Cryptography Reference
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the same as the side distortion at a cost of 100% redundancy when the spread
k = 0 in MDSQ.
Fig. 12.3(b) is an index assignment scheme with a spread of k = 1. There
are only 22 samples, or j
0
,j
1
,,j
21
, which are the valid scalar samples for
transmission. It shows that the quality of side reconstructions needs to be
represented by the small ranges of values in any row or any column depend-
ing on the received description from any one channel. An index assignment
matrix with a higher fraction of occupied cells leads to a quantizer pair with
lower redundancy. From theoretical point of view, the 22 samples can be rep-
resented by log
2
(22)-bit strings. If j
7
is the scalar sample to be transmitted in
Fig. 12.3(b), then after the index assignment step l, we obtain i
1
= 010 and
i
2
= 011. By doing this and when both descriptions are received, the trans-
mitted sample j
7
is determined with probability 1 with Fig. 12.3(b). That
is,
1,
if t =7;
p (j
t
i
1
= 010,i
2
= 011) =
(12.3)
0,
otherwise.
6−log
2
(22)
log
2
(22)
and the redundancy is reduced to
100% = 34.55%. If one of the
channels breaks down, say, Channel 1, then only i
2
= 011 is received. With
the aid of Fig. 12.3(b), we visualize the column containing 011, and estimate
that there are three possible candidates. These are j
7
, j
9
,andj
11
,thatmay
be transmitted with the conditional probabilities:
1
3
,
if t = 7 or 9 or 11;
p (j
t
i
2
= 011) =
(12.4)
0,
otherwise.
The side distortion would be larger than the central distortion, because the
side distortion is the error between the transmitted j
7
and the conditional
expectation in Eq. (12.4),
1
3
(j
7
+ j
9
+ j
11
). Comparing this to the case when
k = 0, the redundancy is greatly reduced, while the side distortion gets some-
what increased.
It is a straightforward task to extend MDSQ to MDVQ. The index assign-
ment of MDVQ becomes more di
cult than MDSQ. Fig. 12.4 demonstrates
that the MDVQ structure with two descriptions. Here, the input X
k
denotes
the small block or code vector. For example, the 44blockforVQoper-
ation. The reconstruction
X
(0)
k
from the central decoder has less distortion
than that from each of the side decoder,
X
(1)
k
X
(2)
k
. In this chapter, we
follow the MDSQ in [27] and MDVQ algorithm in [28], and devise a robust
multi-watermarking algorithm suitable for both error resilient transmission
and copyright protection.
or
12.4 The MDC-Based Single Watermarking Algorithm
The algorithm for single-watermark embedding and extraction with MDC is
described as follows.
