Information Sciences 177 (2007) 4696–4710

www.elsevier.com/locate/ins

A multiple-level visual secret-sharing scheme without imagesize expansion

Yung-Fu Chen a, Yung-Kuan Chan c,*, Ching-Chun Huang b,Meng-Hsiun Tsai c, Yen-Ping Chu d

a Department of Health Services Administration, China Medical University, Taichung, Taiwan, ROCb Department of Information Science, National Chung Hsing University, Taichung, Taiwan, ROC

c Department of Management Information Systems, National Chung Hsing University, Taichung, Taiwan, ROCd Department of Computer and Information Engineering, Tunghai University, Taichung, Taiwan, ROC

Received 25 July 2005; received in revised form 20 March 2007; accepted 7 May 2007

Abstract

In traditional VSS schemes, the size of the share image is substantially expanded since each pixel of the secret image ismapped onto a block consisting of several pixels. In addition, the quality of the reconstructed secret image is normallydegraded in contrast, especially for halftone images. This study proposes a VSS scheme that maps a block in a secret imageonto one corresponding equal-sized block in each share image without image size expansion. Two types of techniques,including histogram width-equalization and histogram depth-equalization, are proposed to generate the correspondingshare blocks containing multiple levels rather than two levels based on the density of black pixels on the blocks for a secretblock. In the former technique, the gray-scale image histogram is obtained by uniformly splitting the range of the pixelgray levels in the secret image, while in the latter the buckets are created so that the area of each bucket is roughly constantby containing approximately the same number of pixels. The proposed schemes significantly improve the quality of thereconstructed secret image compared to several previous investigations.� 2007 Elsevier Inc. All rights reserved.

Keywords: Visual cryptography; Visual secret sharing (VSS); Image security; Halftone image

1. Introduction

A (r,k) visual secret-sharing (VSS) scheme [13] encoding a secret image into k share images, which demon-strate randomly noisy patterns and hide the information about the secret image, are distributed to k recipients.The secret image can easily be decrypted by stacking (‘‘OR’’ operation) any r out of the k (r 6 k) printed trans-parencies of the share images in an arbitrary order displayed by a projector without complicated arithmetic

0020-0255/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2007.05.011

* Corresponding author. Tel.: +886 4 22053366x6315; fax: +886 4 22031108.E-mail addresses: yungfu@mail.cmu.edu.tw (Y.-F. Chen), ykchan@nchu.edu.tw (Y.-K. Chan), floyed@ms17.hinet.net (C.-C. Huang),

mht@nchu.edu.tw (M.-H. Tsai), ypchu@thu.edu.tw (Y.-P. Chu).

Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710 4697

operations, but any r � 1 or fewer share images cannot disclose any information about the secret image. Thescheme for processing binary secret images has been extended to those for processing gray-level [3,7] and colorsecret images [3], respectively, by using halftoning technique to produce binary images so that the traditionalVSS scheme can be applied. Additionally, for color secret image, colors have to be decomposed to individualchannels before the application of halftoning technique [3]. Like their traditional counterparts, every pixel onthe secret image for schemes proposed in [3,7] is still mapped onto an m · n block on each share image, thusexpanding each share and reconstructed image to be m · n times larger than the original secret image.

Traditional VSS scheme is lossy by overlapping share transparencies since only the black pixels can be per-fectly reconstructed but the white pixels [1]. With the fast development of Internet technology, network secu-rity is an important issue for ensuring privacy, authentication, and integrity of network transactions. VSSscheme can also be computation-based without using transparencies and projector by disseminating shareimages created using method similar to pure VSS scheme [6] or VSS combined with watermarking scheme[15] to multiple sites or users. Perfect reconstruction using a computation-based decryption scheme is very crit-ical in recovering secret information for security concern. For binary and halftoning secret images, perfectreconstruction can be done by replacing ‘‘OR’’ with ‘‘XOR’’ operations followed by sub-sampling one pixelfrom each 2 · 2 block of the reconstructed images without size expansion and quality degradation [5]. Perfectreconstruction of gray-scale and color secret images, on the other hand, can be achieved in several similar ways[5,9,11]. Jin et al. [5] proposed a scheme that a pixel in a gray-scale image is first represented with a 3 · 3 micro-block and then expanded into a 6 · 6 block using (2,2)-VSS scheme. An alternative scheme that bit-leveldecomposition technique is first used to decomposing a pixel into 8 bit-planes, (2, 2)-VSS scheme is thenapplied to each bit-plane, and finally share images with gray-scale noise obtained by stacking 8 bit-planes[11]. For encrypting a color secret image, it is needed to decompose the image into 3 color (RGB) channelsbefore the aforementioned procedure has been applied for processing gray-scale images [5,9]. Notice thatthe share images are binary for encrypting binary or halftoning images using the traditional VSS scheme, whilethe share images are gray-scale or color for the encryption of gray-scale or color images. Additionally, all theshare images are expanded in size. For decryption, computation-based logical ‘‘XOR’’ instead of ‘‘OR’’ oper-ations are applied for reconstructing the secret image perfectly without size expansion and quality degradation[5,9,11]. Unfortunately, the share images generated mentioned above are still expanded. To eliminate theshortcoming, the aforementioned schemes have been improved by encrypting the secret image into two shares

