Graphics Reference
In-Depth Information
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Fig. 1.2.
Cubic Bezier-Bernstein curves.
1.4.4 Approximation of Binary Images
Data approximation, for binary images, based on Bezier-Bernstein spline
model is the inverse of drawing mechanism used in computer graphics. So,
instead of supplying the control points from outside, they are extracted from
within images. The extraction, in general, uses the local geometry. As the con-
trol points are viewed as key pixels [26], i.e., knots on the discrete boundary
of objects in the discrete image plane, they are extracted using local discrete
geometry.
Image boundaries, in general, have many discontinuities and we need to
preserve them during an approximation so that the approximated version of
an image boundary does not appreciably change from its original one. It is,
therefore, wise to carry out the polynomial approximation instead of poly-
nomial spline approximation. The main reason is that we do not want to
incorporate smoothness at points where two pieces of boundary segments join
in. Smoothness can appreciably change the shape of a boundary and as a
result, the underlying image may change noticeably. For successful approxi-
mation, one can search for a set of key pixels on contours and, based on them,
decompose the contour into a set of arcs and line segments. Regeneration of
an arc may use vertices of the corresponding Bezier characteristic triangle.
It is possible to eliminate one of the vertices and use an intercept instead.
Regeneration for straight line segments may use Bresenham's algorithm [29]
and Bezier method for generation of arc segments. For regeneration, key pix-
els are considered to be the guiding or control pixels, and their locations are,
therefore, in no way disturbed. This maintains the basic definition or shape
of image boundaries (binary image). To preserve them and to maintain the
connectivity property, sometimes we may need some intermediate operations
(e.g., deletion and shifting of undesirable pixels, generated by Bezier approx-
imation, and insertion of new pixels).
Difference in area as well as the compactness between the input and output
versions of an image may serve as a measure for regeneration error.
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