Image Processing Reference
In-Depth Information
Directed Intervals (DIs)
Extended IA (EIA)
Modal Intervals (MIs)
Generalized IA (GIA)
Parameterized IA
Interval
Arithmetic (IA)
Mean Value Forms (MVFs)
Slopes
Centered Forms (CFs)
Taylor Models (TMs)
Affine Arithmetic (AA)
Fig. 1. Classification of interval-based computations methods.
In MIs (Gardenes, 1985; Gardenes & Trepat, 1980; SIGLA/X, 1999a, 1999b), each element is
composed of one interval and a parameter called "modality" that indicates if the equation of
the MIs holds for a single value of the interval or for all its values. These two descriptions
are used to generate equations that bound the target function. If both descriptions exist and
are equal, the result is exact. Among the publications on MIs, the underlying theoretical
formulation and the justifications are given in (SIGLA/X, 1999a) and the applications,
particularly for control systems, are given in (Armengol, et al., DX-2001; SIGLA/X, 1999b;
Vehí, 1998)
GIA (Hansen, 1975; Tupper, 1996) is based on limiting the regions of the represented
domain using intervals with parameterizable endpoints, such as [1 - 2x, 3 + 4x] with x 
[0,1]. The authors define different types of parameterized intervals (constant, linear,
quadratic, linear, multi-dimensional, functional and symbolic), but their analysis has
focused on evaluating whether the target function is increasing or decreasing, concave or
convex, in the region of interest using constant, linear and polynomial parameters. In the
experiments, they have obtained the areas where the existence of the function is impossible,
but they conclude that this type of analysis is too complex for parameterizations greater
than the linear case.
In the different representations, CFs are based on representing a function as a Taylor Series
expansion with one or more intervals that incorporate the uncertainties. Therefore, all these
techniques are composed of one independent value (the central point of the function) and a
set of summands that incorporate the intervals in the representation.
MVFs (Alefeld, 1984; Coconut_Group, 2002; Moore, 1966; Neumaier, 1990; Schichl &
Neumaier, 2002) are based on developing an expression of a first-order Taylor Series that
bounds the region of interest. The general expression is as follows:
f ( x ) = f ( x 0 ) + f ´( x )( x - x 0 )  f MVF ( I x ) = f ( x 0 ) + f ´( I x ) ( I x - x 0 ) (2)
where x is the point or region where f ( x ) must be evaluated, x 0 is the central point of the
Taylor Series, and I x is the interval that bounds the uncertainty range. The computation of
the derivative is not complex when the function is polynomial, as it is usually the case in
function approximation methods. Since the approximation error is quadratic, this method
does not provide good results when the input intervals are large. However, if the input
intervals are small, it provides better results than traditional IA.
Search WWH ::




Custom Search