using the basis matrices, C0 ¼00

� �;

11

� �� �and C1 ¼

01

� �;

10

� �� �, and applying ‘‘XOR’’ logical operation to

generate share images without expansion [10,12].

Traditional VSS schemes generally employ two basis matrices, CPB and CPW, to encode a secret image into r

share images. For each black (resp. white) pixel in the secret image, one column is selected from CPB (resp. CPW)and permutated for encoding the corresponding blocks in the share images. For example, in a (2,2) VSS schemebased on the basis matrices shown in Fig. 1a, the corresponding column permutation of share and reconstructedblocks are illustrated in Fig. 1b. Fig. 1c demonstrates an example showing a binary secret image, its generatedshare images, and the reconstructed image. Notice that the reconstructed image is noisy compared to the originalsecret image, which indicates that the secret image cannot be completely recovered.

The drawback of the traditional VSS schemes is that the shares are m · n times the size of the secret image.Ito et al. [4] proposed a (2,2) VSS scheme which prevented size expansion so that the secret image and theshares are all the same size. In this scheme, for a black (resp. white) pixel in the secret image, a column CB

(resp. CW) is first randomly chosen from CPB (resp. CPW) as the corresponding pixels in the share images.Let p0 (resp. p1) be the probability that a black pixel on the reconstructed secret image is generated from awhite (resp. black) pixel on the secret image. The contrast of the reconstructed secret image is defined asb ¼j p0 � p1 j.

A similar probabilistic method was recently proposed by Yang [14], who discussed a general (r,k) VSSscheme. In this method, the recognition criterion l0 � 3r0 > l1 � 3r1 þ Nd was demonstrated to discriminatethe black from the white areas, where l0 (resp. l1) and r0 (resp. r1) represent the mean and standard deviationof the white pixels in a black (resp. white) area containing N pixels and Nd ¼ d � N ; 0 < d < ðp0 � p1Þ, indi-cates the minimal difference in white pixels between the white and black areas where the ‘‘black’’ and ‘‘white’’

,0011

0011,

0101

0101,

0110

0110,

1001

1001,

1010

1010,

1100

1100CPW

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧=

.1100

0011,

1010

0101,

1001

0110,

0110

1001,

0101

1010,

0011

1100CPB

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧=

Secret pixel (white) (Black)

Share Block 1

Share Block 2

Reconstructed Block

Secret Image (128*64 pixels)

Share Image 1 (256*128 pixels)

Share Image 2 (256*128 pixels)

Reconstructed Image 256*128 pixels

Enc

rypt

ion

Dec

rypt

ion

Fig. 1. (a) Two Boolean basic code pools, CPW and CPB and (b) their column permutation represented in image blocks. (c) An example oftraditional (2,2) VSS scheme showing a binary secret image, two share images, and its recovered image.

4698 Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710

can be distinguished. Here, p0 and p1 are, respectively, the appearance probabilities of the white pixels in thewhite and black areas of the reconstructed secret image.

In addition to the pixel-wise operation, the block-wise operation was also provided for the non-expansionscheme. In the block-wise operation, the share images are generated block-by-block [2,8], where a block in ashare image corresponds to an equal-sized block in the secret image. We call the block in the secret image asecret block, and the block in the share image a share block. This scheme is different from both the traditionalexpansionary VSS scheme, in which a pixel is mapped onto a block on each share image [3,13], and the non-expansion scheme, where a pixel on a binary secret image is mapped onto a corresponding pixel [4,15].

