## Abstract

We propose a display technique that ensures security of visual information by use of visual cryptography. A displayed image appears as a completely random pattern unless viewed through a decoding mask. The display has a limited viewing zone with the decoding mask. We have developed a multi-color encryption code set. Eight colors are represented in combinations of a displayed image composed of red, green, blue, and black subpixels and a decoding mask composed of transparent and opaque subpixels. Furthermore, we have demonstrated secure information display by use of an LCD panel.

©2004 Optical Society of America

## 1. Introduction

Security has become an important issue as information technology has become increasingly pervasive in our everyday lives. Information security engineering involves securing many different types of information. To deal with the massive amount of data that visual information contains, optical information processing is a promising technique to encrypt image data. There have been many types of optical encryption techniques to secure image information [1–19]. Furthermore, a real-time reconstruction system of a digital hologram [20] enables reconstruction of three dimensional objects encrypted with a high level of security. These encryption techniques ensure data security against unauthorized access to confidential information, such as theft and code-breaking of recorded media, wire-tapping of communication links, and counterfeiting of valuable documents. Transmission of encrypted data with current communication networks may be realized with computational encryption techniques by use of digital wavefront reconstruction [14–19]. By use of such security systems, confidential image data can be encrypted, processed for secure transmission, and decrypted at a remote site. If these techniques are combined with ubiquitous information technologies such as ubiquitous displays [21], it is expected that people can access required information at any place and at any time. In practice, however, access to confidential information will be limited because security risks eventually arise with the display when showing the decrypted information. These risks include wire-tapping of the electrical video signal, peeping at the screen, and other attacks [22]. A secure working space is generally provided by controlling physical user access and by providing partitions to protect the displayed information. In other words, the security of the displayed information is maintained by user authentication and limitation of the viewing zone.

We have reported a secure information display technique for monochrome images by using a decoding mask to view the display [23]. The decoding mask has two functions: as a key for decryption of the encrypted image and as a means to limit the viewing zone of the decrypted image. The encryption is based on visual cryptography, which was originally proposed by Naor and Shamir [24]. Visual cryptography is a shared-secret encryption technique: information of the original secret image is shared between two or more images (shared images). Decryption is accomplished simply by overlaying the shared images, and requires no computation, like some optical logic schemes [25]. Several kinds of algorithms for visual cryptography have been reported, including optimization of the contrast [26], encryption of grey level images [27], analytic construction of color images with color subpixels in every shared image [28], and the application of visual cryptography to data embedding [29, 30]. However, conventional visual cryptography does not limit the viewing zone. In that case, there is a risk that the secret may be obtained by someone peeping at it. In our proposed system, to limit the viewing zone, we place the decoding mask at a certain distance from the display panel, and we arrange the pixels of the encrypted image to maintain a viewing zone of adequate size. It is necessary to construct a suitable code-set to maintain such a limited viewing zone.

This paper describes secure information display of multi-color images by use of a decoding mask based on visual cryptography. A code set is constructed to encrypt multi-color images that are composed of red, green, blue, cyan, magenta, yellow, black, and white pixels. Information on a secret image is shared between two random patterns. One of them is shown on the display panel, and the other one is used as the decoding mask. Each pixel of the displayed image is composed of red, green, blue, and black subpixels. Each pixel of the decoding mask is composed of transparent and black subpixels. The decryption process requires no special computing device and is implemented using only human vision. The displayed image appears as a totally random pattern to anyone looking at it unless that person views it through the decoding mask. When the decoding mask is placed in front of the display panel, the secret image becomes visible within the limited viewing zone. Limitation of the viewing zone has been demonstrated by use of a decoding mask placed in front of an LCD panel.

We illustrate the use of visual cryptography to limit viewing zone in Section 2. In Section 3, we describe the constructed code set to encrypt multi-color images. In Section 4, we present experimental results obtained with a prototype multi-color display system using a decoding mask.

