Biomedical Engineering Reference
In-Depth Information
Goncalves and colleagues
26
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3.2. Features based on multifractal spectrum
Theoretically, the multifractal spectrum of fBm (a representative of mono-
fractal) consists of three geometric parts: the vertical line, the maximum
point and the right slope. The spectral Mode corresponds to the Hurst
exponent and the vertical line is thought to be an inherent feature, which
distinguishes fBm from the multifractal process. However, it is rare to ob-
tain such a perfect spectrum in practice. Due to the error of estimation, the
spectrum generated from an accurately simulated may even deviate from
the theoretical form, as shown in Figure 3. Even with the lack of precise
estimation of the spectrum, the deviation from the vertical line can still
be utilized in the discrimination between the monofractal and multifractal
processes. In Figure 3, two type processes are presented in the multifrac-
tal spectra. One is the fBm and the other is the turbulence measurement,
which is widely believed to be a multifractal process. Comparing with the
turbulence measurement, the fBm is much closer to the vertical line and
this closeness may be quantied by the left Slope of the spectra. Another
important dierence between these two spectra is the width spread of the
spectra. It is obvious that the width spread of the fBm is much smaller
than that of the turbulence measurement.
Despite the existence of the estimation error, the spectrum can be ap-
proximately described by two slopes and one point without loss of the
discriminant information. Alternatively, we can also approximate the spec-
trum by the left Slope, the maximum point and the width spread. A typical
multifractal spectrum, described in this way, is shown in Figure 4.
The left and right slopes can be obtained easily using the linear regres-
sion technique. However, it is not as straightforward to dene the width
spread automatically. The diculties are related to two aspects { one be-
ing how to locate the start and end points of the width spread, while the
other is what to do with the discreteness of the spectrum. It is easy to see
that the former is dicult conceptually, while the latter is dicult compu-
tationally. There are many ways to dene the width spread. In this chapter,
we give one denition of width spread, which we name the broadness of
the spectrum.
Denition 1: Suppose that
1
and
2
are two roots which satisfy the
equation f() + 0:2 = 0 and
1
<
2
, the Broadness of multifractal
spectrum is dened as B =
2
1
, where f() is the spectrum function
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