Abstract

Motivated by the successful printing of a computer-generated hologram using the computer-to-film (CtF) graphic process, we present a further refined technique with increased resolution, applicable in security. The CtF process offers low cost and fast production while persevering high resolution, and it can make every hologram unique. In this paper, we present the improvement of the printing method, with several software modifications and the implementation of security features at different levels of production.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

When printing security features, holograms belong to a group of products known as diffractive optically variable image devices (DOVIDs). They are vital elements in protection against counterfeiting. DOVIDs are manufactured by using expensive and specialized equipment. The production of holograms using unique materials or coatings, which change the polarization of light or selectively filter/scatter the wavelengths of the reference light, can complicate the process of optical copying. However, in addition to the implementation of physical levels of protection, the application of computer-generated holograms (CGHs) in security printing allows the addition of new digital levels of protection that further complicate the process of making counterfeits [1]. When designing security features, it is necessary to ensure that counterfeiting is too expensive in terms of costs, resources, and time [2,3].

In this paper, we will focus on CGH calculation based on a point model cloud [46] for security printing. CGH computation is a well-researched problem [79]. In this paper, we focus on the implementation of security features at different levels in creating and producing a security element. To broaden the printed CGH applications, we present a new way of calculating CGH by first dividing the hologram plane into smaller segments or holo-blocks and calculating each holo-block with a different perspective [10,11] or with a different model altogether.

Since each segment is calculated individually, we control model/image selection, precise object rotation, position, and reconstruction. To achieve the most from a point cloud model, we also present several methods for creating a point model from an input image or an input 3D model.

Alongside the proposed new robust computational method, we applied and tested ordinary computer-to-film (CtF) technology to create an affordable and high-resolution print of CGH, avoiding specialized [12,13] and expensive techniques [14]. Motivated by the successful production of a single perspective (far-field) hologram and its reconstruction using CtF, we extended the reach of our research [15] to involve fine calibration on multiple CtF machines using higher resolution (increased to 3600 dpi) and further refined software with new features.

CGHs manufactured using the CtF process differ from a classic hologram in such a way that the transmittance is binary [16]. Yet, we can retrieve reconstruction with no significant loss in quality. However, when calculating a binary hologram from a gray-scale hologram, we can dynamically change the threshold to achieve a rasterization-type effect and embed an image or a pattern on the surface [17] of the printed hologram. Considering that the CtF process cost is the same when manufacturing different holograms or manufacturing the same one, new applications in security arise. Generating holograms for mass production and making every hologram unique allows this process to be used in security applications [e.g., saving the identification (ID) number in the hologram while the person’s image is embedded with rasterization]. It is possible to combine the holo-blocks and save multiple pieces of information, since every holo-block contains unique data. Limiting a holo-block size to the width of the laser allows for the reconstruction of a single perspective hologram without overlap.

2. METHOD

CGH calculation is a well-known and researched problem. However, using the CtF process for their manufacture enables every produced hologram to be different in size, shape, rasterization, and information that it holds. This opens new methods of information embedding and ways to refine the process further. This section will explain every step in the hologram calculation based on point model calculation—from the input file to the final manufactured hologram, with a further explanation of each step of the process and its possibilities.

A. Model Position and Rotation

The proposed method for the calculation of the CGH uses 3D point cloud models [15]. Point cloud models were used in the previous research but only the 2D projection of a 3D model was saved in the hologram. In this paper, we propose a new way of generating a 3D point model, using an image as the input (A.1), while implementing the existing input of the 3D model. In both cases, the resulting point model has ${N_{{d^\prime}}}$ points with every point defined by radius vector ${\vec r_{pm,i}} = ({x_i},{y_i},{z_i})$ for $i = 1,2,\ldots,{N_{{d^\prime}}}$. For a more explicit representation, we can arrange all of the points from the point cloud model into a matrix $R \in {{\mathbb R}^{3 \times {N_{{d^\prime}}}}}$, containing each point’s coordinates, as

$$R = \left[{\begin{array}{*{20}{c}}{{x_1}}&\;\;\;{{x_2}}&\;\;\;{\,}&\;\;\;{{x_{{N_{{d^\prime}}}}}}\\{{y_1}}&\;\;\;{{y_2}}&\;\;\; \ddots &\;\;\;{{y_{{N_{{d^\prime}}}}}}\\{{z_1}}&\;\;\;{{z_2}}&\;\;\;{\,}&\;\;\;{{z_{{N_{{d^\prime}}}}}}\end{array}} \right].$$

