VISUAL cryptography (VC) is a technique that encrypts a

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1 4336 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 10, OCTOBER 2014 Sharng Vsual Secrets n Sngle Image Random Dot Stereograms Ka-Hu Lee and Pe-Lng Chu Abstract Vsual cryptography schemes (VCSs generate random and meanngless shares to share and protect secret mages. Conventonal VCSs suffer from a transmsson rsk problem because the nose-lke shares wll rase the suspcon of attackers and the attackers mght ntercept the transmsson. Prevous research has nvolved n hdng shared content n halftone shares to reduce these rsks, but ths method exacerbates the pxel expanson problem and vsual qualty degradaton problem for recovered mages. In ths paper, a bnocular VCS (BVCS, called the (2, n-bvcs, and an encrypton algorthm are proposed to hde the shared pxels n the sngle mage random dot stereograms (SIRDSs. Because the SIRDSs have the same 2D appearance as the conventonal shares of a VCS, ths paper tres to use SIRDSs as cover mages of the shares of VCSs to reduce the transmsson rsk of the shares. The encrypton algorthm alters the random dots n the SIRDSs accordng to the constructon rule of the (2, n-bvcs to produce nonpxelexpanson shares of the BVCS. Alterng the dots n a SIRDS wll degrade the vsual qualty of the reconstructed 3D objects. Hence, we propose an optmzaton model that s based on the vsual qualty requrement of SIRDSs to develop constructon rules for a (2, n-bvcs that maxmze the contrast of the recovered mage n the BVCS. Index Terms Vsual cryptography, sngle mage random dot stereograms, transmsson rsk, pxel expanson. I. INTRODUCTION VISUAL cryptography (VC s a technque that encrypts a secret mage nto n shares, wth each partcpant holdng one share; any partcpant wth fewer than k, 2 k n, shares cannot reveal any nformaton about the secret mage. Stackng the k shares reveals the secret mage, whch can be recognzed drectly by the human vsual system [1]. Conventonal shares [1] [4], whch consst of many random and meanngless pxels satsfy the securty requrement for protectng secret contents, but they have a drawback there s a hgh transmsson rsk because nose-lke shares rase the suspcon of attackers, who may ntercept the shares. Thus the rsk both to the partcpants and to the shares ncreases n turn ncreasng the probablty of transmsson falure. Manuscrpt receved September 29, 2013; accepted July 29, Date of publcaton August 7, 2014; date of current verson September 2, Ths work was supported by the Mnstry of Scence and Technology, Natonal Scence Councl of Tawan, under Contract MOST E and Contract MOST H The assocate edtor coordnatng the revew of ths manuscrpt and approvng t for publcaton was Prof. Jana Dttmann. K.-H. Lee s wth the Department of Computer Scence and Informaton Engneerng, Mng Chuan Unversty, Taoyuan 33348, Tawan (e-mal: khlee@mal.mcu.edu.tw. P.-L. Chu s wth the Department of Rsk Management and Insurance, Mng Chuan Unversty, Tape 111, Tawan (e-mal: plchu@mal.mcu.edu.tw. Dgtal Object Identfer /TIP Prevous research nto the Extended Vsual Cryptography Scheme (EVCS provded a meanngful appearance for shares to make the nose-lke shares manageable for partcpants [5] [9]. However, the meanngful shares stll present a rsk of detecton. In EVCSs, the shares that are prnted on transparences stll contan many nose-lke pxels and/or dsplay low-qualty mages. Such shares are easly detected by the naked eye and partcpants who transmt the shares can easly rase the suspcon of potental attackers. Other research nvolves sharng secret mages va hghqualty shares [10] [12]. Zhou et al. proposed a (2, 2-VCS usng the halftonng technque to construct meanngful bnary mages as shares carryng sgnfcant vsual nformaton [10]. The vsual qualty of the halftone s sgnfcantly better than that attaned by extended VC. The shares obtaned usng Zhou et al. s approach can reduce the transmsson rsk of the shares, but that approach exacerbates the pxel expanson problem and the vsual qualty degradaton problem for the recovered mages. Other studes [11], [12] suffer from the same drawbacks as Zhou et al. s method. Gven these drawbacks, the extenson ablty of these approaches could be lmted. Therefore, further research s needed on the current VCSs to fnd an alternatve way to reduce the transmsson rsk problem for partcpants and shares. In 1838, Wheatstone dscovered stereoscopc vson and publshed an explanaton of stereopss (bnocular depth percepton arsng from dfferences n the horzontal postons of mages n the two eyes. When we look at two flat, dssmlar, 2D pctures, our mnd perceves an lluson of 3D depth. In 1960, Julesz developed the random-dot format of the stereogram, n whch the 3D form bypasses the monocular processes and s vsble only when stereoscopc fuson s obtaned. A random-dot stereogram (RDS s a stereo par of mages of random dots, whch when vewed wth the ad of a stereoscope or wth the eyes focused on a pont n front of or behnd the mages, produces a sensaton of depth, wth objects appearng to be n front of or behnd the dsplay level. Tyler and Clarke proposed a stereoscopc technque that allows the stereoscopc presentaton of 3D form from a sngle prnted mage by a random dot pattern. These are known as Sngle Image Random Dot Stereograms (SIRDS, or Random Dot Autostereograms [13]. The appearance of a SIRDS conssts of many random dots that have a smlar appearance wth shares n a VCS. The only dfference s that people can reconstruct the orgnal 3D object va bnocular dsparty from a SIRDS. Hence, hdng a share of a VCS n a SIRDS can reduce suspcon of hdden secrets. Ths property ndcates that the SIRDS s a natural, and the