## 2. Use of visual cryptography to limit viewing zone

The encryption scheme we used is based on visual cryptography, in which a secret image is shared between a displayed image and a decoding mask. An example of encryption of a monochrome image is shown in Fig. 1. There are black (opaque) and white (transparent) pixels in the decoding mask. Decoding of the encrypted image is achieved by overlaying the images and viewing them, that is, without any decoding computation. A pixel in the secret image is represented by 2×2 subpixels in the decoding mask and the displayed image [22]. Every pixel in the secret image is encrypted by a randomly chosen combination of decoding-mask and displayed-image subpixel patterns. Therefore, the decoding mask and the displayed image look like random-dot patterns before they are overlaid.

To limit the viewing zone of the decoded image, a decoding mask that has a reduced pitch is placed away from the displayed image. A top view of the display and the decoding mask is shown in Fig. 2. Each subpixel of the displayed image has a corresponding subpixel in the decoding mask. There is a limited viewing zone where this one-to-one relationship is preserved. The viewing distance *Z*
_{E}, which is the distance of the viewing position from the display, is determined by the distance *Z*
_{M} of the decoding mask from the display panel and is given by *Z*
_{E}=*Z*
_{M}
*P*
_{D}/(*P*
_{D}-*P*
_{M}), where *P*
_{D} and *P*
_{M} are the pitch of the displayed image and that of the decoding mask, respectively.

## 3. Encryption codes for multi-color images

For the sake of securing information display, the required features for the encryption codes include:

1. The secret image cannot be decoded with only one of the displayed image and the decoding mask.

2. Different secret images can be decoded with the same decoding mask.

3. The encryption is compatible with a conventional display such as an LCD panel.

Furthermore, we propose the following conditions in composing the encryption codes for multi-color images:

(a) All colors can be represented with each subpixel pattern of the decoding mask.

(b) All colors can be represented with each subpixel pattern of the displayed image.

(c) The colors of the subpixels of the displayed images are red, green, blue, and black.

To formulate the encryption and decryption process, subpixel values of the displayed images are assigned to multiple colors and the pixel colors are obtained as the sum of the subpixel values of the decoded pixel. The three primary colors that are assigned in condition (c) represent 2^{3}=8 colors in their combinations. The colors can be expressed in 3 bits. We assigned values 2^{0} to red, 2^{1} to green, and 2^{2} to blue and their various sums to the other colors shown in Table 1. The subpixel values of pixels in the decoding mask are 0 or 1 according to the transmittance.

To represent all colors with the displayed-image subpixel patterns, each pixel is composed of at least one red, one green, and one blue subpixel. To represent all colors with the displayed-image subpixel patterns, each pixel is composed of at least one red, one green, and one blue subpixel. In order to represent black, the number *N*
_{MBK} of black subpixels in each mask subpixel pattern must be greater than or equal to the total number of the red, green, and blue image subpixels, that is, *N*
_{MBK}≥*N*
_{DR}+*N*
_{DG}+*N*
_{DB}, where *N*
_{DR}, *N*
_{DG}, and *N*
_{DB} denote the numbers of red, green, and blue subpixels in each displayed-image subpixel pattern, respectively. In order to represent white, the number *N*
_{MTR} of transparent subpixels in each mask subpixel pattern must satisfy *N*
_{MTR}≥*N*
_{DR}+*N*
_{DG}+*N*
_{DB}. Since there is the one-to-one relationship between the displayed-image subpixels and the mask subpixels, each displayed-image subpixel pattern has the same number of subpixels as each mask subpixel pattern, which is expressed by *N*
_{DBK}+*N*
_{R}+*N*
_{G}+*N*
_{B}=*N*
_{MBK}+*N*
_{MTR}, where *N*
_{DBK} is the number of black subpixels in each displayed-image subpixel pattern. By solving the previous two inequalities and the latter equation for *N*
_{DBK}, the inequality *N*
_{DBK}≥*N*
_{DR}+*N*
_{DG}+*N*
_{DB} is obtained; that is, there must be black subpixels numbering not less than the total number of red, green, and blue subpixels. Therefore, the minimum number of subpixels for each displayed-image subpixel pattern is six, where each pixel is composed of one red, one green, one blue, and three black subpixels. Because a code set with a reduced number of subpixels increases the number of effective pixels that can be shown on the display, we construct the encryption code set using this minimum number of subpixels, i.e. six.