A binary-level VSS scheme is generally used to describe a halftone binary image by either pixel-wise orblock-wise operations, and thus greatly degrade the quality of the reconstructed image. In order to improvethe quality of a reconstructed secret image, Lin [8] presented a new visual cryptographic method for encryptinga gray-level image without image size expansion. This method encrypts a secret block with s pixels according

No. of black pixels Share 1 Share 2 Reconstructed

2 ( )

3 ( )

4 ( )

Fig. 2. Chang’s basic codes for m = 2.

Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710 4699

to two basis codes, each holding s/2 white pixels and s/2 black pixels. For example, as shown in Fig. 2, wheres = 2 · 2, after the two basic codes are stacked, the reconstructed secret block may contain 2, 3, or 4 blackpixels.

The aforementioned methods, however, result in unsatisfactory contrast in the reconstructed secret image.To deal with this drawback, this paper proposes a multiple-level visual secret-sharing (MLVSS) scheme whichhas the advantages of contrast enhancement without image size expansion. The MLVSS scheme is a (k,k)scheme which encrypts a grey-level secret image into binary share images without expanding the image. Addi-tionally, the scheme can also be easily implemented on the basis of the traditional VSS schemes. In thisscheme, two techniques, including histogram width-equalization and histogram depth-equalization, are pro-vided to create the corresponding share blocks for each secret block. The former technique classifies the secretblocks by uniformly splitting the range of the pixel gray levels in the secret image, while the latter categorizesthe secret blocks by giving the same frequencies of pixels to each bin. Since both techniques give sharper con-trasts for the reconstructed image, they, especially the histogram depth-equalization technique, can obtainmore conspicuous contours. The depth-equalization technique, however, may occasionally lose some informa-tion in the secret image.

The remainder of this paper is organized as follows. The next section introduces the proposed MLVSSscheme in detail; then Section 2 discusses the security of the scheme. Section 4 describes the histogramwidth-equalization and histogram depth-equalization techniques. The quality of a reconstructed secret imageis analyzed in Section 5. Section 6 addresses the performance of the MLVSS scheme by experiments. Finally, abrief conclusion is given in Section 7.

2. MLVSS scheme without image size expansion

Compared to the traditional VSS schemes, the advantage of the VSS scheme without image size expansionis that the secret image and all share images have the same size. The MLVSS scheme, a (k,k) VSS scheme, firstdivides a secret image into several secret blocks, each containing s = m · n pixels. Then it encodes a secretblock into k share blocks, each of which consists of s/2 black pixels and s/2 white pixels. Hence, the originalsecret image, the reconstructed secret image, and each share image all have the same size.

To build a halftone binary image from a grey-level image, patterns with higher density of black pixels areused to replace the darker regions; whereas, patterns with a lower density of black pixels are applied to sub-stitute in the brighter regions. The MLVSS scheme employs this concept to generate k share blocks for eachsecret block B, so that the reconstructed secret block obtained by stacking up k share blocks is very similar toB visually. The MLVSS scheme computes the average gray-scale intensity of each secret block and adjusts allthe possible average intensities into s/2 + 1 levels. For a secret block, this scheme produces k share blocks,each comprising s/2 black pixels and s/2 white pixels. On one share block, the positions occupied by the blackpixels vary from block to block. As a result, the secret blocks reconstructed by stacking up share blocks havevarious contrasts. In this scheme, each secret block with gray-scale intensity level l; l ¼ 0; 1; . . . ; s=2, corre-sponds to a reconstructed secret block containing s/2 + l black pixels. Therefore, the MLVSS scheme can pro-vide more than two kinds of contrasts for the reconstructed secret blocks.

As shown in Table 1, the MLVSS scheme first arranges all the secret blocks with gray-scale level l in Groupl, symbolized by Gl, and then it produces a basis matrices CPl for Gl. Every element C in CPl is a two-dimen-sional matrix with k rows and s columns. Each row in C maps to a share block which contains s/2 black pixelsas well as s/2 white pixels, so that the ‘‘OR’’ operation for all the k row data results in s/2 + l black pixels ands/2 � l white pixels. For each secret block in Gl, this scheme randomly selects a corresponding matrix elementfrom CPl and sets each row’s data as the content of one share block so that the reconstructed secret blockformed by stacking up all the share blocks has s/2 + l black pixels and s/2 � l white pixels. For example,as shown in Table 1 (s = 2 · 2), there are 4 black pixels and 0 white pixel for CP2 (l = 2) and 3 black pixelsand 1 white pixel for CP1 (l = 1).

