Graphics Reference
In-Depth Information
3
1-d B-B Spline Polynomial and Hilbert Scan
for Graylevel Image Coding
3.1 Introduction
Chapter 3 examines the use of one dimensional Bezier-Bernstein (B-B) poly-
nomial function in image segmentation and image compression. The approx-
imation used here can be viewed as a modification of the standard B-B ap-
proximation. We shall explain the way of approximation in the one dimen-
sional case using graylevel image pixel values. Later on, we shall examine its
feasibility in the area of image coding. To find the justification of such an
approach of approximation, we shall first examine if the conventional way
of approximating an image by Bezier-Bernstein polynomial, in a raster scan,
provides any advantage from the data compression standpoint. For this, one
can consider an entire row (or column) of an image as a single segment for
its approximation. From the approximation theorem of Bernstein [113] it is
evident that, for a given error, the order of the polynomial increases with the
maximum gray value present in the segment. Therefore, if the maximum gray
value in an image is very large, the order of the polynomial also becomes large.
Consequently, it introduces a large number of control or guiding pixels for ap-
proximation. As a result, approximation becomes computationally expensive
and the segment generation also becomes slow. This makes it inconvenient to
use the conventional way of approximating an image for its compression.
We emphasize on the local control of data points (pixels) instead of min-
imizing the global squared error. We can think of an absolute error criterion
to keep the absolute error within a bound during approximation of image
segments. And, for the sake of data compression, of course, one can choose
the second order polynomial function. Approximation is seen to be more ef-
fective on Hilbert scanned images rather than on raster scanned images. This
is because due to the neighborhood property of the Hilbert scan, long homo-
geneous segments are found to be approximated; resulting in fewer numbers
of segments for encoding than that for a raster scanned image. Consequently,
the compression ratio is found to be higher.
Search WWH ::
Custom Search