Image Processing Reference
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The slopes (Moore, 1966; Neumaier, 1990; Schichl & Neumaier, 2002) also use a first-order
Taylor Series expansion, but they apply the Newton's method to recursively compute the
values of the derivatives. Its general expression is as follows:
f ( x ) = f ( x 0 ) + f ´( x )( x - x 0 )  f S ( I S , I x ) = f ( x 0 ) + I S ( I x - x 0 )
(3)
where I S is determined according to the expression (Garloff, 1999):
f(x)
f(x )
0
if x
x
0
xx
I
(4)
0
S
x
if x
x
0
0
It is worth mentioning that slopes typically provide better estimates than MVFs by a factor
of 2, and that the results can be further improved by combining their computation with IA
(Schichl & Neumaier, 2002)
TMs (Berz, 1997, 1999; Makino & Berz, 1999) combine a N -order Taylor Series expansion
with an interval that incorporates the uncertainty in the function under analysis. Its
mathematical expression is as follows:
f TM ( x, I n ) = a n x n + a n -1 x n -1 + ... + a 1 x + a 0 + I n
(5)
where a i is the i -th coefficient of the interpolation polynomial of order n , and I n is the
uncertainty interval for this polynomial. The approximation error has now order N +1, rather
than quadratic as in previous cases. In addition, TMs improve the representation of the
domain regions, which reduces the wrapping effect. The applications of TMs have been
largely studied thanks to the development of the tool COSY INFINITY (Berz, 1991, 1999;
Berz, et al., 1996; Berz & Makino, 1998, 2004; Hoefkens, 2001; Hoefkens, et al., 2001, 2003;
Makino, 1998, 1999). The main features of this tool include the resolution of Ordinary
Differential Equations (ODEs), higher order ODEs and systems, multivariable integration,
and techniques for relieving the wrapping effect, the dimensionality course, and the cluster
effect (Hoefkens, 2001; Makino & Berz, 2003; Neumaier, 2002). Another relevant contributor
in the development of the TMs is the GlobSol project (Corliss, 2004; GlobSol_Group, 2004;
Kearfott, 2004; Schulte, 2004; Walster, 2004), focused on the application of interval
computations to different applications, including systems modeling, computer graphics,
gene prediction, missile design tips, portfolio management, foreign exchange market,
parameter optimization in medical measures, software development of Taylor operators,
interval support for the GNU Fortran compiler, improved methods of automatic
differentiation, resolution of chemical models, etc. (GlobSol_Group, 2004).
There are discussions about the capabilities of TMs to solve the different theoretical and
applied problems. In this sense, it is worth mentioning that "the TMs only reduce the
problem of bounding a factorable function to bounding the range of a polynomial in a small
box centered at 0. However, they are good or bad depending on how they are applied to
solve each problem." (Neumaier, 2002). This statement is also applicable to the other
uncertainty computation methods.
In AA (Comba & Stolfi, 1993; Figuereido & Stolfi, 2002; Stolfi & Figuereido, 1997), each
element or affine form consists of a central value plus a set of noise terms (NTs). Each NT is
composed of one uncertainty source identifier, called Noise Symbol (NS), and a constant
coefficient associated to it. The mathematical expression is:
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