Graphics Reference
In-Depth Information
every pixel position. This is obviously the most stable state of the curve (i.e.,
without any deformation). Homogeneity decreases with the increase of defor-
mation. For the purpose of image compression, we are interested in finding
the homogeneous segments of pixels in an image because such segments can
be approximated with a small amount of error and they do not significantly
produce any smearing effect. From the space curve analogy, homogeneous seg-
ments of pixels are segments with
z
(
x
)
0. However, in practice, it is very
dicult to obtain long segments of pixels with zero gradient everywhere. In
order to circumvent this diculty, we consider the average of the first order
derivative values for a segment of pixels and compute the variance of these
derivative values. Since small value of
z
(
x
) corresponds to small deformation
of the image space curve at a pixel position, its average value should corre-
spond to average deformation and hence, the square root of the variance, i.e.,
the standard error provides a measure for the deformation.
≈
3.5 Image Data Compression
Since we are restricted to one dimensional approximation, we consider both
the Hilbert and raster scanned images for compression. Among the space-
filling curves, note that the Hilbert/Peanno scanned images have already
received attention in image compression due to its neighborhood scanning
behavior.
A. Coding Scheme
An image on a raster scan can be approximated either row wise or column
wise. The one that needs fewer number of segments is selected for coding. For
a Hilbert scanned image, the approximation is along the length of the curve.
We basically encode the approximation parameters of a segment along with
the length of the segment. In the following section, we will be explaining the
bit requirement for the proposed methods of coding.
B. Bit Requirement
Let us consider an image of size
M
×
M
with
L
number of gray levels
{
0
,
1
,
2
,
. Since there may be a number of gray segments resulting in the
process of approximation, each of them can be coded with their corresponding
approximation parameters, namely
v
o
,v
1
,v
2
, and the length of the segment,
n
. Since the positional information of approximation (control parameters of
the Bezier curve) parameters is not taken into account for coding, the size of
the gray segments plays an important part for regeneration of the image. As
the maximum possible size of a segment on a raster scan is M, the maximum
number of bits required for encoding the size of a segment is log
2
M
.Inpar-
ticular, the number of bits required to encode the size of a segment, satisfying
the approximation criterion, depends on the maximum value for a segment
chosen for approximation. In practice, the size of segments is found to be much
less than the length of the raster. The segments, in fact, are found to occur
frequently with the same length. As a result, the probability of occurrence for
···
,
(
L
−
1)
}
Search WWH ::
Custom Search