ISBN 978-952-5726-07-7 (Print), 978-952-5726-08-4 (CD-ROM) Proceedings of the Second Symposium International Computer Science and Computational Technology(ISCSCT 09) Huangshan, P. R. China, 26-28,Dec. 2009, pp. 310-315 Threshold Visual Cryptography Scheme for Color Images with No Pixel Expansion Xiaoyu Wu 1, Duncan S. Wong 2, and Qing Li 2 Department of Computer Science, City University of Hong Kong, Hong Kong, China 1 Email: xiaoyuwu5@student.cityu.edu.hk 2 Email: {duncan,itqli}@cityu.edu.hk Abstract Since the introduction of threshold visual cryptography by Naor and Shamir, there have been many other schemes proposed; some of them support color images with a limited number of color levels while a few others achieve the property of no pixel expansion. However, it is unknown if there is a scheme which can satisfy all the following five commonly desired properties: (1) supporting images of arbitrary number of colors; (2) no pixel expansion; (3) no preprocessing of original images (e.g. dithering or block averaging); (4) supporting k-out-of-n threshold setting; and (5) a tunable number of color levels in the secret share creation process. In this paper, we answer this question affirmatively by proposing a k-out-of-n threshold visual cryptography scheme which satisfies all these properties. In particular, our scheme uses a probabilistic technique for achieving no pixel expansion and generically converts any k-out-of-n threshold visual cryptography scheme for black-andwhite images into one which supports color images. Index Terms Colored Visual Cryptography, Secret Sharing I. INTRODUCTION Visual Cryptography Scheme (VCS), introduced by Naor and Shamir in 1994 [1], is the secret sharing [2] of digitized images. A VCS splits an image into a collection of secret shares which are then printed on transparencies. These shares when separated will reveal no information about the original image (other than the size of it). The image can only be recovered by superimposing a threshold number of shares. This recovery process does not involve any computation. It makes use of the human vision system to perform the pixel-wise OR logical operation on the superimposed pixels of the shares. When the pixels are small enough and packed in high density, the human vision system will average out the colors of surrounding pixels and produce a smoothed mental image in a human s mind. Early VCS are mainly focused on black-and-white secret images [3] [12]. If the original image is not blackand-white, for example, a gray-scale image, dithering [13] is employed to preprocess the original image, that could degrade the image quality. Another issue that is common to most of the previous work is the pixel expansion, which means that each secret share is of size several times bigger than the original image. Two important parameters which govern the quality of reconstructed images are m (pixel expansion rate which represents the loss in resolution from the original image to the shares) 2009 ACADEMY PUBLISHER AP-PROC-CS-09CN005 310 and α (the relative difference in weight between the superimposed shares that come from one color level (e.g. black) and another color level (e.g. white)). For image integrity, a good VCS should bring the value of m close to one (i.e. no pixel expansion) and α as large as possible. In this paper, we propose a new VCS for color images. The scheme has no pixel expansion and allows the original image to have an arbitrary number of colors. The scheme also supports several other desirable properties which are summarized as follows. 1) Supporting images of arbitrary number of colors; 2) No pixel expansion; 3) No preprocessing of original images (e.g. dithering or block averaging); 4) Supporting k-out-of-n threshold setting; and 5) Supporting a tunable number of color levels in the secret sharing process. In our construction, we generically transform any k- out-of-n threshold VCS for black-and-white images (e.g. [1]) to color images. During the transformation, we use a probabilistic technique for achieving no pixel expansion. In addition, we also allow the user of the VCS to choose the number of colors that the reconstructed image will have. We will see that this tunable feature allows the user to control the quality of the reconstructed image. Based on our experimental results, we believe that this feature can help improve the user friendliness of VCS in practice. The rest of the paper is organized as follows. In Sec. II, we review some of the related results in VCS; in Sec. III, we introduce some notations which will be used to describe the new VCS; and in Sec. IV, we propose a new threshold color VCS with no pixel expansion. In Sec. V, we propose a grouping method for tuning the number of colors in the reconstructed image and further discuss how the number of colors that could be chosen during the share creation process in Sec. VI. Finally, we provide a quality comparison between our scheme and other related schemes in Sec. VII. II. RELATED WORK In [1], Naor and Shamir introduced VCS and proposed several constructions, where the generic one supports k- out-of-n threshold setting for black-and-white images. The scheme does not support images of arbitrary number O(k log k). of colors and the pixel expansion rate is log n 2 Since the introduction of VCS, there have been many other schemes proposed [3] [12]. In 2004, Adhikari et al.
