A novel spatial data hiding scheme based on generalized

A novel spatial data hiding scheme based on generalized
Fibonacci sequences
E. Mammia , F. Battistia , M. Carlia , A. Neria , and K. Egiazarianb
a University
b Tampere
Roma TRE, Roma, Italy
University of Technology, Tampere, Finland
ABSTRACT
This paper presents a novel spatial data hiding scheme based on the Least Significant Bit insertion. The bitplane decomposition is obtained by using the (p, r)-Fibonacci sequences. This decomposition depends on two
parameters, p and r. Those values increase the security of the whole system; without their knowledge it is
not possible to perform the same decomposition used in the embedding process and to extract the embedded
information. Experimental results show the effectiveness of the proposed method.
Keywords: LSB insertion, data hiding, (p, r)-Fibonacci sequences, key dependent domain.
1. INTRODUCTION
Multimedia communication is a key-factor in the information technology revolution. It allows to share, to
transmit, and to acquire digital information. In this dynamical scenario, the security and the privacy of the
content is a big concern. A basic requirement is that only the intended recipient should be able to gain access
to the hidden contents of the transmitted data. A common solution to this problem is the use of a ciphering
system to conceal the information content of the message.
Sometimes, it is also desirable to hide to an external observer the whole transmission instance, that can be
even the fact that the transmission is taking place is secret. Several techniques for hiding the sensitive data,
the watermark, into an apparently innocuous document have been recently proposed in literature. Since the
final document tends to appear a plausible one to both humans and machines, the possibility of detecting the
presence of the secret message is becoming harder.
In literature many embedding schemes have been proposed, some of them are working in a transformed
domain (Discrete Cosine Transform (DCT)1 , Discrete Fourier Transform (DFT), Discrete Wavelet Transform
(DWT)2 ), some others operate directly in the spatial domain. In both cases the embedding scheme has to fulfill
three main requirements concerning visibility, robustness, and capacity. Visibility is related to the ability of a
human observer to perceive distortions impacting on quality, as well as to the possibility of detecting hidden data
by some statistical analysis. Robustness refers to the impossibility to remove, modify, or substitute the secret
message once it has been inserted. The capacity of the embedding method is defined as the maximum amount
of information that can be hidden in the cover image for a given visibility level. Those three requirements are
strictly related and the increase of one of them, results in the decrease of the others; a trade-off is often necessary.
In our work we propose an extension of a spatial technique: the Least Significant Bit (LSB) insertion3 -4 - . The classic LSB algorithm allows to obtain an high capacity at the expenses of its robustness. This is
due to the fact that an attacker can easily remove or modify the watermark, when it is hidden in the least
significant bit-plane, without significantly affect the quality of the cover image. In this paper we propose an
alternative bit-plane decomposition based on the (p, r)-Fibonacci sequences. In essence, based on an extension
of the Zeckendorf theorem, we convert the binary representation of an image into the representation using the
(p, r) Fibonacci numbers. The watermark is then inserted by modifying the elements of the bit-planes of the
Fibonacci domain. Since the transformation strictly depends on the actual pair (p, r), greater robustness and
undetectability can be achieved when this pair is kept secret.
5 6
The paper is organized as follows. In Section 2 the Fibonacci representation of a decimal number is presented.
In Section 3, the proposed embedding and extraction schemes are described. The experimental results are
reported in Section 4. Finally in Section 5 we draw our conclusions.
Corresponding author: [email protected], [email protected]
2. THE FIBONACCI P,R SEQUENCES
The easiest way to hide digital data in a host signal is the embedding in the spatial domain. This technique
is characterized by low computational complexity, low cost, and low delay. Furthermore, it allows the selection
of the spatial location to embed the data. This is extremely useful when the perceptual invisibility of the
superimposed watermark on the host data is a strict requirement.
Let us denote with I the original luminance component of the image with values in the range [0, 2L − 1], and
with w = {w0 , w1 , ..., wn−1 } a binary watermark.
