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Figure 2: Pseudospectrum: accumulator derivatives D,, s(k).
As you can see, quadruplicate difference of multiple accumula-
tors 4 - D, 2(k) is a partially convex function on the segment of
the signal presence. This maximum is single and equal to /, more-
over, it is reached on the frame with number 2n (if this maximum
can be reached at all).
Thus, first order regression derivatives behaviour with multiple
memory length recalls spectral decomposition, or rather signal
wavelet transformation. Let us call a multiple-regression pseu-
dospectrum — set of differences of first-order regressive accumu-
lators (7) with multiple characteristics of memory length by a
sequence of powers of two: 1,2,4,8,... (view Figure 2). This
pseudospectrum allows to qualitatively and quantitatively investi-
gate both the duration and amplitude of the input time signal such
as “meander.”
If the maximum of differences between the responses has been
consistently achieved for all accumulators with memory length
N, but for accumulator with memory length n = N +1 predicted
signal maximum was not reached, it means that a constant input
signal had a length of 2N frames, and then began to decrease or
was otherwise dramatically changed.
Similarly, we can make conclusions about the magnitude of the
signal. Cause Dy, 2(2n) = 0.251, for all n whose maximum was
reached,
1=4Dnal2n). (9)
Expected maximum value of Dy 2(k) can be easily found, for
example, for n = 1. Further it should be compared with the
value of differences between accumulators Dj, 2(k) for other n
until maximum on frame k — 2m will be less than all previous
maximums for n « m.
Now consider the problem of determining the sensitivity thresh-
old of the algorithm, detecting the changes of brightness in im-
ages. Figure 3 shows the shape of multiple-regression pseudospec-
trum for the case of shorter time of signal presence on the image
sequence.
Apparently, for lesser duration of the signal, lower frequency
components of pseudospectrum start to move in the negative di-
rection from higher initial values (after a reaction to the passage
of the front edge of the signal) and thus achieve the appropriate
extremum (in this case it will be minimum) at values lower in
magnitude than the specified threshold, based on the expected
drop estimate (8). Figure 3 illustrates it well by the function
D16,2(k) (the lowest frequency component of the presented pseu-
dospectrum). However, this problem can be solved if we jointly
consider a pair of consecutive pseudospectrum components.
Consider previous Dg,2(k) to D16,2(k) pseudospectrum compo-
nent on Figure 3. Since its response to input signal change is
561
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Threshold Y
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pue nh memi n ns Mm e es e aee nae an n v eR, Threshold 2' i
value
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Figure 3: Dynamic brightness threshold correction based on
pseudospectrum.
much faster, it crosses the zero line much earlier, according to
signal disappearance. At this point, the value of current D16,2(k)
component still significantly greater than zero. This value (the
value of the D16,2(k) pseudospectrum component when preced-
ing component Dg.2(k) crosses zero line) is proposed to mem-
orize for each pixel and then to use in dynamic corrections to
the threshold that detects brightness changes. As shown in Fig-
ure 3, detection of the back front of the signal with the threshold
with dynamic correction is successful even in case of significantly
short, compared with the characteristic time of accumulation of
this pseudospectrum component, input signal.
Analysis of the introduced multiple-regression pseudospectrums
is particularly useful in the case of image analysis that studies
moving objects or left/missing items. Since, on the one hand, the
object's motion relative to the background due to the effect of
image pixels obstruction generates in each individual pixel tem-
poral "meander" signal, which has clearly defined leading and
trailing edges (brightness fluctuations over time). On the other
hand, the possibility of signal analysis based on the difference
between the accumulators with multiple memory lets you signifi-
cantly decrease processing time of machine vision systems. Since
estimates of the time signal characteristics must be obtained in-
dependently for each image pixel, in the case of using more com-
plex statistics than the accumulated sums, the necessity to cal-
culate the corresponding parameters estimates of the time signal
directly leads to a huge increase of either computation time, or
use of the program memory, or both.
4 ALGORITHMIC SCHEME
In this section we introduce the algorithmic scheme, which in-
cludes image preprocessing, motion detection and object track-
ing.
Objects detection and tracking are implemented as a modular
three-stage procedure:
I. Detection of moving pixel groups based on pseudospectrum
analysis.
2. Forming of object hypotheses and interframe object track-
ing.
3. Spatiotemporal filtration of object motion parameters.
Let us consider first and second stages of this procedure.
Detection of moving pixel groups is performed as follows:
