Graphics Reference
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or,
2( n
2)
v 1 i
1) 2 ≤|
v 1
|≤
2 max .
(3.14)
( n
Similarly,
B 2 i
max . (3.15)
Therefore, the inequality (3.13) tells th at the function f ( t i )= B 2 i ( t i ) ,i =
1 , 2 ,
min ≤|
B 2
|≤
2 can be approximated by B 2 ( t ) with an error inequality expressed
in equation (3.15).
···
n
3.4.3 Implementation Strategy
It is seen from the previous section that the inequality (3.13) and (3.15) can
be used to approximate a gray tone image segment. During approximation,
it may be the case that the inequality (3.13) does not hold for all values of i
associated with a segment of the image, representing either a row (or a column)
or the entire Hilbert scanned image. Let us consider that the inequality is true
for n o pixels out of n in the segment. Thus the remaining ( n - n o ) pixels can
again be approximated over the interval [0 , 1]. Approximation technique for
a raster scanned image thus may involve decomposition of all the rows (or
columns) into a number of gray segments, while for a Hilbert scanned image,
it may decompose the single piece of segment into segments of different sizes.
The approximation always starts, in either case, with a fixed size, which may
be 16, 32, 64, 128, or 256. Note that the inequality (3.13) is always true for a
segment having three pixels irrespective of the inequality (3.15). The 3-pixel
approximation is, therefore, the smallest segment for approximation. For a
raster scanned image, either the last two pixels or the last pixel of the row (or
column) may remain free. In this case, the same pixels/pixel may be left as it
is or the same pixel may be considered once or twice to ensure approximation.
This is the undesired situation for approximation at the end point. For a raster
scanned image of size M
M , the undesired situation may happen at most
M times (worst case) due to row or column wise approximation while for a
Hilbert scanned image, this undesired situation may happen only once.
Example:
In order to illustrate the method of approximation, let us consider a se-
quence of 38 data points as shown in Table 3.1. The maximum and minimum
errors, max and min , for approximation are 10.0 and 0.000001, respectively.
The approximation partitions the data set into three segments. The beginning
and end point of each partitioned segment are approximated with zero error,
whereas all other data points are approximated with errors between min and
max . Note that the approximation may have much lower error than max .
The partition of data points in Table 3.1 into three segments is controlled by
the equation (3.13). Length of the first partition of the data segment is 11,
whereas the second and third partitions have lengths 7 and 20, respectively.
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