Digital Signal Processing Reference
In-Depth Information
9.2.1.2 Mean and Variance Normalisation
The subtraction of the mean per feature vector component corresponds to an equali-
sation of the first moment of the vector sequence probability distribution. If noise has
also an influence on the variance of the features, according variance normalisation
of the vector sequence can be applied and by that an equalisation of the first two
moments. This is known as MVN. The processed feature vector is obtained by
x
t
−
μ
σ
x
t
=
˜
.
(9.13)
of the standard deviations per feature vector com-
ponents is computed out element-by-element. The new feature vector's components
have zero mean and unity variance.
The division by the vector
σ
9.2.1.3 Histogram Equalisation
HEQ is a popular technique in digital image processing [
40
] where it helps raise
the contrast of images and alleviates the influence of the lighting conditions. In
audio processing, HEQ can improve the temporal dynamics of noise-affected feature
vector components. HEQ extends the principle of CMS and MVN to all moments
of the probability distribution of the feature vector components [
9
,
41
], and by that
compensates non-linear distortions caused by noise.
In HEQ, the histogram of each feature vector component is mapped onto a refer-
ence histogram. The underlying assumption is that noise influence can be described
as a monotonic partly reversible feature transformation. With success depending
on meaningful histograms, HEQ requires several frames for their reliable estima-
tion. A key advantage lending to HEQ's independence of the noise characteristics
is that no assumptions are made on the statistical properties (e.g., normality) of the
noise process.
For HEQ, a transformation
x
˜
=
F
(
x
)
(9.14)
needs to be found for the conversion of the PDF
p
(
x
)
of an audio feature into a
reference PDF
p
˜
(
˜
x
)
=
p
ref
(
˜
x
)
.If
x
is a unidimensional variable with PDF
p
(
x
)
,
a transformation
modifies the probability distribution, such that the new
distribution of the obtained variable
x
˜
=
F
(
x
)
x
can be expressed as
˜
))
∂
(
˜
)
G
x
˜
(
˜
)
=
(
(
˜
p
x
p
G
x
(9.15)
∂
˜
x
with
G
(
˜
x
)
as the inverse transformation corresponding to
F
(
x
)
. For the cumulative
probabilities based on the PDFs, let us consider:
