CONTENTS

13.1 Introduction

While gray-scale images can be readily displayed in computer monitors and other light-emitting displays, they also need to be displayed routinely in other reflective media such as newspaper, magazines, books, and other printed documents. However, in reflective media, the application of ink on the reflective media implies that only 1-bit images (with two tones: black and white) can be displayed. A problem arising from this is that straightforward 1-bit quantization on an image would lose most of the important image details. With such constraints, there is a class of image processing technique called image halftoning that converts an 8-bit image into an 1-bit image, which resembles the 8-bit image when viewed from a distance. Such 1-bit images are called halftone images [1]. Halftone image technologies are widely used in printed matters.

There are two main kinds of halftoning methods: ordered dithering [1] and error diffusion [2]. Ordered dithering uses straightforward 1-bit quantization with fixed pseudo-random threshold patterns to give halftone images with reasonable visual quality. Error diffusion also performs simply 1-bit quantization but allows the 1-bit quantization error to be fed back to the system and thus can achieve higher visual quality than ordered dithering.

Sometimes it is desirable to hide watermarking data in halftone images. Some halftone image watermarks are designed to be fragile and are useful for authentication and tamper detecting of the halftone images. Some halftone image watermarks are designed to be robust and are useful for copyright protection. In some applications, the data are to be embedded into a single halftone image and some special method can be used to read the hidden data. In other applications, visual patterns are hidden in two or more halftone images such that, when they are overlaid, the hidden visual patterns can be revealed. This kindly visual pattern hiding is also called visual cryptography. This chapter is about visual cryptography in error diffused halftone images.

The chapter is organized as follows. Section 13.2 introduces the basic error diffusion technique. Section 13.3 introduces a visual cryptography method for error diffused images called Data Hiding by Stochastic Error Diffusion (DHSED) [3]. Section 13.4 introduces an improved method called Data Hiding by Conjugate Error Diffusion (DHCED) [4]. Section 13.5 gives theoretical and empirical analysis of DHSED and DHCED. At last, Section 13.6 will give a summary of this chapter.

13.2 A Review of Error Diffusion

In this section, we will briefly introduce a halftoning method called Error Diffusion. The method that we will describe is by no means the only way to achieve halftoning, but is a popular approach that gives good visual quality while maintaining reasonable complexity. The error diffusion process converts a multitone image to a halftone one by distributing the error introduced at the current pixel to a neighborhood of yet unprocessed pixels. The neighborhood, as well as the weights in the distribution, is described by a set of positions and weights known as an error kernel. This diffusion of error across a region allows the local intensity of the halftone image to be preserved approximately.

Consider a multitone image with pixels defined over the range of 0 (black) to 255 (white). Let h(k, l) be an error kernel defined over a neighborhood N. For example, the common Steinberg kernel [2] is

defined over the neighborhood N = {(0,1), (1, – 1), (1,0), (1,1)}. Here we have used * to indicate the location of the current pixel. Let (i, j) be the current pixel location. Let x(i, j) be the current multitone pixel to be processed. A modified pixel value u(i, j) will be derived from x(i, j) and 1-bit quantization is applied to u(i, j) to give the output halftoned pixel value y(i, j). Let e(i, j) be the error between u(i, j) and y(i, j). In error diffusion, the error e(i, j) is distributed to future pixels in its neighborhood. For the case of error diffusion using the Steinberg kernel, a portion 716 of e(i, j) will be distributed to (i, j + 1) corresponding to (0,1) in N. Similarly, 316 of e(i, j) goes to (i + 1, j – 1), 516 of it goes to (i + 1, j), and 116 of it goes to (i + 1, j + 1).

For location (i, j), an offset term a(i, j) is defined as

which is the total error propagated from past pixels to the current pixel. For the error at location (i – k, j – l), a portion h(k, l) of it is passed to the current pixel according to the error kernel. The modified pixel value u(i, j) is then defined as

The output halftone value y(i, j) is defined as

and the error e(i, j) is

These steps are summarized in Figure 13.1, in which the block Q(·) is the 1-bit quantization and the block h_k,l is the error kernel applied to past errors in the neighborhood to generate the current offset a(i, j).