Each pixel I(i, j) of the cover image can be represented through its binary representation, as follows:
I(i, j) =
L−1
X
ek · 2k ,
(1)
k=0
where ek are binary digits. In the classical LSB embedding methods, the secret message is inserted into the
least-significant bit planes of the cover image either by replacing or by modifying a subset of them, according
to a specific invertible function. The main advantage of such a technique is that, if L is large enough, the
modification of the LSB plane does not significantly affect the perceived overall image quality. As an example,
the visual impact of performing the embedding in different bit-planes is shown in Figure 1.
(a)
(e)
(b)
(f)
(c)
(d)
(g)
(h)
Figure 1. Visual impact when the LSB is performed in each bit-plane, from the least significant one (a) to the most
significant one (h).
The improvement we introduce in this paper is the embedding of the watermark into the bit planes associated
to the representation of the luminance component in terms of (p, r)-Fibonacci sequences. To illustrate the
method, we recall that a (p, r)-Fibonacci sequence is defined by the following recursive formula:

 0,
1,
Fp,r (n) =
 Pr
n < 0;
n = 0;
j=0 Fp,r (n − 1 − j · p), n > 0.
(2)
Let us notice that each element in the sequence is obtained by adding the previous r elements, taken at distance
p. Some examples of (p, r)-Fibonacci sequences, for r equal to 2 are shown in Table 1. As an extension of
the Zeckendorf theorem7 : ”Each positive integer m can be represented as the sum of distinct numbers in the
sequence of Fibonacci numbers using two no consecutive Fibonacci numbers”, it can be demonstrated that,
by using a (p, r)-Fibonacci sequence, with p ≥ 0, and r > 1, any positive natural number N can be always
represented as follows:
N=
n−1
X
ci Fp,r (i).
(3)
i=p
One drawback presented by the Fibonacci representation is the redundancy. In fact each decimal number can
have more than one representation. To obtain a unique representation of a number, using a (p, r)-Fibonacci
sequence, it is necessary to fulfill the following constraints:
• a valid (p, r)-Fibonacci coefficient vector c must contain less than p − 1 zeros between two ones.
• a valid (p, r)-Fibonacci coefficient vector c cannot contain more than r consecutive groups, being constituted
by one symbol equal to 1 followed by p − 1 symbols equal to 0.
Table 1. Generalized Fibonacci p-r-sequences when r=2, Fp,r=2 (n).
p=1
p=2
p=3
p=4
p=5
n=0
1
1
1
1
1
n=1
1
1
1
1
1
n=2
2
1
1
1
1
n=3
4
2
1
1
1
n=4
7
3
2
1
1
n=5
13
5
3
2
1
n=6
24
8
4
3
2
n=7
44
12
6
4
3
n=8
81
19
9
5
4
To demonstrate the importance of a key-dependent bit-plane decomposition, the Fibonacci representation of
the number 255 is shown when r = 1 (Table 2) and r = 2 (Table 3) for different values of p. It is possible to
notice that different combinations of p and r allow to obtain different representations of the same number. It is
important to underline the relevance of the secret keys p, r; without disclosing these two parameters the hidden
message cannot be recovered as shown in Figure 2.
Table 2. Examples of the cp coefficients corresponding to the representation of the number 255 for the sequence Fp,r=1 (n).
p=2
p=3
p=4
0
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
Table 3. Examples of the cp coefficients corresponding to the representation of the number 255 for the sequence Fp,r=2 (n).
p=2
p=3
p=4
0
0
0
1
0
0
0
0
0
1
1
1
0
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
1
0
0
1
3. PROPOSED EMBEDDING SCHEME
The proposed technique is based on a modified Least Significant Bit insertion scheme. In particular we propose to
perform the bit planes decomposition by using the (p, r)-Fibonacci sequence. To reduce the perceived distortion
on the watermarked image we select the areas to be modified on the base of a Local Activity Index (LAI) that
takes into account the characteristics of the Human Visual System (HVS). In the performed simulations the LAI
is based on the variance brightness.
(a) Original watermark
(d) p=3 r=2
(b) p=0 r=1
(e) p=4 r=4
(c) p=2 r=3
(f) p=2 r=2
Figure 2. Example of extracted watermarks when wrong combinations of p and r (b-e) and the right combination (f)
are used.