In the (k,k) MLVSS scheme in which each secret block contains s pixels, the algorithm BASIS_MA-

TRIX(k, s, l) is designed to randomly generate a Boolean basis matrix for a secret block in Gl. In this algorithm,the symbol � represents the logical exclusive OR operation.

Table 1CPl’s of (2,2) multiple-level VSS scheme

CPl Basis matrices Possible reconstructed secret blocks

CP000110011

� �;

01010101

� �;

01100110

� �;

10011001

� �;

10101010

� �;

11001100

� �� �½0011�; ½0101�; ½0110�;½1001�; ½1010�; ½1100�

� �

CP1

00110101

� �;

00111001

� �;

00110110

� �;

00111010

� �;

01010011

� �;

01011001

� �;

01010110

� �;

01011100

� �;

01100011

� �;

01101010

� �;

01100101

� �;

01101100

� �;

10010011

� �;

10010101

� �;

10011010

� �;

10011100

� �;

10100110

� �;

10100011

� �;

10101001

� �;

10101001

� �;

11000101

� �;

11000110

� �;

11001001

� �;

11001010

� �

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>;

½0111�; ½1011�;½1101�; ½1110�

� �

CP200111100

� �;

01011010

� �;

01101001

� �;

10010110

� �;

10100101

� �;

11000011

� �� �{[1111]}

4700 Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710

Algorithm. BASIS_MATRIX(k, s, l)

01: Randomly select s/2 + l positions p1; p2; . . . ; ps=2þl from the s positions02: For i = 1 to k � 1:03: Randomly select s/2 positions from p1; p2; . . . ; ps=2þl;04: Let Ri be a variable consisting of s bits and set the bits located at the s/2 selected positions in Ri as 1-bits

and others as 0-bits;05: R ¼ R1 � R2 � � � � � Rk�1;06: Let p01; p

02; . . . ; p0h be the positions of all the 0-bits which are located at positions p1; p2; . . . ; ps=2þl in R;

07: Set the bits located at p01; p02; . . . ; p0h of Rk as 1-bits;

08: Set the s/2 � h bits by randomly selecting from the bits located at p1; p2; . . . ; ps=2þl from Rk, except thepositions p01; p

02; . . . ; p0h, as 1-bits, and set the other bits as 0-bits;

09: Regard each member of R1;R2; . . . ;Rk as one row of the basic matrix.

3. Security analysis

When assuming that one holds k 0 of the k share images where 1 6 k 0 < k, one can stack them up to create abinary image I 0. Let B 0 be a block on I 0 in which its corresponding secret block B on the original secret image isin Gl, and let B 0 comprise s/2 + l 0 black pixels and s/2 � l 0 white pixels with 0 6 l 0 6 l. The number of ways toguess the k � k 0 corresponding share blocks, Bk0þ1, Bk0þ2; . . . ;Bk on the other k � k 0 share images is

Cs=2�l0

l�l0 ðCs=2þls=2 Þ

k�k0 þXl�l0

i¼1

ðð�1ÞiCl�l0

i ðCs=2þl�is=2 Þk�k0Þ

!;

so that all can be piled on B 0 to generate a block B00, which has s/2 + l black pixels and s/2-l white pixels.Therefore, the number of ways to make Bk0þ1;Bk0þ2; . . . ;Bk, which will stack up on B 0 to produce a blockon which the number of black pixels does not equal s/2 + l, is

ðCss=2Þ

k�k0 � Cs=2�l0

l�l0 ððCs=2þls=2 Þ

k�k0 þXl�l0

i¼1

ðð�1ÞiCl�l0

i ðCs=2þl�is=2 Þk�k0 ÞÞ:

Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710 4701

For an M · N secret image, when one owns k 0 out of the k share images, the probability of guessing all theexpected reconstructed secret blocks is

Cs=2�l0

l�l0 ððCs=2þls=2 Þ

k�k0 þPl�l0

i¼1

ðð�1ÞiCl�l0

i ðCs=2þl�is=2 Þk�k0 ÞÞ

ðCss=2Þ

k�k0

0BBB@

1CCCA

M�Ns

:

Hence, it is extremely difficult to precisely guess the reconstructed secret image from the k 0 share images.Since B 0 has s/2 + l 0 black pixels and s/2 � l 0 white pixels, whatever the patterns of Bk0þ1;Bk0þ2; . . . ;Bk are,

B00 possesses at least s/2 + l 0 black pixels. To make B00 own s/2 + l black pixels, there must be (l � l 0) of the 0-bits in B 0 which will be OR-ed by some 1-bits in Bk0þ1;Bk0þ2; . . . ;Bk. Other 0-bits in B 0 will be piled up on 0-bitsin Bk0þ1;Bk0þ2; . . . ;Bk. The number of possible combinations of (l � l 0) 0-bits chosen from the s/2 � l 0 0-bits inB 0 is Cs=2�l0

l�l0 . Let b1; b2; . . . ; bl�l0 be the (l � l 0) selected 0-bits. In order to obtain the expected B00, all the 1-bits inBk0þ1;Bk0þ2; . . . ;Bk, can be assigned only to the positions where the b1; b2; . . . ; bl�l0 and the 1-bits of B 0 arelocated. The number of ways to specify all the s/2 1-bits on each Bk0þ1;Bk0þ2; . . . ;Bk in the places selected fromthe positions where b1; b2; . . . ; bl�l0 and the 1-bits of B 0 are located is ðCs=2þl

s=2 Þk�k0 . However, there are

jPl�l0

i¼1 ð�1ÞiCl�l0

i ðCs=2þl�is=2 Þk�k0 j cases where at least one b1; b2; . . . ; bl�l0 is not stacked up on any 1-bit in

Bk0þ1;Bk0þ2; . . . ;Bk. Thus, the number of ways to choose Bk0þ1;Bk0þ2; . . . ;Bk, which can be OR-ed by it B 0 to

generate the expected B00 is Cs=2�l0

l�l0 ððCs=2þls=2 Þ

k�k0 þPl�l0

i¼1 ðð�1ÞiCl�l0

i ðCs=2þl�is=2 Þk�k0 ÞÞ.

On the other hand, if one would directly conjecture the number of black pixels on B from B 0 rather thanguess Bk0þ1;Bk0þ2; . . . ;Bk on other k � k 0 share images, the number of black pixels on B must lies in the intervalbetween s/2 + l 0 and s because B 0 comprises s/2 + l 0 black pixels and s/2 � l 0 white pixels. There areðs=2� l0 þ 1Þ ways to guess the number of black pixels on B; therefore, the probability of guessing the numberof black pixels on B is 1

s=2�l0. Assume that the probability of the secret blocks on I containing s/2 + i black pix-els with i = l 0 to s/2 is uniform. The expected value of the probability guessing the number of black pixels oneach block is

1

s=2þ 1

Xs=2

i¼0

1

s=2þ 1� i

� �6

1þ blog2ðs=2Þcs=2þ 1

:

Let the secret image I consist of M · N pixels and each basic block be made of m · n pixels, then I can bedivided into nb ¼ M�N

m�n secret blocks. The probability of guessing the number of black pixels of all the secret

blocks on I is less than 1þblog2ðs=2Þcs=2þ1

� nb

. For example, let the secret image comprise 512 · 512 pixels and each

basic block contains 2 · 2 pixels for k = 2, and k 0 = 1, then nb ¼ 512�5122�2

¼ 65536. The probability of guessing

the number of black pixels of all the secret blocks on the secret image is less than 23

�65536.

4. Basis matrices of proposed methods

Histograms using binning to approximate data distribution are a popular form of data reduction. A gray-scale histogram of an image partitions the pixel gray levels of the image into disjointed subsets or buckets. Thebuckets are displayed on a horizontal axis. The area of a bucket typically reflects the frequencies of the valuesrepresented by the bucket. Often, buckets represent continuous ranges of the pixel gray levels in the image.Gray-scale histograms are highly effective at approximating both sparse and dense data, as well as highlyskewed and uniform data.

The method for mapping the secret blocks to the basis matrices can affect the quality of the reconstructedsecret image. This paper presents two types of techniques, histogram width-equalization and histogram depth-equalization, to generate the corresponding share blocks for a secret block. In the former technique, the gray-scale image histogram is obtained by uniformly splitting the range of the pixel gray levels in the secret image;that is, the width of each bucket range is identical. Let Max and Min be the maximal and minimal pixel

4702 Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710

gray-scale levels in the secret image. The histogram width-equalization technique partitions the interval [Min, -Max] into s/2 + 1 equal ranges. If the average gray-scale level of a secret block is within the interval of

ðMax�Minðs=2Þþ1

Þ � lþMin; Max�Minðs=2Þþ1

� � ðlþ 1Þ þMin

h iwith 0 6 l 6 s/2, the secret block is arranged into Gl.

