246
Maria Calagna and Luigi V. Mancini
in the first k r i singular values of H i , and then the new singular value matrix is
used to obtain the watermarked block H
i . Embedding a watermark consists of the
following steps:
1. Choose a proper block size and divide the original image that is represented by
the matrix A
M×N into B blocks.
2. For each block H i (i = 1, . . . , B),
• Apply the SVD to the matrix H i and calculate its rank, r i . According to the
rule σ j = 0, for j > r i we represent it as H i U r i Σ r i V
T
r i
.
• Select the most significant singular values of each block (σ 1 , . . . , σ k r i ) for
watermark embedding. We may represent the ith block as
H i
H k r i H r i −k r i
where H k r i is limited to the first k r i singular values and H r i −k r i is related to
the remaining singular values in the ith block.
3. The watermark is embedded as follows:
Σ
k r i
= Σ k r i + αΣ k r i W k r i
where Σ k r i represents the original singular values matrix of order k r i and, Σ
k r i
is
the updated singular values matrix after the watermark embedding. The remaining non-zero singular values σ j (k r i < j ≤ r i ), which are the diagonal entries
of Σ r i −k r i , remain unchanged. Hence, the reconstructed watermarked block H
i is
represented as
H
i =
H
k r i
H r i −k r i
(11.5)
where the sub-blocks H
k r i
and H r i −k r i are given, respectively, by
H
k r i
= U k r i Σ k r i V
T
k r i
+ α · U k r i Σ k r i W k r i V
T
k r i
(11.6)
= H k r i + α · U k r i Σ k r i W k r i V
T
k r i
H r i −k r i = U r i −k r i Σ r i −k r i V
T
r i −k r i
(11.7)
In other words, the ith block is reconstructed by updating its first k r i singular values
according to step 3 of the watermark embedding algorithm and updating the subblocks of H i according to (11.5)–(11.7). Thus, the watermarked image is obtained by
replacing all the original blocks H i with the watermarked blocks H
i , for i = 1, . . . , B.
11.4.3 The Watermarking Extraction
In the extraction phase we take into consideration only the singular values σ j , j =
1, . . . , k r i , of each sub-block H
k r i
, described by (11.6). Given a possible counterfeit
image, whose sub-blocks are indicated with H
k r i
and the singular value matrix Σ k r i of
Maria Calagna and Luigi V. Mancini
in the first k r i singular values of H i , and then the new singular value matrix is
used to obtain the watermarked block H
i . Embedding a watermark consists of the
following steps:
1. Choose a proper block size and divide the original image that is represented by
the matrix A
M×N into B blocks.
2. For each block H i (i = 1, . . . , B),
• Apply the SVD to the matrix H i and calculate its rank, r i . According to the
rule σ j = 0, for j > r i we represent it as H i U r i Σ r i V
T
r i
.
• Select the most significant singular values of each block (σ 1 , . . . , σ k r i ) for
watermark embedding. We may represent the ith block as
H i
H k r i H r i −k r i
where H k r i is limited to the first k r i singular values and H r i −k r i is related to
the remaining singular values in the ith block.
3. The watermark is embedded as follows:
Σ
k r i
= Σ k r i + αΣ k r i W k r i
where Σ k r i represents the original singular values matrix of order k r i and, Σ
k r i
is
the updated singular values matrix after the watermark embedding. The remaining non-zero singular values σ j (k r i < j ≤ r i ), which are the diagonal entries
of Σ r i −k r i , remain unchanged. Hence, the reconstructed watermarked block H
i is
represented as
H
i =
H
k r i
H r i −k r i
(11.5)
where the sub-blocks H
k r i
and H r i −k r i are given, respectively, by
H
k r i
= U k r i Σ k r i V
T
k r i
+ α · U k r i Σ k r i W k r i V
T
k r i
(11.6)
= H k r i + α · U k r i Σ k r i W k r i V
T
k r i
H r i −k r i = U r i −k r i Σ r i −k r i V
T
r i −k r i
(11.7)
In other words, the ith block is reconstructed by updating its first k r i singular values
according to step 3 of the watermark embedding algorithm and updating the subblocks of H i according to (11.5)–(11.7). Thus, the watermarked image is obtained by
replacing all the original blocks H i with the watermarked blocks H
i , for i = 1, . . . , B.
11.4.3 The Watermarking Extraction
In the extraction phase we take into consideration only the singular values σ j , j =
1, . . . , k r i , of each sub-block H
k r i
, described by (11.6). Given a possible counterfeit
image, whose sub-blocks are indicated with H
k r i
and the singular value matrix Σ k r i of