3.1. Embedding procedure
Based on the previous considerations the embedding procedure is performed as follows:
• the luminance component of the cover image I of size N × N is partitioned in k non overlapping blocks bi
with i = 1, . . . , k of size 8 × 8 pixels;
• the LAI of each block bi is computed and the blocks are ranked in ascending order;
• the watermark is inserted in lexicographic order in the least significant bit-plane of the (p,r )-code of the
pixels belonging to the blocks with the highest rank; blocks with high activity usually contain contours,
textures, eventually noise, and they are more robust with respect to embedding, than those with low
activity, that contain mainly flat regions.
• for each bit in bi : if the uniqueness constraints are fulfilled and the Fibonacci (p, r) coefficient vector is
valid, the embedding is performed by substitution in the selected bit-plane otherwise the next bit in the
block is considered;
• if all the suitable bits in bi have been utilized, the next block is considered;
• when the watermark has been entirely inserted in the cover image, the watermarked image is reconstructed
by the Fibonacci’s representation of each blocks.
3.2. Blind extraction
The blind extraction of the watermark requires the knowledge of its length, the set of bit-planes where it has
been inserted, and the secret key pair (p, r). As illustrated in Figure 2, the use of a wrong pair leads to useless
images, even if the other parameters are correct.
It is important to underline that the embedding capacity is content related. In Table 4 the capacity for some
images and for some (p, r)-Fibonacci sequences are reported. For this set, the minimum embedding capacity is
equal to the 40% of the size of the original image.
Table 4. Capacity values (bit) for some example sequences Fp,r (n).
Airfield
Baboon
Barbara
Boat
F16
Hill
Lake
Lena
Peppers
p = 2, r = 2
166983
167306
170382
170205
174537
168690
166910
170713
165793
p = 3, r = 2
143075
151554
153918
154634
158953
153855
149246
152712
148103
p = 3, r = 3
143893
155722
155791
154419
158988
155629
153560
154684
150478
p = 4, r = 3
137945
140342
140526
141184
139738
138365
143099
139847
140450
4. EXPERIMENTAL RESULTS
To verify the effectiveness of the proposed method, several tests have been performed. A set of nine cover images
of size 512 × 512 pixels has been utilized. In the experiments the inserted watermark is a binary matrix of size
128 × 128 pixels. The perceived quality of the watermarked image has been evaluated through the Peak Signal
to Noise Ratio (PSNR):
µ
¶
L2
P SN R(db) = 10 log10
,
M SE
and the Weighted Peak Signal to Noise Ratio (WPSNR):
µ
W P SN R(db) = 10 log10
max(F )2
||N V F (F 0 − F )||2
¶
,
where F and F 0 represent the two images that are compared and NVF is the Noise Visibility Function whose
value is 1 in flat regions and zero in textured regions and along edges (Voloshynovskiy et al. 2001, Watson et al.
1997).
In the following the comparison between the Fibonacci and the classical binary decomposition is performed.
In the first case the parameters p and r allow to choose the number of bit-planes in which the image can be
decomposed. This is possible thanks to the Equation 3. On the other hand, in the binary representation the
number of bit-planes is strictly defined by the cardinality of the numbers to be represented. To compare the
(p, r)-Fibonacci sequences with the binary one we performed some simulations. Tables 5 and 6 report the PSNR
and WPSNR values obtained for Fp,r=1 (n) and Fp,r=2 (n) by varying the bit-plane chosen for the embedding
from the least significant one to the fifth least significant one for the image Peppers.
(a) PSNR(dB)
p=0
p=1
p=2
p=3
p=4
LSB
67,3
67,2
67,3
67,1
67,5
2
61,3
61,1
61,1
61,3
61,6
3
55,2
58,0
57,5
58,0
57,7
(b) WPSNR(dB)
4
49,2
53,3
55,2
55,3
55,6
5
42,9
49,0
51,7
53,7
53,6
p=0
p=1
p=2
p=3
p=4
LSB
79,1
79,1
79,1
79,2
79,1
2
73,2
73,2
73,1
73,1
73,1
3
67,1
69,8
69,6
69,6
69,6
4
60,1
64,8
67,0
67,1
66,9
5
52,4
60,1
63,2
64,9
65,1
Table 5. PSNR (a) and WPSNR (b) values (dB) obtained for five combinations of p and r=1 by varying the bit-plane
chosen for the embedding from the LSB to the fifth LSB for the image Peppers.
