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A robust data hiding scheme in tree-structured Haar transform domain

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AROBUSTDATAHIDINGSCHEMEINTREE-STRUCTUREDHAARTRANSFORMDOMAIN
Michela Cancellaro
1
, Marco Carli
1
, Alessandro Neri
1
and Karen Egiazarian
21
Applied Electronics Dept., University of Roma TRE,Roma, Italy, (carli, neri)@uniroma3.it
2
Institute of Signal Processing, Tampere University of Technology,P.O.Box 553, FIN-33101 Tampere, Finland, karen.egiazarian@tut.ﬁ
ABSTRACT
A robust data hiding method is proposed. A generalizedwavelet transform domain is used to design a private-keyembedding scheme. Speciﬁcally, a binary mask is createdin the Tree Structured Haar Transform (TSH) to selectthe most perceptually signiﬁcant coefﬁcients using level-adaptive thresholding. Experimental results show the ef-fectiveness of the proposed approach.
1. INTRODUCTION
The rapid growth in the production and consumption of the digital multimedia data last two decades has led thescientiﬁc community to deal with issues such as digitalrights management, tampering tools, and more generally,digital content security. A possible solution to these secu-rity issues is given by data hiding. Usually it is performedby embedding digital information in an imperceptible wayinto a digital signal.Several methods have been proposed to hide data into dig-ital images by using spatial or transform features [1], [2],[3]. Among them, a relevant area of research is devotedto embedding methods operating in wavelet domain [4].In fact, this domain allows more freedom in choosing thecoefﬁcients to be modiﬁed; it produces less noticeable ar-tifacts, and the robustness of the embedded image versusattacks is improved.The image decomposition operation results in four im-age
subbands
corresponding to the low pass (LL), the hor-izontal (HL), the vertical (LH), and the diagonal (HH)high pass components. Each band can be further decom-posed using the same algorithm. The possibility of a wideset of coefﬁcients suitable for the embedding, has also ledthe need for a wise selection of them. Several techniqueshave been proposed in literature for selecting the best setof coefﬁcients where to embed the data. A common crite-ria is to balance the robustness versus the perceived qual-ity of the watermarked signal. When the watermark dataare inserted in the low frequency components, the embed-ded information will be less sensitive to image distortionswhile the artifacts introduced by the embedding can beannoying. On the contrary, when the watermark data areinserted into middle and high frequencies the perceivedquality is high but the mark is more fragile.In our work, several combinations to estimate the best setof coefﬁcients to be marked have been tested in order toachieve robustness while preserving the perceived quality.Based on the analysis of the results, the watermark is em-bedded in every
detail
subband but the LL subband. In allthe performed tests, the watermark was a pseudo-randombinary matrix having the dimension of the srcinal image.The embedding domain, we have chosen, is a generaliza-tionoftheclassicalHaarwaveletdomain. Inthefollowingparagraph, the Tree-Structured Haar transform domain isintroduced.
2. TREE-STRUCTUREDHAARTRANSFORM
Classical continuous Haar functions are deﬁned by thedyadic splitting of the time sampling interval.Recently, a generalization of the Haar transform calledtree-structuredHaar(TSH)transformshasbeendeveloped[5]. It is based on changing the rules of the time inter-val splitting. To deﬁne a tree-structured Haar transformwe would need the following deﬁnitions. A rooted tree iscalled
binary
if each node has outdegree at most two. Thelength of the path from the root to the node is called the
depth
of that node. A node is a
splitting node
if it has out-degree two. A binary tree, whose all non-leaf nodes aresplitting nodes, is called full. If all the leaves have samedepth, the tree is called complete. If there is a path withsrcin
a
and end
b
, then
a
is predecessor of
b
and
b
is suc-cessor of
a
. If nodes belongs to adjacent levels of the tree,
b
is child of
a
and
a
is parent of
b
.The edges and nodes of the tree are labeled as follows.If the node has two children (a splitting node) then label
0
is assigned to the left outedge of that node and label
1
to its right outedge. If the node has only one child, thenlabel
2
is assigned to its only outedge. Each node of thetree is indexed by a ternary vector
(
α
1
,α
2
,
...α
k
)
where
α
j
∈
0
,
1
,
2
,j
∈
1
,
2
,...k
; k is the depth of this node and
α
j
are labels of the edges to that node starting from theroot of the tree.Each node
(
α
1
,α
2
,
...α
k
)
is also labeled by the num-ber
ν
(
α
1
,α
2
,...α
k
)
of successors, i.e. the leaves in thesubtree rooted in the node
α
1
,α
2
,
...α
k
. The following in-tervals are associated with the constructed tree:
I
root
=[0
,
1)
,I
0
=
I
(
root,
0)
,I
1
=
I
(
root,
1)
, where
I
0
and
I
1
are
the left and the right sub-intervals of
I
root
correspond-ing to two children of the root. For each non-root, non-terminalnode
(
α
1
,α
2
,
...α
k
)
,
I
(
α
1
,α
2
,
...α
k
) = [
a,b
)
, where
0
≤
a < b <
1
,k
∈
1
,
2
,...,n
−
2
.
