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Audio Signal Classification Rough-Sets based Approach. Outline. Introduction - the research goals Musical instrument acoustics Parameters of sounds and their separability Preprocessing for rough set tools: discretization (quantization) of parameters

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outline
Outline
  • Introduction - the research goals
  • Musical instrument acoustics
  • Parameters of sounds and their separability
  • Preprocessing for rough set tools: discretization (quantization) of parameters
  • Automatic classification and results
  • Summary
the research goals
The Research Goals
  • Motivation – to deal with the problem of the automatic classification of musical data:
    • database searching: there is no possibility to find fragments performed by selected instruments inside files, unless such information is attached to the file
  • Aim – to check if it is possible to recognize sounds on the basis of a limited number of parameters, and reveal these parameters

?

problems
Problems
  • Amount of data in sound files

1 s, Fs=44.1kHz, 16 bits stereo, 176.4 kB

  • Musical instrument sound data are unrepeatable and inconsistent:
    • the sound depends on the articulation, the instrument itself, arrangement of microphones, reverberation, etc.
    • sounds of different instruments can be similar, whereas sounds of one instrument may change significantly within the scale of the instrument
bowed string instruments
Bowed String Instruments
  • articulation:
    • bowed vibrato, muted/not muted,
    • pizzicato (string plucked),
  • sound:
    • body resonances
    • inharmonic partials:

where f1- fundamental (pitch)

    • pizzicato: transients only
woodwind instruments
Woodwind Instruments
  • articulation - vibrato/non vibrato
  • the length of the horn resonator is reduced by holes between the mouthpiece and the end
  • reed instruments – excited by vibrating reeds :
    • single reed: clarinet, saxophone
    • double reed: oboe, English horn, bassoon
  • flute:
    • blowing a stream of air across a hole in the body
brass instruments
Brass Instruments
  • articulation: vibrato, muted/not muted
  • lip-driven
  • mouthpieces only help with tone production
  • long narrow body and extended flaring end - upper modes available
  • mechanical valves
processed data
Processed Data
  • consequent sounds in the musical scale of instruments
  • source - CD: McGill University Master Samples
  • stereo, sampling frequency 44.1 kHz, 16 bits
parameterization frequency domain
Parameterization – Frequency Domain
  • Fourier analysis:
  • example: oboe, 440 Hz

A

partials (harmonics)

f

calculation points for parameters
Calculation Points for Parameters
  • The spectrum changes with time evolution

t - starting transient

qs - quasi-steady state

time envelope of an exemplary sound

parameters of sound
Parameters of Sound
  • fdm– mean frequency deviation for low partials
  • hfd_max=1..5 – a partial with the greatest frequency deviation
  • A1-2 [dB] – amplitude difference between 1st and 2nd partial,
  • h1, h3,4,h5,6,7, h8,9,10, hrest –
  • energy of the selected partials
  • Od, Ev – contents of odd/even partials in the spectrum
  • Br– brightness of the sound:
other parameters
Other Parameters
  • f 1 [Hz] – fundamental
  • |f1max– f1min| – vibrato,
  • dfr – fractal dimension of the spectrum envelope:
        • where N(r) - minimal number of squares r covering the envelope,
  • f1/2 – energy of subharmonic partials in the spectrum
  • qs,te– proportional participation of the quasi-steady state and the ending transient in the total sound time
  • rl – release velocity [dB/s]
separability of parameters
Separability of Parameters
  • criterion:

Di,j

– measure of distances between classesi, j

  • Hausdorff metrics
  • max/min/mean distance between objects
  • from different classes

– measure of dispersion in classi

di

  • mean/max distance between class objects or
    • from the gravity center of the class
  • set of parameters is satisfying if Q>1
metrics
Metrics
  • definition:
  • Euclidean
  • “city”
  • central
separability as a function of metrics
Separability as a Function of Metrics

d1/d2 - mean/max distance

between class objects

d3/d4 - mean/max distance

from the gravity center

D1 - Hausdorff metric

D2/D3/D4 - max/min/mean

distance between objects

from different classes

quantization of parameters
Quantization of Parameters
  • inductive learning methods require a small number of attribute values
  • global methods: simultaneously convert all continuous attributes – large tables
    • Boolean approach (Skowron, Nguyen)
    • cluster analysis (Chmielewski, Grzymala-Busse)
  • local methods: restricted to simple attributes
    • methods usually do not discern between points representing different classes
exemplary local methods
Exemplary Local Methods
  • equal interval width method (EIWM)
  • maximum distance method (MDM)
  • statistical clusterization
foundations of rough set rs based systems 1
Foundations of Rough Set (RS) Based Systems - 1

Let – a decision table

U - a universe - nonempty, finite set of objects

A - a nonempty, finite set of attributes

,

the decision attribute

implies indiscernibility relation IND(B)

reduct - aminimal subset B such that IND(A)=IND(B)

foundations of rs based systems 2
Foundations of RS Based Systems – 2

– lower approximation

of X in A

– upper approximation

of X in A

rough set in A - the family of all subsets of U

having the same lower and upper approximations in A

foundations of rs based systems 3
Foundations of RS Based Systems - 3

- B positive region of A

- the generalized decision inA

B - relative reduct iff B is a minimal subset of A

such that

The relative reduct is such minimal subset of A

which preserves the positive region

rough set based systems
Rough Set Based Systems
  • generated rules

where n - length of the rule

  • a rough measure m of the rule describing concept X

Y – set of all examples described by the rule

exemplary rs based systems
Exemplary RS Based Systems
  • LERS
    • allows unknown attribute values
    • possibility of removing inconsistent examples (i.e. of identical attribute values, but with different decisions)
    • priority of attributes is controlled
  • DataLogic
    • calculates attribute and rule strength
    • quantization of data is available
a proposed system
A Proposed System
  • implemented in Mathematica
  • allows data quantization with number of methods, both local and global
  • ten-fold test included
  • priority of attributes is controlled
  • unnecessary attributes found by reducts and relative calculation
  • the use of produced rules available for whole data sets, not only for singular objects
exemplary reducts
Exemplary Reducts

reduct

relative reduct 1

relative reduct 2

  • up to 70% correct recognition obtained in RS tests
  • parameters 60,61,62 and 41,44,30,55 are the most significant
summary 1
Summary (1)
  • the huge amount of data contained in digital sound representation requires parametrization as preprocessing
  • a great number of parameters is a consequence of the variety of musical instruments and differences in their sounds
  • inconsistency of the data implies soft computing techniques for automatic classification
  • quantization is necessary as preprocessing for RS algorithms
summary 2
Summary (2)
  • an appropriate choice of the quantization requires many experiments
  • rough set algorithms allow the evaluation of the significance of parameters
  • composition of parameters in RS reducts confirms that the evolution of the sound must be taken into account during parametrization
  • the use of learning algorithms allows finding rules for managing classification
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