Hiding data in images by optimal moderately-significant-bit replacement

(1)

Pruuf’qf’ Theorem (ii): Note that motion of a test point T from thc central point C along the line from this point to one of the outer points, say P, does not affect the sum of the absolute dis- tances to thcsc two points (i.e. CT

+ PT

remains constant). How- ever, it does increase the distances to the other two points, Q, R (i.e. QT

+

R T is increased). Sinlilar arguments apply for motions towards either of thc other two points, or for motions in inter- mediate directions. This proves that C is already at the L , mini- mum.

1. If we move a fourth point in a circular orbit around a region containing three points, we go from a Case (i) situation to a Case (ii) situation, then back again, and so on for a total of six changes, and it is straightforward to show that the L, minimum moves once around the triangle formed from the stationary three points, pausing for a while at each of these points. If the circle is at infin- ity, it is easily shown that the L , minimum spends equal times along the sides of the triangle and at the three points. Thus there is in general approximately equal probability of the L, minimum being situated on a sample as lying in the space between the threc points.

2. If one of the four points moves slightly out of thc plane of the other three, the situation will not change substantially, and Cases (i) and (ii) are retained. However, this situation will not be retained as the fourth point moves an indefinitely large distance from the plane of the other three points. In fact, if it moves to a position a large distance away, the L, minimum will be forced into a modified Case (i) situation, and will lie in the space between points: space precludes providing a proof of this result here.

The theorems presented here, and the ensuing discussion, have shown that there is a significant probability in the case of four points in both 2D and 3D that the L, minimum will not be situ- ated at any of the sample points. There would appear to be no reason why the case of four sampling points is particularly special; in any case, we can always find instances when larger numbers of points give Ll minimum positions well away from all the sampling points (we merely select a symmetrical arrangement of points for which there is no point at the centre of symmetry, and the latter will always be the position of the Ll minimum), In general; this means that a significant error is quite likely to arise if the L, min- imum is constrained to just those positions occupied by sampling points.

Concluding reniurks: A general view is that the Ll minimum must lie within, or on, an innermost triangle (in 2D), tctrahedron (in 3D), or polyhedron (in nD). In I D there will be an upper bound on the error of the L , minimuin of at most half the distance between sampling points, and a minimum bound of zero. In the case of four points in 2D, a better estimate of the upper bound is 1/43 of the distance between sampling points, and in the case of four points in 3D, a better estimate of the upper bound is 4(3/8) of

this distancc (proofs of these results are based on the radii of the circumscribing circles for equilateral triangles and circumscribing spheres for regular tctrahedra). Proofs of these statements and a fuller discussion of the problem will be presented at a latcr date. Meanwhile, the basic aim of this Letter has been achieved - of showing that rcstricting the output multichannel median to one of the input vector samples is likely to introduce significant error.

0 IEE 2000

Electronics Letters Online No: 20001465

Dol:

10. lO4~/eI:2O00146S

E.R. Davics (Machine Vision G‘ruup. Department of Physics, Royd HOhWaJi, Univer.vity ? f London, E,yIiam, Surrey, TW2O OEX, United Kingdom)

E-mail: E.R.Davis@rhbnc.ac.uk

Two comments on this result are in order:

13 September 2000

References

1 DAVIES, ER.: ‘Machine vision: thcory, algorithms, practicalities’ (Academic Press 1997), 2nd edn.

2 ASTOLA, I., HAAvisro, I)., and NEUVO, Y.: ‘Vector nicdian filters’, Proc. lEEE3 1990, 18, (4), pp. 678-689

3 ROGAZKONI. c.s., and TESCFIIONI, A.: ‘A new approach io vector median filtering based on space filling curves’, IEEE Trans. Imaye Process., 1997, 6, (7), pp. 1025-1037

Hiding data in images by optimal

moderately-significant-bit replacement

Ran-Zan

W a n g , Chi-Fang Lin a n d Ja-Chen

Lin

A data hiding technique for the storage and transmission of important data is proposed. It embeds the important diitir in the moderately-si~iilificant-hit of an image, and applies a global substitution step and a local pixel adjustment proccss to reduce any image degradaiion. Experimental results show that the visual quality of the resulting image is acceptable.

Introduction: Hiding important data such as inilitary information or personal financial documents in images has been a popular research topic [l]. The goal is to make the embedded data invisible to the grabbers under the covcr of the host image, i.e. to make the host image, after processing, as similar as possible to the original

host image.

A very simple and direct method is to hidc the important data in some bits of each pixel of the host image, Least-significant-bit (LSB) substitution is a common way to do this. Many articles [2, 31 have addressed approaches related to LSB substitution. Although embedding data in LSB introduces small distortion to the host image, the embedded data is more easily lost when thc resulting image is used at a later date. (For example, in order to stave space, increase the transmission rate, or accelerate the image processing speed, some software discard thc LSB (and make the image become 7 bits per pixel instead of 8 bits). Storing data in thc LSB is therefore less safe.) In this Letter, we describe data being embedded in the moderately-significant-bit (MSB) of the host image, an approach seldom seen in the literature.

genetic algorithm

n

-

1

-temporal image (1 bit/pixel) important image perturbed image

(1 bit/pixel) (1 bit/pixel) rePlaiement

host image residual image optimal substitution resulting ima e (8 bits/pixel) (1 biffpixel) image (8 bitdpixel) (8 bits/pixely

a

-

z

resulting image residual image important image

(8 bits/pixel) (1 bit/pixel) (1 biffpixel)

b Fig. 1 Block dingmms of proposed methud a Embedding stage

b Extraction stage

Proposed datu hiding scheme: A mcthod to embed important data in the MSB of an imagc is proposed. The system consists of two stages: the embedding stage that embeds the important data in the host image, and the extraction stage that retrieves thc important data from an image. Block diagrams of the method are shown in Fig. 1.

Let E be the important data of M x N bits. We may treat E as a ‘binary’ image (1 bit per pixel) of A4 x N pixels. To embed E in a grey-scale (8 bits per pixel) host image H containing M x N pixels, the following three steps are conducted sequentially.

Step 1: The cipher proms,r. A polynomial transfoimation [4] is applied to cipher the important image E to obtain a perturbed image

E

as follows. Assume that the pixels in E are numbered sequentially froin 1 to M x N . Pixel at location

x

of B is trans- posed to a new locationflx) by thc following equation:

f(z) = (ko

+

kl x z

+

k2 x 52

+

’ ’ ’

+

k,. x z?) wherc g c d ( S i , M x N ) = 1 0

<

i

5

T m d

mod(M x N )

(2)

gcd(k,, k j ) = 1 for i

#

j ₍₁₎ In this equation, ki are the keys, and gcd(.) means the greatest common divisor.

Step 2: Optimal substitution process. The MSB (the fifth bit) of all pixels of the host image are extracted to form a (binary) residual image R. Both the perturbed image E‘ and the residual image R are divided into non-overlapping blocks of size 4 x 1. Note that all four bit blocks must be one of the following 16 kinds I = (0000, 0001,

...,

1111). Let P be a permutation operator on I, i.e.

P: I

+

I be a one-to-one mapping from and onto

I.

According to

P,

each 4 x 1 block of E’ is transfonncd to a new 4 x 1 block, E‘

is thus transformed to a new temporal image E”. The optimal sub-

stitution image Z can be obtained by replacing R (the fifth bit plane of H) by E”.

To define the permutation operator P in the above, there are 16! possible choices. A genetic algorithm (GA) [5] is developed (below) to search for a good P from the 16! possible choices. In our GA, each

P

is expressed a s a chromosome G containing 16 genes as follows:

G = 9091

. . .

gi5 = goooogoooi

. . .

giiii

where goooo means that, when P is applied, the block 0000 is replaced by a four bit block goooo. The operators in the G A are designed as follows:

[Crossover] Given two chromosomes GI = popI...p7...p15 and G2 =

qoq, ...q7...qis, the offspring are G; = !~0p1...~7qa...q15 and C; = qoq l...q7p7p8...p,s. The chromosomes G; and G ; obtained as abovc may be invalid, i.e. some genes might occur more than once (e.g.

p5 = q9), and a validation process is thus designed to correct them.

[Mutation] Given a chromosome G = g,gi...g15, two of the 16 genes of G are selected randomly and replaced by each other.

(2)

b a

d

Fig. 2 Illustration of experimental re,sult.s

U Host image ‘Lena’

h ‘Binary’ important image ‘Jct’

c Resulting image from simple MSB substitution method

d Resulting image Z (after slcps 1 - 3) of proposcd mcthod

Step 3: Pixel arljlrstment process. To improve the image quality, we add a post-processing which will not touch the fifth bit of each pixel of the image Z (hence will not hurt the important data), but will slightly modify the other bits of each pixel of Z to improve

visual quality. Let p and p‘ be the corresponding (8 bit) grey val- ues of a pixel of H and Z, respectively, and

6

be the value of the last three bits (bits 6 - 8) in p’. If p zp’, then either (i) p’ = p - 8 or (ii) p’ = p

+

8 (because the only difference between H a n d Z is the fifth bit plane).

Case 1: when p‘ = p - 8. If 6 t 4, then the value 8 - 6 - 1 is added

top‘. If 6 < 4 and if the fourth bit of p’ is 0, then the fourth bit of p’ is changed to 1, and the value 6 is subtracted from p‘. Do nothing otherwisc.

Case 2: when p’ = p

+

8. If

6

< 4, then the value 6 is subtracted from p’. If 6 t 4 and if the fourth bit ofp’ is 1, then the fourth bit of p’ is changed to 0, and the value 8 -

6

- 1 is added to p’. Do

nothing otherwise.

To extract the important data from an image, the data in the bit plane where the important data were embedded are extracted, and a reverse-substitution step and a decipher process are conducted to reveal the important data.

E.xperimenta1 results: The 256 grey-value host image ‘Lena’ and the binary important image ‘Jet’ tested in this experiment are shown in Figs. 2a and b, respectively. The important image is embedded in the fifth bit plane of the host image. Figs. 2c and d are the resulting images of the simple replacement method (i.e. replace directly) and the proposed method, respectively. The PSNR of the two images in Figs. 2c and dare 33.02 and 38.75dB, respectively. It can be seen that the image in Fig. 2d is much better.

Conclu.sion: Moderately-significant-bit replacement is seldom used in data hiding, since there will be degradation of imagc quality (see Fig. 2c). However, we have shown that with careful design, such as the use of optimal substitution process and local pixel adjustment, MSB can still be used (see Fig. 2 4 as an alternative choice for the storage and transmission or important data.

0 IEE 2000

Electronics Lellers Online No: 20001429 DOI: IO. 1049/el:20001429

Ran-Zan Wang and Ja-Chcn Lin (Department of’ Chtnputer arid Information Science, Nntionol Chiao TunR University, Hsinchu, 300, Taiwan, Republic of China)

E-mail: rzwang~cis.nctu.ed~i.tw

Chi-Fang Lin (Deprrrtment .f Computer. EnEineering and Science, Yurm-Ze University, Tuoyuan, 320, Taiwan. Republic of’ China)

References

28 Septemher 2000

1 PETITCOLAS, F A.P., ANDERSON, R J., and KUHN, M.G.: ‘Information

hiding - a survey’, Proc. IEEE, 1999, 87, (7), pp. 1062-1078

2 VAN SCIIYNDEL, R.G., TIRKEL, A z., and OSBORNE, c.F.: ‘A digital walemark’. Inl. ConT. linage Processing, 1994, pp. 86-90 3 CHRN:T.S., CHANG, c.c., and HWANG, M.s.: ‘A virtual imagc

cryptosystem based upon vector quantization’, IEEE Trans. Imnge Process., 1998, 7, (lo), pp. 1485-1488

4 R I T ~ E , M.Y.: ‘Cryptography and sccurc communications’ (McGraw- Hill, Singapore, 1994)

5 HOLLAND, J.H.: ‘Adaptation in natural and artificial systems’ (University of Michigan Press, Aim Arbor, MI, 1975)

Improved lossless compression

of

general

data

S.

Keating

and J.

Pelly

New algoriihms lor lossless compression or general data are presented. They are based on adaptive lossless data compression (ALDC) but offer improved compression, typically 24% belter Cor image data. The algorithms are simple to implement and arc capable of high data throughput, whilst maintaining compatibility with legacy ALDC bit streams.

Introduction: Lossless data compression is used extensively in data storage and communication. Popubr algorithms are frequently based on thc work of Lempel and Ziv [l]; one of these is adaptive lossless data compression (ALDC) [2]. These algorithms give good compression of many types of data. However, better compression can be achieved for image data using algorithms that are more specific; these generally include some form of modelling of the image data such as differential pulse code modulation (DPCM). This Letter describes algorithms for compressing gcneral data, which use a single architecture for all data types. The algorithms have been namcd SZIP and DZIP. The architecture is based on