[5] proposed a VCS which has less pixel expansion than that in [1]. In [14], Yang proposed another one which achieves no pixel expansion. The scheme only supports black-and-white images. In 2007, Chen et al. extended the results to gray-scale images and proposed a gray-scale VCS [15] with no pixel expansion. However, the scheme does not support the general k-out-of-n threshold setting. In addition, it also needs to perform block averaging (i.e. preprocessing) on the original image before carrying out the secret sharing. Another gray-scale VCS without pixel expansion was proposed by Chan et. al [16] in 2004. The scheme also needs preprocessing by dithering and adjusting the graylevel of the original image. The general k-out-of-n threshold setting is not supported either. For color VCS, [17] [22], Hou s schemes [17] are considered to be the first set of color VCS. All the schemes in [17] have the pixel expansion of 4 and do not support the general k-out-of-n threshold setting and dithering is required for preprocessing the original image. In 2005, Hou and Tu proposed a new color VCS [23]. The scheme also supports k-out-of-n threshold setting with no pixel expansion. Dithering is still required for preprocessing the original image before secret sharing. III. PRELIMINARIES The k-out-of-n threshold color VCS proposed in this paper supports original images of any number of color levels. Without loss of generality, we herewith assume that the color of the original image is represented by the conventional 24-bit color primitives, R (red), G (green) and B (blue), each has 256 levels (i.e. 8-bits), that is, for each pixel of the original image, the color quality is represented by three bytes of values; and each byte specifies the intensity of the corresponding color primitive: R, G and B. In the following, we introduce some notations which will be used in the rest of the paper. Consider a generic k-out-of-n threshold VCS for blackand-white images (e.g. [1]), suppose the pixel expansion rater is m, we use an n m Boolean Matrix S (below) to denote the secret sharing process, namely n rows corresponding to n shares and m columns corresponding to the colors (1 for black; 0 for white/transparent) of the m pixels of each share. S 0,0 S0,1 K S 0, m 1 S = M S n 1,0 S n 1,1 K S n 1, m 1 where S i,j {0, 1}. Depends on the actual black-andwhite VCS, the pixel expansion rate m varies. For example, if the Naor-Shamir k-out-of-n VCS [1] is applied, m = log n 2 O(k log k). A k-out-of-n black-and-white VCS typically consists of two n m Boolean Matrices B 0 and B 1, which correspond to the white and black of a pixel in the original image, respectively. Let C b = {matrices obtained by permuting the columns of B b } where b = 0, 1. The secret sharing of the original image is performed pixel-by-pixel. For each pixel in the original image, if the color is white (resp. black), one n m Boolean Matrix in C 0 (resp. C 1 ) is randomly pixel and used for creating the n shares. Due to the page limitation, we refer readers to [24] for details. For 3-out-of-4 black-and-white VCS, below is an example of the base Boolean Matrices: 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 = 1 = 0 1 0 0 1 1 B B 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 1 Here, the pixel expansion is 6. In our scheme described in the next section, we will see how to convert this scheme to a threshold color VCS with no pixel expansion (i.e. the pixel expansion rate would be 1). IV. A NEW K-OUT-OF-N COLOR VCS We now describe the VCS which supports all the five properties listed in Sec. I. Along with the scheme description, we use Lena image for illustration. The scheme consists of four steps: 1. Histogram Generation: Three histograms representing the intensity distribution of R, G and B color primitives of the original image are first generated. In the histogram for R (resp. G or B), the horizontal axis represents the intensity of R (resp. G or B) ranging from 0 to 255; and the vertical axis represents the number of pixels of each intensity value. Fig. 1 shows the original Lena image and Fig. 2 shows the R, G and B components of it. Fig. 3 shows the three histograms generated in this step. 2. Color Quality Determination: As we can see in the previous step, each color component has 256 levels of intensity. In our scheme, we can let the user choose the number of intensity levels that the reconstructed secret image will have. In this step, the user is to determine this intensity level with the purpose of maximizing the quality of the reconstructed image. (Please refer to Sec. V and VI for details of choosing the number of levels.) Let N be the number of levels of reconstructed image where N=N R N G N B (N X denotes the number of levels of the primitive X { R, G, B}). Suppose user would like to show a 64- color reconstructed image, then he may choose the number of levels for R, G and B as follows: 64 (N) =4 (N R 4 (N G ) 4 (N B ). Remark: the color levels of the reconstructed image for R, G and B do not need to be the same. 3. ing: For each color primitive X { R, G, B}, we create N X groups on the histogram of X. To do so, we specify the boundary color intensity between every pair of adjacent groups as K 0,, K. In other words, we N X 2 divide the histogram of X into N X regions, i.e. [0, K 0 ), [K 0, K 1 ),, [ K, 255]. The principle of dividing is to N X 2 make each of these N X regions to have the same size in area. So for each group, there will be an equal of pixels in the image that fall into each group/region. In Sec. V, we further discuss the reasons behind choosing this approach 311
