Direct Binary Search Based Algorithms for Image Hiding

Transcription

1 1 Xia ZHUGE, 2 Koi NAKANO 1 School of Electron and Information Engineering, Ningbo University of Technology, No.20 Houhe Lane Haishu District, , Ningbo, Zheiang, China zhugexia2@163.com *2 Department of Information Engineering, Hiroshima University, Kagamiyama, Higashi-Hiroshima , Japan nakano@cs.hiroshima-u.ac.p Abstract Digital halftoning is an important process to convert a continuous-tone image into a binary image with pure black and white pixels. The Direct Binary Search (DBS) is one of the well-known halftoning methods that can generate high quality binary images. The main contribution of this paper is to present a DBS based halftoning method that conceals a binary image into two binary images. More specifically, three distinct gray scale images are given, such that one of them should be hidden in the other two gray scale images. Our DBS based halftoning method generates three binary images that reproduce the tone of the corresponding original three gray scale images. Quite surprisingly, the secret binary image can be seen by overlapping the other two binary images. In other words, the secret binary image is hidden in two public binary images. Also, it is very hard to guess the secret images using only one of the two public images, and both of these two public images are necessary to get the secret image. Another contribution of this paper is to extend our DBS based halftoning method to hide one image and more than one image into more than two images. The resulting images show that our DBS based halftoning method hides and recovers the original images. Hence, our DBS based halftoning technique can be used for watermarking as well as amusement purpose. Keywords: Image Processing, Printing, Halftoning, Direct Binary Search, DBS, Image Hiding, Watermarking, Amusement 1. Introduction Halftoning is an important process to convert a continuous-tone image into a binary image [6] that eye perceives as a continuous-tone image. This process is necessary when printing a monochrome or color image by a printer with limited number of ink colors. It is required to generate a binary image that reproduces the tone and the details of the original gray scale image. Many kinds of halftoning methods have been proposed: screening, error diffusion, and search-based iterative methods. In screening, the binary value of the halftone image is obtained by comparing pixels values of the continuous-tone image with the corresponding threshold values [1]. In Error Diffusion [4, 11, 15], the pixelby-pixel comparison with a threshold is also required, but after a pixel is binarized, the rounding error is propagated to its unprocessed neighboring pixels according to some fixed ratios. Search-based iterative methods [2, 6, 7, 13, 14] are to find the best values of binary pixels which can minimize a perceived [2, 13, error between the continuous-tone image and the halftone image. The Direct Binary Search (DBS) 14] is one of the search-based iterative methods, which generates good quality binary images. Digital watermarking is to embed messages into digital contents (audio, video, images, text) which can be detected or extracted later. One role of digital watermarking is to embed copyright information of the content which is supposed not to be visible, but can be retrieved by electronic devices [9][16]. Another role of digital watermarking is known as steganography, which hides messages in content without typical citizens or public authorities noticing its presence, and ust special recipients can decode the hidden messages. Many methods for digital watermarking via halftone images have been proposed. One kind of these methods is to embed an invisible watermark into a halftone image and the hidden information can be extracted with some special procedure. Goldschneider and Riskin [5] proposed an embedding method using scalar quantizers or vector quantizers to embed a low bit-depth halftone image into a higher bitdepth halftone image. Also the hidden image can be extracted from the higher bit-depth image simply International Journal of Digital Content Technology and its Applications(JDCTA) Volume6,Number23,December 2012 doi: /dcta.vol6.issue

2 by masking some bits off. Baharav and Shaked [3] used a sequence of two different dither matrices to hide watermark into images in halftoning process. And the order of the used matrices is the binary representation of the watermark. Wang and Knox [12] embedded digital watermark in halftone screen. When this screen is used to halftone an image, the watermark is embedded in the image automatically. Kacker and Allebach [8] divided images into blocks and in each block got DCT coefficients, and watermark is hidden depending on the magnitude of the DCT coefficient. After that, the watermarked image is halftoned using the Direct Binary Search. The above watermarking ways hide information in a single image and the hidden information should be extracted by corresponding decoding ways. Another kind of digital watermarking methods is to embed information in two or more than two halftone images. And when these halftone images are overlapped, the embedded visible pattern appears. Soo-Chang Pei [11] proposed such a kind of method by applying the regular error diffusion and the proposed noise-balanced error diffusion to the unwatermarked part and the watermarked part respectively of a gray scale image, and we can see the watermark when this halftone image is overlapped with another halftone image obtained from the same gray scale image using ust the regular error diffusion. But the hidden visible patterns were ust reproduced in black areas and didn t contain enough details, so they cannot take enough information. The main contribution of this paper is to present a DBS-based halftoning method for three images of the same size, say, A, B and C. Our DBS-based halftoning method generates three binary images A ', B ' and C ' that reproduce original gray scale images A, B and C respectively. Further, C ' is the overlapped image of A ' and B '. In other words, a pixel of C ' is white if and only if both of the corresponding pixels of A ' and B ' are white. Figure1 shows examples of images A ', B ' and C '. Thus, if A ' and B ' are printed in transparent sheets, then we can see C ' by superposing them together. Also, we cannot guess C ' if only one of the two images A ' and B ' is available. Both A ' and B ' are necessary to get secret image C '. Figure 1. Three binary images A ', B ' and C ' satisfying Condition 2 Our proposed halftoning method has several possible applications. For example, it can be used for watermarking as follows. Image A ' is a public image and image C ' is an image to be hidden. Image C ' can be a hidden image that indicates the copyright information of image A '. Then, image B ' can be used as a secret key to see image C '. Also, our halftoning method can be used for amusement purpose as follows. A ' and B ' can be parents pictures and C ' can be a picture of their child. We can see their child by pasting parent pictures. In other words, child picture is hidden in his/her parent pictures. Quiz is given using an image A ', and three options are given as three images B ' 1, B ' 2 and B ' 3. If B ' is a correct answer for the quiz, then we can see some special image by overlapping A 2 ' and B '

3 Our second contribution is to extend our DBS-based halftoning method to hide one image into more than two images, and to hide more than one images into more than two images. This paper is organized as follows. Section 2 reviews the Direct Binary Search (DBS) method and presents the idea behind our halftoning method. Section 3 shows our DBS-based halftoning algorithm. Section 4 gives our experiment results. Section 5 extends our halftoning method to generate more than three binary images. Section 6 offers concluding remarks. 2. Halftoning and Image Hiding Suppose that an original gray scale image A ( a i, ) of size n n is given, where a denotes the intensity level at position ( )(0 n 1) taking a real number intensity in the range [0, 1]. Although we assume that images are square for simplicity, it is easy to generalize them to non-square images. The goal of halftoning is to find a binary image A' ( a' i ) of the same size that reproduces, the original image A, where each a ' i, is either 0(black) or 1(white). Direct Binary Search (DBS) is one of the well-known halftoning methods. The idea of DBS is to find a binary image A ' whose proected image onto human eyes is very close to the original gray scale image A. The proected image is computed by applying a Gaussian filter which approximates the characteristic of the human visual system. Let V ( v kl, ) denote a Gaussian filter, i.e. a 2-dimensional symmetric matrix of size ( 2 +1)( 2 +1), where each non-negative real number vkl, ( k, l ) is determined by a 2-dimensional Gaussian distribution such that their sum is 1. In other words, kl, k 2 2 l v c e 2 2 (1) where is a parameter of the Gaussian distribution and c is a fixed real number to satisfy v, kl, 1. Let A'' ( a'' kl ) be the proected gray scale image of a binary image A' ( a' ) obtained by applying the Gaussian filter as follows: (2) a'' v a' (0 n 1) k, l i k, l kl, We can say that a binary image A ' is an approximation of original gray scale image A if the difference between A and A '' is small enough. Hence, we define the error between A and A '' as follows. The error e at each pixel location (, i ) is defined by and the total error is defined by e a a 2 ( '' ) (3) Error A A (, '') e 0 n 1 (4) The idea of DBS is to find a binary image A ' which can make the Error( A, A '') sufficiently small. For this purpose, the DBS repeats improvement of binary image A '. The value of a particular pixel a ' i, is modified by toggling ( a' 1 a' ) and swapping (exchange the value with its 8 neighboring pixels). One of these operations (toggling and swapping) which decreases the total error 459

