Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 1 (15) ISSN 1843-6188

ON HIGH CAPACITY DIFFERENCE EXPANSION REVERSIBLE WATERMARKING

Iulian UDROIU, Nicoleta ANGELESCU, Henri COANDA, Dinu COLTUC

Valahia University Targoviste E-mail: iudroiu@valahia.ro, dnicoleta@valahia.ro, coanda@valahia.ro, coltuc@valahia.ro

Abstract: In order to increase the hiding capacity provided in a single embedded level, this paper revisits the basic difference expansion (DE) version and focuses on two directions. The first direction is the increase of the amount of data embedded into each selected coefficient. The second one is the increase of the number of embedded coefficients. In a single embedding level, the proposed DE scheme can provide bit-rates of around 2bpp. Such bit-rates are obtained with the classical DE scheme by multiple embedding (in 4-5 levels), i.e., at a higher computational cost (the mathematical complexity linearly increases with the number of embedding levels). Experimental results on standard test images are provided.

Keywords: difference expansion, reversible watermarking, high-capacity.

1. INTRODUCTION

This paper discuss the problem of high capacity reversible watermarking algorithms. Our approach considers as a starting point the difference expansion (DE) algorithm and proposes high capacity expansions. Difference Expansion, introduced by Tian, [1], is the most popular approach to reversible watermarking. Most of the reversible watermarking schemes proposed so far are either modified versions of Tian’s DE, or inspired by it (see [2], [3], [4], [6], [7], etc.). Even if Tian’s DE provides high capacity and a good capacity/distortion ratio, there is a continuous interest in reversible watermarking schemes with improved capacity, lower distortion or lower mathematical complexity.The basic DE version performs one level of integer Haar wavelet transform on the cover image. Some high-frequency (HF) coefficients are selected and shifted to the left to free the LSBs in order to embed 1 bit of data/coefficient. Other HF coefficients are embedded by LSB substitution, as long as the substitution does not generate overflow or underflow. The marked pixels are obtained by inverse transform of the low-frequency coefficients and of the modified HF ones. A location map is used to indicate, at detection, the selected coefficients. The location map is lossless compressed and inserted within the watermark.For natural images, the first level of embedding of DE watermarking provides almost 0.5 bpp, i.e., its theoretical upper bound. This is due to the high correlation existing between pixels. Thus, data can be

embedded into almost all the HF coefficients and, meantime, the location map can be very efficiently compressed. More hiding capacity can be obtained by multiple levels of embedding. The capacity decreases with the embedding level. Each level of watermarking increases the difference between pixels. Thus, the correlation between pixels decreases. As an immediate consequence, less HF coefficients are available for data embedding and the size of the compressed map increases, as well. Generally, by 4-5 embedding levels, embedding bit-rates around 2 bpp can be expected.High capacity reversible data embedding is of interest for several data hiding applications. For instance, in stereo imaging, by embedding into one frame of the stereo pair the information needed to recover the other frame, the transmission/storage requirements are halved [8]. Furthermore, the content of the image remains available and, at detection, if the embedding bit-rate is high enough, the stereo pair is exactly recovered.This paper aims at increasing the capacity provided in a single embedding level by the DE reversible watermarking. This paper revisits the basic DE version and focuses on two directions. The first direction is the increase of the amount of data embedded into each selected coefficient. The second one is the increase of the number of embedded coefficients.The outline of the paper is as follows. The increase of the amount of data embedded per coefficient from 1 bit to an integer in [0, n -1], n ≥ 2, is discussed in Section II. The increase of the number of embedded coefficients and the resulted algorithms are presented in Sections III and IV, respectively. Experimental results on standard test images are provided in Section V. Finally, the conclusions are drawn in Section V.

2. EMBEDDING MORE DATA

The HF coefficients of the integer Haar transform are exactly the differences between pairs of adjacent pixels. The shifting to the left by 1 position is a simple multiplication by 2. Tian’s DE embeds 1 bit into a pixel pair by increasing two times the difference between adjacent pixels. To embed more than 1 bit, more room should be produced. Let n be an integer, n ≥ 2. In the sequel, we show that by increasing n times the difference

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Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 1 (15) ISSN 1843-6188

between a pair of adjacent pixels, an integer w in [0, n-1] can be embedded.Let ix , 1+ix ∈ [0,255] be a pair of pixels and let l and h

be their average and difference, respectively:

l=

++

21ii xx

; h= ix - 1+ix (1)

The modified DE transform is defined as: h ( )1+−= iin xxn (2)The embedding of integer w [ ]( )1,0 −∈ nw into the expanded difference is:

( ) wxxnwhh iinw +−=+= + 1 (3)

Finally, the watermarked pixels are:

−=

++= + 2

;2

1 '1

' wi

wi

hlxhlx (4)

Given the watermarked pixels '1

' , +ii xx , the message w

should be extracted and the original pixels 1, +ii xx should be recovered. The embedded data can be extracted as:

w = ( )'1

'+− ii xx mod n (5)

The proof is immediate:

,22

1'1

' wnhwnhwnhxx ii +=

++

++− + with

[ ]1,0 −∈ nw (we used that aaa =

++

21

2).Since

[ ]1,0 −∈ nw , it can be obtained from the pixel difference as the integer reminder after division by n. Furthermore, h and l are obtained as:

nwxxh ii −−= +

'1

',

+= +

2

'1

'ii xxl (6)

Once h, l available, the original pixels are recovered as:

−=

++= + 2

;2

11

hlxhlx ii (7)

It should be noticed that Tian’s scheme is recovered forn = 2, when w is one bit. By increasing n times the difference between a pair of pixels, an integer data in the range [0,n-1] is reversibly embedded into the pair of pixels. Provided that no overflow/underflow appears (i.e., the pair is expandable), the proposed DE approach can embed log 2 (n) bits into each transformed pair.At detection, the transformed pairs should be retrieved. A straightforward method is the use of a location map. For each pair of pixels, the corresponding bit of the location map shows if the pair was transformed or not. In order to reduce its size, the location map is compressed. The compressed location map can be embedded in some predetermined locations (for instance first or last rows of the image) by LSB substitution. Meantime, the substituted bits should be

saved and embedded (together with the watermark) into the transformed pixel pairs. Obviously, the pairs of pixels where the map is embedded cannot be used for data embedding by DE expansion and furthermore, the substituted bits (which have exactly the size of the map) should be embedded by DE expansion into the remaining pairs. The procedure is very simple, but unless the case of a negligible size compressed map, it is not efficient.Tian’s procedure, [1], uses for data embedding not only expandable pairs. For nonexpandable pairs, a bit of data is substituted into the LSB of h, while the LSB of h is saved within the watermark. The pairs are called changeable if the substitution of the LSB of h does not generate overflow or underflow at the inverse transform of equations (7) and they are called nonchangeable, otherwise. The expandable pairs are also changeable and they remain changeable after data embedding by DE expansion. Similarly, regardless the value of the substituted bit, the changeable pairs remain changeable after substitution. The watermark (including the compressed map and the string of substituted LSBs) is embedded into the selected expandable pairs and into the remaining changeable pairs. At detection, the nonchangeable pairs are detected. The watermark is extracted by concatenating the LSBs of pixel differences. The watermark is decoded and then, the information needed to recover the original image (location map and string of saved LSBs) becomes available.With the modified DE transform, the LSB substitution is replaced by the addition of an integer d∈ [0,n-1] as follows:

dnhnhd +

= (8)

Furthermore, the embedded pixel pair "1

", +ii xx is obtained

by equations (7), where wh is replaced by dh .A pixel pair is changeable, if for any d ∈ [0, n- 1], the embedded pair "