When each pixel of the displayed image is composed of 6 subpixels in 2 rows and 3 columns, the subpixel pattern of a pixel of the displayed image is written as a 2×3 matrix **D**:

Subpixel pattern **M** of the corresponding pixel of the decoding mask that contains 3 transparent subpixels and 3 opaque subpixels is also written as a 2×3 matrix:

The decoding process, that is, viewing the displayed image through the decoding mask, is expressed by multiplying the matrices. There are two types of matrix multiplication, **M**
^{T}
**D** and **MD**
^{T}, which are expressed by the following equations.

and

The pixel value *P*
_{0} of a pixel of the decoded image is expressed by the trace of the matrix multiplication, as follows:

where Tr(·) denotes the trace of the matrix. Non-diagonal elements of **M**
^{T}
**D** correspond to horizontal shifts of the decoding mask. Non-diagonal elements of **MD**
^{T} correspond to vertical shifts of the decoding mask.

The number of subpixel patterns in the decoding mask is _{6}C_{3}=20. The subpixel patterns of the decoding mask are

$\begin{array}{cccc}{\mathbf{M}}_{1}=\left(\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\end{array}\right),& {\mathbf{M}}_{2}=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 0\end{array}\right),& {\mathbf{M}}_{3}=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\end{array}\right),& {\mathbf{M}}_{4}=\left(\begin{array}{ccc}0& 0& 1\\ 1& 1& 0\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{M}}_{5}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\end{array}\right),& {\mathbf{M}}_{6}=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\end{array}\right),& {\mathbf{M}}_{7}=\left(\begin{array}{ccc}1& 1& 0\\ 0& 0& 1\end{array}\right),& {\mathbf{M}}_{8}=\left(\begin{array}{ccc}0& 0& 0\\ 1& 1& 1\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{M}}_{9}=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 1\end{array}\right),& {\mathbf{M}}_{10}=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 1\end{array}\right),& {\mathbf{M}}_{11}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 1& 1\end{array}\right),& {\mathbf{M}}_{12}=\left(\begin{array}{ccc}0& 1& 0\\ 1& 1& 0\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{M}}_{13}=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\end{array}\right),& {\mathbf{M}}_{14}=\left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\end{array}\right),& {\mathbf{M}}_{15}=\left(\begin{array}{ccc}0& 1& 1\\ 0& 0& 1\end{array}\right),& {\mathbf{M}}_{16}=\left(\begin{array}{ccc}0& 1& 1\\ 0& 1& 0\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{M}}_{17}=\left(\begin{array}{ccc}1& 0& 1\\ 0& 0& 1\end{array}\right),& {\mathbf{M}}_{18}=\left(\begin{array}{ccc}1& 0& 1\\ 1& 0& 0\end{array}\right),& {\mathbf{M}}_{19}=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\end{array}\right),& {\mathbf{M}}_{20}=\left(\begin{array}{ccc}1& 1& 0\\ 1& 0& 0\end{array}\right),\end{array}$

The number of subpixel patterns in the displayed image is _{6}P_{3}=120. The subpixel patterns of the displayed image are expressed by the following equations

$\begin{array}{cccc}{\mathbf{D}}_{1}^{w}=\left(\begin{array}{ccc}{a}_{w}& {b}_{w}& {c}_{w}\\ 0& 0& 0\end{array}\right),& {\mathbf{D}}_{2}^{w}=\left(\begin{array}{ccc}0& {b}_{w}& {c}_{w}\\ {a}_{w}& 0& 0\end{array}\right),& {\mathbf{D}}_{3}^{w}=\left(\begin{array}{ccc}{a}_{w}& 0& {c}_{w}\\ 0& {b}_{w}& 0\end{array}\right),& {\mathbf{D}}_{4}^{w}=\left(\begin{array}{ccc}0& 0& {c}_{w}\\ {a}_{w}& {b}_{w}& 0\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{D}}_{5}^{w}=\left(\begin{array}{ccc}{a}_{w}& 0& 0\\ 0& {b}_{w}& {c}_{w}\end{array}\right),& {\mathbf{D}}_{6}^{w}=\left(\begin{array}{ccc}0& {b}_{w}& 0\\ {a}_{w}& 0& {c}_{w}\end{array}\right),& {\mathbf{D}}_{7}^{w}=\left(\begin{array}{ccc}{a}_{w}& {b}_{w}& 0\\ 0& 0& {c}_{w}\end{array}\right),& {\mathbf{D}}_{8}^{w}=\left(\begin{array}{ccc}0& 0& 0\\ {a}_{w}& {b}_{w}& {c}_{w}\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{D}}_{9}^{w}=\left(\begin{array}{ccc}0& 0& {b}_{w}\\ 0& {a}_{w}& {c}_{w}\end{array}\right),& {\mathbf{D}}_{10}^{w}=\left(\begin{array}{ccc}0& 0& {b}_{w}\\ {a}_{w}& 0& {c}_{w}\end{array}\right),& {\mathbf{D}}_{11}^{w}=\left(\begin{array}{ccc}0& {a}_{w}& 0\\ 0& {b}_{w}& {c}_{w}\end{array}\right),& {\mathbf{D}}_{12}^{w}=\left(\begin{array}{ccc}0& {b}_{w}& 0\\ {a}_{w}& {c}_{w}& 0\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{D}}_{13}^{w}=\left(\begin{array}{ccc}{a}_{w}& 0& 0\\ {b}_{w}& 0& {c}_{w}\end{array}\right),& {\mathbf{D}}_{14}^{w}=\left(\begin{array}{ccc}{a}_{w}& 0& 0\\ {b}_{w}& {c}_{w}& 0\end{array}\right),& {\mathbf{D}}_{15}^{w}=\left(\begin{array}{ccc}0& {a}_{w}& {b}_{w}\\ 0& 0& {c}_{w}\end{array}\right),& {\mathbf{D}}_{16}^{w}=\left(\begin{array}{ccc}0& {a}_{w}& {c}_{w}\\ 0& {b}_{w}& 0\end{array}\right),\end{array}$ $\begin{array}{cccc}{\mathbf{D}}_{17}^{w}=\left(\begin{array}{ccc}{a}_{w}& 0& {b}_{w}\\ 0& 0& {c}_{w}\end{array}\right),& {\mathbf{D}}_{18}^{w}=\left(\begin{array}{ccc}{a}_{w}& 0& {c}_{w}\\ {b}_{w}& 0& 0\end{array}\right),& {\mathbf{D}}_{19}^{w}=\left(\begin{array}{ccc}{a}_{w}& {b}_{w}& 0\\ 0& {c}_{w}& 0\end{array}\right),& {\mathbf{D}}_{20}^{w}=\left(\begin{array}{ccc}{a}_{w}& {c}_{w}& 0\\ {b}_{w}& 0& 0\end{array}\right),\end{array}$

where case number *w* and parameters *a*_{w}
, *b*_{w}
, and *c*_{w}
are shown in Table 2. Since there are 120 different displayed image patterns, in total there are 2400 combinations of the mask patterns and the displayed patterns.

For each mask pattern, there are 120 possible combinations with different displayed image patterns. Table 3 is a summary of calculations of the trace of the matrix multiplications according to eq. (5) for each mask pattern. All eight colors can be represented. The calculation result is written as Proposition A: for any *u*∈{1, 2,…, 20} and any secret pixel value *S*∈{0, 1,…, 7}, there exist *v*∈{1, 2,…, 20} and *w*∈{1, 2,…, 6} that satisfy *S*=Tr(${{\mathbf{M}}_{u}}^{\mathrm{T}}$
${{\mathbf{D}}_{v}}^{w}$
). This proposition shows that the constructed code set satisfies condition (a) mentioned above.

On the other hand, for each displayed image pattern, there are 20 possible combinations with different mask patterns. Table 4 is a summary of the combination results in terms of *w*, *a*_{w}
, *b*_{w}
, and *c*_{w}
that are shown in Table 2. All eight colors can be represented. The results give a Proposition B: for any *v*∈{1, 2,…, 20}, *w*∈{1, 2,…, 6}, and *S*∈{0, 1,…, 7}, there exist *u*∈{1, 2,…, 20} that satisfy *S*=Tr(${{\mathbf{M}}_{u}}^{\mathrm{T}}$
**D**
_{v}^{w}
). The Proposition B shows that the constructed code set satisfies condition (b).