Fig. 1. The Original Lena Image. Fig. 2. The RGB Component Images of Lena Fig. 4. Histograms Illustrating the 4 4 4 Color Levels Fig. 3. Histograms of the RGB Component Images for grouping. Fig. 4 shows the histograms after grouping. Here we use different color intensities to represent different groups. 4. Share Creation: The last step is to create the secret shares. To do so, we apply the following method to each of the primitive color independently. First, take a k-outof-n black-and-white VCS, for example, the Naor-Shamir VCS [1]. For the base Boolean Matrices B b (for b=0,1), denote them as: B b =[B b 0, B b 1,, B b m-1] That is, B b i denotes column i (0 i m-1) of B b. Second, for each color primitive X { R, G, B}, we carry out the following steps for each of the pixels in the original image. For each pixel, 1) suppose the color intensity of the pixel with respect to color primitive X falls into the k-th group (where 0 k N X 1). We compute a probability value P=k/(N X -1) which determines the likelihood of going through one of the following steps. 2) With the probability P, we carry out the following two steps: We look into B 0 and randomly pick a column, for example, B 0 j where 0 j m 1. Consider B 0 j as an n-bit vector. For the first bit, we assign the black color (i.e. 0 color intensity) if the bit is 1, otherwise we assign red color (i.e. 255 color intensity). This continues until we have assigned colors to this pixel for all the n shares. 3) With the probability 1-P, we carry out similar steps to the above, but change B 0 to B 1. Determined by the value of P, in Table I, we summarize the probability distribution of B 0 and B 1 for individual groups. Since the columns in B 0 (resp. B 1 ) are chosen uniformly at random, the chance of picking any particular column in B 0 (resp. B 1 ) for k will be k/((n X 1)m) (resp. (1 k/(n X 1))/m). TABLE I THE CHANCE OF USING B 0 OR B 1 FOR INDIVIDUAL GROUPS DURING SHARE CREATION (X {R,G,B}) Probability of using B 0 Probability of using B 1 0 0 1 M M M k k/( N X -1) 1- k/( N X -1) M M M (N X -1) 1 0 312
Finally, we superimpose the i-th R share with the i-th G share as well as the i-th B share, for i=1,, n, to form the final i-th share which consists of the corresponding R, G, B components. A. Example Suppose we want to create a 64-level 3-out-of-4 set of secret shares such that each color component has four groups, that is N R = N G = N B = 4. After dividing the R, G, B components of the original image into four groups (i.e. ing), we carry out the Share Creation by first computing the probability value for each group. Table II shows the probability distribution of the individual columns of the base Boolean Matrices B 0 and B 1 of the 3- out-of-4 black-and-white VCS described in Sec. III. Column 2 of Table II specifies the pixel color of the four secret shares when the i-th column of B 0 or B 1 is chosen (0 represents coloring the pixel of the corresponding secret share to the primitive color; 1 represents coloring it to the black color). Column 3 to 6 indicate the probabilities of choosing the i-th column if the pixel color in the original image is in one of these four groups. V. GROUPING METHODS - TUNABLE NUMBER OF COLOR LEVELS IN RECONSTRUCTED IMAGES In the scheme description above, after generating the histogram for each primitive color and deciding the number of groups, we do the grouping (step 3) by dividing the histogram into several regions so that each region contains the same number of pixels. There are many other ways of doing the grouping. As an example, one can evenly divide the histogram into N X regions for each primitive color X {R,G,B} based on the color intensities, that is, making N X equal-width regions: [0, 255/ N X ), [255/ N X, 255 2/ N X ),,[255 (N X 1)/ N X, 255]. Fig. 5 shows the histogram of R, G and B of Lena image when N R = N G = N B = 4. From the figure, we can TABLE II THE CHANCE OF USING ANY ONE PARTICULAR COLUMNS OF B 0 OR B 1 FOR DIFFERENT GROUPS DURING SHARE CREATION The possibility of choosing the i-th column of matrix M Shares 0 1 2 3 i=1, M=B 0 0000 0 1/18 1/9 1/6 i=2, M=B 0 0000 0 1/18 1/9 1/6 i=3, M=B 0 1110 0 1/18 1/9 1/6 i=4, M=B 0 1101 0 1/18 1/9 1/6 i=5, M=B 0 1011 0 1/18 1/9 1/6 i=6, M=B 0 0111 0 1/18 1/9 1/6 i=1, M=B 1 1000 1/6 1/9 1/18 0 i=2, M=B 1 0100 1/6 1/9 1/18 0 i=3, M=B 1 0010 1/6 1/9 1/18 0 i=4, M=B 1 0001 1/6 1/9 1/18 0 i=5, M=B 1 1111 1/6 1/9 1/18 0 i=6, M=B 1 1111 1/6 1/9 1/18 0 Fig. 5. Histograms with 4 Equal-Width Regions see that most of the pixels of Lena