1", +ii xx ∈ [0, 255]. As for equations (11),

overflow and underflow are checked against the embedding of maximum and minimum values of d. Thus, when h ≥ 0 (x 1+≥ ii x ), the overflow should be checked on the first pixel of the pair against the embedding of 1−= nd and the underflow on the second pixel at the embedding of 0=d , while when h < 0, one should check the underflow on the first pixel and the overflow for the second one. For a changed pair, "

1", +ii xx ,

the embedded data is extracted as:

( ) nhnxxd dii modmod"1

" =−= + (9)

Equation (9) is identical to equation (5), i.e., the embedded data is extracted with the same equation as for the pairs embedded by DE expansion.Instead of saving the simple LSB of h, an integer

[ ]1,0 −∈ nc should be saved,

−=nhnhc . At detection,

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ISSN 1843-6188 Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 1 (15)

once c is available, the original h is immediately

recovered as:

+=

nhnch d . The average is

( ) 2"1

"++= ii xxl and the original pixel values are

recovered with equations (7). Further details are given in [10].

3. EMBEDDING INTO MORE PAIRS

The basic DE considers disjoint pairs. For an image having P pixels, the maximum number of disjoint pair is P/2. If one considers pairs having some common elements, the total number of pairs increases. Such a pairing is done in [9]. Image pixels are indexed according to a scanning ordering and the pairing is performed by grouping the pixels located at i and i+1 ( )1,..,1,, 1 −++ Pixx ii . Compared with the disjoint pairing, this dense one doubles the number of pairs.The basic DE scheme cannot work directly with the dense pixel pairing. For the dense pairing, once a pair is transformed and embedded, both pixels are modified. One out of the two modified pixels is further considered in another pair, either an expandable, or a changeable one and, so on. A problem is that, in order to decide if the current pair is expandable, one should know the exact values of both pixels, i.e., the data embedding into the pixel of the formerly processed pair should be already finished. This is not possible since one cannot embed data until the entire information to be included into the watermark (map, integer codes) is available. Another problem is at detection. In order to detect if a pair is changeable or not, both original pixel values should be known. One of the two pixels is part of another pair from which the embedded data has already been extracted, but the recovery of the original values may not be possible. The recovery can be done only after the entire watermark is completely extracted and decoded.From the above comments, it follows that in order to use a non-disjoint pairing, at least two modifications should be done. First, data embedding principle should be modified to allow a late data embedding. Second, a different scheme to identify, at detection, the changeable/nonchangeable pairs should be introduced.The late data embedding can be solved by replacing the embedding into both pixels of equation (4) by the embedding into a single pixel, namely the one that will not be further considered for computation into another pair. Let 1, +ii xx be the current pair (pair i). By considering the pairing defined above for a processing of pairs in the increasing order of i,x i and x 1+i appear to be involved in the computation of pairs i-1, i and i, i+1, respectively. Let us consider the following embedding of w into the pair i:

lxwnhlhwlx ii =++=+= +'

1' ; (10)

The embedding can be regarded as composed of 2 steps, a difference expansion transform of the pair 1, +ii xx and then, the embedding of w into the first pixel by simple addition. Since the first pixel is not further involved in other computation, the embedding can be delayed until the entire information needed to recover the original pixels is available.With the modified transform, overflow or underflow should be checked against the embedding of the maximum/minimum values of w only for the first pixel of a transformed pair:

2550 ' ≤≤ ix (11)

Similarly, for h > 0, the overflow at the embedding of 1−= nw is checked.

