following steps:
Input: The image A with
NuM
pixels and the watermark W of length n.Output: The watermarked image A'.
• Apply the DWT decomposition to the origi- nal image A, thus leading to the frequency bands: LL, LH, HL, HH;
• Apply the SVD transform to each band; • Apply the SVD transform to the watermark
represented as an image;
• Embed the watermark, by adjusting the singular values in the frequency bands λi ,
n
i
1"
, with the singular values of thewatermark image λwi ,
wi
1"n
. The em- bedding rule is: λi* λi αλwi . Substitute the singular values matrix, containing the updated singular values calculated in the previous step, into the SVD decomposition of each frequency band;• Apply the inverse DWT (IDWT) transform to produce the watermarked image. The watermark extraction is possible by re- versing the previous steps: the DWT transform is applied to each frequency band and the watermark singular values are computed as:
α λ λ
λwi ( *i i)/ .
The watermark image is represented as the product of the orthogonal matrices Uw , Vw and the singular values matrix, 6w which contains the
singular values λwi as diagonal elements: ' w w w V U W 6 (9) In particular, embedding the watermark in the LL frequency band is robust against typical low- SDVV ¿OWHULQJ LQFOXGLQJ *DXVVLDQ EOXUULQJ DQG JPEG compression, while embedding the water-
mark in the HH frequency band is robust against W\SLFDOKLJKSDVV¿OWHULQJLQFOXGLQJVKDUSHQLQJ and histogram equalization. Then, embedding the ZDWHUPDUNLQDOOIUHTXHQF\EDQGVLVVXI¿FLHQWO\ robust to a large class of attacks.
Here, we illustrate a DWT-SVD watermarking scheme. However, a full discussion on SVD trans- form will be provided in the following section.
SVD-Based Watermarking
Recently, some researchers have used the
SVD domain for the watermark embedding.
Advantages related to this transform concern
the robustness properties of the singular
values, as they remain invariant even if they
undergo further processing. The SVD is a
powerful tool that transforms a generic matrix
as a product of three matrices, with interesting
properties for several data processing applica-
tions. The SVD of a matrix
A(mun)(mdn)is
GH¿QHGDVWKHSURGXFWRIWKUHHVSHFLDOPDWUL-
ces: two orthogonal matrices,
(m m)U u
,
V(nun)such that
UUT VVT I, and a diagonal
matrix,
6 diag(σ1,σ2,"σm), whose main
entries
σi(i
1"m)are called singular values.
Singular values are in decreasing order. The
columns of U and V are referred as the left
and right singular vectors, respectively, of A.
In technical terms:
T wV U A 6 (10)Singular values play an important role in transforming elements from an m-dimensional vector space into elements of an n-dimensional vector space. A vector
x
in m (n) may be expressed in terms of the left (right) singular vectors. The geometrical interpretation of SVD is that this decomposition dilates or contracts some components of x according to the magnitude of corresponding singular values. Given a matrixA, a singular value and a pair of singular vectors DUHGH¿QHGDVDQRQQHJDWLYHVFDODUıDQGWZR non-zero vectors u and v, such that:
u
Av
σ
(11) that is, the images under A of the singular vectorsm v v
v1, 2," are the vectors:
m mu u
u σ σ
σ1 1, 2 2,"
Equation (11) is equivalent to:
v u
AT σ (12) We remind the author two noticeable properties of SVD, which are at the basis of state-of-the-art watermarking algorithms: approximation and
perturbation.
Approximation. Let the full SVD decompo- sition of matrix A be given by equation (10). For some
k
r
, with r the rank of the matrix, it is possible to approximate A with Ak:T k k k k U V A 6 (13)
where Uk is a
nuk
matrix, whose columns are JLYHQE\WKH¿UVWk left singular of the SVD of A,that is u1,u2,"uk form an orthonormal basis
for the range of A and, VkT is a
kun
matrix,ZKRVHURZVDUHJLYHQE\WKH¿UVWk right singu- lar vectors of the SVD of A, that is v1,v2,"vk
form an orthonormal basis for the kernel of A.