image on primitive color B are grouped into the second region. After superimposition, these pixels will show no difference in the reconstructed image. From this example, one can see that our approach of making the groups with same number of pixels per region rather than with the equal color-intensity interval is for improving the color level difference among pixels which have different color levels in the original image. VI. DISCUSSIONS ON DETERMING THE NUMBER OF COLOR LEVELS In this section, we discuss how to choose the number of color levels (i.e. N=N R N G N B ) in the reconstructed secret image. The scheme supports an arbitrary number of color levels which affects the quality of reconstructed image in a significant way. We observe that the number of color levels to be chosen depends on the number of colors that the original image has. We first classify the original images into two categories: in category 1, the number of levels on a particular primitive color is small, for example, less than 4; and in category 2, the number is large, say at least 4. For images in category 1, if the original image on a particular primitive color X {R,G,B} is OriginalN X, since OriginalN X in this case is small, there is no need to try with different color levels. Hence we should set N X to OriginalN X. Images that fall into this category could be some text and logos. Fig. 6 shows the original image and reconstructed image of Logo of Mac with color levels set to 2 (N R ) 2 (N G ) 2 (N B ). For images in category 2, one may try the color level N X from a small value, say 2 or 4, to the full level 313
Fig. 8. The Original Image of Alice Fig. 6. The Original and Reconstructed Image of Mac Logo (Color Levels: 8 = 2 2 2) OriginalN X. Based on our experimental results shown below, we observe that for photos or color cartoon images with large number of color levels, trying these three values (namely 2, 4 or OriginalN X ) for the value of N X can already attain one of the best results in the reconstructed image. In Fig. 9 we can see that the reconstructed image of Alice in the Wonderland (Fig. 8) with 2 2 2 levels has the sharpest image but limited number of colors while the full level version, i.e. 256 256 256, which has the same number of colors as the original one, looks blurry. Image with 4 4 4 shows the best result with abundant colors and clear figure. Fig. 7 and Fig. 11 show similar results as that in Fig. 9. Fig. 13 shows the reconstructed image of Gray scale (21 levels) (original image: Fig. 12) with 4 levels and full level (i.e. 21 levels). Note that in this image, the values of R, G and B components are the same for every pixel. We can see that full level version gives better result as the gradual change in the gray intensity can better be chosen than that of the 4 level. Fig. 9. Reconstructed image of Alice with 2 2 2, 4 4 4, N N N levels Fig. 10. The Original Image of F22-Raptor Fig. 11. Reconstructed image of F22-Raptor with 2 2 2, 4 4 4, N N N levels VII. QUALITY COMPARISON In this section, we compare our VCS with eight other schemes: Naor-Shamir (NS in short) (the first VCS), Hou (the first colored VCS), Yang (a probabilistic method for gray images), Chan et al. (no pixel expansion for gray images), Hou-Tu (HT in short) (no pixel expansion for color images), Shyu, Chen et al. (multiple-level and no pixel expansion for gray images) and Yang-Chen (YC in short) (a probabilistic method for color images). Table III shows the comparison of the schemes, where C is the number of colors of the original image, m is the pixel expansion rate and m 1 =log n 2 O(k log k). For the column Level, it indicates whether there is a limitation on the number of color levels of the original image. Compared with other VCS, the new scheme proposed in this paper supports color images and lets users choose the number of color levels in the reconstructed images based on their Fig. 12. The Original Image of Gray (21-Level) Fig. 13. The Reconstructed Image of Gray (21-Level) with 4 and 21 Levels desired image quality. Besides, the original image does not need to preprocess such as dithering, which would degrade the quality of reconstructed images. Furthermore, the scheme does not have pixel expansion. When Fig. 7. Reconstructed image of Lena with 2 2 2, 4 4 4, N N N levels 314
Colored TABLE III COMPARISON Expansion rate compared with Chen et al. s [15], we can see that their scheme only supports gray scale images. Also, their scheme does share creation based on the average color intensity of a block of pixels, and the number of color levels of the reconstructed image depends on the block size. The larger the block is, the more levels the reconstructed image has. However, more levels also mean that the more pixels would have the color intensity averaged, thus the quality is degraded. Yang-Chen s [22] scheme also uses the probabilistic method and support color images. It has a fixed expansion rate 3. The scheme does not support tunable color levels for the reconstructed images. VIII. 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