In the encryption process, one of the patterns **M**
_{u}
, *u* ∈{1, 2,…, 20} is chosen at random for each pixel. To encrypt a secret pixel of pixel value *S*, one of the displayed image patterns ${\mathbf{D}}_{v}^{w}$
that satisfies *S*=Tr=(${{\mathbf{M}}_{u}}^{\mathrm{T}}$
${\mathbf{D}}_{v}^{w}$
) is selected at random. Note that the existence of such a displayed pattern is guaranteed by Proposition A. The obtained decoding mask and the encrypted displayed image have random textures that independently disclose no information of the secret image.

As shown in Table 3 and Table 4, there are redundancies in encrypting a pixel value by use of the code set that contains all possible subpixel patterns. These redundancies suggest that there is much room to extract the essential codes. This is important for practical use, particularly with portable digital devices, to construct a code set that is composed of the minimum number of subpixel patterns because of the limited amount of available memory. Furthermore, a fixed horizontal order of red, green, and blue subpixels is suitable for use with LCD panels in which subpixels of the three primary colors are located in consecutive columns. Table 5 shows the extracted code set that has no redundancy. The code set satisfies the following Propositions C and D:

Proposition C: For any *u*∈{1, 2,…, 8}, any *S*∈{0, 1,…, 7}, and any *w*∈{1, 2,…, 6}, there exists one and only one *v*∈{1, 2,…, 8} that satisfies *S*=Tr=(${{\mathbf{M}}_{u}}^{\mathrm{T}}$
${\mathbf{D}}_{v}^{w}$
).

Proposition D: For any *v*∈{1, 2,…, 8}, any *w*∈{1, 2,…, 6}, and any *S*∈{0, 1,…, 7}, there exists one and only one *u*∈{1, 2,…, 8} that satisfies *S*=Tr=(${{\mathbf{M}}_{u}}^{\mathrm{T}}$
${\mathbf{D}}_{v}^{w}$
).

Proposition C indicates that different secret images can be decoded with the same decoding mask by changing the displayed images. Propositions C and D therefore satisfy the conditions (a) and (b). Thus, the pixel value of a decoded pixel cannot be determined without knowledge of both the decoding mask pattern and the displayed image pattern of the pixel.

Since every pixel in the secret image is encrypted by a randomly chosen combination of decoding-mask and displayed-image subpixel patterns, the probability of getting the secret image is expressed by the reciprocal of the number of possible key patterns [30]. Even in the extracted code set, there are eight possible subpixel patterns for each pixel. When a secret image contains *K* pixels, the number of possible decoding-mask subpixel patterns is 8^{K}. The number of possible decoding-mask subpixel patterns increases exponentially with the number of pixels. For example, when a secret image with 256×256 pixels is encrypted, the number of possible decoding-mask subpixel patterns is 8^{256×256}; that is approximately 10^{59185}. The probability of getting the secret image is nearly zero. It is impossible to determine the secret image without knowledge of both the decoding mask and the displayed image because information of the secret image is shared between the decoding-mask and displayed-image subpixel patterns.

## 4. Experiments

To represent multi-color images, we have utilized the extracted code set that is determined with *w*=1. Figure 3 shows the code set used for our experiments. Each pixel of the secret image is expanded into 6 subpixels. Because the aspect ratio of the codes is 3:2, a secret image is stretched in the vertical direction by 150 % before encryption to maintain the proper aspect ratio. Decoding mask patterns are selected according to a random sequence generated by a computer program. Based on the code set shown in Fig. 3, the subpixel patterns of the displayed image are determined to represent the secret image.