Consider the special case of X being of constant intensity A in a neighborhood around (i, j). As error diffusion can preserve the average image intensity, the probability distribution of a halftone pixel y(i, j) is

such that

Typically, the percentages of white and black pixels in X are A/255 • 100% and (255 – A)/255 • 100% respectively, distributed evenly in X. In a way, one can argue that

such that

Another common error diffusion kernel is the Jarvis kernel [5]

which has a larger support than the Steinberg kernel. A typical image, Lena, is shown in Figure 13.2. The corresponding halftone images generated by Steinberg and Jarvis kernels are shown in Figures 13.3 and 13.4 respectively. It can be observed that different error diffusion kernels give rise to different textures in the halftone images. In general, Jarvis gives images with higher contrast while the Steinberg kernal gives smoother texture. Both are capable of generating halftone images that mimic the original images when viewed afar, though tiny details of the original image such as Lena’s hair tend to be masked by the halftone image texture generated by the error kernel.

The size of all the images in this chapter are 512 × 512. Due to limited space, all remaining halftone figures are generated by the Jarvis kernel only, though the methods described in the chapter are applicable to any error diffusion kernels.

13.3 Data Hiding by Stochastic Error Diffusion (DHSED)

Data Hiding for halftone images is quite different from that for multitone images due to the fact that halftone pixels can take on only two values: 0 and 255. They contain high frequency noise but resemble the original multitone images when viewed afar. As such, normal data hiding techniques such as least significant bit (LSB) embedding technique [6] would not work on them because the resulting stego-images will be effectively the watermark image and would not resemble the original multitone images even when viewed afar. Thus, it is necessary to develop special data hiding techniques for halftone images. Although several data hiding technologies for halftone images have been proposed before, Data Hiding by Stochastic Error Diffusion (DHSED) is the first visual cryptography method based on error diffusion.

DHSED is a method that embeds a binary secret pattern into two halftone images derived from the same underlying multitone image. The binary pattern should be revealed when the two halftone images are superimposed. The idea of DHSED is to stochastically create a texture phase shift between the two halftone images at locations where the watermark is ”active” or black (binary pattern value being zero). The resulting mismatch allows the watermark to become visible while maintaining the original halftone background. Let X be the original multitone image and W the binary secret pattern of the same size. Let Y₁ be a halftone image obtained by applying regular error diffusion to X. Let Y₂ be the halftone image obtained through DHSED. The problem, then, is to obtain Y₂ such that W is revealed when Y₁ and Y₂ are overlaid. We denote the individual pixels at location (i, j) of both halftone images as y₁(i,j) and y₂(i,j), respectively.

The second image Y₂ is generated by applying regular error diffusion to certain areas in X, but with different error conditions. These areas are obtained by referencing both Y₁ and W. Let W_b be the collection of the locations of all the black pixels in W, and W_w the collection of the white pixel locations. In constructing Y₂, we will force the pixel value at all locations belonging to W_w to be identical to Y₁. In other words, values at these locations are merely copied from Y₁ to Y₂. That is,

For the remaining pixels in W_b, Y₂ needs to look natural and thus DHSED applies error diffusion with the same error kernel. However, DHSED seeks to make Y₂ different from Y₁ statistically so that when they are overlaid, pixels in W_b would tend to be darker. To achieve this, DHSED first morphologically dilate, W_b with a structuring element D consisting of a (2L +1) × (2L + 1) matrix. We denote the dilated W_b as C.

which can be interpreted as a L-pixel expansion of W_b in all directions.

For the pixels outside C, DHSED copies Y₂ from Y₁ using (13.10) but forces the error for Y₂ to be zero, i.e. e₂(i, j) = 0 for (i, j) ∉ C. Note that the error for Y₁ are nonzero in general for the same locations.

Let E = C – W_b = C ∩ W_w be the ”border” of the secret pattern, obtained by removing W_b from the expanded region C. For the pixels in E, (13.1) and (13.2) are still applied while (13.3) and (13.4) are not. (13.10) will be used to replace (13.3) since E ⊂ W_w and Y₂ pixels in W_w are copied from Y₁. We will use (13.12) to replace (13.4).

which is basically (13.4) with a limiter. As Y₂ pixels in E are copied from Y₁ with artificial zero error outside C, there are chances that u₂(i, j) – y₂(i, j) is outside ± 127. The limiter would then help to make the e₂(i, j) reasonable.

For the pixels in W_b, DHSED uses regular error diffusion to generate Y₂ so that region W_b in Y₂ still has the same characteristic texture as regular error diffusion. But the ”phase” of the texture will be different compared to the corresponding region in Y₁ since the error outside the region C is different in Y₁ and Y₂.

