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we characterize the class of curves that are amenable to this generic approach
of analysis [40].
5.5.1 Characterizing Properties of the Class
A planar curve with one or two unknown control parameters may be rep-
resented by the equations f(x,y,z) = 0 or f(x,y,a,b) = 0 respectively where
x, y denote the special variables and a, b denote the unknown control pa-
rameters. Without any loss of generality, we can assume that the segment of
the curve in the first quadrant in a rectangular mesh defined by 0 6 x 6 n,
0 6 y 6 n. Moreover, there is no grid line that does not intersect the curve.
To be precise, we assume the following:
Assumption 0: The curve is continuous and differentiable in the
region of interest.
Our next assumption about the class of curves is that the equation
f(x,y,a,b) = 0 may be rearranged to write the following four equivalent
equations x = f x (y,a,b), y = f y (x,a,b), a = f a (x,y,b), and b = f b (x,y,a).
We call it the separability property of the curve. It is important to note that
to solve the reconstruction problem using the I R scheme, the curve has to
possess this separability property.
Assumption 1. The curve possesses the separability property.
Now let us formalize the concept of monotone curves in our context. In
the following definition, x = (x 1 ,..,x n ) denotes a vector of n variables. Also,
x i denotes the vector of (n−1) variables (x 1 ,..,x i−1 ,x i+1 ,...,x n ).
Definition 5.5. A function g(x) is monotone decreasing with respect to x i
if and only if x i > x
i implies g(x 1 ,..,x i ,...,x n ) < g(x 1 ,..,x
i ,...,x n ) for any
choice of values (in the region interest) for the other (n− 1) variables.
Similarly, g(x) may be defined as monotone increasing with respect to x i .
In other words, g(x) is monotone decreasing if and only if
∂g
∂x i < 0 over our
interval of attention.
Definition 5.6. A function g(x) is monotone if and only if ∀i,1 6 i 6 n,
g(x) is either monotone increasing or monotone decreasing with respect to x i .
Hitherto we refer to g as an increasing (a decreasing) function if g is monotone
increasing (decreasing).
Definition 5.7. We say that a curve given by f(x) = 0 is monotone if and
only if the function f is monotone.
Assumption 2. The curve is monotone.
As we shall see in Theorem 5.11, if f is monotone and x i = f i (x i ), then
the function f i is also monotone for all i, 1 6 i 6 n and vice versa.
Then it follows from the above discussions that we can capture the mono-
tonicity of a particular curve f(x) = 0, if we know the nature of the mono-
tonicity of the individual functions f i ,1 6 i 6 n. This can be easily depicted
by a matrix, which we shall call the monotonicity matrix (MM).
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