Then
I
(
α
1
,α
2
,
...α
k
,
0)
= [
a,a
+
ν
(
α
1
,α
2
,...αk,
0)
ν
(
α
1
,α
2
,...αk
)
(
b
−
a
))
,I
(
α
1
,α
2
,
...α
k
,
1)
= [
a
+
ν
(
α
1
,α
2
,...αk,
0)
ν
(
α
1
,α
2
,...αk
)
(
b
−
a
)
,b
)
.
(1)For each non-splitting, non-terminal node
(
α
1
,α
2
,
...α
k
)
,
I
(
α
1
,α
2
,
...α
k
,
2)
=
I
(
α
1
,α
2
,
...α
k
)
.
(2)
I
(
α
1
,α
2
,
...α
k
,
0)
and
I
(
α
1
,α
2
,
...α
k
,
1)
split the interval
I
(
α
1
,α
2
,
...α
k
)
into two non-intersecting sub-intervals in theproportion of the numbers of leaves that are successors of these nodes. The set of TSH functions is deﬁned from theconstructed tree according to the following equations:
TSH
root,
0
(
t
) =
1
√
N
,
if
t
∈
[0
,
1)
TSH
root,
1
(
t
) =
ν
(1)
Nν
(0)
,
if
t
∈
I
0
−
ν
(0)
Nν
(1)
,
if
t
∈
I
1
(3)where
ν
(0)
and
ν
(1)
are the labels of the left and rightchildren of the root of the tree, respectively. To each non-root splitting node with index
(
α
1
,α
2
,
...α
k
)
it is associ-ated a basis function
TSH
(
α
1
,α
2
,
...α
k
)
(
t
)
,t
∈
[0
,
1)
,k
∈
1
,
2
,...,n
−
1
deﬁned by
TSH
(
α
1
,α
2
,...,α
k
)
(
t
) =
p
(
k
)
,
for
t
∈
I
(
α
1
,α
2
,...,α
k
,
0)
q
(
k
)
,
for
t
∈
I
(
α
1
,α
2
,...,α
k
,
1)
0
,
otherwise(4)where
p
(
k
) =
ν
(
α
1
,α
2
,...,α
k
,
1)
ν
(
α
1
,α
2
,...,α
k
)
ν
(
α
1
,α
2
,...,α
k
,
0)
,
(5)
q
(
k
) =
−
ν
(
α
1
,α
2
,...,α
k
,
0)
ν
(
α
1
,α
2
,...,α
k
)
ν
(
α
1
,α
2
,...,α
k
,
1)
.
(6)The set of TSH functions form a set of orthogonal func-tions [6].The system of classical Haar functions (normalized) is aparticular case of this system, when
ν
(
α
1
,α
2
,...,α
k
) = 2
n
−
k
,ν
(
α
1
,α
2
,...,α
k
,
1) =
ν
(
α
1
,α
2
,...,α
k
,
0) = 2
n
−
k
−
1
.
(7)It corresponds to the complete full binary tree. AnotherparticularcaseofthesystemcorrespondstotheFibonacci-Haar functions.In this case:
ν
(
α
1
,α
2
,...,α
k
) =
φ
p
(
n
−
k
)
,ν
(
α
1
,α
2
,...,α
k
,
0) =
φ
p
(
n
−
k
−
1)
,ν
(
α
1
,α
2
,...,α
k
,
1) =
φ
p
(
n
−
k
−
p
)
,
(8)where
φ
p
(
i
)
is the Fibonacci
(
p,i
)
number (the case of
p
= 1
stands for the classical Fibonacci sequence 1, 1, 2,Figure 1. Decomposition matrices for the TSH matrix forthe tree shown.3, 5, 8, 13, ...).The discrete TSH functions for a given tree are deﬁned bysampling the continuous TSH functions at N points,
jN
,for
j
∈
0
,
1
,...,N
−
1
. The
(
N
×
N
)
orthogonal matrix,T, whose rows are these functions, is the TSH transformmatrix for a given tree.The TSH transform for an input vector
f
is
g
=
T
×
f.