The overlaying operation is equivalent to applying the logical AND operation between images Y₁ and Y₂. Since the pixels in region W_w of Y₁ and Y₂ have been forced to be identical, the overlaid pixel values are simply the regular error diffused pixel values. However, in region W_b, although the texture in Y₁ and Y₂ maintains the same characteristic, there is an artificially introduced phase shift such that collocated Y₁ and Y₂ pixels tend to be statistically independent. As a result, the overlaying operation tends to give darker local intensity thus revealing the secret pattern W. A more detailed analysis of this will be given in Section 13.4.

Using Lena as the test image and Figure 13.5 as the secret binary pattern W, Figure 13.4 is Y₁ and Figure 13.6 is Y₂ generated using DHSED with respect to Y₁ and W. A threshold L = 5 is used. Note that Figure 13.6 looks like Figure 13.4, which verifies that DHSED can give halftone images with good visual quality. Figure 13.7 shows the image obtained by overlaying Figure 13.6 and Figure 13.4. The secret pattern W is clearly visible in Figure 13.7 verifying that DHSED is an effective visual cryptography method.

13.4 Data Hiding by Conjugate Error Diffusion (DHCED)

DHSED as outlined in the previous section is both computationally and conceptually simple, but suffers from three major problems. First, Y₁ and Y₂ must be obtained from the same X. In other words, DHSED cannot embed a binary secret pattern in two halftone images obtained from two different multitone images. Second, when the images are overlaid, the contrast of the revealed secret pattern is relatively low. Third, occasional boundary artifacts may happen in Y₂ especially towards the right and bottom sides of the secret pattern where error is not diffused properly across the W_b boundary. Such boundary artifacts can lower the visual quality of Y₂ considerably.

In this section, we will introduce another method called Data Hiding by Conjugate Error Diffusion (DHCED) that addresses these three problems.

The block diagram of DHCED is shown in Figure 13.8. Unlike DHSED, the problem now contains two original multitone images X₁ and X₂, where X₁ and X₂ may or may not be identical. Two halftone images Y₁ and Y₂ are to be generated from X₁ and X₂, respectively such that when Y₁ and Y₂ are overlaid, the binary secret pattern W is revealed. Similar to DHSED, Y₁ is generated using regular error diffusion on X₁. Then the DHCED process in Figure 13.8 is used to generate Y₂.

To explain the detailed operation of DHCED, we assign logical ”1” to pixel value 255 and logical ”0” to pixel value 0. The conjugate of a pixel is then equivalent to logical NOT, or toggling of the value. Recall that W_w and W_b are the collections of white and black pixel locations in W, respectively. As Y₁ is already generated by regular error diffusion, the secret pattern W at location (i, j) can be naively inserted in Y₂ as follows

where (·)^c denotes logical NOT. This is equivalent to logical XNOR between Y₁ and W. The resulting Y₂ will look like X₁ in W_w and the negative of X₁ in W_b because

But Y₂ should resemble X₂ instead of X₁ and thus this naive method does not work. Even if X₁ and X₂ are the same image, this naive method does not work in W_b as Y₂ should resemble X₁ instead of the negative of X₁.

Although this naive method does not work, DHCED follows a similar logic except that it favors, instead of forces, Y₂ to take on the values in (13.13). In other words, it treats the value in (13.13) as the favored value of y₂(i, j).

Consider Figure 13.8 and any location (i, j) ∈ W_b. Basically, DHCED applies error diffusion on x₂(i, j) including (13.1) to calculate α₂(i, j), (13.2) to calculate u₂(i,j) and, in the first Q(·) block, (13.3) to calculate the trial halftone value y₂(i, j). In the N(·) block, the trial value is compared with the favored value that is obtained by XNOR of w(i, j) and y₁(i, j). If they are equal, no change needs to be done to x₂(i, j) and u₂(i, j) such that u′₂(i, j) = u₂(i, j). If they are not equal, DHCED considers the possibility of forcing trial y₂(i, j) to toggle to achieve the favored value. Recognizing that forced toggling is equivalent to applying a distortion to x₂(i, j) followed by regular error diffusion, DHCED will execute the forced toggling only if the required distortion is not excessive.

Here are the details. Note that forcing the trial halftone value to toggle is equivalent to adding an offset value Δu(i, j) to u₂(i, j). If the trial value should be toggled from 0 to 255, then u₂(i, j) < 128 initially and we need an offset Δu(i, j) such that the resulting value, which we called u′₂(i, j), is u₂(i, j) ≡ u₂(i,j) + Δu(i, j) ≥ 128. Thus, there is a lower bound for Δu(i, j): Δu(i, j) ≥ 128 – u₂(i, j) > 0.