At detection, no modifications appear with the modified transform for watermark extraction. For original image recovery, no modifications appear for h, while l is simply the second of the pair, '

1+= ixl . Once l and h are available, the DE transform is immediately inverted by equations (7) and the original pixel 1+ix is recovered.The basic DE scheme does not embed data into nonchangeable pairs. At detection, the nonchangeable pairs are directly identified and the embedded data is extracted from the remaining (expandable and changeable) pairs. Knowledge if a pair was changed or expanded is necessary only for recovering the original values. Since this is not the case for the dense pairing, we discuss next a modified version.The basic idea is to use the divisibility by n of the pixel difference, h, to detect if a pair was embedded or changed. If the expandable pairs are embedded only with integer codes [ ]1,1 −∈ nw , the difference is no more divisible by n. Meantime, the nonexpandable pairs (either changeable or nonchangeable ones) are adjusted in order to have the difference between pixels divisible by n. While the basic principle of detection by divisibility is inspired from [9], we introduce original elements to deal with underflow and to treat in a symmetric manner both changeable and nonchangeable pairs.The adjustment to ensure the divisibility for the nonexpendable pairs is performed on the second pixel. Let 1, +ii xx be a nonexpandable pair and let the integer correction code 1c be defined as:

hnhnc −

=1 (12)

If 11 cxi >+ , let us replace 1+ix by 11 cxi −+ .The difference between the pixels cxx ii −+ 1, is divisible by n:

.)( 111

=−

+=+=−− + n

hnhnhnhchcxx ii If 11 cxi <+ ,

let 2c be:

−=nhnhc2 (13)

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Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 1 (15) ISSN 1843-6188

Furthermore, let us replace 1+ix by 21 cxi ++ . As above, the difference is divisible by n.

=

+−=−− + n

hnnhnhhcxx ii 21 .Obviously, since

11 cxi <+ , no overflow appears. Either by subtracting 1cor by adding 2c , the nonexpandable pair is adjusted to ensure divisibility by n. The adjusted pixel 1+ix is further used with the pair i+1. Meantime, in order to recover the original value, a map should be created to keep trace if the pair was corrected by addition or by subtraction. Details on the algorithm are given in the next Section.

4. HIGH CAPACITY ALGORITHM

A. MarkingThe marking consists in two major stages: pairs expansion/adjustment and watermark embedding. In the first stage of the algorithm, the expandable pairs are transformed in order to increase n times pixel difference and a location map, tM , is created. The nonexpandable pairs are adjusted as described above in order to ensure the divisibility by n and an underflow map, uM , is created. The processing is done sequentially, pair by pair, starting with the first pair in the scanning order.The processing steps of the first marking stages follow. For :1,..,1: −Pi

1) compute ;,2 1

1+

+ −=

+= ii

ii xxhxxl

2) if pair 1, +ii xx is embeddable (equation (11) is fulfilled):a) nhlxi += and lxi =+ 1 ;b) ( ) ( ) .0,1 == iMiM ut

3) if pair 1, +ii xx is not embeddable, and 11 cxi <+ , with

;1 hnhnc −

=

a) ;111, cxx ii −= ++ ;

b) save 1c into the sequence of correction codes cC ;

c) ( ) ( ) .0,0 == iMiM ut

4) if pair 1, +ii xx is not expandable and 11 cxi <+

a) ;2

−=nhnhc

b) ;211 cxx ii += ++

c) save 2c into cC ;d) ( ) ( ) .1,0 == iMiM ut

In the second stage of the marking, the watermark is created and embedded. Besides the message to be embedded, the watermark should contain the underflow map uM and the sequence cC . The map is lossless

compressed. The sequence of bits obtained by concatenating the message data and the additional information is further encoded as a string of integers

[ ]1,1 −∈ nw j . Besides the watermark, a header, containing at least the integer n, should be created. The header can be hidden by LSB substitution into some predefined locations, while the overwritten bits are saved into the watermark. Finally, each jw is further embedded with equation (10) by simple addition into the first pixel of an expanded pair.

B. Watermark Detection and Original Image RecoveryAt detection, the watermark is extracted, decoded and finally, the original image is recovered. Watermark extraction proceeds sequentially, following the same pairing, but in reverse order.

