Bernstein Polynomial and B´ezier-Bernstein Spline - Bezier and Splines in Image Processing and Machine Vision - page 16

Definition

A function f ( x ), constant in [ k 1 ,k 2 ], in the discrete plane is said to have a
key pixel P at x = c (where c =( k 1 + k 2 ) / 2or( k 1 + k 2 +1) / 2 corresponding
to even and odd values of ( k 1 + k 2 )) provided δ 1 ,δ 2 ∈{ 0 , 1 } exist such that

in both the intervals [( k 1 − δ 1 ) ,k 1 ] and [ k 2 , ( k 2 + δ 2 )] either f ( c ) >f ( x )

or f ( c ) <f ( x ) when k 1 = k 2 = c ; the definition is applicable for Figure
1.5 where δ 1 = δ 2 = 1. Note that the foregoing definition corresponds to
Figures 1.4 and 1.5, where key pixels lie on a horizontal sequence of pixels
for the interval [ k 1 ,k 2 ]of x . Similarly, key pixels can also be defined for a
vertical sequence of pixels for the interval [ k 1 ,k 2 ]of y .

Contour Approximation

Let k 1 ,k 2 , ,k p be P key pixels on a contour. The segment (geometrical

entity, GE) between two key pixels can be classified as either an arc or a
straight line. If the distance of each pixel from the line joining the two key
pixels is less than a pre-specified value, say δ , then the segment is considered
to be a straight line (Figure 1.6(c)); otherwise, it is an arc. The arc may again
be of two types, with all the pixels either lying on both sides (Figure 1.6(a))
or lying on the same side (Figure 1.6(b)) of the line joining the key pixels. We
denote the GE in Figure 1.6(c) by L (line) and that in Figure 1.6(b) by CC
(curve). GE in Figure 1.6(a), therefore, is nothing but a combination of two
CCs meeting at a point Q (point of inflection). Key pixels on the contour of a
two-tone picture can hence be used to decompose the contour into two types
of GEs, namely, arcs and lines.
Consider Figure 1.7, where the curve CC in Figure 1.6(b) is enclosed within
a right triangle ABC . AC , the line joining k j and k j +1 , is the hypotenuse,
whereas AB and BC are the two other sides.
Proposition 1 justifies that the arc CC will always be confined within
a right triangle ABC . A line DF is drawn parallel to the hypotenuse AC
and passing through the pixel E of maximum displacement with respect to
AC . The sub-triangles, ADE and CFE , so constructed may be taken as the
characteristic triangles to approximate the curve CC by the quadratic Bezier
approximation technique. Information preservation of Bezier characteristic
triangles with the key pixels forms the basis of the underlying concept of the
generation scheme.

···

Proposition 1

In the discrete plane, all pixels on the arc between two key pixels remain
always on or inside a right triangle, with the line joining the key pixels as the
hypotenuse. The other two sides of the right triangle are the horizontal and
vertical lines through the key pixels.