Likewise, if we want to toggle from 255 to 0, then u₂(i, j) ≥ 128 initially and we need an offset Δu(i, j) such that u′₂(i, j) ≡ u₂(i, j) + Δu(i, j) ≤ 127. There is thus an upper bound for Δu(i, j): Δu(i, j) ≤ 127 – u₂(i, j) < 0.

The smallest Δu(i, j) (in terms of magnitude) needed to achieve toggling is

We further note that adding a distortion Δu(i, j) to u₂(i,j) may be interpreted as a distortion to the original multitone image pixel x₂(i, j). Defining Δx(i, j) = Δu(i, j), the input to the normal quantizer u′₂(i, j) may be written as

The interpretation of (13.18) is that the output halftone image using DHCED in fact represents X′2, not X₂, in the sense of that it can be obtained from X′2 directly using standard error diffusion. Thus, |Δx| can be treated as a measure of the distortion introduced by the DHCED process. To control this distortion, we define a threshold T that determines whether or not the pixel should be toggled, allowing a trade-off between distortion and visual quality of the watermarked halftone image. If |Δx| is less than T, the pixel toggling will be performed, and vice versa. If T decreases, the visual quality of the watermarked image Y₂ will improve at the price of lower contrast of the secret pattern when the two halftone images are superimposed.

Consider any location (i, j) ∈ W_w. If X₁ and X₂ are the same image, DHCED would copy y₁ (i, j) to y₂ (i, j) so that Y₂ values are effectively obtained by applying error diffusion to X₁. The error e₂ (i, j) will be computed as in normal error diffusion. When Y₁ and Y₂ are overlaid, the regular error diffused value y₁(i, j) will be revealed. The overlaying operation will reveal a local intensity similar to the local intensity of X₁, which is typical for regular error diffusion.

If X₁ and X₂ are different, DHCED would not force y₂(i, j) to be identical to y₁(i, j) at (i, j) ∈ W_w. Instead, it merely treats y₁(i, j) as the favored value of y₂(i, j). And DHCED performs the same operation as in W_b.

The proposed DHCED for the case of identical X₁ and X₂ is simulated. Using Lena as the test image and Figure 13.5 as the secret binary pattern W, Figure 13.4 is Y₁ and Figure 13.9 is Y₂ generated using DHCED with respect to Y₁ and W. A threshold of T – 10 is used. Note that Figure 13.9 looks like Figure 13.4 which verifies that DHCED can give halftone images with good visual quality. Figure 13.10 shows the image obtained by overlaying Figure 13.9 and Figure 13.4. The secret pattern W is clearly visible in Figure 13.7 verifying that DHSED is an effective visual cryptography method.

Comparing DHSED and DHCED, it can be observed that DHSED in Figure 13.6 has strong boundary artifacts along the embedded watermark especially on the right and bottom edges of W. With DHCED, the boundary artifacts are reduced considerably. In addition, the watermark in Figure 13.10 is significantly more visible than Figure 13.7 in terms of contrast, when Y₁ and Y₂ are overlaid.

DHCED for the case of different X₁ and X₂ is also simulated. Here Lena is used as X₁ and Pepper is used as X₂. The Y₁ with respect to X₁ is simply the one in Figure 13.4. The DHCED generated Y₂ from X₂ with respect to Y₁ and W is shown in Figure 13.12. A threshold of T = 10 is used. The overlaid image of Y₁ and Y₂ is shown in Figure 13.13. As expected, traces of both Lena and Pepper can be observed in the overlaid image. More importantly, the watermark W is revealed also, though the contrast of W in Figure 13.13 is not as good as in Figure 13.10.

13.5 Performance Analysis

In this section, we give an indepth analysis of both DHSED and DHCED. We will show theoretically why DHCED has better performance than DHSED.

Consider the case when X₁ and X₂ are the same image X. Consider a rectangular region R of constant intensity A in X. Suppose that the left half of R is in the white region W_w of W and the right half in the black region W_b of W. We assume that error diffusion is effective in both the left and right halves of R such that the average image intensity is preserved. Thus, the probability distribution of a halftone pixel y₁ (i, j) in the region R of Y₁ is

such that the expected value is

for (i, j) ∈ R. Typically, the percentage of white and black pixels in Y₁ in R are A/255 • 100% and (255 – A)/255 • 100% respectively, distributed evenly in R.