Figure 4 shows examples of a decoding mask and encrypted images. When the decoding mask in Fig. 4 (a) is overlaid on the displayed image Fig. 4 (b), the decoding result in Fig. 4 (c) is obtained. The decoded result shown here is obtained by overlaying two images with a computer graphics application. Although the decoding mask and the displayed image look like random patterns, the decoded image shows the secret image “RGB”. Figure 4 (d) is an unsuccessfully decoded image by relatively shifting and then overlaying the mask and the displayed image; in this case, the mask was shifted to the right by three subpixels, that is, one pixel of the secret image. The unsuccessfully decoded image corresponds to the image viewed outside the viewing zone when the decoding mask is placed away from the displayed image. A rightward shift of the decoding mask, for example, corresponds to a leftward shift of the viewing position in the display setup shown in Fig. 2. The decoded result obtained by a shifted mask looks like a random pattern, which shows no information about the secret image. Figure 4 (e) is another displayed image encrypted for the same decoding mask pattern. When the decoding mask in Fig. 4 (a) is overlaid on Fig. 4 (e), the decoded result “CMY”, shown in Fig. 4 (f), is obtained. Thus, different secret images can be decoded with the same decoding mask.

For the purpose of experimentally demonstrating the proposed secure display technique, we have developed a prototype multi-color display system with a decoding mask. The decoding mask was printed by an inkjet printer and was placed in front of an LCD panel. The pitch of the LCD was 0.248 mm. To observe the decoding by optical processing of coded subpixel patterns, relatively large subpixels were used for the experiments: the size of each subpixel of the displayed image was set to 2.48 mm×2.48 mm, i.e. each subpixel was formed of 10×10 dots of the LCD. Each red, green, and blue subpixel was composed of a 6×6 dot light emitting area at the center, surrounded with a 2-dot margin of black dots. The subpixel pitch of the decoding mask was 2.47 mm×2.47 mm. The distance of the decoding mask from the LCD panel was adjusted so that the viewing distance was 1.5 m. The viewed images are shown in Fig. 5. The images viewed through the decoding mask represent the secret images. The displayed images, which are also shown on the LCD of the laptop computer in the same figure, look like random patterns. There is some light leakage observed, in particular when displaying a black image, between the subpixels at the bottom left. The leakage was due to distortion of the decoding mask and will be removed in future by improving the holder of the decoding mask, for example, by sandwiching the mask between transparent flat plates.

Images viewed at different viewing points are shown in Fig. 6. When viewed from relatively close to the ideal viewing position (10 cm to the left of the center at a distance of 1.5 m from the LCD panel), the secret image “RGB” was visible, as shown in Fig. 6(b). When viewed from the left side (20 cm to the left of the center at a distance of 1.5 m) and viewed from the right side (40 cm to the right of the center at a distance of 1.5 m), the secret image “RGB” was not perceived. When viewed at a close range (5 cm to the right at 40 cm from the LCD) and at a long range (5 cm to the left at 2 m from the LCD), the secret image was not perceived.

The viewing zone is limited according to crosstalk and disappearance of subpixels of the displayed image that are viewed through the corresponding aperture subpixels of the decoding mask. Careful observation of Fig. 6(a) shows that there are red, green, and blue regions that result from the disappearance of the subpixels of the secret “RGB”, and white regions that result from crosstalk of the subpixels in the originally black regions. Since the subpixel patterns of the decoding mask were selected at random, such crosstalk and disappearance occur stochastically at every pixel when viewed off the center. Then, the viewed images Figs. 6(c), (d), and (e) appear as gray images that are darker than the white image shown in Fig. 5(d). Thus, the prototype display showed that there is a limited viewing zone of the decoded image.

## 5. Conclusions

We have proposed a secure information display technique by using multi-color visual cryptography. We have constructed a multi-color visual cryptography code set and developed a prototype of the proposed secure display. Secret images were perceived only when viewed through the decoding mask at the viewing position. The experimental results showed that there is a limitation on the viewing zone and the limitation results from crosstalk and disappearance of subpixels. The size and design of the viewing zone is an interesting issue relating to visual perception of partially decoded images. The decoding process is a kind of optical processing based on spatial encoding and it needs no computer calculation, which means that no decrypted data exists in a computer system. Thus, the proposed display technique is secure against theft of the decrypted data and eavesdropping of the display signals, and provides a limited viewing zone.

## Acknowledgments

This research was partially supported by the Ministry of Education, Culture, Sports, Science, and Technology, Grant-in-Aid for Encouragement of Young Scientists, 14750031 and a research grant from The Mazda Foundation.

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