2 best, canddate to serve as a cover mage for a share of a conventonal VCS. We are nterested n developng a novel technque for sharng vsual secrets usng SIRDSs. In ths study, a 2-out-of-n bnocular VCS, called the (2, n-bvcs, s proposed to provde non-expanded and hghqualty cover mages for shares of the VCS to reduce the rsk of ntercepton durng the transmsson phase. The proposed (2, n-bvcs shares a bnary secret mage wth n partcpants (n 2; when any two partcpants stack ther transparences, the encrypted secret s revealed. The shares of the (2, n-bvcs are hdden n n SIRDSs to reduce susceptblty to attackers durng the transmsson phase. The proposed encrypton procedure conssts of two phases. In the frst phase, the procedure generates n SIRDSs by usng exstng autostereogram generaton programs. In the second phase, the procedure alters the random dots n the SIRDSs accordng to a constructon rule of the (2, n-bvcs for hdng the bnary secret mage. The constructon rule s a gudelne for hdng the secret mage n the SIRDSs securely. However, alterng the dots n a SIRDS wll nterfere wth the human bran s ablty to perceve the orgnal 3D objects n the SIRDSs and degrade the vsual qualty of the reconstructed 3D objects. Hence, we adopt an optmzaton approach to fnd constructon rules for the (2, n-bvcs such that the encrypton process yelds a secure BVCS and the contrast of the recovered secret mage can be maxmzed, subject to the vsual qualty of the SIRDSs. The remander of ths paper s organzed as follows. Secton II, provdes a bref survey of prevous works. Secton III, revews the threshold VCSs. Secton IV states the problem of the (2, n-bvcs. The proposed encrypton procedure, optmzaton model, and encrypton algorthms are presented n Secton V. In Secton VI, the performance of the proposed scheme s evaluated by experments. Fnally, concludes ths work n Secton VII. II. PREVIOUS WORK From the perspectve of research methodology, research nto the VCSs wth meanngful shares can be classfed nto two approaches: cryptography approaches and embedded approaches. The cryptographc approach uses a set of bass matrces [5], [6] or an algorthm [7], [9] to smultaneously encrypt a VCS and provde a meanngful appearance for the shares of the VCS. The former method requres desgnng a set of bass matrces for a specfc VCS, and suffers from the pxel expanson problem. The random-grd-based (RG-based approach (an algorthmc method nvolves constructng VCSs and EVCSs [7], [9]. The man dea behnd the RG-based EVCS algorthm approach s that t encrypts a secret mage to the shares accordng to a gven probablty p and stamps cover mages on the shares wth (1 p probablty. The encrypton of the secret mage can use any exstng RG-based VCS. By adjustng probablty p, the algorthm can tune the vsual qualtes between the recovered mage and the shares of an EVCS. Chen et al. and Guo et al. proposed RG-based (2, 2- and(k, k-evcss, respectvely. Chen et al. s approach must use a par of complementary mages as cover mages. Fg. 1. An example of Zhou et al. s approach. (a Sharng a black secret pxel, (b sharng a whte pxel. Guo et al. s approach does not need to adopt complementary mages as cover mages, but the vsual qualty of the shares s reduced when probablty p s too small or too large. The embedded approach tres to stamp coverng mages n the shares of a VCS [8] or to hde shares behnd coverng mages [10] [12]. Zhou et al. proposed a halftone VCS that can construct (2, 2-EVCSs va complementary coverng shares [10]. Frst, they prepared a par of complementary halftone mages, I and I, as covers of nosy shares. Halftone mage I s obtaned by applyng any halftonng method on a gray-level mage. Halftone mage I s obtaned by reversng all black/whte pxels of mage I to whte/black pxels. Second, a secret pxel s encoded as m sub-pxels (called secret nformaton pxels for each share; the sub-pxels are randomly selected from two bass matrces (.e., C 0 and C 1 of the conventonal (2, 2-VCS. These sub-pxels are used to modfy the Q 1 Q 2 halftone cell n both shares, I and I. Zhou developed a vod and cluster algorthm to select m postons n the halftone cells to embed the m secret nformaton pxels. Hence, the secret mage s revealed by the secret nformaton pxels when the shares are stacked together. In Fg. 1, a secret pxel s shared to two 4 4 halftone cells n shares I and I. If the secret pxel s black, two sub-pxels for each share, [ 01] and [ 10], are randomly selected from C 1. The postons for embeddng the secret nformaton pxels are marked A and B. As shown n Fg. 1(a, sub-pxels [ 01] (.e., a whte pxel and a black pxel were embedded nto postons A and B of the halftone cell n share I. Sub-pxels [ 10] were embedded nto share I. In ths way, the stacked halftone cell wll reveal a black secret pxel. Another example for sharng a whte pxel s shown n Fg. 1(b. Applyng Zhou s approach, the sze of a halftone cell must be greater than or equal to the pxel expanson factor. The vsual qualty of the halftone shares mproves as the sze of a halftone cell ncreases; however, there s a tradeoff between the vsual qualty of the meanngful shares and the vsual qualty of the recovered mages. Zhou s approach can be extended to an arbtrary access structure, but t may requre dstrbutng several mages to partcpants. The proposed approach can be classfed nto the embedded approach. There are two major dfferences between ths study and the prevous research. Frst, ths study adopts SIRDSs as coverng mages of VCSs. Second, ths study presents new constructon rules for sharng a secret mage rather than usng conventonal bass matrces or RG-based algorthms.

3 III. PRELIMINARY OF THE THRESHOLD PROBVCSS In ths study, shares of sze-nvarant VCSs (SIVCS are hdden n n SIRDSs to share a bnary secret mage usng the proposed BVCS. Therefore, we frst revew the basc concepts of SIVCSs. A SIVCS, also called a probablstc VCS (ProbVCS, frst proposed by Ito et al. n 1999 [2], reled on exstng bass matrces of conventonal VCSs. Afterward, Yang proposed general constructon rules for (2, n- and(n, n-probvcss n 2004 [3]. Both Ito et al. and Yang proved that a ProbVCS s as secure as a conventonal VCS. The decrypton process of SIVCSs drectly stacks shared mages; so, we assume that black and whte pxels are represented as Boolean 1 and 0, respectvely. Therefore, the stackng operaton for shared mages nvolves an OR -ed Boolean operaton for each pxel. To state the concept of SIVCSs formally, we refer to our prevous paper [4] and present the followng defntons. Defnton 1: Let code collectons μ n denote the set of all n-tuple 0/1 column vectors v n wth Hammng weght, where 0 n. That s, the pxel dstrbuton pattern for code collecton μ n s B(n W. Defnton 2: Code set Ɣ n ={μ n 0 n} denotes all code collectons of n-tuple vectors. Defnton 3: The two sets C 0 and C 1 comprse the codebook of (k, n-probvcs to encpher whte and black secret pxels, respectvely. Let all code collectons μ n, where the chosen probablty f 0 (μ n = 0(f1 (μ n = 0, 0 n, comprse the codebook C 0 (C 1 ; these code collectons are arranged by n ascendng order. The overall chosen probablty f 0 ( f 1 for encryptng whte (black secret pxels s 1. That s, 0 n f 0 (μ n = 0 n f 1 (μ n = 1. Defnton 4: Let F 0 (F 1 denote the chosen-probablty set, whch conssts of all chosen probabltes of code collectons μ n n codebook C 0 (C 1.Thereare μ n column vectors n code collectons μ n ; these column vectors have the same chosen probabltes f 0 (μ n / μ n ( f 1 (μ n / μ n for whte (black secret pxels. Example 1: The constructon rule for a (2, 3-ProbVCS chooses a code collecton from the code set Ɣ 3 = {μ 3 0,μ3 1,μ3 2,μ3 3 } to encrypt a secret pxel. All code collectons μ 3 are shown as follows: μ 3 0 = 0, μ3 1 = 0, 1, μ 3 2 = 1 1, 1 0, 0 1, and μ3 3 = Assume we used chosen-probablty sets F 0 ={0.5, 0, 0, 0.5} and F 1 = {0, 0, 1, 0} to construct the (2, 3-ProbVCS. The encrypton process randomly selects one column vector from C 0 (C 1 to share a whte (black secret pxel, where C 0 = { μ 3 0,μ3 3 } and C 1 = { μ 3 2 }. One of the major metrcs for evaluatng the performance of VCSs s the contrast of the recovered mage. For a recovered (bnary mage, the contrast of the mage s the dfference n the blackness of the recovered pxels from the black and whte secret pxels. Defnton 5: Parameter G (r,n denotes the probablty of the appearance of a black pxel n a recovered mage, whch s yelded by stackng r shares n a (k, n-vcs, f the shared pxels are encoded by μ n.g(r,n can be calculated as G (r,n = 1 r 1 j=0 ( 1 n j. (1 For example, for a (2, 5-VCS, shared pxels are encoded by 2B3W; that s, μ 5 2. The probablty of the appearance of black stackng pxels s G (2,5 2 = 1 ( 1 2 ( = 0.7. The formal defntons for contrast and blackness for Prob- VCSs are gven below. Defnton 6: In a (k, n-probvcs, the appearance probabltes of black pxels for reconstructed whte and black secret pxels are p 0 = n =1 f 0 ( μ n (k,n G and p 1 = n=1 f 1 ( μ n (k,n G, respectvely. Contrast α of the (k, n- ProbVCS can be defned as Equaton (2. Blackness β of the recovered mage s formulated as Equaton (3 α = p 1 p 0 = n ( f 1 ( μ n =1 n β = p 1 = f 1 ( μ n =1 f 0 ( μ n (k,n G (2 G (k,n (3 Defnton 7: Assume α TH (α TH > 0 s the threshold for a human vsual system to detect a dfference n an mage. When r shares are stacked, the soluton to the (k, n-probvcs s consdered feasble f the followng condtons are satsfed: 1. 1 r < k, n =0 (( f 1 ( μ n f 0 ( μ n For r = k, contrast α α TH. G( μ (r,n = Condton 1 s the securty condton that restrcts access to a secret n any forbdden set. Condton 2 ensures that the blackness of recovered black secret pxels s hgher than that of recovered whte secret pxels n a recovered mage whch stacks at least k shares. If α α TH, a human vsual system can detect a dfference n contrast from the recovered mage. If α TH s large enough, a human vsual system can dstngush between the recovered black and whte secret pxels to obtan the secret mages. IV. PROBLEM STATEMENT Although both SIRDSs and shares of VCSs have the same nose-lke appearance, the pxel dstrbutons for a set of SIRDSs and for shares of a specfc VCS are qute dfferent. The pxel dstrbuton among shared pxels must obey the constructon rules or codebooks of the VCS. Shared pxels mean that a set of pxels shares the same secret pxel n a VCS. In Example 1, the codebook (.e., C 0 and C 1 and the chosen-probablty sets (.e., F 0 = {0.5, 0, 0, 0.5} and F 1 = {0, 0, 1, 0} are used to construct (2, 3-ProbVCS,

4 TABLE I THE PIXEL DISTRIBUTION PROBABILITIES FOR THREE SIRDSS AND SHARES OF VCSs where C 0 = 0, 1 and C 1 = 1, 0, Hence, the pxel dstrbuton patterns n the resultant shares comply wth C 0 and C 1. If the encrypton process selects column vector [ 101] T from C 1 for sharng a black secret pxel, shares 1 and 3 wll get a black pxel and share 2 wll get a whte pxel, and the pxel dstrbuton pattern for the shares wll be 2B1W. The pxel dstrbuton pattern, B(n W, ndcates there are black pxels and n whte pxels dstrbuted among n shared pxels. The probablty of each pxel dstrbuton pattern for the (2, 3-ProbVCS s lsted n Table I. Notaton d, s called the pxel densty of a share (or a SIRDS, denotes the frequency of appearance of black pxels n a share (or n a SIRDS. In ths example, pxel densty d of each share s 2 / 3. On the other hand, n a SIRDS, the mage contans many random-dot patterns that perodcally repeat n the horzontal drecton; the stereopss of the objects arses from dfferences n the horzontal postons of the mage. The pxel dstrbuton n n SIRDSs that were generated ndependently s totally ndependent. Suppose each SIRDS has the same pxel densty d, the probablty of pxel dstrbuton pattern B(n W can be calculated as followng: ( P,n d n = d (1 d n. (4 The pxel dstrbuton among three SIRDSs s lsted n Table I. In general, whle all SIRDSs are stacked, each pxel dstrbuton pattern wll unformly appear n the stacked mage. Hence, t s almost mpossble to reveal any meanngful nformaton by stackng two shares together. In ths study, we try to alter pxels n SIRDSs such that the altered SIRDSs can share secret mages the same way as VCSs. In the followng, we wll nvestgate whether the altered-pxels n a SIRDS wll nterfere wth the vsual effect of stereopss n the SIRDS. Fg. 2 llustrates an example for alterng pxels n a SIRDS. The depth map, as shown n Fg. 2(a, s used to create the SIRDS n Fg. 2(b. In ths paper, all autostereograms can be vewed n the wall-eyed vewng. The terms wall-eyed s a condton where eyes do not pont n the same drecton when lookng at an object. Wall-eyed vewng requres the two eyes to adopt a relatvely parallel angle. Hence, t s nformally known as parallel-vewng. Fg. 2(c shows the verfcaton Fg. 2. An example of hdng a secret mage n a SIRDS. (a The depth map, (b the SIRDS of Fg. 2(a, (c the verfcaton mage of Fg. 2(b (ε = 90 pxels, (d the locaton map, (e the altered SIRDS after hdng Fg. 2(d, (f the verfcaton mage of Fg. 2(e (ε = 90 pxels. mage of a stereopss n the SIRDS. The verfcaton mage, whch s a computer-generated 2D mage, can dsclose the stereopss n a SIRDS n 2D format. The verfcaton mage for a SIRDS can be generated as follows. Defnton 8 (Verfcaton Image Generaton Rule: Assume p x,y denotes pxel (x, y, n a SIRDS, and each pxel ( pv x,y n the verfcaton mage can be produced by operaton pv x,y = p x,y p x ε,y,wherep x ε,y = 0fx <ε. Parameter ε s the separaton parameter of the SIRDS and logcal operator represents the XOR operaton. Suppose the color of the orgnal pxels n the SIRDS, as shown n Fg. 2(b, wll be altered, black boxes n a locaton map, as shown n Fg. 2(d, ndcates that the pxels n whch regons of the SIRDS could be altered. The color of the orgnal pxels wthn two black boxes s randomly altered n varous probabltes (20% and 50% for the top and bottom boxes, respectvely. Fg. 2(e demonstrates the SIRDS after alterng the orgnal pxels. By vewng Fg. 2(e bnocularly, we perceve that addtonal stereopss appear n Fg. 2(e. The stereopss (.e., two boxes n the bottom of Fg. 2(e s clearer than the stereopss n the top of Fg. 2(e. In Fg. 2(f, the verfcaton mage of Fg. 2(e shows the same result. From the above example, we have the followng observatons: altered pxels could be dsclosed by the verfcaton mage of a SIRDS and wll nterfere wth the stereopss n the SIRDS. The degree of nterference s drectly proportonal to the number of altered pxels. Ths observaton ndcates that the altered SIRDSs can be detected by a human vsual system, thus makng t dffcult to share extra nformaton n a RDS. Hence, there are a tradeoff between keepng the vsual qualty of stereopss n a SIRDS and producng a hgh-qualty VCS. When we try to construct

5 Fg. 3. The two-phase encrypton process of (2, n-bvcs. a specfc VCS from a set of SIRDSs, t may be necessary to alter a large number of random pxels to obey the pxel dstrbuton rule of the VCS. However, these altered pxels can be perceved as an lluson of 3D depth and these nterfere wth the orgnal stereopss n the SIRDS. Based on the above observaton, we formulated a mathematcal optmzaton model to fnd an optmum soluton to share a secret mage n SIRDSs where the objectve s to maxmze contrast under the constrant of the vsual qualty of SIRDSs. Usng ths model, dealers can adjust the vsual qualty of SIRDSs to obtan the best dsplay qualty of the recovered mages. V. THE (2, N-BVC SCHEME A. The Two-Phase Encrypton Procedure In ths study, we propose a (2, n-bvcs for sharng a bnary secret mage n n SIRDSs. The proposed two-phase encrypton process s shown n Fg. 3. In the frst phase, n depth maps are used to produce n SIRDSs usng the autostereogram generator that adopts Thmbleby s algorthm [13]. In the proposed (2,n-BVCS, each depth map has the same mage sze and all generated SIRDSs have the same pxel densty d. In the second phase, accordng to constructon rules for (2, n-bvcs, pxels n the n generated SIRDSs are altered to share a bnary secret mage for the SIRDSs by the (2, n-bvcs encryptor. Based on the above-mentoned observaton, the encryptor tres to reduce the number of altered pxels n a SIRDS to mnmze the amount of nterference ntroduced nto the resultant share. The constructon rules generator, based on gven parameters, n and d, of each SIRDS, generates gudelnes for alterng pxels n the SIRDSs. The encryptor alters pxels only wthn a specfc regon, whch s called the encrypton regon, where black secret pxels appear. Due to the altered pxels could be dsclosed n the verfcaton mage of a SIRDS. To preserve the securty condton for each share of the BVCS, the encrypton regon wll be enlarged to cover neghbors of the black secret pxels. Hence, a locaton map, whch s a bnary mage as shown n Fg. 2(d n Secton IV, s used to ndcate the encrypton regons (.e., the black regons n the map for the secret mage n the BVCS. At the end of the second phase, n mage-sze-nvarant shares are generated for the (2, n-bvcs. B. The Constructon Rules Generator 1 The Basc Idea of the (2, n-bvcs Constructon: The constructon rules generator generates constructon rules based on pxel densty d of SIRDSs and a gven access structure of the BVCS. The rules are used to construct the BVCS by alterng pxels n the SIRDSs; therefore, the constructon rules smultaneously satsfy the condtons of the VCSs and cannot dsclose any nformaton related to the secret mage n the verfcaton mage of a SIRDS. Defnton 9: The constructon rules of a BVCS consst of two (n + 1 (n + 1 matrces, M 0 and M 1, for sharng whte and black secret pxels n a secret mage. Modfcaton probablty m 0, j (m1, j,0 n, 0 j n, s an element of M 0 (M 1 and s used to alter the dstrbuton pattern of n pxels that have the same coordnates n n SIRDSs from B(n W to jb(n jw. Moreover, 0 n m0, j = 0 n m1, j = 1, 0 j n. In the second phase, the encryptor adopts probabltes m 0, j and m 1, j to change the orgnal pxel dstrbuton probablty P,n d of SIRDSs for encryptng a secret mage n n SIRDSs. The orgnal pxel dstrbuton probablty for each pattern s therefore altered and becomes a functon of m 0, j and m 1, j. Defnton 10: Notatons Pa 0 and P1 a denote the alteraton probabltes of the orgnal pxels for sharng whte and black secret pxels, respectvely, n a SIRDS. Pa 0 and P1 a can be calculated as Pa 0 = n ( n =0 Pd,n j=0 δn, j m0, j (5 Pa 1 = n ( n =0 Pd,n j=0 δn, j m1, j (6 where δ, n j = j / n represents the pxel alteraton probablty for each share when the pxel dstrbuton pattern of the n SIRDSs s changed from B(n W to jb(n jw. Defnton 11: Assume 0 (μ n j and 1 (μ n j denote the dstrbuton probabltes of μ n j (.e., pattern jb(n jw for whte and black secret pxels, 0 j n. Functons 0 (μ n j and 1 (μ n j ndcate the total dstrbuton probabltes of code collecton μ n j among n resultant shares for sharng whte and black secret pxels n (2, n-bvcs. Hence, pxel dstrbuton probablty sets 0 and 1 can be calculated as ( 0 μ n j = n =0 Pd,n m0, j, (7 ( 1 μ n j = n =0 Pd,n m1, j, (8 ( where P,n d n = d (1 d n. Accordng to the altered pxel dstrbutons, 0 (μ n j and 1 (μ n j, the probabltes of the appearance of black pxels for reconstructed whte and black secret pxels n the (2, n-bvcs are p 0 = n =1 f 0 (μ n G(2,n and p 1 = n=1 1 (μ n G(2,n. Contrast ᾱ and blackness β of the recovered mage of the (2, n-bvcs can be defned as ᾱ = p 1 p 0 = n ( 1 ( μ n =1 β = p 1 = n 1 ( μ n =1 f 0 ( μ n G (2,n (9 G (2,n. (10 Based on the defnton of (k, n-probvcs, formal defnton of the (2, n-bvcs problem s gven n Defnton 12. Defnton 12: The soluton to the (2, n-bvcs conssts of two modfcaton matrces M 0 and M 1 that are used to alter the pxel dstrbuton n n SIRDSs.

6 Assume the pxel densty of each SIRDS s d and α TH (α TH >0 s the threshold for a human vsual system to detect a dfference n blackness n an mage. When two shares are stacked, the soluton s consdered feasble f the followng condtons are satsfed: 1 (Securty condton P 0 a = P1 a. 2 (Securty condton n =1 0 ( μ n (1,n G = 3 (Contrast condton ᾱ = n ( 1 ( μ n =1 n =1 1 ( μ n (1,n G = d. 0 ( μ n (2,n G α TH. Condtons 1 and 2 are the securty condtons of the (2, n-bvcs. Condton 1 ensures that shared pxels n each share have the same probablty of beng altered, regardless of whch pxels were used for sharng whte or black secret pxels. In other words, the resultant share cannot avod leakng a secret n ts verfcaton mage. Condton 2 guarantees that the pxel densty of each resultant share equals that of the orgnal SIRDS. Expressons n =1 0 ( μ n (1,n G and n=1 1 ( μ n (1,n G represent the pxel densty of whte and black shared pxels, respectvely. Condton 3, whch s called the contrast condton, ensures that the secret can be revealed when two shares are stacked. Example 2: Assume a (2, 2-BVCS s constructed by alterng pxels n two SIRDSs. The pxel densty of each SIRDS s d = Hence, by Equaton (5, the pxel dstrbuton probabltes for patterns 0B2W, 1B1W, and 2B0W are , 0.375, and , respectvely. The encryptor uses matrces M 0 and M 1 as constructon rules to yeld the (2, 2-BVCS, where M 0 = 1/6 2/3 1/6 and M 1 = /9 8/9 Pxel alteraton probabltes δ, 2 j,0 2and0 j 2, are lsted as matrx 2,where = The pxel dstrbuton probablty among the resultant shares of the (2, 2-BVCS can be calculated usng Equatons (7 and (8. For example, 0 (μ 2 0 = /6 = and 0 (μ 2 1 = /3 = From Equatons (5 and (6, we have Pa 0 = 2=0 P, ( 2 j=0 δ, 2 j m0, j = and P1 a = The frst securty condton s satsfed. From Equaton (1, the appearance probablty vectors for one share and two shares are G (1,2 =[0, 1/2, 1] and G (2,2 = [0, 1, 1], respectvely. Hence, the pxel densty of the resultant share s 2=1 1 (μ 2 G(1,2 = Clearly, the second securty condton s also satsfed. Fnally, based on Equaton (9, the contrast of the (2,2-BVCS s ᾱ = 2 =1 ( 1 (μ 2 0 (μ 2 G(2,2 = TABLE II LIST OF GIVEN PARAMETERS AND DECISION VARIABLES 2 The Optmzaton Model of the (2, n-bvcs: The (2, n-bvcs problem s formulated here as an optmzaton model. Both constants d and n are gven and we determne modfcaton matrces M 0 and M 1 for hdng whte and black secret pxels n n SIRDSs. The objectves of ths problem are to maxmze the contrast of recovered secret mages and to mnmze the alteraton probablty of each SIRDS under the vsual qualty and securty constrants. The gven parameters, decson varables, and formulatons are lsted n Table II. Objectve Functon: max. ᾱ = n ( 1 ( μ n =1 0 ( μ n (2,n G mn.pa 1 = n ( n =0 Pd,n j=0 δn, j m1, j (P1 Subject to: Pa 0 = P1 a (C1 n 0 ( μ n (1,n G = d (C2 =1 n =1 1 ( μ n (1,n G = d (C3 Pa 1 P a,max (C4 0 n m0, j = 1, 0 j n (C5

7 0 n m1, j = 1, 0 j n (C6 0 m 0, j 1, 0, j n (C7 0 m 1, j 1, 0, j n (C8 The frst objectve of the proposed model s to maxmze the contrast of the recovered mage n the (2, n-bvcs. The contrast value s the most mportant performance metrc n VCSs; hence t s the major objectve of the model. The other objectve of ths model s to mnmze the ntroduced nterference for the SIRDSs; hence the model mnmzes the alteraton probablty of each SIRDS. The second objectve s related to the vsual qualty of the SIRDSs. Constrants (C1 (C3 are the securty condton of the (2, n-bvcs. Constrant (C4 s the vsual qualty constrant that lmts the maxmum alteraton probablty for each SIRDS (share so that t s no more than P a,max. By tunng P a,max, nterference n the stereopss n each SIRDS can be mantaned at an acceptable level. Constrants (C5 and (C6 lmt the overall modfcaton probablty of each pxel dstrbuton pattern for sharng whte and black secret pxels, respectvely. Constrants (C7 and (C8 lmt the range of decson varables m 0, j and m 1, j. The orgnal mathematcal model (P1 can be solved by exstng optmzers or by a customzed algorthm. In our prevous studes on VC-related problems, we successfully developed smulated-annealng-based (SA-based algorthms to solve the constructon problems of VCSs [4], [8]. We also developed a SA-based algorthm to solve mathematcal model P1 by modfyng our prevous algorthms. Gven space lmtatons, we omt the detals of that algorthm n ths paper. Example 3: Assume a (2, 3-BVCS s constructed by alterng pxels n three SIRDSs. Parameters P a,max and d are 0.15 and 0.5, respectvely. By Equaton (4, we have P 0.5 P 0.5 3,3 0,3 = = and P0.5 1,3 = P0.5 2,3 probablty vectors for one share and two shares are G (1,3 = The appearance = [0, 1/3, 2/3, 1] and G (2,3 = [0, 2/3, 1, 1], respectvely. Applyng model (P1 to the (2, 3-BVCS and solvng the mathematcal model, we can obtan matrces M 0 and M 1 as constructon rules to yeld the (2, 3-BVCS, where M 0 = and M1 = Based on the (2, 3-BVCS scenaro, we can verfy the feasblty of soluton M 0 and M 1. Alteraton probabltes Pa 0 = ( 3 =0 P, j=0 δ, 3 j m0, j = 0.15 and Pa 1= 0.15; hence, Constrants (C1 and (C4 are satsfed. Constrants (C2 and (C3 can be verfed by evaluatng expresson 3=1 0 ( μ 3 (1,3 G = 3 =1 1 ( μ 3 (1,3 G = 0.5. Clearly, Constrants (C5 (C8 are also satsfed. Fnally, the optmal contrast of the (2, 3-BVCS s ᾱ = 3=1 ( f 1 ( μ 3 0 ( μ 3 (2,3 G = C. The (2, n-bvcs Encryptor Next, we desgn an encrypton algorthm for the BVCS encryptor. Based on the modfcaton rule for a gven BVCS, the algorthm alters pxels on n SIRDSs, ST 1,..., ST n, to share a bnary secret SE. The man dea of the encrypton algorthm s as follows. Notaton C x,y = [p ST 1 x,y... p ST n x,y ] denotes a collecton of pxel colors for n SIRDSs ST 1,...,ST n at coordnate (x, y. Notaton H(C x,y denotes the Hammng weght of C x,y.ifpxelpx,y L on locaton map L s a whte pxel (.e., px,y L = 0, pxelsps 1 x,y,..., p S n x,y on the resultant shares S 1,...,S n are the same as pxels p ST 1 x,y,..., p ST n x,y on SIRDSs ST 1,...,ST n, respectvely. Otherwse, resultant pxels p S 1 x,y,..., p S n x,y could be dfferent from pxels p ST 1 x,y,..., p ST n x,y accordng to secret mage SE and the modfcaton rules. When the color of secret pxel p SE x,y s whte (.e., pse x,y = 0, the encrypton algorthm uses modfcaton rule m 0 H(C x,y,0,..., m 0 H(C x,y,n to alter the colors of pxels pst 1 x,y,..., p ST n x,y to yeld resultant pxels p S 1 x,y,..., p S n x,y. On the contrary, the encrypton algorthm uses modfcaton rule m 1 H(C x,y,0,..., m 1 H(C x,y,n to yeld resultant pxels ps 1 x,y,..., p S n x,y. Eventually, the dstrbuton of the resultant pxels, B(n W, wll be altered to jb(n jw, where = H(C x,y and 0 j n. The encrypton algorthm for the (2,n-BVCS s lsted n Table III. The nput mages nclude n SIRDSs ST 1,..., ST n, one secret mage SE, and one locaton map L. The output mages are n resultant shares S 1,..., S n. Notaton px,y I denotes pxel colors of mage I n coordnate (x, y, I {ST 1,...,ST n, SE, L,S 1,...,S n }. In Step 1, resultant shares S 1,...,S n are ntalzed to SIRDSs ST 1,...,ST n, respectvely. Steps 3 10 embed a secret pxel at coordnate (x, y where locaton map L contans a black pxel. Step 3 calculates the Hammng weght of p ST 1 x,y,..., p ST n x,y.step6 determnes the number of black pxels n n resultant shares, b S, accordng to random number ρ, whch s generated n Step 4, b ST, constructon rules M 0 and M 1 as well as the color of secret pxel c. In ths study, we adopt the well-known proportonate selecton method, roulette wheel selecton, for selectng b S. Based on the gven parameters c and b ST,weuse a set of modfcaton probabltes m c b ST,0, mc b ST,1,..., mc b ST,n to determne the number of black pxels n the resultant shares. Steps 7 and 9 alter the pxel dstrbuton of n resultant shares from b ST B ( n b ST Wtob S B ( n b S W. In Step 7, when b S > b ST, the algorthm randomly alters b S b ST whte pxels of n shares n (x, y. On the contrary, when b S < b ST,the algorthm randomly alters b ST b S black pxels of n shares n (x, y n Step 9. Fnally, Step 11 outputs the resultant shares S 1,...,S n. Example 4: Assume matrces M 0 and M 1 are the constructon rules for yeldng the (2, 3-BVCS, where M 0 = and M1 = Collecton C x,y = [101] shows there are black, whte, and black pxels on ST 1, ST 2, and ST 3 n (x, y. Hence, the Hammng weght of C x,y,h(c x,y, s 2. For embeddng a whte secret pxel n (x, y, the canddate modfcaton

8 TABLE III THE ENCRYPTION ALGORITHM FOR THE BVCS TABLE IV THE BEST CONTRAST (% FOR (2, n-bvcs (P a,max = 40% TABLE V THE CORRESPONDING BLACKNESS (% FOR THE (2, n-bvcs LISTED IN TABLE IV (P a,max = 40% probablty set s [ ]. When ρ = 0.5, collecton C x,y = [101] wll be altered to C x,y = [111]. For embeddng a black secret pxel n (x, y where collecton C x,y = [100], the canddate modfcaton probablty set s [ ]. Whenρ = 0.9, collecton C x,y =[100] wll be altered randomly to C x,y =[110] or C x,y =[101] randomly. VI. EXPERIMENTAL RESULTS In ths secton, we dscuss a seres of experments that were conducted to assess the performance of the proposed (2, n-bvcss. We also present some demonstratons of the mplementaton results for observng the vsual effects of the BVCSs. Fnally, we compare the propertes of ths study wth prevous approaches. A. Performance Evaluaton Frst, we assess the performance of the proposed algorthm from a quanttatve vewpont. In ths experment, we solve the (2,n-BVCS, 2 n 10, optmzaton problem subject to varous pxel alteraton probabltes of SIRDSs P a,max. The values of P a,max range between 10% and 40%. Pxel densty d of SIRDSs ranges between 40% and 80% n evaluatng how dfferent values of d affect performance. In ths study, contrast ᾱ of the recovered mages, whch s defned n objectve functon P1, s the major performance metrc. The second performance metrc s the alteraton probablty of a SIRDS, whch s the second goal of the optmzaton model. In general, when the contrast of an mage s fxed, the vsual qualty of the mage s proportonal to the blackness of the mage. So, we take the blackness of the recovered mage as the thrd performance metrc. Table IV and Table V lst the best contrast and the correspondng blackness of the recovered mages n varous (2,n- BVCSs; n each scheme, P a,max s set to 40%. From these tables, we make the followng observatons. Frst, pxel densty d of SIRDSs wll affect the contrast value of the recovered mages n a (2,n-BVCS. The peak contrast value of ths scenaro can be found when the gven value of d s between 50% and 60%. When d s outsde the range, the contrast value s nversely proportonal to the blackness of SIRDSs. Second, the blackness of the recovered mages s proportonal to the value of d. When d 60%, the blackness of the recovered mages can reach 80%. Hence, pxel densty d s an effectve parameter for adjustng the blackness of the recovered mages. Thrd, n both cases where the gven pxel denstes of SIRDSs are 40% and 60%, the recovered mages have the same contrast value, but the latter case has a hgher blackness for the recovered mages. The other scenaros (.e., P a,max values 10%, 20%, and 30% also agree wth ths result. Therefore, we omt the frst case n the remander of the experments. Fourth, both the contrast values and the blackness values of the recovered mages decrease n (2, n-bvcss as n ncreases. Ths characterstc s the same as for conventonal (2, n-vcss. Table VI shows that the alteraton probablty of SIRDSs reaches ts peak when the gven P a,max s larger than the peak value. For example, the peak alteraton probablty of SIRDSs n the (2, 2-BVCS s no more than 25% when the gven P a,max 25%. Table VI verfes the effectveness of the second goal of the optmzaton model. The performance of (2, n-bvcss under other scenaros (.e., P a,max values 10%, 20%, and 30% shows the same trends as lsted n Table IV and Table V. Therefore, we lst only the ranges of the performance results of these scenaros n Table VII and Table VIII. The actual value of the alteraton probablty of SIRDSs n each scenaro s equal to the gven value of P a,max untl parameter P a,max reaches the peak value

9 TABLE VI THE PEAK ALTERATION PROBABILITY (% OF EACH SIRDS IN THE (2, n-bvcs LISTED IN TABLE IV (P a,max = 40% Fg. 4. The secret mage and the locaton map that s used n Experment-I, (a the secret mage ( pxels, (b the locaton map. TABLE VII THE RANGES OF THE BEST CONTRAST OF THE (2, n-bvcss WHILE PIXEL DENSITY (d OF SHARES RANGES BETWEEN 50% AND 80% Fg. 5. The generated SIRDS 1 (the depth map and ts correspondng V-share are shown Fg. 2(a and Fg. 2(c, respectvely. 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 scenaro as lsted n Table VI. Table VII ndcates that the best contrast value s proportonal to the gven value of P a,max and that t reaches ts peak when the actual alteraton probablty reaches ts peak. The more alteratons there are n a SIRDS, the more nterference s ntroduced nto a SIRDS, whch wll lead to degradaton of the vsual qualty of the SIRDS. Hence, there s a tradeoff between the vsual qualty of the recovered mages and the vsual qualty of the SIRDS. By tunng parameter P a,max, we can get the desred vsual qualty for the (2, n-bvcs. B. Demonstratons and Dscussons In ths subsecton, we evaluate the vsual effects of the proposed algorthm by observng mplementaton results of (2, n-bvcss. 1 Experment-I: Experment-I nvestgates the performance of a (2, 2-BVCS. The bnary secret mage and ts locaton map are shown n Fg. 4. In the frst phase, the depth map, as shown n Fg. 2(a, s used to produce two dfferent SIRDSs (SIRDS 1 s shown as Fg. 5 usng the autostereogram generator. The pxel densty of the SIRDSs (d s set to 0.5. In the second phase, the generated SIRDSs, the secret mage, and the locaton map are used to yeld two shares, as shown n Fg. 6, of the (2, 2-BVCS. The constructon rules for the (2, 2-BVCS are found by the proposed optmzaton model based on the parameters P a,max = 25% and d = 0.5. The best contrast of the recovered mage s 0.5; the constructon rules are as follows: M 0 = and M 1 = All mages used and generated n the same experment n ths secton are n the same dmenson. Gven the space lmtatons n ths paper, each mage s reduced to a sutable sze. The orgnal mages and more results of ths study are avalable on the followng webste: Fg. 6 ndcates that the alteraton probabltes of the shares and the contrast of the recovered mage are very close to ther theoretcal values. Note that the (2, 2-BVCS and the (2, 2-VCS have the same optmal contrast values (.e., ᾱ = 0.5 for the recovered mages, a characterstc that has not been acheved prevously n research that provdes meanngful shares for VCSs. As shown n Fg. 2(a and Fg. 4(a, the secret BVCS overlaps wth the ball n the depth map. Compared wth Fg. 2(c and Fg. 6(c, alteratons for hdng BVCS cannot be dsclosed by the verfcaton mage of a share (herenafter V-share, because all of the alteratons occurred wthn the

10 Fg. 7. The mages that s used n Experment-II the (2, 4-BVCS, (a (d depth maps 1 4 ( pxels, (e the secret mage, (f the locaton map. Fg. 6. The mplementaton results of (2, 2-BVCS n Experment I, (a share 1 (Pa 1 = 25.1%, (b share 2 (P1 a = 25.0%, (c V-share 1, (d the recovered mage (contrast = 0.5 area of the 3D object n the SIRDS. By vewng shares (Fg. 6(a and Fg. 6(b bnocularly, the 3D object (.e., a ball clearly emerges from the background. On contrary, a part of the secret content n the rghtmost sde of the secret mage, as shown n Fg. 4(a, cannot be hdden n 3D objects n a SIRDS. Fg. 6(a and Fg. 6(b show that a 3D object (a semellptcal emerges from the background n the rghtmost sde of the share. The extra 3D object was ntroduced by alterng random dots n the background of the orgnal SIRDS. Hence, the 3D percepton of the extra object s dmmer than that of the 3D ball n the shares. Because of the nterference that occurs n the background of the SIRDS, t can be dsclosed by ts V-share, as shown n Fg. 6. Although the extra object can be sensed, t reveals nothng about the secret mage. Eventually, the share satsfes the frst securty condton of (2, n-bvcss. 2 Experment-II:: Experment-II uses four dfferent SIRDSs as cover mages of a (2, 4-BVCS. The depth maps, the secret mage, and ts locaton map are shown n Fg. 7. Two experments, Set-A and Set-B, are performed on the (2, 4-BVCS. The constructon rules of Set-A are based on the parameters P a,max = 30% and d = 0.6. The best contrast of the recovered mage s 0.3. Some of the mplementaton results of experment Set-A are shown n Fg. 8. Fg. 8(e shows that the contrast value and the blackness of the recovered mage can reach 0.29 and 90%, respectvely. The (2, 4-BVCS can produce excellent vsual qualty for the recovered mages. By observng four shares n Fg. 8, the 3D scenes can be perceved from each share, but nterference also can be detected n the shares. Wthn the nterference area (.e., the area of the black box n Fg. 7(f, the 3D objects become unclear. From Fg. 8(a to Fg. 8(d, note that the nterference n each share s not the same. A rectangle area can be detected n the center of share 2. In the other shares n Fg. 8, the nterference s unclear. Fg. 9 shows the results of experment Set-B, wth parameters P a,max = 15% and d = 0.5. The best contrast of the recovered mage s reduced to 21.25%. In ths experment, the alteraton probablty P a,max s only a half that found n Set-A. The rectangle box n the center of share 2 (as shown n Fg. 9(b becomes very dm. Comparng Fg. 9(b and Fg. 8(b, the 3D scene n Fg. 9(b s clearer than n Fg. 8(b because the nterference s reduced n experment Set-B. That means the vsual qualty of the shares and the vsual qualty of the recovered mage can be balanced by tunng parameters P a,max and d. The above experments prove the effectveness of adjustng parameters P a,max and d n the proposed (2, n-bvcss. In addton, the verfcaton mage (e.g., V-share 2 n Fg. 8(f fals to reveal the nterference on the share because of the complcated content (.e., rch depth level and less area of background and the depth map of the SIRDS. Ths secton demonstrates the mplementaton results for the (2, 2- and (2, 4-BVCSs. These results show that the proposed approach can produce hgh vsual qualty shares and hgh-contrast recovered mage for (2, n-bvcss. Hence, both the transmsson rsk for the VCS and the vsual qualty degradaton of the recovered mage can be reduced. Fnally, there are two observatons that can be made from the above experments. Frst, the proposed BVCS s sutable for use wth varous knds of depth maps to generate SIRDSs to cover the shares. The 3D objects of the shares retan excellent vsual qualty f the BVCS adopts complcated content SIRDSs as covers. On the contrary, f the SIRDSs contan a large area