The generation of Y₂ using DHCED is effectively the application of error diffusion to the equivalent noisy multitone image. And the error diffusion should be effective, as usual. The distortion Δx(i, j) introduced by DHCED is equally likely to be positive and negative and its magnitude is bounded by T. The average intensity in R should still be approximately A. Thus, the probability distribution of a halftone pixel y₂(i, j) in the region R of Y₂ is approximately

such that

for (i, j) ∈ R_A.

Let y(i, j) = y₁(i, j) ∩ y₂(i, j) be the output pixel obtained by overlaying Y₁ and Y₂. The y(i, j) would be white only if both y₁(i, j) and y₂(i, j) are white. For the proposed DHCED, the y₁(i, j) and y₂(i, j) are designed to be dependent.

In the left half of R, which is in W_w, DHCED would simply copy y₁(i, j) as y₂(i, j) such that P[y₂(i, j) = y₁(i, j)] = 1. Then

such that

In other words, y(i, j) = y₁(i, j)) = y₂(i, j) for the region W_w and the overlay operation does not change the halftone pixel values at all.

In the right half of R which is in W_b, the y₁(i, j) and y₂(i, j) are favored to be conjugate to each other (y₂(i, j) ≈ y₁(i, j)) such that y₁(i, j) and y₂(i, j) tend not to be black at the same time. To investigate the behavior of y(i, j), we consider two different cases: (1) 255 ≥ A > 127 and (2) 127 ≥ A ≥ 0.

For the case of 255 ≥ A > 127, there are more white pixels than black pixels in R of Y₁. The percentage of black pixels in the right half of R of Y₁ is about (255 – A)/255 • 100%. For example, if A = 190, about 25% of the pixels in the right half of R of Y₁ should be black. As y₁(i, j) and y₂(i, j) tend not to be black at the same time, the black pixels in the right half of R of Y₂ tend to be at different locations from those in Y₁. Consequently, the percentage of black pixels in the right half of R of Y tends to be doubled to 2 • (255 – A)/255 • 100%. In our example of A = 190, the approximately 25% black pixels in the right half of R of Y₁ and those of Y₂ tend to be at different locations such that there are about 50% black pixels in the right half of R of Y. Thus,

such that

In other words, the expected value should increase approximately linearly with A, from 1 (for A = 128) to 255 (for A = 255). The difference between the average intensity of the left and right halves of R for 255 ≥ A > 127 is

For the case of 127 ≥ A ≥ 0, there are fewer white pixels than black pixels in the right half of R of Y₁. Again the y₁(i, j) and y₂(i, j) tend not to be black at the same time in DHCED. This implies that the black pixels in the right half of R of Y₂ tend to fill up all the white pixel locations in the right half of R of Y₁, leading to all y(i, j) being black. Thus, P(y(i, j) = 0) ≈ 1 and

for 127 ≥ A ≥ 0. And the difference between the average intensity of the left and right halves of R is, for 127 ≥ A ≥ 0,

Consequently, the contrast between the left and right halves of R of Y can be expressed as, for 255 ≥ A > 127,

and, for 127 ≥ A ≥ 0,

To summarize, for any (i, j) in the right half of R that is in W_b, the expected value of y(i, j) is

The difference in average intensity of the left half and right half of R is

and the contrast between the left half and right half of R is

To verify these, we simulate DHCED using an artificial image called ”Ramp” shown in Figure 13.14 as X = X₁ = X₂, in which the image intensity decreases gradually and linearly from 255 at the top row to 0 at the bottom row. The hidden pattern to be embedded is called ”Column” and is black at the center and white on the left and right, as shown in Figure 13.15. The DHCED generated Y₁ and Y₂ are shown in Figures 13.16 and 13.17, respectively. The row-wise average intensity of Y₁ and Y₂ are plotted against the average intensity of the corresponding row in X in Figure 13.22. As expected, the row-wise average intensity of Y₁ and Y₂ are very similar to the corresponding intensity in X, which verifies (13.21) and (13.24).

Figure 13.18 is the overlaid image Y. We note that in the W_w region, Y is identical to Y₁ and Y₂, with intensity decreasing from top to bottom. In the Wb region, the intensity of Y is as high as X at the top. But as the intensity of X decreases from the top to bottom, we observe that the intensity of Y decreases at a fast pace to about the middle of the image (where intensity is about 127), and remains low in the lower half of the image.

