1-d B-B Spline Polynomial and Hilbert Scan for Graylevel Image Coding - Bezier and Splines in Image Processing and Machine Vision - page 71

3.7 Results and Discussion

Here, we have made an attempt to demonstrate an application of 1-dimensional
quadratic Bezier-Bernstein polynomial approximation in coding gray tone
Hilbert and raster scanned images. Drawbacks in using the conventional way
of approximation were examined and a modification was then introduced in
order to make it useful for image data compression. Based on the modified
concept, two different algorithms have been formulated. Both the algorithms
have been examined to compress 256 256 (8 bits) gray tone images following

the Hilbert and raster scan. The performance of the algorithms on the Hilbert
scanned images is found to be better than that on the raster scanned images.
This is due to the neighborhood property of the Hilbert scan. More precisely,
the Hilbert curve always passes through the neighborhood pixels, and since
the neighborhood pixels are, in general, strongly correlated, the approxima-
tion is done over longer segments. Over such long segments, the variation in
pixel intensity is low. As a result, arc approximation is not as economical
as the line segment approximation (in terms of approximation parameters).
Consequently, lower compression ratio or larger number of bits/pixel is re-
quired. But the line segment approximation reduces the PSNR value com-
pared to that for arc segment approximation. On the other hand, for raster
scanned images, the quality of the reconstructed images is disturbed when
the maximum length of segment exceeds a certain value. Short segments, in
general, are found to produce better quality for the reconstructed images. Ta-
ble 3.2 shows the results on compression and quality for 256

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256 8-bit raster
scanned images for Algorithm 1, while Table 3.3 provides the results for the
corresponding Hilbert scanned images. The approximation uses both the line
and arc segments. Tables 3.4 and 3.5 indicate the performance of Algorithm
2 for the raster and Hilbert scanned images. Finally, the comparison for the
algorithm due to Kamata et al. [86] is shown in Table 3.6.
Note that Algorithm 1 in the raster scan mode may produce smearing for
large values of max , because with the increase in the value of max ,the
possibility of long homogeneous segments of pixels satisfying the approxima-
tion criterion increases. As a result, visual disparity may arise. This fact is
also true for Algorithm 2 in the raster mode for larger values of the standard
error. Figure 3.5 shows this smearing effect for Algorithm 1 and Algorithm 2
in the raster scan mode. The line segment approximation in the raster mode
also affects the reconstructed quality for high values of max .
For the 8-bit Lena and Girl images, compression is found to be higher
in the Hilbert scan mode compared to that in the raster scan mode. From
the Tables 3.4, and 3.5, it is seen that Algorithm 2 also behaves in the same
way as Algorithm 1. Higher compression is found to occur in the Hilbert scan
mode. Figure 3.6 shows two different decoded images for Lena and Girl images
for Algorithm 1, while Figure 3.7 shows the results of the decoded images
for Algorithm 2 due to Hilbert scan. Comparison with Kamata's algorithm
(Figures 3.8 and 3.9) shows that the proposed algorithms perform better for

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