In Table 5 it is possible to compare the behavior of the binary decomposition (corresponding to the case when
p = 0, r = 1) versus some examples of Fp,r (n). The obtained results show a common trend for the PSNR and
(a) PSNR(dB)
p=1
p=2
p=3
p=4
p=5
LSB
67,2
67,4
67,1
67,0
67,1
2
61,2
61,1
61,1
61,1
61,3
3
55,4
57,7
57,9
57,8
57,8
(b) WPSNR(dB)
4
50,1
53,2
55,0
55,1
54,7
5
44,8
49,2
52,0
57,4
52,7
p=1
p=2
p=3
p=4
p=5
LSB
79,1
79,1
79,1
79,2
79,1
2
73,3
73,1
73,2
73,0
73,2
3
67,1
69,6
69,6
69,7
69,6
4
61,6
65,0
66,9
67,0
66,9
5
55,0
60,1
63,1
64,9
65,2
Table 6. PSNR (a) and WPSNR (b) values (dB) obtained for five combinations of p and r=2 by varying the bit-plane
chosen for the embedding from the LSB to the fifth LSB for the image Peppers.
WPSNR. An increase in the perceptual importance of the bit-plane where the embedding is performed, results in
a degradation in the values of both the quality metrics. It is possible to notice that the classical LSB technique
is equivalent to the Fibonacci LSB one in terms of PSNR and WPSNR, if the embedding is performed in the
least significant bit-plane. When increasing the perceptual importance of the selected bit-plane, the Fibonacci
case outperforms the classical one. In particular, when we consider the fifth least significant bit-plane, the PSNR
gains 11dB with respect to the classical LSB in the case Fp=5,r=1 (n). This suggests that different bit-planes
other than the least significant one can be used. Figure 3 shows the resulting watermarked images when the
binary representation is used (b) and when the (p, r)-Fibonacci sequences are used (c-d).
(a) Original image
(b) Fp=0,r=1 (n)
(c) Fp=2,r=1 (n)
(d) Fp=2,r=2 (n)
Figure 3. Example of watermarked images when different (p, r)-Fibonacci sequences are used.
Table 6 shows the results obtained for some Fp,r (n). In this case the PSNR and WPSNR are computed
when the watermark is inserted in the least significant bit-plane. Finally, in Table 7, the PSNR and WPSNR
are evaluated when the maximum capacity of the cover image is exploited.
Table 7. PSNR(dB) and WPSNR(dB) values between the original image and five watermarked images when the maximum
embedding capacity is exploited.
PSNR
p = 2, r = 2
p = 3, r = 2
p = 3, r = 3
p = 4, r = 3
WPSNR
p = 2, r = 2
p = 3, r = 2
p = 3, r = 3
p = 4, r = 3
Airfield
51,12
51,16
51,14
51,14
Airfield
65,98
65,41
65,81
65,66
Baboon
51,16
51,13
51,14
51,16
Baboon
66,38
66,33
66,38
66,31
Barbara
51,14
51,16
51,14
51,17
Barbara
66,28
66,36
66,33
66,29
Boat
51,15
51,14
51,15
51,15
Boat
66,26
66,28
66,24
66,23
F16
51,16
51,13
51,14
51,15
F16
66,22
66,03
66,11
65,88
Hill
51,14
51,12
51,16
51,16
Hill
66,29
66,27
66,31
66,38
Lake
51,14
51,14
51,13
51,12
Lake
66,19
66,27
66,27
66,27
Lena
51,13
51,14
51,16
51,13
Lena
66,19
66,19
66,33
66,19
Peppers
51,13
51,13
51,15
51,15
Peppers
65,99
66,02
66,12
66,05
5. CONCLUSIONS
In this paper we have introduced a novel data hiding scheme based on the (p, r)-Fibonacci sequences. We propose
a modification of the traditional LSB embedding technique. The luminance component of each pixel of the image
is decomposed by using two secret parameters p and r. This key-dependence increases the secrecy of the whole
system while maintaining a low computational complexity. The watermarked image presents less artifacts than
the ones obtained in the classical LSB scheme as demonstrated by the PSNR and WPSNR values.
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