Image Processing Reference
In-Depth Information
Definition 5.8. We define a monotonicity matrix (MM) as an n×n matrix
whose rows correspond to the functions f
i
,1 6 i 6 n, and the columns corre-
spond to variables x
i
,1 6 i 6 n, respectively. The entry in a cell for such a
matrix is either 'I' or 'D'. MM(i,j) ='I' (or 'D') indicates that the function
f
i
is increasing (decreasing) with respect to the variable x
j
.
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Although the monotonicity matrix fully describes the nature of mono-
tonicity of a curve, the enormous number of MMs depicting different kinds of
monotone curves is a severe drawback for the easy handling of MMs. More-
over, such a characterization of a monotone curve is not obtained directly from
the equation of the curve. The next theorem, however, offers a solution.
Theorem 5.11. For any separable function f(x), we have
∂f
i
∂x
j
> 0, if and
only if
∂f
∂x
i
.
∂f
∂x
j
< 0 for 1 6 i,j 6 n,i = j [40].
Proof: Since all other variables are treated as constants, we may write f(x) =
0 as f(x
i
,x
j
) = 0. Restated differently, x
i
= f
i
(x
j
). We know that
∂f
i
∂x
j
=
−(
∂f
∂x
i
)/(
∂f
∂x
j
).
Hence,
∂f
i
∂x
j
> 0 if and only if
∂f
∂f
∂x
j
and
have opposite signs.
∂x
i
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The above theorem helps us to construct the monotonicity matrix if we
define a partial derivative sign vector (PDSV).
Definition 5.9. A PDSV of a function f(x) is an n-tuple where the i-th
element stands for the sign of the partial derivative
∂f
∂x
i
. If π is a PDSV then
π
i
, the i-th component of π, is '+' if
∂f
∂x
i
is positive and '-' if it is negative.
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If we consider a function f(x,y,a), then every PDSV π of f is a three-
tuple. π
1
,π
2
,π
3
denote the signs of
∂f
∂x
,
∂f
∂y
and
∂f
∂a
, respectively. Since each of
these π
i
may be either a '+' or a '-' there may be at most eight PDSVs for
the given function.
Every PDSV corresponds to a unique MM in the following manner. Let π
be the given PDSV.
′
′
MM(i,j)
=
I
if π
i
= π
j
′
′
=
D
if π
i
= π
j
.
This is a direct consequence of the definition of MM(i,j) and Theorem 5.11.
Clearly then, there can be no more than 2
n
(number of n-tuple PDSVs)
distinct MMs defining the monotone functions. We use the following proper-
ties of PDSVs to identify equivalent MMs (with respect to our analysis). We
assume that f is a d-dimensional curve with k = n−d unknown parameters,
i.e., f(x
l
,x
2
,..,x
d
,a
1
,..,a
n−d
) = 0.
Property 1: (Complementation) The complement π
c
of a π, where a '+'
