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try faster alternative algorithms. For some algorithms this function is
disabled.</h6>
<h2><a name=SUBBAND></a>Multiresolution Subband Decomposition - Independent
Component Analysis (MSD-ICA)</h2>
<p>By definition, standard <st1:City w:st="on"><st1:place w:st="on">ICA</st1:place></st1:City>
algorithms are not able to estimate statistically dependent sources, that is,
when the assumption of independence does not hold. In many cases, however, we
may be able to reconstruct the original sources using simple preprocessing
techniques and to estimate mixing and separating matrices, even if the sources
are not independent. </p>
<p>The <b>ICALAB</b> Toolbox enables blind separation of sources for a wide
class of signals that do not satisfy the independence assumption. This can be
achieved by applying second order statistics (SOS), exploiting spatio-temporal
decorrelation (STD) of sources, or applying linear predictability (LP) and
smoothness criteria (see the <a href="#book">book</a>) and for some preprocessing,
such as: differentiation, high- low-pass filtering, sparsification or subband
decomposition. </p>
<p>Moreover, each unknown source can be modeled or represented as a sum of
narrow-band sub-signals (components). Provided that for some of the sub-bands
(at least one) sub-components are mutually independent or temporally
decorrelated, suitably designed sub-band filters can be used in the
preprocessing stage to extract mixture of them assuming that these sub-bands
can be identified by <i>a priori</i> knowledge. The standard <st1:City w:st="on"><st1:place
w:st="on">ICA</st1:place></st1:City> or BSS algorithms for such transformed
(filtered) mixed signals can then be applied. In one of the simplest case, the
source signals can be modeled or decomposed into their low- and high- frequency
components. In practice, the high frequency components are often found to be
mutually independent. In this case, we can use a High Pass Filter (HPF) to
extract high frequency sub-components and then apply any standard <st1:City
w:st="on"><st1:place w:st="on">ICA</st1:place></st1:City> algorithm to such
preprocessed sensor (observed) signals. </p>
<p>In the version 3 <b>ICALAB</b>, some optional preprocessing has been
implemented. To use this option, click on the <tt><span style='font-size:10.0pt'>Preprocessing</span></tt>
button at the main <b>ICALAB</b> window. This option is particularly useful for
blind separation of dependent or correlated source signals or images, such as
faces or natural images, where you will notice significant improvements in the
performance of the algorithms. In the preprocessing stage, more sophisticated
methods, such as band pass filters or wavelet transforms, can also be applied.
Optimal choice of a transformation depends on a specific application and
optimal parameters are problem dependent. Experiments are necessary to choose
the optimal parameters. </p>
<h2><a name=Preprocessing></a>Preprocessing</h2>
<p>Click on the <tt><span style='font-size:10.0pt'>Preprocessing</span></tt>
button in order to perform preprocessing of sensor data. In the first step two
window appear (Fig. A), when you can select different preprocessing techniques.</p>
<p>You can choose one from the following options: </p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>1.<span style='font-size:7.0pt'>
</span>No preprocessing.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>2.<span style='font-size:7.0pt'>
</span>Differentiation (first and second order).</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>3.<span style='font-size:7.0pt'>
</span>Highpass filtering (Butterworth filter) with adjustable cutoff
frequency.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>4.<span style='font-size:7.0pt'>
</span>Averaging with adjustable number of cascades of first order low-pass
filters.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>5.<span style='font-size:7.0pt'>
</span>Lowpass filtering (Butterworth filter) with adjustable cutoff frequency.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>6.<span style='font-size:7.0pt'>
</span>Subband decomposition and selection: See below. </p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>7.<span style='font-size:7.0pt'>
</span>IIR/FIR filter design tool with numerous options. See below.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>8.<span style='font-size:7.0pt'>
</span>User-defined preprocessing function. Edit the file <span class=style3>preprocessing_user.m</span>
accordingly. </p>
<p class=MsoNormal align=center style='text-align:center'><img border=0
width=432 height=476 id="_x0000_i1042" src="icalab_files/image016.gif"></p>