4 mostly is chosen to modify the pixel value a ' i,. DBS repeats this process in a raster order until no more improvement is possible. Suppose that we are given three gray scale images A, B and C of size n n each. Our goal is to generate corresponding binary images A ', B ' and C ' such that Condition 1 binary images A ', B ' and C ' reproduce the tone of original gray scale images A, B and C, and Condition 2 C ' is the overlapped images of A ' and B ', that is c' min( a', b' ) for all i and (0 n 1). In other words, if either a ' i, and b ' i, is 0 (i.e. black) then c ' i, is also 0. If both a ' i, and b ' i, is 1 (i.e. white) then c ' i, is also 1. Figure 1 shows examples of three binary images A ', B ' and C ' satisfying Condition 2. Suppose that images A ' and B ' are printed in transparent sheets. Clearly, we can see the image C ' if these transparent sheets are pasted together. 3. Our DBS-based Algorithm The main purpose of this section is to show our algorithm that generates three binary images A ', B ' and C ' satisfying Conditions 1 and 2 for given original gray scale images A, B and C. Our algorithm consists of two steps as follows: Step 1 Adust the intensity levels of original gray scale images A, B and C. Step 2 Using our new DBS-based technique, generate three binary images A ', B ' and C ' from adusted gray scale images of A, B and C Adusting Intensity Levels of Original Gray Scale Images We will show how the intensity levels of original gray scale images are adusted. Let R be a small region of m pixels in an n n image, and RA ( '), RB ( ') and R( C ') be corresponding subimages of A ', B ' and C ' respectively. Clearly, if RA ( ') and RB ( ') has x and y black pixels respectively, then the number of black pixels in R( C ') must be in the range [max( x, y), min( m, x y)] to satisfy Condition 2. Let [, ]( ) be the range of the number of black pixels in R( A ') and R( B '). If both R( A ') and R( B ') have black pixels, then the number of black pixels in R( C ') is in the range [,min( m,2 )]. Similarly, if both R( A ') and R( B ') have black pixels, then R( C ') can have black pixels in the range [,min( m,2 )]. Thus, if min( m, 2 ), we can guarantee that RC ( ') can have any number black pixels in the range [,min( m,2 )]. Hence, we select m 2 and 3m 4 to maximize the range of the number of black pixels of R( A '), RB ( ') and R( C '). In this case, RA ( ') and R( B ') has [ m 2,3m 4] black pixels and R( C ') has [3m 4, m ] black pixels. Thus, it makes sense to adust the intensity levels of A and B in the range [0.25,0.5], and that of C in the range [0,0.25]. Hence we perform linear conversion of the intensity x such that function f( x) x is used for adusting intensity levels of A and B, and function g( x) x 4 is used for C. Figure 2 shows the graph of functions f and g. Clearly, the value of f is in the range [0.25,0.5] and that of g is [0,0.25]. 460

5 Figure 2. Functions f and g used to adust intensity levels 3.2. Using an Error-Diffusion-Based Algorithm for Generating Three Binary Images In this section we show our halftoning method based on Direct Binary Search (DBS). We assume that the intensity levels of three original gray scale images have been adusted using the linear conversion in the previous section. Let A ( a i, ), B ( b i, ) and C ( c i, ) be the adusted gray scale images. We will show how halftone images A' ( a' ), B' ( b' ) and C' ( c' ) are computed using our new DBS-based algorithm. Similar to the conventional DBS algorithm, pixels in binary images A ', B ' and C ' are determined in raster order. Recall that, in conventional DBS algorithm, the value of binary pixel is determined such that the error between the proected image and the original gray scale image is small enough. Our key idea is to extend this policy for three binary images such that binary values of a ' i,, b ' i and, c ' i are, selected to make the sum of the total errors of these three binary images small enough. The details are spelled out as follows. It should be clear that, to satisfy Condition 2, possible 3-tuple value ( a', b', c ' ) is one of (0,0,0), (0,1,0), (1,0,0) and (1,1,1). Let A'' ( a'' ), B'' ( b'' ) and C'' ( c'' ) be the proected images of A ', B ' and C ' respectively. The sum of the total errors of the three images can be defined as Error( A, A'') Error( B, B '') Error( C, C '') 0 n 1 ( a a'' ) ( b b'' ) ( c c'' ) (5) We selected a 3-tuple value ( a', b', c ' ) for the three pixels of images A, B and C at the same location that minimizes the sum of the total errors. This process is repeated on each 3-tuple pixel in raster order until no more improvement is possible for the whole three images. 4. Experimental results In this section we give a set of our experimental results of hiding an image in two distinct images. Figure 3 shows the resulting three binary images A ', B ' and C ' generated by our DBS-based algorithm. All of these three images have the same size of pixels. Image A ' and image B ' are obtained directly by our DBS-based halftoning method proposed in Section 3, which have the image C ' hidden in both of them in the halftoning process. Image C ' is the overlapped result of images A ' and B '. In images A ' and B ', we cannot see any hiding trace, and from image C ' we see the hidden image can be recovered clearly. 461