Figure 1. Test images: Lena, Mandrill, Pirate, Woman

Thus, for 1−= Pi to i=1, one computes ( )1+−= iii xxd mod n. If 0=id , no data was embedded. The pair was changed in order to obey the divisibility. Otherwise, the pair contains a watermark code. The embedded code is subtracted:

iii dxx −= (14)

and thus, ix recovers the value before data embedding. The extracted watermark code is stored into the watermark sequence, ij dw = .Meantime, the location map tM is recovered by setting

( ) 1=iM t if 0≠id , and ( ) 0=iM t , otherwise. Once the entire sequence of watermark codes has been extracted, the watermark is decoded. The correction codes, cC , and the underflow map, uM , become available, too.Furthermore, the recovery of the original pixels proceeds sequentially, pair by pair, for 1−= Pi to 1=i . For each pair 1, +ii xx , one has 3 possibilities:1) Embeddable pair ( ) )1( =iM t . The corresponding code

ij dw = was already extracted. The original l and h are

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ISSN 1843-6188 Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 1 (15)

computed and the graylevels before expansion, ix and 1+ix are recovered with equations (7). The recovered

value of 1+ix is the original image pixel value, while ix take place in the restoration of the next pair, .,1 ii xx −

2) Nonembeddable pair ( ( ) ( ) 0,0 == iMiM ut ). The pair was adjusted by subtraction of a correction code computed by equation (12). The corresponding code, c, is available from cC and the original pixel value is recovered as: cxx ii += ++ 11 .3) Nonembeddable pair ( ( ) ( ) 1,0 == iMiM ut ). As above, the pair was adjusted, but in case of underflow. The corresponding code is subtracted in order to recover the original pixel value: cxx ii −= ++ 11 .

Figure 2. Embedding bit-rate with respect to n in a single embedding level

C. CapacityAn integer [ ]1,1 −∈ nw means ( )1log2 −n bits inserted into each embeddable pair. Besides the message, one should store a map and a string of integers in[ ]1,0 −n , one for each nonembeddable pair (log 2 n bits/pair). Let P, E be the total number of pixel pairs and the number of expandable pairs, respectively and let M be the size in bits of the compressed underflow map uM . The space available for message insertion is:

( ) ( ) MnepnEC −−−−= 22 log1log (This means an embedding bit-rate of :

( ) ( ) bppP

MnEPP

nEB +−−−= 22 log1log (15)

The upper-bound of the bit-rate (for PE ≈ ), maxB , is ( )1log2max −= nB bpp.

The embedding bit-rate depends on the integer n. Thus, for n = 3, the bit-rate is bounded by 1 bpp, almost 1.5 bits for n = 4, while for n = 5 by 2 bpp and, so on. The increase of n increases the difference between the transformed pixels. This yields a decrease of the number of pairs obeying the conditions to be embedded, hence a decrease of the embedding bit-rate.

5. EXPERIMENTAL RESULTS

We investigate next the embedding bit-rate in a single embedding level with respect to n. The test images used in our experiments, Lena, Mandrill, Pirate, Woman1, are presented in Fig. 1.The results for the modified DE scheme are plotted in Fig. 2. The highest bit-rates are obtained for Woman (which has large uniform areas), while the lowest ones for Mandrill (which has large textured areas). For ≤n 5, the embedding bit-rates are rather close to the upper-bound of ( )1log2 −n . For n = 3, the bit-rates are almost 1 bpp (slightly lower for Mandrill. The approximation is still good for n = 4 and fair for n = 5.

Figure 3. Embedded images for maximum bit-rate of the high capacity DE (first embedding level): Woman 2.58 bpp,

Lena 2.30 bpp, Pirate 1.96 bpp, Mandrill 1.33 bpp.

The approximation is very good for Woman and worse for Mandrill. For n > 6, the bit-rates still increases, but far from the limit of ( )1log 2 −n . The maximum bit-rates are: 2.58 bpp (n = 9) for Woman at a PSNR of 17.03 dB, 2.30 bpp (n = 9) for Lena at a PSNR of 16.16 dB, 1.96 bpp (n = 8) for Pirate at a PSNR of 15.3 dB and 1.33 bpp (n = 6) for Mandrill at a PSNR of 14.14 dB.The scheme provides high capacity in a single embedding level. The higher the embedding bit-rate, the more distorted the marked image. At maximum embedding capacity, the marked images (Fig. 3) are of poor quality. However the reversibility of the marking ensures, at detection, the exact recovery of the original.

