4336 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 10, OCTOBER 2014

Sharing Visual Secrets in Single ImageRandom Dot Stereograms

Kai-Hui Lee and Pei-Ling Chiu

Abstract— Visual cryptography schemes (VCSs) generaterandom and meaningless shares to share and protect secretimages. Conventional VCSs suffer from a transmission riskproblem because the noise-like shares will raise the suspicionof attackers and the attackers might intercept the transmission.Previous research has involved in hiding shared content inhalftone shares to reduce these risks, but this method exacerbatesthe pixel expansion problem and visual quality degradationproblem for recovered images. In this paper, a binocular VCS(BVCS), called the (2, n)-BVCS, and an encryption algorithm areproposed to hide the shared pixels in the single image randomdot stereograms (SIRDSs). Because the SIRDSs have the same2D appearance as the conventional shares of a VCS, this papertries to use SIRDSs as cover images of the shares of VCSsto reduce the transmission risk of the shares. The encryptionalgorithm alters the random dots in the SIRDSs according tothe construction rule of the (2, n)-BVCS to produce nonpixel-expansion shares of the BVCS. Altering the dots in a SIRDS willdegrade the visual quality of the reconstructed 3D objects. Hence,we propose an optimization model that is based on the visualquality requirement of SIRDSs to develop construction rules fora (2, n)-BVCS that maximize the contrast of the recovered imagein the BVCS.

Index Terms— Visual cryptography, single image random dotstereograms, transmission risk, pixel expansion.

I. INTRODUCTION

V ISUAL cryptography (VC) is a technique that encrypts asecret image into n shares, with each participant holding

one share; any participant with fewer than k, 2 ≤ k ≤ n,shares cannot reveal any information about the secret image.Stacking the k shares reveals the secret image, which can berecognized directly by the human visual system [1].

Conventional shares [1]–[4], which consist of many randomand meaningless pixels satisfy the security requirement forprotecting secret contents, but they have a drawback—thereis a high transmission risk because noise-like shares raise thesuspicion of attackers, who may intercept the shares. Thusthe risk both to the participants and to the shares increasesin turn increasing the probability of transmission failure.

Manuscript received September 29, 2013; accepted July 29, 2014. Date ofpublication August 7, 2014; date of current version September 2, 2014. Thiswork was supported by the Ministry of Science and Technology, NationalScience Council of Taiwan, under Contract MOST-103-2221-E-130-010and Contract MOST-103-2410-H-130-019. The associate editor coordinat-ing the review of this manuscript and approving it for publication wasProf. Jana Dittmann.

K.-H. Lee is with the Department of Computer Science and Informa-tion Engineering, Ming Chuan University, Taoyuan 33348, Taiwan (e-mail:khlee@mail.mcu.edu.tw).

P.-L. Chiu is with the Department of Risk Management and Insurance, MingChuan University, Taipei 111, Taiwan (e-mail: plchiu@mail.mcu.edu.tw).

Digital Object Identifier 10.1109/TIP.2014.2346026

Previous research into the Extended Visual CryptographyScheme (EVCS) provided a meaningful appearance forshares to make the noise-like shares manageable forparticipants [5]–[9]. However, the meaningful shares stillpresent a risk of detection. In EVCSs, the shares that areprinted on transparencies still contain many noise-like pixelsand/or display low-quality images. Such shares are easilydetected by the naked eye and participants who transmit theshares can easily raise the suspicion of potential attackers.

Other research involves sharing secret images via high-quality shares [10]–[12]. Zhou et al. proposed a (2, 2)-VCSusing the halftoning technique to construct meaningful binaryimages as shares carrying significant visual information [10].The visual quality of the halftone is significantly better thanthat attained by extended VC. The shares obtained using Zhouet al.’s approach can reduce the transmission risk of the shares,but that approach exacerbates the pixel expansion problemand the visual quality degradation problem for the recov-ered images. Other studies [11], [12] suffer from the samedrawbacks as Zhou et al.’s method. Given these drawbacks,the extension ability of these approaches could be limited.Therefore, further research is needed on the current VCSs tofind an alternative way to reduce the transmission risk problemfor participants and shares.

In 1838, Wheatstone discovered stereoscopic vision andpublished an explanation of stereopsis (binocular depth per-ception) arising from differences in the horizontal positions ofimages in the two eyes. When we look at two flat, dissimilar,2D pictures, our mind perceives an illusion of 3D depth.In 1960, Julesz developed the random-dot format of thestereogram, in which the 3D form bypasses the monocularprocesses and is visible only when stereoscopic fusion isobtained. A random-dot stereogram (RDS) is a stereo pairof images of random dots, which when viewed with the aid ofa stereoscope or with the eyes focused on a point in front ofor behind the images, produces a sensation of depth, withobjects appearing to be in front of or behind the displaylevel. Tyler and Clarke proposed a stereoscopic technique thatallows the stereoscopic presentation of 3D form from a singleprinted image by a random dot pattern. These are known asSingle Image Random Dot Stereograms (SIRDS), or RandomDot Autostereograms [13].

The appearance of a SIRDS consists of many random dotsthat have a similar appearance with shares in a VCS. The onlydifference is that people can reconstruct the original 3D objectvia binocular disparity from a SIRDS. Hence, hiding a shareof a VCS in a SIRDS can reduce suspicion of hidden secrets.This property indicates that the SIRDS is a natural, and the

best, candidate to serve as a cover image for a share of aconventional VCS. We are interested in developing a noveltechnique for sharing visual secrets using SIRDSs.

In this study, a 2-out-of-n binocular VCS, called the(2, n)-BVCS, is proposed to provide non-expanded and high-quality cover images for shares of the VCS to reduce the riskof interception during the transmission phase. The proposed(2, n)-BVCS shares a binary secret image with n participants(n ≥ 2); when any two participants stack their transparences,the encrypted secret is revealed. The shares of the (2, n)-BVCSare hidden in n SIRDSs to reduce susceptibility to attackersduring the transmission phase. The proposed encryption pro-cedure consists of two phases. In the first phase, the proceduregenerates n SIRDSs by using existing autostereogram gener-ation programs. In the second phase, the procedure alters therandom dots in the SIRDSs according to a construction ruleof the (2, n)-BVCS for hiding the binary secret image. Theconstruction rule is a guideline for hiding the secret image inthe SIRDSs securely. However, altering the dots in a SIRDSwill interfere with the human brain’s ability to perceive theoriginal 3D objects in the SIRDSs and degrade the visualquality of the reconstructed 3D objects. Hence, we adoptan optimization approach to find construction rules for the(2, n)-BVCS such that the encryption process yields a secureBVCS and the contrast of the recovered secret image can bemaximized, subject to the visual quality of the SIRDSs.

The remainder of this paper is organized as follows.Section II, provides a brief survey of previous works.Section III, reviews the threshold VCSs. Section IV statesthe problem of the (2, n)-BVCS. The proposed encryptionprocedure, optimization model, and encryption algorithms arepresented in Section V. In Section VI, the performance ofthe proposed scheme is evaluated by experiments. Finally,concludes this work in Section VII.

II. PREVIOUS WORK

From the perspective of research methodology, researchinto the VCSs with meaningful shares can be classifiedinto two approaches: cryptography approaches and embeddedapproaches.

The cryptographic approach uses a set of basis matrices[5], [6] or an algorithm [7], [9] to simultaneously encrypta VCS and provide a meaningful appearance for the sharesof the VCS. The former method requires designing a setof basis matrices for a specific VCS, and suffers from thepixel expansion problem. The random-grid-based (RG-based)approach (an algorithmic method) involves constructing VCSsand EVCSs [7], [9]. The main idea behind the RG-basedEVCS algorithm approach is that it encrypts a secret imageto the shares according to a given probability p and stampscover images on the shares with (1 − p) probability. Theencryption of the secret image can use any existing RG-basedVCS. By adjusting probability p, the algorithm can tune thevisual qualities between the recovered image and the sharesof an EVCS. Chen et al. and Guo et al. proposed RG-based(2, 2)- and (k, k)-EVCSs, respectively. Chen et al.’s approachmust use a pair of complementary images as cover images.

Fig. 1. An example of Zhou et al.’s approach. (a) Sharing a black secretpixel, (b) sharing a white pixel.

Guo et al.’s approach does not need to adopt complementaryimages as cover images, but the visual quality of the shares isreduced when probability p is too small or too large.

The embedded approach tries to stamp covering images inthe shares of a VCS [8] or to hide shares behind coveringimages [10]–[12]. Zhou et al. proposed a halftone VCSthat can construct (2, 2)-EVCSs via complementary coveringshares [10]. First, they prepared a pair of complementaryhalftone images, I and I

′, as covers of noisy shares. Halftone

image I is obtained by applying any halftoning method ona gray-level image. Halftone image I

′is obtained by revers-

ing all black/white pixels of image I to white/black pixels.Second, a secret pixel is encoded as m sub-pixels (calledsecret information pixels) for each share; the sub-pixels arerandomly selected from two basis matrices (i.e., C0 and C1)of the conventional (2, 2)-VCS. These sub-pixels are used tomodify the Q1×Q2 halftone cell in both shares, I and I

′. Zhou

developed a void and cluster algorithm to select m positionsin the halftone cells to embed the m secret information pixels.Hence, the secret image is revealed by the secret informationpixels when the shares are stacked together. In Fig. 1, a secretpixel is shared to two 4 × 4 halftone cells in shares I and I

′.

If the secret pixel is black, two sub-pixels for each share, [ 0 1 ]and [ 1 0 ], are randomly selected from C1. The positions forembedding the secret information pixels are marked A and B.As shown in Fig. 1(a), sub-pixels [ 0 1 ] (i.e., a white pixeland a black pixel) were embedded into positions A and B ofthe halftone cell in share I. Sub-pixels [ 1 0 ] were embeddedinto share I

′. In this way, the stacked halftone cell will reveal a

black secret pixel. Another example for sharing a white pixelis shown in Fig. 1(b). Applying Zhou’s approach, the sizeof a halftone cell must be greater than or equal to the pixelexpansion factor. The visual quality of the halftone sharesimproves as the size of a halftone cell increases; however,there is a tradeoff between the visual quality of the meaningfulshares and the visual quality of the recovered images. Zhou’sapproach can be extended to an arbitrary access structure, butit may require distributing several images to participants.

The proposed approach can be classified into the embeddedapproach. There are two major differences between this studyand the previous research. First, this study adopts SIRDSs ascovering images of VCSs. Second, this study presents newconstruction rules for sharing a secret image rather than usingconventional basis matrices or RG-based algorithms.

III. PRELIMINARY OF THE THRESHOLD PROBVCSS

In this study, shares of size-invariant VCSs (SIVCS) arehidden in n SIRDSs to share a binary secret image using theproposed BVCS. Therefore, we first review the basic conceptsof SIVCSs.

A SIVCS, also called a probabilistic VCS (ProbVCS), firstproposed by Ito et al. in 1999 [2], relied on existing basismatrices of conventional VCSs. Afterward, Yang proposedgeneral construction rules for (2, n)- and (n, n)-ProbVCSs in2004 [3]. Both Ito et al. and Yang proved that a ProbVCSis as secure as a conventional VCS. The decryption processof SIVCSs directly stacks shared images; so, we assume thatblack and white pixels are represented as Boolean 1 and 0,respectively. Therefore, the stacking operation for sharedimages involves an “OR”-ed Boolean operation for eachpixel.

To state the concept of SIVCSs formally, we refer to ourprevious paper [4] and present the following definitions.

Definition 1: Let code collections μni denote the set of all

n-tuple 0/1 column vectors vn with Hamming weight i , where0 ≤ i ≤ n. That is, the pixel distribution pattern for codecollection μn

i is iB(n − i)W. �Definition 2: Code set �n = {μn

i |0 ≤ i ≤ n} denotes allcode collections of n-tuple vectors. �

Definition 3: The two sets C0 and C1 comprise the codebookof (k, n)-ProbVCS to encipher white and black secret pixels,respectively. Let all code collections μn

i , where the chosenprobability f 0(μn

i ) �= 0 ( f 1(μni ) �= 0), 0 ≤ i ≤ n, comprise

the codebook C0 (C1); these code collections are arrangedby i in ascending order. The overall chosen probability f 0

( f 1) for encrypting white (black) secret pixels is 1. That is,∑0≤i≤n f 0(μn

i ) = ∑0≤i≤n f 1(μn

i ) = 1. �Definition 4: Let F0 (F1) denote the chosen-probability set,

which consists of all chosen probabilities of code collectionsμn

i in codebook C0 (C1). There are |μni | column vectors in

code collections μni ; these column vectors have the same

chosen probabilities f 0(μni )

/|μni | ( f 1(μn

i )/|μn

i |) for white(black) secret pixels. �

Example 1: The construction rule for a (2, 3)-ProbVCSchooses a code collection from the code set �3 ={μ3

0, μ31, μ

32, μ

33} to encrypt a secret pixel. All code collections

μ3i are shown as follows:

μ30 =

⎧⎨

⎩

⎡

⎣000

⎤

⎦

⎫⎬

⎭, μ3

1 =⎧⎨

⎩

⎡

⎣100

⎤

⎦ ,

⎡

⎣010

⎤

⎦ ,

⎡

⎣001

⎤

⎦

⎫⎬

⎭

μ32 =

⎧⎨

⎩

⎡

⎣110

⎤

⎦ ,

⎡

⎣101

⎤

⎦ ,

⎡

⎣011

⎤

⎦

⎫⎬

⎭, and μ3

3 =⎧⎨

⎩

⎡

⎣111

⎤

⎦

⎫⎬

⎭.

Assume we used chosen-probability sets F0 = {0.5, 0, 0, 0.5}and F1 = {0, 0, 1, 0} to construct the (2, 3)-ProbVCS. Theencryption process randomly selects one column vector fromC0 (C1) to share a white (black) secret pixel, where

C0 ={μ3

0, μ33

}and C1 =

{μ3

2

}. �

One of the major metrics for evaluating the performance ofVCSs is the contrast of the recovered image. For a recovered

(binary) image, the contrast of the image is the difference inthe blackness of the recovered pixels from the black and whitesecret pixels.

Definition 5: Parameter G(r,n)i denotes the probability of the

appearance of a black pixel in a recovered image, which isyielded by stacking r shares in a (k, n)-VCS, if the sharedpixels are encoded by μn

i . G(r,n)i can be calculated as

G(r,n)i = 1 −

∏r−1

j=0

(

1 − i

n − j

)

. (1)

�For example, for a (2, 5)-VCS, shared pixels are encoded

by 2B3W; that is, μ52. The probability of the appearance of

black stacking pixels is G(2,5)2 = 1− (

1 − 25

)× (1 − 2

4

) = 0.7.The formal definitions for contrast and blackness for Prob-

VCSs are given below.Definition 6: In a (k, n)-ProbVCS, the appearance prob-

abilities of black pixels for reconstructed white and blacksecret pixels are p0 = ∑n

i=1 f 0(μn

i

) · G(k,n)i and p1 =

∑ni=1 f 1

(μn

i

) · G(k,n)i , respectively. Contrast α of the (k, n)-

ProbVCS can be defined as Equation (2). Blackness β of therecovered image is formulated as Equation (3)

α = p1 − p0 =∑n

i=1( f 1 (

μni

) − f 0 (μn

i

)) · G(k,n)

i (2)

β = p1 =n∑

i=1

f 1 (μn

i

) · G(k,n)i (3)

�Definition 7: Assume αTH (αTH > 0) is the threshold for a

human visual system to detect a difference in an image. Whenr shares are stacked, the solution to the (k, n)-ProbVCS isconsidered feasible if the following conditions are satisfied:

1. ∀1 ≤ r < k,∑n

i=0

(( f 1

(μn

i

) − f 0(μn

i

)) · G(μ

(r,n)i )

)=

0.2. For r = k, contrast α ≥αTH.

Condition 1 is the security condition that restricts accessto a secret in any forbidden set. Condition 2 ensures that theblackness of recovered black secret pixels is higher than thatof recovered white secret pixels in a recovered image whichstacks at least k shares. If α ≥ αTH, a human visual systemcan detect a difference in contrast from the recovered image.If αTH is large enough, a human visual system can distinguishbetween the recovered black and white secret pixels to obtainthe secret images.

IV. PROBLEM STATEMENT

Although both SIRDSs and shares of VCSs have the samenoise-like appearance, the pixel distributions for a set ofSIRDSs and for shares of a specific VCS are quite different.The pixel distribution among shared pixels must obey theconstruction rules or codebooks of the VCS. Shared pixelsmean that a set of pixels shares the same secret pixel ina VCS. In Example 1, the codebook (i.e., C0 and C1) andthe chosen-probability sets (i.e., F0 = {0.5, 0, 0, 0.5} andF1 = {0, 0, 1, 0}) are used to construct (2, 3)-ProbVCS,

TABLE I

THE PIXEL DISTRIBUTION PROBABILITIES FOR THREE SIRDSS

AND SHARES OF VCSs

where

C0 =⎧⎨

⎩

⎡

⎣000

⎤

⎦ ,

⎡

⎣111

⎤

⎦

⎫⎬

⎭and C1 =

⎧⎨

⎩

⎡

⎣110

⎤

⎦ ,

⎡

⎣101

⎤

⎦ ,

⎡

⎣011

⎤

⎦

⎫⎬

⎭.

Hence, the pixel distribution patterns in the resultant sharescomply with C0 and C1. If the encryption process selectscolumn vector [ 1 0 1 ]T from C1 for sharing a black secretpixel, shares 1 and 3 will get a black pixel and share 2 will geta white pixel, and the pixel distribution pattern for the shareswill be 2B1W. The pixel distribution pattern, iB(n − i)W,indicates there are i black pixels and n − i white pixelsdistributed among n shared pixels. The probability of eachpixel distribution pattern for the (2, 3)-ProbVCS is listed inTable I. Notation d , is called the pixel density of a share (or aSIRDS), denotes the frequency of appearance of black pixelsin a share (or in a SIRDS). In this example, pixel density dof each share is 2

/3.

On the other hand, in a SIRDS, the image contains manyrandom-dot patterns that periodically repeat in the horizontaldirection; the stereopsis of the objects arises from differencesin the horizontal positions of the image. The pixel distributionin n SIRDSs that were generated independently is totallyindependent. Suppose each SIRDS has the same pixel densityd , the probability of pixel distribution pattern iB(n − i)W canbe calculated as following:

Pdi,n =

( ni

)× di × (1 − d)n−i . (4)

The pixel distribution among three SIRDSs is listed in Table I.In general, while all SIRDSs are stacked, each pixel distribu-tion pattern will uniformly appear in the stacked image. Hence,it is almost impossible to reveal any meaningful informationby stacking two shares together.

In this study, we try to alter pixels in SIRDSs such thatthe altered SIRDSs can share secret images the same wayas VCSs. In the following, we will investigate whether thealtered-pixels in a SIRDS will interfere with the visual effectof stereopsis in the SIRDS.

Fig. 2 illustrates an example for altering pixels in a SIRDS.The depth map, as shown in Fig. 2(a), is used to create theSIRDS in Fig. 2(b). In this paper, all autostereograms can beviewed in the wall-eyed viewing. The terms “wall-eyed” is acondition where eyes do not point in the same direction whenlooking at an object. Wall-eyed viewing requires the two eyesto adopt a relatively parallel angle. Hence, it is informallyknown as parallel-viewing. Fig. 2(c) shows the verification

Fig. 2. An example of hiding a secret image in a SIRDS. (a) The depthmap, (b) the SIRDS of Fig. 2(a), (c) the verification image of Fig. 2(b)(ε = 90 pixels), (d) the location map, (e) the altered SIRDS after hidingFig. 2(d), (f) the verification image of Fig. 2(e) (ε = 90 pixels).

image of a stereopsis in the SIRDS. The verification image,which is a computer-generated 2D image, can disclose thestereopsis in a SIRDS in 2D format. The verification imagefor a SIRDS can be generated as follows.

Definition 8 (Verification Image Generation Rule): Assumepx,y denotes pixel (x, y), in a SIRDS, and each pixel ( pvx,y)in the verification image can be produced by operationpvx,y = px,y ⊕ px−ε,y , where px−ε,y = 0 if x < ε. Parameterε is the separation parameter of the SIRDS and logical operator⊕ represents the XOR operation. �

Suppose the color of the original pixels in the SIRDS, asshown in Fig. 2(b), will be altered, black boxes in a locationmap, as shown in Fig. 2(d), indicates that the pixels in whichregions of the SIRDS could be altered. The color of theoriginal pixels within two black boxes is randomly alteredin various probabilities (20% and 50% for the top and bottomboxes, respectively). Fig. 2(e) demonstrates the SIRDS afteraltering the original pixels. By viewing Fig. 2(e) binocularly,we perceive that additional stereopsis appear in Fig. 2(e). Thestereopsis (i.e., two boxes) in the bottom of Fig. 2(e) is clearerthan the stereopsis in the top of Fig. 2(e). In Fig. 2(f), theverification image of Fig. 2(e) shows the same result.

From the above example, we have the following observa-tions: altered pixels could be disclosed by the verificationimage of a SIRDS and will interfere with the stereopsis inthe SIRDS. The degree of interference is directly proportionalto the number of altered pixels.

This observation indicates that the altered SIRDSs can bedetected by a human visual system, thus making it difficult toshare extra information in a RDS. Hence, there are a tradeoffbetween keeping the visual quality of stereopsis in a SIRDSand producing a high-quality VCS. When we try to construct

Fig. 3. The two-phase encryption process of (2, n)-BVCS.

a specific VCS from a set of SIRDSs, it may be necessaryto alter a large number of random pixels to obey the pixeldistribution rule of the VCS. However, these altered pixelscan be perceived as an illusion of 3D depth and these interferewith the original stereopsis in the SIRDS.

Based on the above observation, we formulated a mathe-matical optimization model to find an optimum solution toshare a secret image in SIRDSs where the objective is tomaximize contrast under the constraint of the visual qualityof SIRDSs. Using this model, dealers can adjust the visualquality of SIRDSs to obtain the best display quality of therecovered images.

V. THE (2, N)-BVC SCHEME

A. The Two-Phase Encryption Procedure

In this study, we propose a (2, n)-BVCS for sharing a binarysecret image in n SIRDSs. The proposed two-phase encryptionprocess is shown in Fig. 3. In the first phase, n depth mapsare used to produce n SIRDSs using the autostereogram gen-erator that adopts Thimbleby’s algorithm [13]. In the proposed(2,n)-BVCS, each depth map has the same image size and allgenerated SIRDSs have the same pixel density d .

In the second phase, according to construction rules for(2, n)-BVCS, pixels in the n generated SIRDSs are alteredto share a binary secret image for the SIRDSs by the(2, n)-BVCS encryptor. Based on the above-mentionedobservation, the encryptor tries to reduce the number of alteredpixels in a SIRDS to minimize the amount of interferenceintroduced into the resultant share. The construction rulesgenerator, based on given parameters, n and d , of each SIRDS,generates guidelines for altering pixels in the SIRDSs. Theencryptor alters pixels only within a specific region, which iscalled the encryption region, where black secret pixels appear.Due to the altered pixels could be disclosed in the verificationimage of a SIRDS. To preserve the security condition for eachshare of the BVCS, the encryption region will be enlarged tocover neighbors of the black secret pixels. Hence, a locationmap, which is a binary image as shown in Fig. 2(d) inSection IV, is used to indicate the encryption regions (i.e., theblack regions in the map) for the secret image in the BVCS.At the end of the second phase, n image-size-invariant sharesare generated for the (2, n)-BVCS.

B. The Construction Rules Generator

1) The Basic Idea of the (2, n)-BVCS Construction: Theconstruction rules generator generates construction rules basedon pixel density d of SIRDSs and a given access structureof the BVCS. The rules are used to construct the BVCS by

altering pixels in the SIRDSs; therefore, the construction rulessimultaneously satisfy the conditions of the VCSs and cannotdisclose any information related to the secret image in theverification image of a SIRDS.

Definition 9: The construction rules of a BVCS consist oftwo (n + 1) × (n + 1) matrices, M0 and M1, for sharingwhite and black secret pixels in a secret image. Modificationprobability m0

i, j (m1i, j ), 0 ≤ i ≤ n, 0 ≤ j ≤ n, is an element of

M0 (M1) and is used to alter the distribution pattern of n pixelsthat have the same coordinates in n SIRDSs from iB(n − i)Wto jB(n − j)W. Moreover,

∑0≤i≤n m0

i, j = ∑0≤i≤n m1

i, j = 1,∀0 ≤ j ≤ n. �

In the second phase, the encryptor adopts probabilities m0i, j

and m1i, j to change the original pixel distribution probability

Pdi,n of SIRDSs for encrypting a secret image in n SIRDSs.

The original pixel distribution probability for each pattern istherefore altered and becomes a function of m0

i, j and m1i, j .

Definition 10: Notations P0a and P1

a denote the alterationprobabilities of the original pixels for sharing white and blacksecret pixels, respectively, in a SIRDS. P0

a and P1a can be

calculated as

P0a =

∑n

i=0Pd

i,n ×(∑n

j=0δn

i, j × m0i, j

)(5)

P1a =

∑n

i=0Pd

i,n ×(∑n

j=0δn

i, j × m1i, j

)(6)

where δni, j = |i − j |/n represents the pixel alteration proba-

bility for each share when the pixel distribution pattern of then SIRDSs is changed from iB(n − i)W to jB(n − j)W. �

Definition 11: Assume f 0(μnj ) and f 1(μn

j ) denote thedistribution probabilities of μn

j (i.e., pattern jB(n − j)W) forwhite and black secret pixels, 0 ≤ j ≤ n. Functions f 0(μn

j )

and f 1(μnj ) indicate the total distribution probabilities of code

collection μnj among n resultant shares for sharing white and

black secret pixels in (2, n)-BVCS. Hence, pixel distributionprobability sets f 0 and f 1 can be calculated as

f 0(μn

j

)=

∑n

i=0Pd

i,n×m0i, j , (7)

f 1(μn

j

)=

∑n

i=0Pd

i,n×m1i, j , (8)

where Pdi,n =

( ni

)× di × (1 − d)n−i . �

According to the altered pixel distributions, f 0(μnj ) and

f 1(μnj ), the probabilities of the appearance of black pix-

els for reconstructed white and black secret pixels in the(2, n)-BVCS are p0 = ∑n

i=1 f 0(μni ) · G(2,n)

i and p1 =∑n

i=1 f 1(μni ) · G(2,n)

i . Contrast α and blackness β of therecovered image of the (2, n)-BVCS can be defined as

α = p1 − p0 =∑n

i=1( f 1 (

μni

) − f 0 (μn

i

)) · G(2,n)

i (9)

β = p1 =∑n

i=1f 1 (

μni

) · G(2,n)i . (10)

Based on the definition of (k, n)-ProbVCS, formal definitionof the (2, n)-BVCS problem is given in Definition 12.

Definition 12: The solution to the (2, n)-BVCS con-sists of two modification matrices M0 and M1 thatare used to alter the pixel distribution in n SIRDSs.

Assume the pixel density of each SIRDS is d and αTH(αTH>0) is the threshold for a human visual system to detecta difference in blackness in an image. When two shares arestacked, the solution is considered feasible if the followingconditions are satisfied:

1) (Security condition) P0a = P1

a .2) (Security condition)

n∑

i=1

f 0 (μn

i

) · G(1,n)i =

n∑

i=1

f 1 (μn

i

) · G(1,n)i = d.

3) (Contrast condition)

α =∑n

i=1( f 1 (

μni

) − f 0 (μn

i

)) · G(2,n)

i ≥αTH. �

Conditions 1 and 2 are the security conditions of the(2, n)-BVCS. Condition 1 ensures that shared pixels in eachshare have the same probability of being altered, regardlessof which pixels were used for sharing white or black secretpixels. In other words, the resultant share cannot avoid leakinga secret in its verification image. Condition 2 guaranteesthat the pixel density of each resultant share equals that ofthe original SIRDS. Expressions

∑ni=1 f 0

(μn

i

) · G(1,n)i and

∑ni=1 f 1

(μn

i

) · G(1,n)i represent the pixel density of white and

black shared pixels, respectively. Condition 3, which is calledthe contrast condition, ensures that the secret can be revealedwhen two shares are stacked.

Example 2: Assume a (2, 2)-BVCS is constructed by alter-ing pixels in two SIRDSs. The pixel density of each SIRDSis d = 0.75. Hence, by Equation (5), the pixel distributionprobabilities for patterns 0B2W, 1B1W, and 2B0W are 0.0625,0.375, and 0.5625, respectively. The encryptor uses matricesM0 and M1 as construction rules to yield the (2, 2)-BVCS,where

M0 =⎡

⎣1 0 0

1/6 2/3 1/60 0 1

⎤

⎦ and M1 =⎡

⎣0 1 00 1 00 1/9 8/9

⎤

⎦.

Pixel alteration probabilities δ2i, j , 0 ≤ i ≤ 2 and 0 ≤ j ≤ 2,

are listed as matrix �2, where

�2 =⎡

⎣0 0.5 1

0.5 0 0.51 0.5 0

⎤

⎦.

The pixel distribution probability among the resultant shares ofthe (2, 2)-BVCS can be calculated using Equations (7) and (8).For example, f 0(μ2

0) = 0.0625 × 1 + 0.375 × 1/6 = 0.125and f 0(μ2

1) = 0.375 × 2/3 = 0.25.From Equations (5) and (6), we have P0

a =∑2i=0 P0.75

i,2 ×(∑2

j=0 δ2i, j×m0

i, j ) = 0.0625 and P1a = 0.0625.

The first security condition is satisfied. From Equation (1), theappearance probability vectors for one share and two sharesare G(1,2)

i = [0, 1/2, 1] and G(2,2)i = [0, 1, 1], respectively.

Hence, the pixel density of the resultant share is∑2i=1 f 1(μ2

i ) · G(1,2)i = 0.75. Clearly, the second security

condition is also satisfied. Finally, based onEquation (9), the contrast of the (2,2)-BVCS isα = ∑2

i=1 ( f 1(μ2i ) − f 0(μ2

i )) · G(2,2)i = 0.125. �

TABLE II

LIST OF GIVEN PARAMETERS AND DECISION VARIABLES

2) The Optimization Model of the (2, n)-BVCS: The(2, n)-BVCS problem is formulated here as an optimizationmodel. Both constants d and n are given and we determinemodification matrices M0 and M1 for hiding white and blacksecret pixels in n SIRDSs. The objectives of this problem areto maximize the contrast of recovered secret images and tominimize the alteration probability of each SIRDS under thevisual quality and security constraints. The given parameters,decision variables, and formulations are listed in Table II.

Objective Function:

max. α =∑n

i=1( f 1 (

μni

) − f 0 (μn

i

)) · G(2,n)

i

min.P1a =

∑n

i=0Pd

i,n ×(∑n

j=0δn

i, j × m1i, j

)(P1)

Subject to:

P0a = P1

a (C1)n∑

i=1

f 0 (μn

i

) · G(1,n)i = d (C2)

n∑

i=1

f 1 (μn

i

) · G(1,n)i = d (C3)

P1a ≤ Pa,max (C4)

∑

0≤i≤nm0

i, j = 1, ∀0 ≤ j ≤ n (C5)

∑

0≤i≤nm1

i, j = 1, ∀0 ≤ j ≤ n (C6)

0 ≤ m0i, j ≤ 1, ∀0 ≤ i, j ≤ n (C7)

0 ≤ m1i, j ≤ 1, ∀0 ≤ i, j ≤ n (C8)

The first objective of the proposed model is to maximizethe contrast of the recovered image in the (2, n)-BVCS. Thecontrast value is the most important performance metric inVCSs; hence it is the major objective of the model. Theother objective of this model is to minimize the introducedinterference for the SIRDSs; hence the model minimizes thealteration probability of each SIRDS. The second objectiveis related to the visual quality of the SIRDSs. Constraints(C1)–(C3) are the security condition of the (2, n)-BVCS.Constraint (C4) is the visual quality constraint that limits themaximum alteration probability for each SIRDS (share) so thatit is no more than Pa,max . By tuning Pa,max , interference in thestereopsis in each SIRDS can be maintained at an acceptablelevel. Constraints (C5) and (C6) limit the overall modificationprobability of each pixel distribution pattern for sharing whiteand black secret pixels, respectively. Constraints (C7) and (C8)limit the range of decision variables m0

i, j and m1i, j .

The original mathematical model (P1) can be solved byexisting optimizers or by a customized algorithm. In ourprevious studies on VC-related problems, we successfullydeveloped simulated-annealing-based (SA-based) algorithmsto solve the construction problems of VCSs [4], [8]. Wealso developed a SA-based algorithm to solve mathematicalmodel P1 by modifying our previous algorithms. Given spacelimitations, we omit the details of that algorithm in this paper.

Example 3: Assume a (2, 3)-BVCS is constructed by alter-ing pixels in three SIRDSs. Parameters Pa,max and d are0.15 and 0.5, respectively. By Equation (4), we have P0.5

0,3 =P0.5

3,3 = 0.125 and P0.51,3 = P0.5

2,3 = 0.375. The appearance

probability vectors for one share and two shares are G(1,3)i =

[0, 1/3, 2/3, 1] and G(2,3)i = [0, 2/3, 1, 1], respectively.

Applying model (P1) to the (2, 3)-BVCS and solving themathematical model, we can obtain matrices M0 and M1 asconstruction rules to yield the (2, 3)-BVCS, where

M0 =

⎡

⎢⎢⎣

1 0 0 00.6 0.4 0 00 0 0.4 0.60 0 0 1

⎤

⎥⎥⎦ and M1 =

⎡

⎢⎢⎣

0 1 0 00 0.73 0.26 00 0.26 0.73 00 0 1 0

⎤

⎥⎥⎦.

Based on the (2, 3)-BVCS scenario, we can verify thefeasibility of solution M0 and M1. Alteration probabilitiesP0

a = ∑3i=0 P0.5

i,3 ×(∑3

j=0 δ3i, j×m0

i, j

)= 0.15 and P1

a = 0.15;hence, Constraints (C1) and (C4) are satisfied. Constraints(C2) and (C3) can be verified by evaluating expression∑3

i=1 f 0(μ3

i

) · G(1,3)i = ∑3

i=1 f 1(μ3

i

) · G(1,3)i = 0.5.

Clearly, Constraints (C5)–(C8) are also satisfied. Finally,the optimal contrast of the (2, 3)-BVCS is α =∑3

i=1

(f 1

(μ3

i

) − f 0(μ3

i

)) · G(2,3)i = 0.23. �

C. The (2, n)-BVCS Encryptor

Next, we design an encryption algorithm for the BVCSencryptor. Based on the modification rule for a given BVCS,

the algorithm alters pixels on n SIRDSs, ST1,. . ., STn , toshare a binary secret SE. The main idea of the encryptionalgorithm is as follows. Notation Cx,y = [pST1

x,y . . . pSTnx,y ]

denotes a collection of pixel colors for n SIRDSs ST1,. . ., STn

at coordinate (x, y). Notation H(Cx,y) denotes the Hammingweight of Cx,y . If pixel pL

x,y on location map L is a white

pixel (i.e., pLx,y = 0), pixels pS1

x,y, . . ., pSnx,y on the resultant

shares S1,. . ., Sn are the same as pixels pST1x,y ,. . ., pSTn

x,y onSIRDSs ST1, . . ., STn , respectively. Otherwise, resultant pixelspS1

x,y,. . ., pSnx,y could be different from pixels pST1

x,y ,. . ., pSTnx,y

according to secret image SE and the modification rules. Whenthe color of secret pixel pSE

x,y is white (i.e., pSEx,y = 0), the

encryption algorithm uses modification rule m0H(Cx,y),0

, . . .,

m0H(Cx,y),n

to alter the colors of pixels pST1x,y ,. . ., pSTn

x,y to

yield resultant pixels pS1x,y, . . ., pSn

x,y. On the contrary, theencryption algorithm uses modification rule m1

H(Cx,y),0, . . .,

m1H(Cx,y),n

to yield resultant pixels pS1x,y, . . ., pSn

x,y. Eventu-ally, the distribution of the resultant pixels, iB(n − i)W,will be altered to jB(n − j)W, where i= H(Cx,y)and 0 ≤ j ≤ n.

The encryption algorithm for the (2,n)-BVCS is listed inTable III. The input images include n SIRDSs ST1, . . .,STn , one secret image SE, and one location map L. Theoutput images are n resultant shares S1, . . ., Sn . NotationpI

x,y denotes pixel colors of image I in coordinate (x, y),I ∈ {ST1, . . . ,STn, SE, L,S1, . . . ,Sn}. In Step 1, resultantshares S1, . . ., Sn are initialized to SIRDSs ST1, . . ., STn ,respectively. Steps 3–10 embed a secret pixel at coordinate(x, y) where location map L contains a black pixel. Step 3calculates the Hamming weight of pST1

x,y , . . ., pSTnx,y . Step 6

determines the number of black pixels in n resultant shares,bS, according to random number ρ, which is generated inStep 4, bST, construction rules M0 and M1 as well as thecolor of secret pixel c. In this study, we adopt the well-knownproportionate selection method, roulette wheel selection, forselecting bS. Based on the given parameters c and bST, we usea set of modification probabilities mc

bST,0, mc

bST,1, . . ., mc

bST,nto determine the number of black pixels in the resultant shares.Steps 7 and 9 alter the pixel distribution of n resultant sharesfrom bSTB

(n − bST

)W to bSB

(n − bS

)W. In Step 7, when

bS > bST, the algorithm randomly alters bS −bST white pixelsof n shares in (x, y). On the contrary, when bS < bST, thealgorithm randomly alters bST −bS black pixels of n shares in(x, y) in Step 9. Finally, Step 11 outputs the resultant sharesS1, . . ., Sn .

Example 4: Assume matrices M0 and M1 are the construc-tion rules for yielding the (2, 3)-BVCS, where

M0 =

⎡

⎢⎢⎣

1 0 0 00.8 0.2 0 00 0 0.2 0.80 0 0 1

⎤

⎥⎥⎦ and M1 =

⎡

⎢⎢⎣

0 0 1 00 0.87 0.13 00 0.47 0.53 00 0 1 0

⎤

⎥⎥⎦.

Collection Cx,y = [ 1 0 1 ] shows there are black, white,and black pixels on ST1, ST2, and ST3 in (x, y). Hence,the Hamming weight of Cx,y , H(Cx,y), is 2. For embeddinga white secret pixel in (x, y), the candidate modification

TABLE III

THE ENCRYPTION ALGORITHM FOR THE BVCS

probability set is [ 0 0 0.2 0.8 ]. When ρ = 0.5, collectionCx,y = [ 1 0 1 ] will be altered to Cx,y = [ 1 1 1 ]. Forembedding a black secret pixel in (x, y) where collectionCx,y = [ 1 0 0 ], the candidate modification probability setis [ 0 0.87 0.13 0 ]. When ρ = 0.9, collection Cx,y = [ 1 0 0 ]will be altered randomly to Cx,y = [ 1 1 0 ] or Cx,y = [ 1 0 1 ]randomly.

VI. EXPERIMENTAL RESULTS

In this section, we discuss a series of experiments thatwere conducted to assess the performance of the proposed(2, n)-BVCSs. We also present some demonstrations of theimplementation results for observing the visual effects of theBVCSs. Finally, we compare the properties of this study withprevious approaches.

A. Performance Evaluation

First, we assess the performance of the proposed algorithmfrom a quantitative viewpoint. In this experiment, we solvethe (2,n)-BVCS, 2 ≤ n ≤ 10, optimization problem subjectto various pixel alteration probabilities of SIRDSs Pa,max .The values of Pa,max range between 10% and 40%. Pixeldensity d of SIRDSs ranges between 40% and 80% in eval-uating how different values of d affect performance. In thisstudy, contrast α of the recovered images, which is defined inobjective function P1, is the major performance metric. Thesecond performance metric is the alteration probability of aSIRDS, which is the second goal of the optimization model.In general, when the contrast of an image is fixed, the visualquality of the image is proportional to the blackness of theimage. So, we take the blackness of the recovered image asthe third performance metric.

TABLE IV

THE BEST CONTRAST (%) FOR (2, n)-BVCS (Pa,max = 40%)

TABLE V

THE CORRESPONDING BLACKNESS (%) FOR THE (2, n)-BVCS

LISTED IN TABLE IV (Pa,max = 40%)

Table IV and Table V list the best contrast and the corre-sponding blackness of the recovered images in various (2,n)-BVCSs; in each scheme, Pa,max is set to 40%. From thesetables, we make the following observations. First, pixel densityd of SIRDSs will affect the contrast value of the recoveredimages in a (2,n)-BVCS. The peak contrast value of thisscenario can be found when the given value of d is between50% and 60%. When d is outside the range, the contrastvalue is inversely proportional to the blackness of SIRDSs.Second, the blackness of the recovered images is proportionalto the value of d . When d ≥ 60%, the blackness of therecovered images can reach 80%. Hence, pixel density dis an effective parameter for adjusting the blackness of therecovered images. Third, in both cases where the given pixeldensities of SIRDSs are 40% and 60%, the recovered imageshave the same contrast value, but the latter case has a higherblackness for the recovered images. The other scenarios (i.e.,Pa,max values 10%, 20%, and 30%) also agree with this result.Therefore, we omit the first case in the remainder of theexperiments. Fourth, both the contrast values and the blacknessvalues of the recovered images decrease in (2, n)-BVCSs as nincreases. This characteristic is the same as for conventional(2, n)-VCSs.

Table VI shows that the alteration probability of SIRDSsreaches its peak when the given Pa,max is larger than thepeak value. For example, the peak alteration probability ofSIRDSs in the (2, 2)-BVCS is no more than 25% when thegiven Pa,max ≥ 25%. Table VI verifies the effectiveness of thesecond goal of the optimization model.

The performance of (2, n)-BVCSs under other scenarios(i.e., Pa,max values 10%, 20%, and 30%) shows the sametrends as listed in Table IV and Table V. Therefore, we listonly the ranges of the performance results of these scenariosin Table VII and Table VIII. The actual value of the alterationprobability of SIRDSs in each scenario is equal to the givenvalue of Pa,max until parameter Pa,max reaches the peak value

TABLE VI

THE PEAK ALTERATION PROBABILITY (%) OF EACH SIRDS IN THE

(2, n)-BVCS LISTED IN TABLE IV (Pa,max = 40%)

TABLE VII

THE RANGES OF THE BEST CONTRAST OF THE (2, n)-BVCSS WHILE

PIXEL DENSITY (d ) OF SHARES RANGES BETWEEN 50% AND 80%

TABLE VIII

THE RANGES OF THE BLACKNESS OF THE (2, n)-BVCSS WITH PIXEL

DENSITY (d ) OF SHARES RANGES BETWEEN 50% AND 80%

of each scenario as listed in Table VI. Table VII indicates thatthe best contrast value is proportional to the given value ofPa,max and that it reaches its peak when the actual alterationprobability reaches its peak. The more alterations there are ina SIRDS, the more interference is introduced into a SIRDS,which will lead to degradation of the visual quality of theSIRDS. Hence, there is a tradeoff between the visual qualityof the recovered images and the visual quality of the SIRDS.By tuning parameter Pa,max , we can get the desired visualquality for the (2, n)-BVCS.

B. Demonstrations and Discussions

In this subsection, we evaluate the visual effects of theproposed algorithm by observing implementation results of(2, n)-BVCSs.

1) Experiment-I: Experiment-I investigates the performanceof a (2, 2)-BVCS. The binary secret image and its location

Fig. 4. The secret image and the location map that is used in Experiment-I,(a) the secret image (512 × 280 pixels), (b) the location map.

Fig. 5. The generated SIRDS 1 (the depth map and its corresponding V-shareare shown Fig. 2(a) and Fig. 2(c), respectively).

map are shown in Fig. 4. In the first phase, the depth map, asshown in Fig. 2(a), is used to produce two different SIRDSs(SIRDS 1 is shown as Fig. 5) using the autostereogramgenerator. The pixel density of the SIRDSs (d) is set to 0.5.In the second phase, the generated SIRDSs, the secret image,and the location map are used to yield two shares, as shownin Fig. 6, of the (2, 2)-BVCS. The construction rules for the(2, 2)-BVCS are found by the proposed optimization modelbased on the parameters Pa,max = 25% and d = 0.5. The bestcontrast of the recovered image is 0.5; the construction rulesare as follows:

M0 =⎡

⎣1 0 0

0.5 0 0.50 0 1

⎤

⎦ and M1 =⎡

⎣0 1 00 1 00 1 0

⎤

⎦.

All images used and generated in the same experi-ment in this section are in the same dimension. Giventhe space limitations in this paper, each image is reducedto a suitable size. The original images and more resultsof this study are available on the following website:http://www.csie.mcu.edu.tw/~khlee/bvcs/bvcs.htm.

Fig. 6 indicates that the alteration probabilities of the sharesand the contrast of the recovered image are very close totheir theoretical values. Note that the (2, 2)-BVCS and the(2, 2)-VCS have the same optimal contrast values (i.e., α =0.5) for the recovered images, a characteristic that has notbeen achieved previously in research that provides meaningfulshares for VCSs.

As shown in Fig. 2(a) and Fig. 4(a), the secret “BVCS”overlaps with the ball in the depth map. Compared withFig. 2(c) and Fig. 6(c), alterations for hiding “BVCS” cannotbe disclosed by the verification image of a share (hereinafterV-share), because all of the alterations occurred within the

Fig. 6. The implementation results of (2, 2)-BVCS in Experiment I,(a) share 1 (P1

a = 25.1%), (b) share 2 (P1a = 25.0%), (c) V-share 1, (d) the

recovered image (contrast = 0.5)

area of the 3D object in the SIRDS. By viewing shares(Fig. 6(a) and Fig. 6(b)) binocularly, the 3D object (i.e., aball) clearly emerges from the background. On contrary, a partof the secret content in the rightmost side of the secret image,as shown in Fig. 4(a), cannot be hidden in 3D objects in aSIRDS. Fig. 6(a) and Fig. 6(b) show that a 3D object (a semi-elliptical) emerges from the background in the rightmost sideof the share. The extra 3D object was introduced by alteringrandom dots in the background of the original SIRDS. Hence,the 3D perception of the extra object is dimmer than that of the3D ball in the shares. Because of the interference that occursin the background of the SIRDS, it can be disclosed by itsV-share, as shown in Fig. 6. Although the extra object can besensed, it reveals nothing about the secret image. Eventually,the share satisfies the first security condition of (2, n)-BVCSs.

2) Experiment-II:: Experiment-II uses four differentSIRDSs as cover images of a (2, 4)-BVCS. The depth maps,the secret image, and its location map are shown in Fig. 7.Two experiments, Set-A and Set-B, are performed on the(2, 4)-BVCS. The construction rules of Set-A are based onthe parameters Pa,max = 30% and d = 0.6. The best contrast

Fig. 7. The images that is used in Experiment-II—the (2, 4)-BVCS,(a) ∼ (d) depth maps 1 ∼ 4 (800 × 600 pixels), (e) the secret image,(f) the location map.

of the recovered image is 0.3. Some of the implementationresults of experiment Set-A are shown in Fig. 8.

Fig. 8(e) shows that the contrast value and the blacknessof the recovered image can reach 0.29 and 90%, respectively.The (2, 4)-BVCS can produce excellent visual quality for therecovered images. By observing four shares in Fig. 8, the 3Dscenes can be perceived from each share, but interference alsocan be detected in the shares. Within the interference area(i.e., the area of the black box in Fig. 7(f)), the 3D objectsbecome unclear. From Fig. 8(a) to Fig. 8(d), note that theinterference in each share is not the same. A rectangle areacan be detected in the center of share 2. In the other sharesin Fig. 8, the interference is unclear.

Fig. 9 shows the results of experiment Set-B, with para-meters Pa,max = 15% and d = 0.5. The best contrast of therecovered image is reduced to 21.25%. In this experiment, thealteration probability Pa,max is only a half that found in Set-A.The rectangle box in the center of share 2 (as shown inFig. 9(b)) becomes very dim. Comparing Fig. 9(b) andFig. 8(b), the 3D scene in Fig. 9(b) is clearer than in Fig. 8(b)because the interference is reduced in experiment Set-B. Thatmeans the visual quality of the shares and the visual qualityof the recovered image can be balanced by tuning parametersPa,max and d . The above experiments prove the effective-ness of adjusting parameters Pa,max and d in the proposed(2, n)-BVCSs. In addition, the verification image (e.g.,V-share 2 in Fig. 8(f)) fails to reveal the interference on theshare because of the complicated content (i.e., rich depth leveland less area of background) and the depth map of the SIRDS.

This section demonstrates the implementation results forthe (2, 2)- and (2, 4)-BVCSs. These results show that theproposed approach can produce high visual–quality sharesand high-contrast recovered image for (2, n)-BVCSs. Hence,both the transmission risk for the VCS and the visual qualitydegradation of the recovered image can be reduced. Finally,there are two observations that can be made from the aboveexperiments. First, the proposed BVCS is suitable for use withvarious kinds of depth maps to generate SIRDSs to cover theshares. The 3D objects of the shares retain excellent visualquality if the BVCS adopts complicated content SIRDSs ascovers. On the contrary, if the SIRDSs contain a large area

Fig. 8. The implementation results of the (2, 4)-BVCS in Experiment-II with Set-A parameters Pa,max = 30% and d = 0.6, (a) ∼ (d) shares 1 ∼ 4(800 × 600 pixels), (e) the recovered image (contrast = 0.29), (f) V-share 2.