6 Binary image A ' : Airplane Binary image B ': Sailboat Binary image C ': Lenna Figure 3. Three binary images Airplane, Sailboat and Lenna generated by our DBS-based halftoning algorithm 5. Our DBS-based Halftoning Method for Generating More Than Three Binary Images The main purpose of this section is to extend our halftoning method to generate more than three binary images for hiding one image into N ( N 2) images, and for hiding M( M 1) images into N( N 2, N M) images Hiding One Image in N( N 2) Images Suppose we are given a gray scale image G ( g i, ) and other NN ( 2) gray scale images H1 ( h1 ),, HN ( hn ) of the same size n n. Our goal is to get their corresponding binary images G' ( g' ), H1' ( h1' ),, HN' ( hn' ), and to make image G ' hidden in images H1',, HN'. Condition 1 and Condition 2 given in Section 2 should be modified as follows: 462

7 Condition 1 binary images G', H1',, HN ' reproduce the tone of original gray scale images GH, 1,, HN, and Condition 2 G ' is the overlapped image of H1',, HN', that is g' min( h1',, hn' ) for all i and (0 n 1). In order to satisfy Conditions 1 and 2, we need to do the following two steps: Step 1 Adust the intensity levels of these N 1 gray scale images. Similarly to the way that we have discussed in Section 3.1, we perform linear conversion of the intensity x such that function f ( x) ( N 1) x 2 N ( N 1) 2N is used for adusting intensity levels of H1,, HN, and function g() x ( N 1) x/2n is used for G. As N increases, the range of adusted intensity levels increases and the increase rate decreases. Step 2 Using the DBS-based technique, generate N 1 binary images G', H1',, HN ' from the adusted gray scale images GH, 1,, HN. Possible ( N 1)-tuple values ( g', h1',, hn' ) should be chosen to satisfy Condition 2. Let G'' ( g'' ), H1'' ( h1'' ),, HN '' ( hn '' ) be the proected images of G', H1',, HN ' respectively. The sum of the total errors of the N 1 images can be defined as: Error( G, G '') Error( H1, H1'') Error( HN, HN '') 0 n 1 ( g g '' ) ( h1 h1'' ) ( hn hn '' ) (6) We select a ( N 1 )-tuple value ( g ', h1',, hn' ) for the N 1 pixels of images GH, 1,, HN at the same location that minimizes the sum of the total errors. This process is repeated on each ( N 1)-tuple pixel in raster order until no more improvement is possible for the whole N 1 images. Here if we take N 2, the previous case is included. Here we show the experimental results for N 3. Figure 4 shows the graph of functions f and g. Figure 5 shows the four binary images G ', H 1', H 2' and H 3' generated by our DBS-based algorithm. H 1', H 2' and H 3' are generated directly by our DBS-based halftoning method, and G ' is the superposed result of H 1', H 2' and H 3'. We can see this set of images is brighter than the set given in Figure 3. As the number of public images N increases, the range of adusted intensity levels of G, H 1, H 2, H 3 increases, so the corresponding generated binary images have higher contrast. Figure 4. Functions f and g used to adust intensity levels ( N 3 ) 463

