) =min(|A|, |B|)|A ∪B|M11(A,B) =max(|A|, |B|)|A ∪B|M12(A,B) =min(|A|, |B|)max(|A|, |B|)The classical set-theoretic operations that are used in thesemeasures, can be generalized to fuzzy sets by means of tri-angular operations (t-operations). An increasing, associa-tive and commutative [0, 1]2 → [0, 1] mapping is called atriangular norm (t-norm) T if it satisfies T (x, 1) = x for allOnly the first pages of this paper are presented here because of copyright restrictions. If you would like to obtain the full text, please e­mail to Klaas.Bosteels@UGent.be.

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Fuzzy audio similarity measures based on spectrum histograms and fluctuation patterns

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