Image Processing Reference
In-Depth Information
(between 0 and 1). Figures 8.a-c show the averaged mean values computed by the estimator
for the previous three lengths. It can be observed that the interval-based estimators tend to
obtain slightly better results than the ones of the numerical simulation, although they are
roughly of the same order of magnitude. Figures 8.d-f show the variances of these
computations. In this case, all the results are approximately equal, and the values decrease
(i.e. they become more precise) with longer simulations. Figures 8.g-i show the mean of the
variance of the interval-based simulations estimator. It can be observed that when the
intervals have small widths, the ideal values are obtained, but when the interval widths are
comparable to the variance of the distribution (approximately from 1/4 of its value) the
computed values increase significantly the variance of the estimator. Figures 8.j-l show the
evolution of the variance estimator. The results are approximately equal in all cases, and
decrease with the longer simulations.
Therefore, interval-based simulations tend to reduce the edges of the PDFs and to equalize the
other parts of the distribution according to the interval widths. If no additional operation is
performed, the edges of the PDFs may change significantly, particularly in uniform
distributions. However, since these effects are known, they can possibly be compensated.
When using normal signals, the mean and variance of the MC method are similar to the ones
obtained in numerical simulations, but the mean of the variance tends to grow for widths
above 1/8 of the variance. However, since the improvement in the computed accuracy is
small, it does not seem to compensate the increased complexity of the process.
3.4 Discussion on interval-based simulations
Section 3.1 has revealed the importance of using EIA in the interval-based simulation of DSP
systems, particularly when they contain feedback loops. It has also shown that traditional IA
provides overestimated results due to the cancellation problem. Although the analysis has
been performed through a simple example, it can be shown that this problem occurs in most
IIR realizations of order equal or greater than two. If there are no dependencies, IA provides
the same results than AA, but AA is recommended to be used in the general case. In
interval-based simulations of quantized systems, the affine forms must be modified to
include all the possible values of the quantization operations without increasing the number
of noise terms. The proposed approach solves the overestimation problem, and allows
performing accurate analysis of linear systems with feedback loops.
Another important conclusion is that, since the propagation of uncertainties in AA is
accurate for linear computations, the features of AA perfectly match with the requirements
of the interval-based simulations of digital filters and transforms.
Section 3.2 has evaluated the effects of including one or more uncertainties in a deterministic
signal. In addition to determining the maximum and minimum bounds of the variations of
the signals in the frequency domain, the analyses have shown the position of the largest
uncertainties. Since these amplitudes are not equal, the noise at the output of the FFT does
not seem to be white. Moreover, its effect seems to be dependent on the position of the
uncertainties in the time domain. The analyses based on interval computations have
detected this effect, but they must be combined with statistical techniques to verify the
results. A more precise understanding of these effects would help to recover weak signals in
environments with low signal-to-noise ratios.
In Section 3.3 the effects of using intervals or extended intervals of a given width in the
Monte-Carlo method instead of the traditional numerical simulations has been analyzed. In
the first part, the results show that this type of processing softens the edges and the peaks of
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