(9)The classical Haar transform can be computed efﬁ-ciently with an algorithm of linear complexity in whichtheHaartransformationmatrixisdecomposedintoaprod-uct of sparse matrices. This reduced the complexity fromthe
(
N
−
1)
2
addition and
N
2
multiplication operationsof the direct implementation to
2(
N
−
1)
addition and
N
multiplication operations. In [5], the fast tree-structuredHaar transform algorithm was developed. It requires only
2(
N
−
1)
additions and
3
N
−
2
multiplications to performany one-dimensional TSH transform. An example of theTSH matrix decomposition (resulting in the fast computa-tional algorithm) is shown in Figure 1.Two-dimensional TSH transform matrix can be deﬁned asa tensor product of one-dimensional transform matrices.In other words, similar to the classical case, transforma-tion by rows of an input image is followed by a transfor-mationbyitscolumns. Thiswillleadtothewaveletpackettype of decomposition. In order to obtain a logarithmictree decomposition, a step-by-step transform has to be ap-plied as it is demonstrated in Figure 2. Figure 3 shows anexample of three-level decomposition. As it can be noted,this is a wavelet-like transform since it follows the sameprinciple as the Haar transform. The innovation is the ir-regular decomposition of the image constructed from the
Figure 2. Logarithmic tree decomposition scheme in two-dimensional case.Figure 3. A TSH decomposition of Lena (
256
×
256
pix-els).particular tree. As it will be seen in the next Section, itis possible to use this innovation as a steganographic keyof a system, that is to keep secret a key that permits to anauthorized user to build the correct tree, i.e. the correctdecomposition.
3. DISCONTINUITYPOINTVECTORANDTSHTRANSFORM
To reduce the computational complexity and to gain inhigh capacity embedding by a key-dependent watermark-ing algorithm, we propose to use a secret key to generatethe tree in the TSH domain. In the previous section, ithas been shown that the TSH transform of an image is ob-tained according to a particular tree used. Given the valueof the root of the tree, it is split according to a vector, the
Discontinuity Point Vector
(DPV), composed by the nodesrepresenting the points in which a decomposition has to beexecuted.An example in Figure 4 is presented to clarify the process.In this example, the root of the tree has value
8
. Let usanalyze the DPV [3, 6, 4, 2]. The root, labeled by
8
, de-scribes a vector of length
8
with the elements labeled from1 to 8. The DPV indicates that the ﬁrst decomposition willFigure 4. Construction of a tree from the DiscontinuityPoint vector: (a)the tree is constructed and after (b)it iscompleted.be present at the third node, that is
8
is divided in propor-tion
3 + 5
; the second decomposition is at sixth node, i.e.in the part that has
58
of the value of the root; so
5
will bedecomposed in
3 + 2
. As the second decomposition, thethird element of the vector indicates that there is a split atthe fourth node; since
48
>
38
the cut is in the part that has
58
of the value of the root, in particular in the
35
part i.e.
3
is divided in
1 + 2
as
1
correspond to the fourth node.Finally the last element of the vector is
2
that is simply acut in the second node, i.e.
3
at level
1
is decomposed in
2+1
. Now the tree can be completed to obtain the tree inFigure 4(b).Note that the DPV
[3
,
2
,
6
,
4]
will result in exactly thesame tree, but the the DPV
[2
,
4
,
6
,
3]
will result to theone in Figure 5(a) and the DPV
[2
,
3
,
6
,
4]
in the one inFigure 5(b). If, for example, the DPV is
[4]
the corre-sponding tree, once complete, it describes the classicalHaar decomposition and the DPV
[5
,
3
,
2]
once completeis a Fibonacci tree. Both these trees are shown in Figures5(c) and 5(d). Once they are made complete it is possibleto decompose a vector that has the length equal to the la-bel at the root of the tree according to the decompositionof the fast algorithm explained in the previous chapter. Ina bi-dimensional case, the root of the tree has value equalto the dimension of an image; once the tree is completed,it can be applied to all the vector-rows and after to all thevector-columns of the image. In other words, it is possibleto decompose an image according to the TSH transform just from the knowledge of the Discontinuity Points vec-tor. Thus, if this vector is not available it is impossible toperform the correct decomposition.In this paper the DPV is used to generate the domain inwhich a watermark will be embedded. A secure and ro-bust algorithm has been realized avoiding the high com-putational complexity of Fridrich’s method [6], [7].