To show the exact behavior of Y in the W_b region, we compute the average intensity in the W_b for each row of Y and plot it against the average intensity of the corresponding row in X in Figure 13.19. Three such curves are obtained with 3 values of T, namely T = 5, 10, 15. Also shown in the figure is the theoretical behavior as predicted by (13.35). It can be observed from the figure that as T increases, the average intensity curves appear to converge to the theoretical curve, as expected.

We also compute the difference between the average intensity in the W₅ and the W_w regions for each row of Y, and plot this Δintensity against the average intensity of the corresponding row in X in Figure 13.20. Similarly, the contrast is computed and plotted in Figure 13.21. Also shown in the two figures are the theoretical behavior predicted by (13.36) and (13.37). It can be observed that as T increases, the Δintensity curves and the contrast curves appear to converge to the corresponding theoretical curves, as expected.

Next, we analyze DHSED and make a comparison with DHCED. Since DHSED forces Y₁ and Y₂ to be identical in W_w, both y₁(i, j) and y₂(i, j) are identical in the left half of R and the probability distribution is

such that E[y(i, j)] = A. For any (i, j) in W_b, the corresponding pixel values in Y₁ and in Y₂ are error diffused with relative random phase. Thus, the local intensity for Y₁ and Y₂ is

and y₁(i, j) and y₂(i, j) are approximately independent with

such that

Thus, the intensity of DHSED is expected to be greater than or equal to that of DHCED, with equality if A = 255. The difference in average intensity is

The contrast is

We also simulate DHSED on Ramp to verify the equations above. The hidden pattern is still the Column in Figure 13.15. The DHSED generated Y₁ and Y₂ are shown in Figures 13.16 and 13.23, respectively. In Figure 13.28, the row-wise average intensity of Y₁ and Y₂ are plotted against the average intensity of the corresponding row in X. As expected, they are very similar to X, which verifies (13.39).

Figure 13.24 is the overlaid image Y. Similar to DHCED, the Y of DHSED is identical to Y₁ and Y₂ in the W_w region, with intensity decreasing from top to bottom. In the W_b region, the intensity of Y is as high as X at the top row. But as the intensity of X decreases towards the bottom of the image, the intensity of Y decreases faster in the center than on the two sides. In other words, the intensity of Y decreases faster in W_b than in W_w, making W_b visible in Y.

Similar to DHCED, to show the exact behavior of DHSED in W_b in Y, we compute the average intensity in the W_b for each row of Y and plot it against the average intensity of the corresponding row in X in Figure 13.25. Three curves are obtained for 3 values of L, namely L = 1, 5, 10. Also shown is the theoretical curve according to (13.42). It can be observed that the 3 empirical curves match the theoretical curve very well. This also suggests that the choice of L does not have much effect on the intensity of W_b in Y.

We also compute the difference between the average intensity in W_b and W_w regions for each row of Y, and plot this Δintensity against the average intensity of the corresponding row in X in Figure 13.26. Similarly, the contrast is computed and plotted in Figure 13.27. Also shown in the two figures are the theoretical behavior predicted by (13.43) and (13.44). It can be observed that the empirical Δintensity curves and contrast curves are similar to the corresponding theoretical curves, as expected. Again, the choice of L has little effect.

Comparing DHCED and DHSED, they have the same Y₁. For their Y₂, their pixel values are identical in W_w but different in W_b. On overlaying the corresponding Y₁ and Y₂, the Y of both DHCED and DHSED are identical in W_w, but DHCED has lower E[y(i, j)] in W_b than DHSED such that the black patterns of W would look darker in DHCED than in DHSED as predicted by Figure 13.29. And, the contrast of the revealed W in DHCED is higher than that in DHSED as predicted by Figure 13.30.

13.6 Summary

In this chapter, we introduce two ways to achieve steganography in halftone images, namely DHSED and DHCED. Both methods can embed a binary secret pattern into two halftone images that come from the same multitone image. When the two halftone images are overlaid, the secret pattern is revealed. DHCED can further embed a binary secret pattern into two halftone images from two different multitone images. DHSED operates by introducing different stochastic phases in the two images. DHCED operates by favoring certain conjugate values for each pixel and taking on the values only if the implied distortion is small enough. Both theoretical analysis and simulation results suggest that DHCED can give better contrast of the revealed secret pattern than DHSED.