International Journal of Signal Processing, Image Processing and Pattern Recognition
Vol.8, No.3 (2015)
Copyright ⓒ 2015 SERSC 189
watermark has better robustness when the image is to small perturbations, its singular value
won't have big change; (b) the singular value reflects the intrinsic characteristics of image
characteristics (energy), embedding watermark image singular value, the image of the visual
effect will not have too big effect, and it ensures the good image quality; (c) digital image can
effectively resist geometric attacks to enhance the robustness of the watermark for the
singular value of geometric distortion invariance. Therefore, using singular value
decomposition as a digital image watermark embedding and extraction method has good
practicability. For example, we use
matrix to represent the image. The singular
value decomposition (SVD) of matrix A is defined as follows: A = USVT. In the formula,
,
are orthogonal matrix;
is diagonal matrix and its non
diagonal elements are zero, on the diagonal elements of the
[
12
... 1 ,1,2,...,
r r m
im
] is the rank of matrix and it is equal to the number of
non-zero singular value,
is the singular values of A.
A. Method of the Singular Value Decomposition
Singular value decomposition is to use the characteristic value or singular value as
orthogonal basis in spatial orthogonal decomposition of signal characteristics to enhance the
coherence energy and suppress interference signals. Let us set the two-dimensional record
section for X, word number is m and the sampling points is n, the m * n order the singular
value decomposition (SVD) of matrix X can be turned into m * m order orthogonal array U
and m * n order diagonal matrix Σ and n * n order orthogonal array of V. The singular value
decomposition is a kind of analysis tool commonly used in numerical matrix. For any m * n
matrix B, it can be resolved in the following:
(1)
In the formula, U is consists of the characteristic value of the XXT vector; V is composed
of the characteristic value of the XTX vector; Σ consists of singular values, singular values by
the order of the matrix on the main diagonal and the number of non-zero singular values is the
same as the matrix of rank and the rest of the location of the elements is zero, so the 2D data
matrix X of the covariance matrix
can be represented by the correlation matrix
,
.
(2)
Obviously, the diagonal elements of the matrix are the autocorrelation values of zero. If
is the energy or the variance of each word and each unrelated, r = e, the above
matrix can be simplified into diagonal matrix
1
2
2
20
02
y
n
e
e
kk
e
(3)
If the square root of the characteristic value of matrix is the singular value of matrix, which
indicates that the strength of the energy related to the size of the singular value
, or the sum
of characteristic value
reflects the sum of signal energy
(4)
In the formula
1 1, 2,...,diag sigma sigma sigma r
,
,r = rank( A). The
Singular value σ likes the characteristic value, and
decreases rapidly. In most cases, the top