8 Binary image H 1' : Airplane Binary image H 2' : Sailboat Binary image H 3' : Pepper Binary image G ' : Zelda Figure 5. Four binary images Airplane, Sailboat, Pepper and Zelda generated by our DBS-based halftoning algorithm 5.2. Hiding M( M 1) Images in N( N 2, N M) Images Essentially, hiding MM ( 1) gray scale images, we say G1 ( g1 ),, GM ( gm ), into NN ( 2, N M) gray scale images, we say H1'' ( h1'' ),, HN'' ( hn'' ), is to hide each of these M images in two or several of those N images, and these M secret images share one or more than one of those N public images. For example, M 2, N 3, we can hide G 1 into H 1 and H 2, and hide G 2 into H 2 and H 3. We can also hide G 1 into H 1, H 2 and H 3, and hide G 2 into H 1, H 2 and H 3. The steps are the same as we discussed above. Firstly, we adust the intensity levels of the M N images into proper ranges. Then, we use the DBS-based technique to generate corresponding binary images G1'' ( g'' i ),, M '' ( M '' ), 1'' ( 1'' ),, N '' ( N '' ), G g i, H h i, H h from the adusted M N i, gray scale images. Possible ( M N )-tuple values ( g1',, gm', h1',, hn' ) should be chosen to satisfy Condition 2. Let G1'' ( g1'' ),, GM '' ( gm '' ), H1'' ( h1'' ),, HN '' ( hn '' ) be the 464

9 proected images of G1',, GM', H1',, HN' respectively. The sum of the total error of the M N images can be defined to be: Error( G1, G1'') Error( GM, GM '') Error( H1, H1'') Error( HN, HN '') 0 n 1 ( g1 g1'' ) ( gm gm '' ) ( h1 h1'' ) ( hn hn '' ) (7) The ( M N )-tuple value ( g1',, gm', h1',, hn' ) for the M N pixels of images G1,, GM, H1,, HN at the same location that minimizes the sum of the total errors. This process is repeated on each ( M N )-tuple pixel in raster order until no more improvement is possible for the whole M N images. 6. Concluding Remark In this paper, we have presented a Direct-Binary-Search-based algorithm that conceals a binary image into two binary images, and extended this algorithm to hide one or more than one binary image into more than two binary images. Our image hiding method can hide and recover images well, and as the number of public images increases, both the secret and public images have higher contrast. 7. References [1] B. E. Bayer, An optimum method for two-level rendition of continuous-tone pictures, IEEE international conference on communications-icc, vol.1, pp.11-15, June 11-13, [2] M. Analoui and J. Allebach, Model-based halftoning by direct binary search, In Proc. SPIE/IS&T Symposium on Electronic Imaging Science and Technology, vol. 1666, pp , [3] Z. Baharav and D. Shaked, Watermarking of dither halftoned images, IS&T/SPIE Conference, Security and Watermarking of Multimedia Contents, Vol.3957, pp , January, [4] R. Floyd and L. Steinberg, An adaptive algorithm for spatial gray scale, SID 75 Digest, Society for information display, pp , [5] J. Goldschneider and E. Riskin, Embedded multilevel error diffusion, IEEE Trans. on Image Processing, vol. 6, no.7, pp , July, [6] Y. Ito and K. Nakano, FM screening by the local exhaustive search, with hardware acceleration, In Proc. of Workshop on Advances in Parallel and Distributed Computational Models (CD-ROM of International Parallel and Distribution Processing Symposium), April, [7] Y. Ito and K. Nakano, FM screening by the local exhaustive search, with hardware acceleration, International Journal of Computer Science, vol.16, no.1, pp , February, [8] D. Kacker and P. Alleach, Joint halftoning and watermarking, IEEE Transactions on Signal Processing, vol.51, no.4, pp , April, [9] S. Katzenbeisser and F. Petitcolas (Editors). Information hiding techniques for steganagraphy and digital watermarking. Artech House, Eastern Europe, [10] D. Lau and G. Arce. Modern Digital Halftoning. Marcel Dekker, New York, [11] S. Pei and J. Guo, Data hiding in halftone images with noise-balanced error diffusion, IEEE Signal Processing letters, vol.10, no.12, pp , December, [12] S. Wang and K. Knox, Embedding digital watermarks in halftone screens, Proceedings of SPIE, Security and watermarking of multimedia contents II, Vol.3971, pp , [13] X. Zhuge, Y. Hirano and K. Nakano, A new hybrid multitoning based on the direct binary search, In Proc. of International Multi Conference of Engineers and Computer Scientists, vol. I, IMECS 2008, March, 2008, pp , Hong Kong. [14] Xia Zhuge, Koi Nakano, Clipping-Free Halftoning and Multitoning Using the Direct Binary Search, IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences, vol.e92-a, no.4, pp , April,

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