Image Processing Reference
In-Depth Information
f
AA
(
i
) =
x'
=
x
c
+
x
0
0
+
x
1
1
+
2
x
2
+ ... +
n
x
n
(6)
where
x'
represents the affine form,
x
c
is the central point, and each
i
and
x
i
are the NS and
its associated coefficient. In AA the operations are classified in two types: affine and non-
affine operations. Affine operations (addition and constant multiplication) are computed
without error, but non-affine operations need to include additional NTs to provide the
bounds of the results. The main advantage of AA is that it keeps track of the different noise
symbols and cancels all the first-order uncertainties, so it is capable of providing accurate
results in linear sequences of operations. In nonlinear systems, AA obtains quadratic
convergence, but the increment of the number of NTs in the nonlinear operations makes the
computations less accurate and more time-consuming. A detailed analysis of the
implementation of AA and a description of the most relevant computation algorithms is
given in (Stolfi & Figuereido, 1997).
Among other applications, AA has been successfully used to evaluate the tolerance of circuit
components (Femia & Spagnuolo, 2000), the sizing of analog circuits (Lemke, et al., Nov.
2002), the evolution of deformable models (Goldenstein, et al., 2001), the evaluation of
polynomials (Shou, et al., 2002), and the analysis of the Round-Off Noise (RON) in Digital
Signal Processing (DSP) systems (Fang, 2003; López, 2004; López et al., 2007, 2008), etc.
Modified AA (MAA) (Shou, et al., 2003) has been proposed to accurately compute the
evolution of the uncertainties in nonlinear descriptions. Its mathematical expression is as
follows:
(7)
k
2
2
k
f
( e
) x'
x
x e
x e
x e
x e e
x e
...
x
MAA
i
c
00
11
2
0
3
0
1
41
n
i
i,k
It is easy to see that MAA is an extension of AA that includes the polynomial NTs in the
description. Thus, it is capable of computing the evolution of higher-order uncertainties that
appear in polynomial descriptions (of a given smooth system), but the number of terms of
the representation grows exponentially with the number of uncertainties and the order of
the polynomial description. Thus, in this case it is particularly important to keep the number
of NTs of the representation under a reasonable limit.
Obviously, the higher order NTs are not required when computing the evolution of the
uncertainties in LTI systems, so MAA is less convenient than AA in this case.
3. Interval-based analysis of DSP systems
This Section examines the variations of the properties of the signals that occur in the
evaluation of the DSP systems when Monte-Carlo Simulations (MCS) are performed using
Extensions of IA (EIA) instead of the traditional numerical simulations. The simulations
based on IA and EIA can handle the uncertainties and nonlinearities associated, for
example, to the quantization operations of fixed-point digital filters, and other types of
systems in the general case.
The most relevant advantages of using EIA to evaluate DSP systems can be summarized in
the following points:
1.
It is capable of managing the uncertainties associated with the quantization of
coefficients, signals, complex computations and nonlinearities.
2.
It avoids the cancellation problem of IA.
3.
It provides faster results than the traditional numerical simulations.