In the histogram depth-equalization technique, the buckets are created so that the area of each bucket isroughly constant, that is each bucket contains approximately the same number of pixels. Let ni be the numberof pixels with color intensity i in an M · N secret image. In this secret image, the technique assigns a secretblock with average color c to group Gj, if j 6 c 6 j + 1 and M�N

ðs=2Þþ1� j 6

Pci¼0ni <

M�Nðs=2Þþ1

� ðjþ 1Þ.

5. Quality analysis of reconstructed secret image

For a secret image with a small color range [Min,Max], the histogram width-equalization technique canprovide a reconstructed image of good quality, since it abridges the histogram width; that is, it results in amore accurate color intensity for the secret image. Fig. 3 shows image Zelda and its gray-scale histogram.As shown in this figure, Min = 19 and Max = 206, and no pixel settles beyond these two extremities. The his-togram width-equalization technique can offer a reconstructed image with quality similar to histogram width-equalization technique, as demonstrated in Figs. 8a and 9a.

For a secret image with a significantly skewed color histogram, most of its secret blocks are assigned toa small number of color levels, and only a few secret blocks are mapped to most color levels. Take the

Fig. 3. (a) Gray-scale image of Zelda, (b) its halftone image, and (c) its gray-scale histogram.

Fig. 4. (a) Gray-scale image of Tiffany, (b) its halftone image, and (c) its gray-scale histogram.

Fig. 5. (a) Gray-scale image of Airplane, (b) its halftone image, and (c) its gray-scale histogram.

Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710 4703

image Tiffany in Fig. 4a as an example. In this image, Min = 85, Max = 251, and about 96% of the pixelsare between 157 and 238 gray-scale values. The histogram depth-equalization technique uniformly arrangesall pixels of an image to all gray-scale levels and provides a better performance for this kind of secret imagethan the histogram width-equalization technique, as compared between Figs. 8b and 9b. However, it may

Fig. 6. (a) Gray-scale image of Toys, (b) its halftone image, and (c) its gray-scale histogram.

Fig. 7. Share images of the secret image shown in Fig. 3a.

Fig. 8. Images reconstructed by using histogram width-equalization technique: (a) Zelda, (b) Tiffany, (c) Airplane, and (d) Toys.

4704 Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710

damage some important information on the image. For example, in the image Airplane shown in Fig. 5a, onlya few pixels are similar to gray-scale levels of the pixels of the empennage. Hence, its information may be lostin the reconstructed secret image. However, the histogram width-equalization technique can abate thephenomenon.

Fig. 9. Images reconstructed by using histogram depth-equalization technique: (a) Zelda, (b) Tiffany, (c) Airplane, and (d) Toys.

Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710 4705

A secret block with s pixels is mapped to a reconstructed secret block containing s/2 to s black pixels cor-responding to s � s/2 + 1 = s/2 + 1 gray-scale intensities. Each of the possible gray-scale intensities is called acolor factor. Consider an M · N secret image which can be divided into M�N

s secret blocks. Every share blockconsists of s/2 black pixels and s/2 white pixels; the reconstructed secret block comprises s/2 to s black pixels.

4706 Y.-F. Chen et al. / Information Sciences 177 (2007) 4696–4710

There are M�Ns � ðs=2þ 1Þ ¼ M�N

2þ M�N

s color factors in the reconstructed secret image. For a same-recon-structed secret image, the more color factors it has, the better its quality. Hence, a reconstructed secret imagewith a better quality can be obtained for a smaller s.

6. Experimental results and discussions

The first experiment encoded the secret images shown in Figs. 3a, 4a, 5a, and 6a by using the histogramwidth- and histogram depth-equalization techniques, where s = (m · n) is (2 · 2), (3 · 3), and (4 · 4), respec-tively, and k is 2. Fig. 7 shows the two share images of the secret image displayed in Fig. 3a; Figs. 8 and 9 showtheir corresponding reconstructed secret images.