4. DATAEMBEDDINGINTSHDOMAIN
In this paper, we have generalized the algorithm intro-duced by Kim [8]. The embedding scheme, based onthe level-adaptive thresholding, shows improved robust-
Figure 5. Example of construction of trees from the Dis-continuity Point vectors: (a),(b) arbitrary tree; (c)classicalHaar tree; (d)a Fibonacci tree.ness when adapted in the TSH transform.Embedding the watermark in the most signiﬁcant coef-ﬁcients allows to create a mask to whom the watermark adapts itself. In other words, when the image is trans-formed according to the TSH transform, a mask is createdby analyzing the signiﬁcant coefﬁcients, i.e. those that aregreater than the thresholds computed separately for eachsubband.The mask has binary values: it is equal to
1
if the coef-ﬁcient correspondent (in the same position) in the trans-formed image takes part to the embedding and it is equalto
0
otherwise. This mask is multiplied (pixel-by-pixel)by the watermark and the resulted adapted-watermark isembedded. Tests have shown that the robustness of themethod relies on the uniqueness of the mask: if, in thedetection, the image is not the watermarked one with thesrcinal watermark, it is not possible to extract from it themask used in the embedding since the mask extracted willbe different with respect to the attack executed and to theinserted watermark.The embedding procedure can be summarized as follows:1. THS transform of the srcinal image;2. computation of the
multi-level
thresholds set for se-lecting the signiﬁcant coefﬁcients;3. watermark insertion by using additiveor multiplica-tive rule;4. inverse TSH of the watermarked data.Insertion process is implemented in four steps:
•
TSH transform of the srcinal image according tothe Discontinuity Point vector;
•
calculation of multi-level thresholds in order to se-lect the signiﬁcant coefﬁcients;
•
watermark insertion;
•
inverse TSH transform of the coefﬁcients with wa-termarks.The watermark is a pseudo-random binary matrix
{−
1
,
1
}
that has the dimension of the srcinal image generatedfrom a private key held by authorized users
w
i,j
∈ {−
1
,
1
}
.
(10)To ﬁnd the perceptually signiﬁcant transform coefﬁ-cientsthatindicatewhichcoefﬁcientofthewatermarkwillbe embedded, for each subband the threshold is computedas
T
b
= 2
⌊
log
C
b
⌋−
1
,
(11)where
C
b
is the maximum coefﬁcient of the subband
b
and
⌊
x
⌋
represents the largest integer which is not greater than
x
. Coefﬁcients greater than the threshold correspond tothe values
1
in the mask
M
. Thresholds have been com-puted in a different way with respect to [8] in order toembed a larger watermark. Even if the subbands have nota spatial correlation from a level to the following one (thispropriety is used by all classic watermarking methods inDWT domain), these coefﬁcients show a probabilistic cor-relation between the subbands at different levels.The watermark is embedded only to the selected coefﬁ-cients as
C
′
i,j
=
C
i,j
+
αC
i,j
w
i,j
M
i,j
(12)where
C
i,j
is the selected coefﬁcient,
w
i,j
is the water-mark and
α
is the scale factor that is different accordingto the level of the examined subband. In this case a smallscale factor is used in the LL while for other subbandsthe scale factor is properly tuned according to the fact thatas the mean of wavelet coefﬁcients is reduced by a half as one level goes up, the scale factor is increased twice.Scale factors of
0
.
1
,
0
.
2
, and
0
.