) , , ( 1 2 k k diag σ σ "σ 6 . Equation (13) repre-
sents the best approximation of order k of the original A matrix among all the matrices of order
i, with
ik
. The 2-norm error of the approxi- mation related to equation (13) is equal to σk1 . The reader is referred to Golub and Van Loan (1996) for proof of this statement. This result is relevant for image compression. In fact, the SVD of an image returns the image coding in terms ofn basis images uivTi , which are weighted by the singular values σi. It holds that the approximation
of A with the reduced SVD of order k, Ak, is the optimal solution to minimize the Frobenius norm of the error matrix, that is, no other transform has better energy compaction than the reduced SVD of order k. The key idea is that the SVD captures the best possible basis vectors for the image, rather than using any compatible basis. Generally, k is such that
k
r
, where r is the rank of the matrix. Since rLVGH¿QHGDVWKHQXPEHURIOLQHDUO\LQGH- pendent columns (rows, respectively), it provides a measure for the redundancy of the matrix.Perturbation. Important properties of singu- lar values concern their accuracy and sensitivity. Small perturbations in the original matrix lead to small changes in the singular values. Consider a perturbation (
δA
) to the original matrix A, which produces (AδA
) as a result; now, we show the problem of the perturbation analysis by taking into consideration the singular value matrixȈ: V A A UT( δ ) δ6 6 (14)The matrices U and V preserve norms since they are two orthogonal matrices. Thus, it fol- lows that:
A
δ
δ6
(15)Related Work on SVD
SVD is a powerful tool for image processing, especially image compression and LPDJH¿OWHU- ing. Image coding by SVD goes back to the 1970s: Andrews and Patterson (1976) describe the statistical properties of singular values and singular vectors of images. In this work, the SVD by blocks is employed. Experimental results show that singular value mean and variance vary in a large range, which indicates the need for variable bit coding as a function of the singular value index. As for the singular vectors, they are well behaved in their range and tend to have an increasing number of zero-crossings as a function of the singular vectors index.
SVD may be combined with vector quantiza- tion (VQ), which is a low bit-rate compression technique (Chung, Shen, & Chang, 2001; Yang & Lu, 1995). Yang and Lu introduce two methods for image coding: iterative (I) and fast (F) SVD- VQ that lead to different bit-rates. Chung et al. introduce a novel SVD-VQ scheme for information hiding, which produces images with a high level of quality and good compression ratio.
Another interesting property of the SVD is related to the singular values ordering. The higher the singular values are, the more is the energy packed into them. Since the singular values are LQGHFUHDVLQJRUGHULWKROGVWKDWWKH¿UVWVLQJXODU value of an image contains more information (signal components), while the remaining lower singular values are associated to the noise com- ponents. This property is at the basis of image ¿OWHULQJLWLVSRVVLEOHWRUHGXFHQRLVHLQWKHLPDJH E\WDNLQJLQWRFRQVLGHUDWLRQRQO\WKH¿UVWVLQJXODU values whose magnitude is upon a given threshold. Muti, Bourennane, and Guillaume (2004) discuss the equivalence of SVD and simultaneous row and column principal components analysis (PCA) and DGRSWWKH69'WUDQVIRUPIRULPDJH¿OWHULQJ,Q particular, the SVD transform leads to an image with higher quality. The singular values distribu- tion changes according to the image content. For example, their magnitude covers a large range in the case of a diagonal line, while the energy is FRQFHQWUDWHGLQWKHYHU\¿UVWVLQJXODUYDOXHLQ the case of a vertical line. This means that the more the principal directions of an image are aligned with the row and column directions, the lower is the number of singular values required to GHVFULEHLPDJHFRQWHQW7KXVLPDJH¿OWHULQJFDQ be improved by making some image directions to be aligned with row or column directions, that is, by means of image rotation.