The results indicate that both proposed techniques can provide a better quality for the reconstructed secretimage when s ¼ ðm� nÞ ¼ ð2� 2Þ. When given a bigger s, the reconstructed secret image looks vaguer since itcontains fewer color factors. In addition, when s = 2 · 2, each secret block relates to one of three gray-scaleintensities. Thus, each secret block belongs to G0, G1, or G2. For a brighter region in the secret image, the cor-responding Boolean basis matrices of the secret blocks in this region were selected mostly from CP0. On theother hand, for a darker region the corresponding codes were mostly chosen from CP2. This choice will causea block-effect phenomenon in the reconstructed secret image. Hence, even though this image can offer morecolor factors, it may lose some delicate textures.

Mostly, the histogram depth-equalization technique provides a better quality of reconstructed secret imagethan the histogram width-equalization technique, especially for an image with most of its pixel gray-scalesranging only within a small interval. For example, in the image shown in Fig. 4a, about 96% of the pixelgray-scales are located at the interval [157, 238]. The reconstructed images of Tiffany displayed in Fig. 9 areclearer than those of the same image in Fig. 8. However, for the image in Fig. 3a, whose pixel gray-scalesare almost uniformly distributed over a wide interval [19, 206], the qualities of its reconstructed images shownin Figs. 8 and 9 appear very similar. Furthermore, as shown in Fig. 9, if the information in an image isdescribed in gray-scale by the pixels which are similar to only a few pixels in the image, the information, such

Fig. 10. Images reconstructed by using scheme proposed by Ito et al.: (a) Zelda, (b) Tiffany, (c) Airplane, and (d) Toys.

Fig. 11. Images reconstructed by using Yang’s (2,2) ProbVSS scheme: (a) Zelda, (b) Tiffany, (c) Airplane, and (d) Toys.

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as the empennage of image Airplane and the toys in image Toys, may be lost by using the histogram depth-equalization technique.

The second experiment compared the performances of our schemes with the schemes proposed by Yang[14], Ito et al.[4], Chang [2], and Lin [8]. Figs. 10 and 11 show the images reconstructed by using the schemesproposed by Ito et al. [4] and Yang [14], respectively. These images look vague, especially those in Fig. 11,whose contents are not easy to recognize. Figs. 12 and 13 displayed the images reconstructed by Chang’sand Lin’s schemes with m · n = 2 · 2, 3 · 3, and 4 · 4, respectively.

The experimental results also show that Lin’s scheme performs better than Chang’s for a small s; on thecontrary, the latter scheme performs better when given a larger s. The histogram depth-equalization tech-nique obtains a reconstructed image with quality better than those obtained by Chang’s and Lin’s schemesfor any size of s, especially when most of the pixel colors are distributed over merely a small interval, suchas in the case of image Tiffany. However, some information in an image, characterized by pixels which aresimilar to only a small portion of pixels in gray-scales on the image, may be lost, as demonstrated in Fig. 9for the empennage in image Airplane and the toys in image Toys.

7. Conclusions

This paper introduced a MLVSS scheme which encodes a grey-level secret image into binary halftoneshare images. For each secret block, the MLVSS scheme creates an equal-sized block for every share. Thisscheme improves the quality of the reconstructed secret image without expanding the image size by allo-cating the blocks into multiple levels rather than two levels based on the density of the black pixels in ablock. Additionally, the proposed scheme can be easily implemented via the traditional VSS scheme.

The proposed scheme presents two techniques, histogram width-equalization and histogram depth-equal-ization, to map a secret block to its corresponding share blocks. The depth-equalization technique providesa more distinct contour, especially for an image with most of its pixel gray-scales ranging within a small inter-val. However, some information may be lost when the number of pixels describing the information is only an

Fig. 12. Images reconstructed by using Chang’s scheme: (a) Zelda, (b) Tiffany, (c) Airplane, and (d) Toys.

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extremely small part of the histogram area of the secret image. The experimental results also show that thehistogram depth-equalization technique provides a reconstructed secret image with quality better than thoseproposed by previous investigations [2,4,8,14].

Fig. 13. Images reconstructed by using Lin’s scheme: (a) Zelda, (b) Tiffany, (c) Airplane, and (d) Toys.

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Acknowledgements

This work was supported in part by the National Science Council of Taiwan under Grant Nos. NSC92-2622-E-212-024-CC3 and NSC93-2213-E-212-038. The authors also wish to express their appreciation toDr. Cheryl Rutledge for her editorial assistance.

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