4
are used for level
3
,
2
,
1
respectively.The detection process is the inverse process of the em-bedding. The srcinal watermark and the DiscontinuityPoint vector are used to detect the watermark. Watermark detection process is implemented in three steps:
•
TSH transform of the given image according to theDiscontinuity Point vector
•
calculation of multi-level thresholds in order to se-lect the signiﬁcant coefﬁcients
C
′′
;
•
calculation of the correlation.Thresholds for every subband are calculated exactly likein the embedding part due to the similarity between thesrcinal host image and the watermarked image: the per-ceptuallysigniﬁcantwaveletcoefﬁcientsforthewatermarkedimage will be close to those used in the
embedder
. Thenormalized correlation coefﬁcient is computed betweenthe embedded watermark
V
=
w
i,j
M
i,j
and watermarkedcoefﬁcients
C
′′
,
ρ
=
i
j
V
i,j
−
V
C
′′
i,j
−
C
′′
i
j
V
i,j
−
V
2
C
′′
i,j
−
C
′′
2
(13)This quantity is bounded by
−
1
and
+1
. The greater is
ρ
,the more conﬁdent of the existence of the watermark thedetector is. This value is compared with a threshold com-puted in a probabilistic way applying 500 different marksto the srcinal image: if the correlation is greater than thethreshold, then the watermark is detected. In the experi-ments it was computed even the correlation between theextracted watermark and the watermarked coefﬁcients: nodifference was noted with respect to the correlation de-ﬁned above since, applying
500
different watermarks, thesrcinal mask was always identiﬁed. Without using thismask, it is not possible to extract the srcinal watermark.
5. TESTS
Tests were executed on grayscale
256
×
256
images. Todecompose the image, a Discontinuity Point vector equalto [107,3,189] was used. A third level decomposition wasrealized.Since the results were conform between each other, justthe graphs that refer to one of these images (Lena) areshown. The algorithm in which the LL of the last levelis excluded from the embedding is distinguished from thecase in which it is involved in the embedding. As it can beseen, the second algorithm is much more robust than theﬁrst one as it was expected since the LL contains most of energy and most of information of the srcinal image.The correlation for
500
different random watermarks isshown in Figures 6 and 8 together with the response of the detector in the case of gaussian noise attack (Figures7, 9. Other attacks considered are: equalization attack,rotation-cropping attack and JPEG compression.All the results are shown in table 6 and in table 7. Fromthese results it is clear how the size of the watermark in-creases with the texture of the image. In ”Airplane”
2118
coefﬁcients are used to embed the watermark while in”Stream and Bridge” the number of coefﬁcient taking partto the embedding is
12817
. As it is expected, at the sametime the PSNR decreases proportionally.
6. CONCLUSION
In this paper a robust data hiding method has been pro-posed. It is based private-key embedding scheme per-formed in the Tree Structured Haar Transform domain.The designed algorithm allows secure and robust embed-ding while presenting low computational complexity. Ex-perimental results show the effectiveness of the proposedapproach.
7. REFERENCES
[1] A. Piva M. Barni, F. Bartolini, “Improved wavelet-based watermarking through pixel-wise masking,”
IEEE Transaction on Image Processing
, vol. 10, no.5, pp. 783–791, may 2001.[2] J. Bloom I. Cox, M. Muller,
Digital Watermark-ing
, Society of Photo-Optical Instrumentation Engi-neers(SPIE) Bellingham, WA, USA.[3] Roman Tzschoppe Joachim J. Eggers, Robert Bamland Berd Girod, “Scalar costa scheme for informa-tion embedding,”
IEEE Transaction on InformationTheory
, vol. 51, no. 4, pp. 1003–1019, april 2003.[4] Andreas Uhl Peter Meerwald, “A survey of wavelet-domain watermarking algorithms,” 2000.[5] J. Astola K. Egiazarian, “Tree-structured haar trans-form,”
Journal of Mathematical Imaging and Vision
,pp. 269–279, 2002.[6] R.J.Simard J. Fridrich, A.C. Baldoza, “Robust digi-tal watermarking based on key-dependent basis func-tion,”
Information hiding: second international work-shop, volume 1525 of Lecture notes in computer sci-ence
, april 1998.[7] J. Fridrich, “Key-dependent random image trans-forms and their application in image watermarking,”
Proceedings of the 1999 International Conference on Imaging Science, Systems and Technology, CISST’99
, june 1999.[8] Young Shik Moon Jong Ryul Kim, “A ro-bust wavelet-based digital watermarking using level-adaptive thresholding,”
IEEE
, 1999.

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