Fig. 9. The implementation results of the (2, 4)-BVCS in Experiment-II with Set-B parameters Pa,max = 15% and d = 0.5, (a) ∼ (b) shares 1 ∼ 2(800 × 600 pixels), (c) the recovered image (contrast = 0.21).

of background, the interference for the shares can be reducedby carefully selecting the hidden place for the secret image.Second, altering dots on a SIRDS degrades the visual qualityof the SIRDS. The factors that affect the degradation includethe number of altered dots, the place where the dots werealtered, and the contents of the depth map that was used togenerate the SIDRS. Hence, the visual quality of 3D sceneson shares can be enhanced by tuning these factors.

C. Comparison

Table IX shows a comparison of the properties betweenthis study and previous works [9]–[12]. The RG-basedapproach [9] can remove the drawback of pixel expansion,but it cannot provide high visual–quality shares and recoveredimages.

The embedded approaches [10]–[12] can generate halftoneshares with high visual quality, hence these approaches can

TABLE IX

COMPARISON OF THE STUDY AND PREVIOUS APPROACHES

reduce the suspicion of an encrypted secret. To retain theaspect ratio of the images during the halftoning process, thesize of a halftone cell should be a square number (e.g., 4, 9,16,. . .), in which case the maximum contrasts of the recovered

images using the embedded approach will be reduced to 1/4,1/9, 1/16,. . .. These approaches introduce serious pixel expan-sion, which reduces the contrast value of the recovered image.On the contrary, the maximum contrasts of the (2, n)-BVCSwith parameters Pa,max = 25% and d = 0.5 are 0.5, 0.33,0.27, 0.25, 0.23, 0.22, and 0.21, respectively. These resultsare very close to the best contrast values for conventional(2, n)-VCSs in [4]. Based on the above discussion, the per-formance of the proposed (2, n)-BVCS is superior to theperformance of previous studies in terms of pixel expansionand contrast of the recovered image.

The visual quality of meaningful shares can be measuredby the peak signal-to-noise ratio and the universal qualityindex [14], but other factors also affect human’s visual percep-tion. For example, the halftone shares with rich contents havebetter visual perception than shares with a simple binary cover,even if they have the same assessment index. The SIRDSs and2D images have total different visual perception; therefore, acomparison of the visual quality between the SIRDS sharesand the halftone shares may not fair. By observing the SIRDSshares in Section VI-B, we believe that the visual quality ofthe SIRDS shares is sufficient to cover shares of VCSs from atransmission security perspective. In other words, the proposedBVCS is an excellent scheme for reducing the suspicion of anencrypted secret.

VII. CONCLUSION

This study proposed a (2, n)-BVCS and developed a newmethod for hiding a size-invariant (2, n)-VCS in n SIRDSs.This work explored the possibility of hiding a share of a VCSin SIRDSs that are printed on transparencies. We developed amathematical model that defines a set of construction rules sothat the recovered images of (2, n)-BVCSs have the highestcontrast under the constraint of the interference introducedinto the SIRDSs. Using this mathematical model, a desiredvisual quality for shares and recovered images can be foundby adjusting parameters Pa,max and d . The best contrastfor the recovered images in (2, n)-BVCSs, 2 ≤ n ≤ 10,ranges between 0.5 and 0.2, and can produce clear recoveredimages for a (2, n)-BVCS. The experimental results prove theeffectiveness and the flexibility of the proposed (2, n)-BVCSs.In the near future, we plan to extend this study to explore newmethods for hiding a (k, n)-VCS in n SIRDSs.

ACKNOWLEDGMENT

Hereby, the authors appreciate the anonymous reviewers fortheir valuable comments.

REFERENCES

[1] M. Naor and A. Shamir, “Visual cryptography,” Advances inCryptology—EUROCRYPT (Lecture Notes in Computer Science).New York, NY, USA: Springer-Verlag, 1995, pp. 1–12.

[2] R. Ito, H. Kuwakado, and H. Tanaka, “Image size invariant visualcryptography,” IEICE Trans. Fundam. Electron., Commun., Comput.Sci., vol. E82-A, no. 10, pp. 481–494, 1999.

[3] C. N. Yang, “New visual secret sharing schemes using probabilis-tic method,” Pattern Recognit. Lett., vol. 25, no. 4, pp. 481–494,Mar. 2004.

[4] P. L. Chiu and K. H. Lee, “A simulated annealing algorithm for generalthreshold visual cryptography schemes,” IEEE Trans. Inf. ForensicsSecurity, vol. 6, no. 3, pp. 992–1001, Sep. 2011.

[5] G. Ateniese, C. Blundo, A. D. Santis, and D. R. Stinson, “Extendedcapabilities for visual cryptography,” Theoretical Comput. Sci., vol. 250,nos. 1–2, pp. 143–161, Jan. 2001.

[6] D. Wang, F. Yi, and X. Li, “On general construction for extendedvisual cryptography schemes,” Pattern Recognit., vol. 42, no. 11,pp. 3071–3082, Nov. 2009.

[7] T.-H. Chen and K.-H. Tsao, “User-friendly random-grid-based visualsecret sharing,” IEEE Trans. Circuits Syst. Video Technol., vol. 21,no. 11, pp. 1693–1703, Nov. 2011.

[8] K.-H. Lee and P.-L. Chiu, “An extended visual cryptography algorithmfor general access structures,” IEEE Trans. Inf. Forensics Security, vol. 7,no. 1, pp. 219–229, Feb. 2012.

[9] T. Guo, F. Liu, and C. Wu, “k out of k extended visual cryptogra-phy scheme by random grids,” Signal Process., vol. 94, pp. 90–101,Jan. 2014.

[10] Z. Zhou, G. R. Arce, and G. D. Crescenzo, “Halftone visual cryptog-raphy,” IEEE Trans. Image Process., vol. 15, no. 8, pp. 2441–2453,Aug. 2006.

[11] Z. Wang, G. R. Arce, and G. D. Crescenzo, “Halftone visual cryptog-raphy via error diffusion,” IEEE Trans. Inf. Forensics Security, vol. 4,no. 3, pp. 383–396, Sep. 2009.

[12] F. Liu and C. Wu, “Embedded extended visual cryptography schemes,”IEEE Trans. Inf. Forensics Security, vol. 6, no. 2, pp. 307–322,Jun. 2011.

[13] H. W. Thimbleby, S. Inglis, and I. H. Witten, “Displaying 3D images:Algorithms for single-image random-dot stereograms,” Computer,vol. 27, no. 10, pp. 38–48, Oct. 1994.

[14] W. Zhou and A. C. Bovik, “A universal image quality index,” IEEESignal Process. Lett., vol. 9, no. 3, pp. 81–84, Mar. 2002.

Kai-Hui Lee received the Ph.D. degree in electronicengineering from the National Taiwan University ofScience and Technology, Taipei, Taiwan, in 2002.He is a Professor with the Department of ComputerScience and Information Engineering, Ming ChuanUniversity, Taoyuan, Taiwan.

His current research interests include visual cryp-tography, wireless networks, and network resourcemanagements.

Pei-Ling Chiu received the Ph.D. degree in informa-tion management from National Taiwan University,Taipei, Taiwan, in 2007. She is a Professor withthe Department of Risk Management and Insurance,Ming Chuan University, Taipei, Taiwan.

Her research focuses on visual cryptography, wire-less sensor networks, and optimizing technologies.