11 Fg. 8. The mplementaton results of the (2, 4-BVCS n Experment-II wth Set-A parameters Pa,max = 30% and d = 0.6, (a (d shares 1 4 ( pxels, (e the recovered mage (contrast = 0.29, (f V-share 2. Fg. 9. The mplementaton results of the (2, 4-BVCS n Experment-II wth Set-B parameters Pa,max = 15% and d = 0.5, (a (b shares 1 2 ( pxels, (c the recovered mage (contrast = of background, the nterference for the shares can be reduced by carefully selectng the hdden place for the secret mage. Second, alterng dots on a SIRDS degrades the vsual qualty of the SIRDS. The factors that affect the degradaton nclude the number of altered dots, the place where the dots were altered, and the contents of the depth map that was used to generate the SIDRS. Hence, the vsual qualty of 3D scenes on shares can be enhanced by tunng these factors. TABLE IX C OMPARISON OF THE S TUDY AND P REVIOUS A PPROACHES C. Comparson Table IX shows a comparson of the propertes between ths study and prevous works [9] [12]. The RG-based approach [9] can remove the drawback of pxel expanson, but t cannot provde hgh vsual qualty shares and recovered mages. The embedded approaches [10] [12] can generate halftone shares wth hgh vsual qualty, hence these approaches can reduce the suspcon of an encrypted secret. To retan the aspect rato of the mages durng the halftonng process, the sze of a halftone cell should be a square number (e.g., 4, 9, 16,..., n whch case the maxmum contrasts of the recovered

12 mages usng the embedded approach wll be reduced to 1/4, 1/9, 1/16,... These approaches ntroduce serous pxel expanson, whch reduces the contrast value of the recovered mage. On the contrary, the maxmum contrasts of the (2, n-bvcs wth parameters P a,max = 25% and d = 0.5 are 0.5, 0.33, 0.27, 0.25, 0.23, 0.22, and 0.21, respectvely. These results are very close to the best contrast values for conventonal (2, n-vcss n [4]. Based on the above dscusson, the performance of the proposed (2, n-bvcs s superor to the performance of prevous studes n terms of pxel expanson and contrast of the recovered mage. The vsual qualty of meanngful shares can be measured by the peak sgnal-to-nose rato and the unversal qualty ndex [14], but other factors also affect human s vsual percepton. For example, the halftone shares wth rch contents have better vsual percepton than shares wth a smple bnary cover, even f they have the same assessment ndex. The SIRDSs and 2D mages have total dfferent vsual percepton; therefore, a comparson of the vsual qualty between the SIRDS shares and the halftone shares may not far. By observng the SIRDS shares n Secton VI-B, we beleve that the vsual qualty of the SIRDS shares s suffcent to cover shares of VCSs from a transmsson securty perspectve. In other words, the proposed BVCS s an excellent scheme for reducng the suspcon of an encrypted secret. VII. CONCLUSION Ths study proposed a (2, n-bvcs and developed a new method for hdng a sze-nvarant (2, n-vcs n n SIRDSs. Ths work explored the possblty of hdng a share of a VCS n SIRDSs that are prnted on transparences. We developed a mathematcal model that defnes a set of constructon rules so that the recovered mages of (2, n-bvcss have the hghest contrast under the constrant of the nterference ntroduced nto the SIRDSs. Usng ths mathematcal model, a desred vsual qualty for shares and recovered mages can be found by adjustng parameters P a,max and d. The best contrast for the recovered mages n (2, n-bvcss, 2 n 10, ranges between 0.5 and 0.2, and can produce clear recovered mages for a (2, n-bvcs. The expermental results prove the effectveness and the flexblty of the proposed (2, n-bvcss. In the near future, we plan to extend ths study to explore new methods for hdng a (k, n-vcs n n SIRDSs. [3] C. N. Yang, New vsual secret sharng schemes usng probablstc method, Pattern Recognt. Lett., vol. 25, no. 4, pp , Mar [4] P. L. Chu and K. H. Lee, A smulated annealng algorthm for general threshold vsual cryptography schemes, IEEE Trans. Inf. Forenscs Securty, vol. 6, no. 3, pp , Sep [5] G. Atenese, C. Blundo, A. D. Sants, and D. R. Stnson, Extended capabltes for vsual cryptography, Theoretcal Comput. Sc., vol. 250, nos. 1 2, pp , Jan [6] D. Wang, F. Y, and X. L, On general constructon for extended vsual cryptography schemes, Pattern Recognt., vol. 42, no. 11, pp , Nov [7] T.-H. Chen and K.-H. 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Bovk, A unversal mage qualty ndex, IEEE Sgnal Process. Lett., vol. 9, no. 3, pp , Mar Ka-Hu Lee receved the Ph.D. degree n electronc engneerng from the Natonal Tawan Unversty of Scence and Technology, Tape, Tawan, n He s a Professor wth the Department of Computer Scence and Informaton Engneerng, Mng Chuan Unversty, Taoyuan, Tawan. Hs current research nterests nclude vsual cryptography, wreless networks, and network resource managements. ACKNOWLEDGMENT Hereby, the authors apprecate the anonymous revewers for ther valuable comments. REFERENCES [1] M. Naor and A. Shamr, Vsual cryptography, Advances n Cryptology EUROCRYPT (Lecture Notes n Computer Scence. New York, NY, USA: Sprnger-Verlag, 1995, pp [2] R. Ito, H. Kuwakado, and H. Tanaka, Image sze nvarant vsual cryptography, IEICE Trans. Fundam. Electron., Commun., Comput. Sc., vol. E82-A, no. 10, pp , Pe-Lng Chu receved the Ph.D. degree n nformaton management from Natonal Tawan Unversty, Tape, Tawan, n She s a Professor wth the Department of Rsk Management and Insurance, Mng Chuan Unversty, Tape, Tawan. Her research focuses on vsual cryptography, wreless sensor networks, and optmzng technologes.