6. CONCLUSIONS

The basic Tian’s DE reversible watermarking scheme is modified in order to increase its embedding capacity provided in a single level of embedding. Two directions are followed: the simple modification of the transform and the modification of the pairing. The original

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Scientific Bulletin of the Electrical Engineering Faculty – Year 11 No. 1 (15) ISSN 1843-6188

transform doubles the difference between a pair of pixels, while the modified one increases n times the difference, with integer 2≥n . Instead of a bit-rate of up to 0.5 bpp provided by the basic DE version, by this simple modification the bit-rate increases up to

2/log2 n bpp. Furthermore, by increasing the number of pairs considered for data embedding, a scheme providing, in a single level of embedding, up to

( )1log2 −n bpp is proposed. For natural test images with large uniform areas, the embedding bit-rates provided in a single embedding level are greater than 2 bpp. Similar embedding bit-rates can be obtained with the basic DE version in 4-5 embedding levels, hence at considerably higher mathematical complexity. (The mathematical complexity linearly increases with the number of embedding levels).The parameter n controls the capacity of the watermarking. At high embedding bit-rates (> 2bpp), the quality of the marked images is poor. However, it should be noticed that, at detection, the embedded data is extracted and the original image is exactly recovered. Since distortions increases with capacity, n should be selected to match the required capacity of the application at hand. A fine tuning of the distortions versus embedding capacity can be further obtained by using, together with n, of a threshold control scheme, i.e., by transforming and embedding only the pairs whose pixel difference is less than a threshold.

7. ACKNOWLEDGMENTS:

This work was supported by UEFISCDI, Grant ID 2200/2008.

8. REFERENCES

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[2] Alattar, A. M.: Reversible Watermark Using the Difference Expansion of a Generalized Integer Transform, IEEE Trans. on Image Processing, 2004, 13, (8), pp. 1147-1156.

[3] Kamstra, L. and Heijmans, H. J. A. M.: Reversible Data Embedding Into Images Using Wavelet Techniques and Sorting, IEEE Trans. on Image Processing, 2005, 14, pp. 2082–2090.

[4] Thodi, D. M. and Rodriguez, J.J.: Expansion Embedding Techniques for Reversible Watermarking, IEEE Trans. on Image Processing, 2006, 15, pp. 721-729.

[5] Coltuc, D., Chassery, J.-M.: Very Fast Watermarking by Reversible Contrast Mapping, 2007, IEEE Signal Processing Letters, 15, pp. 255–258.

[6] Wang, X., Shao, C., Xu, X., Niu, X.: Reversible Data-Hiding Scheme for 2-D Vector Maps Based on Difference Expansion, IEEE Trans. On Information Forensics and Security, 2007, 2, pp. 311-319.

[7] Kim, H. J., Sachnev, V., Shi, Y. Q., Nam, J., Choo, H.-G.: A Novel Difference Expansion Transform for Reversible Data Embedding, IEEE Trans. on Information Forensics and Security, 2008, 3, pp. 456-465.

[8] Coltuc, D. and Caciula I.: On Stereo Embedding by Reversible Watermarking: Further Results, Proc. Int. Symposium on Signals, Circuits and Systems, Iasi, Romania, 2009, pp. 121-124.

[9] Coltuc, D.: Improved Capacity Reversible Watermarking, Proc. Int. Conf. Image Processing, 2007, pp. 249-252.

[10] Coltuc, D.: Modified Versions of Tian’s Difference Expansion Reversible Watermarking, Proc. Int. Conf. Image Processing, 2009, pp.4225-4228.

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