<h6><b>Fig. A</b> The two windows for choosing parameters of preprocessing and
visualizing the mixed signals.</h6>
<p>Every preprocessing procedure is performed before the <st1:City w:st="on"><st1:place
w:st="on">ICA</st1:place></st1:City> algorithm and after pressing of the
button <tt><span style='font-size:10.0pt'>OK</span></tt>. Preprocessing options
remain active as long as you do not change the mixing matrix, the noise level
or one of the preprocessing options. </p>
<p> </p>
<h3>IIR/FIR filter design tool</h3>
<p>This option allows comprehensive design of IIR and FIR filters with
visualization of parameters. The Fig. B below shows the options, which include:</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>1.<span style='font-size:7.0pt'>
</span>Setting of frequency range, auto-detection of significant part of the
signal spectrum, based on spectrogram (95% of the signal power). You can select
<st1:PlaceType w:st="on"><tt><span style='font-size:10.0pt'>Range</span></tt></st1:PlaceType><tt><span
style='font-size:10.0pt'> of <st1:PlaceName w:st="on">Interest</st1:PlaceName></span></tt>
or <st1:place w:st="on"><st1:PlaceName w:st="on"><tt><span style='font-size:
10.0pt'>Full</span></tt></st1:PlaceName><tt><span style='font-size:10.0pt'> <st1:PlaceType
w:st="on">Range</st1:PlaceType></span></tt></st1:place>.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>2.<span style='font-size:7.0pt'>
</span>Filter type: lowpass, highpass, bandpass, <span class=GramE>bandstop</span>.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>3.<span style='font-size:7.0pt'>
</span>IIR filters: Butterworth, Chebyshev I & II, Elliptic</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>4.<span style='font-size:7.0pt'>
</span>FIR filters: Window-based, Least-Squares, Equiripple.</p>
<p class=MsoNormal style='margin-left:36.0pt;text-align:justify;text-indent:
-36.0pt;line-height:18.0pt'>5.<span style='font-size:7.0pt'>
</span>Filter order, visualization of filter impulse response, phase or
magnitude characteristics and other parameters.</p>
<p class=MsoNormal align=center style='text-align:center'><span
style='color:blue'><img border=0 width=440 height=483 id="_x0000_i1043"
src="icalab_files/image017.gif" alt="IIR/FIR design tool"></span></p>
<h6>Fig. B. The <span class=GramE>window for IIR/FIR filter</span> design. <span
class=GramE>Adjusting of <st1:place w:st="on"><st1:PlaceType w:st="on"><tt>Range</tt></st1:PlaceType><tt>
of <st1:PlaceName w:st="on">Interest</st1:PlaceName></tt></st1:place> and
visualization of filter characteristics.</span></h6>
<p> </p>
<h3>Subband decomposition and selection</h3>
<p>This option provides a powerful preprocessing method for the ICA/BSS. The
subband transform decomposes the input signal into several subbands by applying
the corresponding bandpass filters. The figure shows the subband decomposition
structure and the frequency responses for the filters. The number of subbands
and the specific filter (Butterworth, Chebyshev I/II, Elliptic) can be selected
by the user.</p>
<p class=MsoNormal align=center style='text-align:center'><span
style='color:blue'><img border=0 width=440 height=142 id="_x0000_i1044"
src="icalab_files/image018.gif" alt="Subband filtering"></span></p>
<h6><b>Fig. C </b>Subband filtering.</h6>
<p>Let be <i>m </i>signal mixtures <span class=GramE><i>x<sub>i</sub>(</i></span><i>k)</i>;
(<i>i</i>= 1, ..., <i>n</i>). Let <i>L</i> be the number of subbands. Then,
every mixture <span class=GramE><i>x<sub>i</sub>(</i></span><i>k) </i>is
decomposed into <i>L</i> subsignals <i>x<sub>i</sub><sup>(l)</sup>(k)</i>; (<i>l</i>
= 1, ..., <i>L</i>). It is expected that if we select one or preferably several
subsignal(s) from them (including the original mixture <span class=GramE><i>x<sub>i</sub>(</i></span><i>k)</i>
(denoted as <i>x<sub>i</sub><sup>(0)</sup>(k)</i> in the figure B), based on an
appropriate criterion, we can achieve better separation.</p>
<p>You can set parameters listed as follows: </p>
<p style='margin-left:36.0pt;text-indent:-36.0pt'><span style='font-size:10.0pt;
font-family:Symbol'>·</span><span style='font-size:7.0pt'>
</span>Number of subbands: This parameter corresponds to <i>L</i> in the above
figure.</p>
<p style='margin-left:36.0pt;text-indent:-36.0pt'><span style='font-size:10.0pt;
font-family:Symbol'>·</span><span style='font-size:7.0pt'>
</span>Filter name: To construct a bank of filters, you can choose a filter
from Butterworth, Chebyshev I/II and Elliptic.</p>
<p style='margin-left:36.0pt;text-indent:-36.0pt'><span style='font-size:10.0pt;
font-family:Symbol'>·</span><span style='font-size:7.0pt'>
</span>Order of the filter </p>
<p style='margin-left:36.0pt;text-indent:-36.0pt'><span style='font-size:10.0pt;
font-family:Symbol'>·</span><span style='font-size:7.0pt'>
</span>Number <em>L</em> of subbands to be selected </p>
<p style='margin-left:36.0pt;text-indent:-36.0pt'><span style='font-size:10.0pt;
font-family:Symbol'>·</span><span style='font-size:7.0pt'>
</span>Subband selection criterion: It can be chosen the following cost <span
class=GramE>functions :</span> <i>l</i><sub>1</sub>-norm, <i>l</i><sub>p</sub>-norm
or kurtosis </p>
<p class=MsoNormal align=center style='text-align:center'><img border=0
width=329 height=69 id="_x0000_i1045" src="icalab_files/image019.gif"
alt="l-p norm">; <i>p </i>= 0.5 or 1 </p>
<p class=MsoNormal align=center style='text-align:center'><img border=0
width=334 height=69 id="_x0000_i1046" src="icalab_files/image020.gif"
alt=kurtosis></p>
<p class=MsoNormal><span class=GramE>where</span> <i>x<sub>i</sub><sup>(l)</sup>(k)</i>
is normalized so that it has zero mean and unit variance. The user can define <span
class=GramE>his own</span> cost function by editing the <span class=style3>useralg_pre.m</span>
file </p>
<p>It is possible to view the spectrum of the data (FFT) by pressing the <tt><span
style='font-size:10.0pt'>View FFT</span></tt> button in the preprocessing
window. </p>
<p class=MsoNormal align=center style='text-align:center'><span
style='color:blue'><img border=0 width=440 height=306 id="_x0000_i1047"
src="icalab_files/image021.gif" alt="View FFT"></span></p>
<h6><span class=GramE><b>Fig. D. </b>Spectrum plot of the signal.</span> The
signals can be divided into pages. The user can change the number of samples
displayed in each page by replacing the value by an arbitrary integer
number. This allows the user to employ the zoom facility on the visualized
data. The pages can be changed by clicking on the arrows.</h6>
<p class=MsoNormal align=center style='text-align:center'><span
style='color:blue'><img border=0 width=440 height=558 id="_x0000_i1048"
src="icalab_files/image022.gif" alt="Filtering parameters"></span></p>
<h6><span class=GramE><b>Fig. E. </b>Subband decomposition and selection of the
filtering parameters.</span> The green field in the plot represents the part of
signal spectrum which is not significant for <span class=GramE>analysis,</span>
the user can change this frequency range manually or detect it automatically
(detection based on signal power). The subbands are selected to span the
significant part of signal band. </h6>
<p class=MsoNormal style='text-align:justify'>The user can choose the <st1:place
w:st="on"><st1:PlaceType w:st="on"><tt><span style='font-size:10.0pt'>Range</span></tt></st1:PlaceType><tt><span
style='font-size:10.0pt'> of <st1:PlaceName w:st="on">Interest</st1:PlaceName></span></tt></st1:place>,
display the <tt><span style='font-size:10.0pt'>Prototype</span></tt> filter or <tt><span
style='font-size:10.0pt'>All Filters</span></tt>. Other options are similar to
those presented in previous subsection.</p>
<p>If you check <tt><span style='font-size:10.0pt'>Display channel signals</span></tt>
and select nodes manually you can display a subband signal for each mixture
channel and check the value of the cost function. The detail of this option
will be explained below. </p>
<p> </p>
<h3>Display channel signals and select nodes manually </h3>
<p>If this option is active (see Fig. E below), after you click the <tt><span
style='font-size:10.0pt'>APPLY</span></tt> button, the following two windows
appear: </p>
<p class=MsoNormal align=center style='text-align:center'><span
style='color:blue'><img border=0 width=440 height=358 id="_x0000_i1049"
src="icalab_files/image023.gif" alt="Subband selection"></span></p>
<h6><span class=GramE><b>Fig. F. </b>Subband selection.</span></h6>
<p class=MsoNormal align=center style='text-align:center'><span
style='color:blue'><img border=0 width=440 height=253 id="_x0000_i1050"
src="icalab_files/image024.jpg" alt="Subband selection"></span></p>
<h6><b>Fig. G. </b>Subband signals.</h6>
<p>The lower figure (in Fig. F) illustrates the value of the chosen criterion
for each subband. </p>
<p>If the mixture i.e. in original, observed data (ROOT in the figure) does not
contain higher frequency components, those components are automatically
discarded, because the cost function is normalized by the variance of filtered
signals, which implies that even if the amplitudes are negligibly small, the
value of the cost function can be large. The cost values for those spurious or
undesirable subbands are indicated by <span style='background:yellow'>yellow</span>
bars. The selected subband(s) is/ are indicated by <span style='background:
red'>red</span> bar(s). </p>
<p>Although the <b>ICALAB</b> can select the subband(s) automatically, the ⌨️ 快捷键说明
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