1. Point Model of an Input Image

Let ${I^{\rm{in}}} \in {{\mathbb R}^{n \times m}}$ be the input gray-scale image and let the ${N_d}$ be a desired number of points in the final 3D point model. In this model, the pixel probability $p(i,j)$ that a point $(i,j)$ will be placed on a pixel depends on the pixel intensity as

$$p(i,j) = 1 - {I^{\rm{in}}}(i,j),$$
where ${I^{\rm{in}}}(i,j)$ represents $(i,j)$ pixel of the input image. In this model, a dark pixel (with a value 0) represents an object in the image, and a white pixel (with a value 1) represents the background. Therefore, the sum of all pixel probabilities should equal ${N_d}$. To achieve this, we re-scale the input image ${I_{\rm{in}}}$ using the tricubic interpolation [18] with scaling factor $S$ calculated as
$$S = \frac{{{N_d}}}{{\sum\nolimits_{i = 1,j = 1}^{n,m} {p(i,j)}}}$$
and retrieve new re-scaled image $I_{\rm{in}}^{\rm{rs}} \in {{\mathbb R}^{n^\prime \times m^\prime}}$, where $n^\prime = \lceil n/\sqrt{S}\rceil$ and $m^\prime = \lceil m/\sqrt{S}\rceil$. Let us note that after this step, the final number of points ${N_{{d^\prime}}}$ will differ slightly to the desired number ${N_d}$. To be certain that the equation ${N_{{d^\prime}}} = {N_d}$ holds, we can choose a threshold for the image $I_{\rm{in}}^{\rm{rs}}$ such that the final black-and-white image has exactly ${N_d}$ pixels with value zero. While the semi-stochastic method of point model calculation will yield more natural-looking gray-scaled images, both ways (semi-stochastic and the fixed threshold) will perform about the same when presented with black-and-white images. Note that the fixed threshold value will have a shorter computational time, which can be necessary if the point model differs for each printed hologram. An example of a point model can be seen in Fig. 1.
 figure: Fig. 1.

Fig. 1. Example of point cloud model creation from an input image with (a) an input gray-scale image, (b) semi-stochastic point model calculation visualization, and (c) the fixed threshold calculation visualization. For both examples, we used ${N_d} = 2000$.

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 figure: Fig. 2.

Fig. 2. Fast Fourier transform (FFT) reconstruction of holograms (a) without the noise (whole domain), (b) cropped reconstruction without the noise, and (c) reconstruction with the implemented noise. Note that the noise added to the cloud point model is exaggerated for better representation. The repeating line artifact is drastically lowered compared to the FFT reconstruction without the noise.

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If there is no restriction where ${N_{{d^\prime}}} = {N_d}$, but rather ${N_{{d^\prime}}} \approx {N_d}$, a semi-stochastic method can be used, and a point will be placed on a pixel depending on its pixel probability (a pixel with an intensity equal to 0.32 has a pixel probability of 0.68, which means there is a 68% chance that a point will be placed in that position). Using a fixed seed method makes the solution deterministic, repeatable, and adds another layer of security to the final product. Note that if the CGH is used on an ID card, the seed for the semi-stochastic calculation can be linked to the ID number, date of issue, or any other connected value.

This method will yield a reconstruction with fewer points on the location of the gray parts on the input image (Fig. 1), and the reconstruction will appear more natural than a black-and-white image reconstruction. We can recalculate the position of the points until the final number of points ${N_{{d^\prime}}}$ is close enough to the desired number ${N_d}$. The reference point is placed in the center of the image to appoint a radius vector to each pixel and save values into a matrix ${R_{{im}}}$. Let us note that the final row of the matrix ${R_{{im}}}$ contains only zeros since the image has only 2D values.

B. Noise Introduction and Point Model Size

Reconstruction of a point model that has points aligned on a grid creates artifacts on the reconstructed image (Fig. 2). This is emphasized when creating point models from images (since the pixels are formed in a grid of rows and columns) and from the 3D model since the points are placed on the edges.

To compensate for this problem, we introduce semi-stochastic noise to the matrices ${R_{\rm{im}}}$.

Let us define matrix $U \in {{\rm R}^{3 \times {N_{{d^\prime}}}}}$ as a uniform random matrix, with values in the range $(- 1,1)$, and the maximum physical size of the final model as ${D_{\rm{im}}}$. As we want a point model created from images, we can define a half-width of a pixel ${w_p}$ as

$${w_p} = \frac{{{D_{\rm{im}}}}}{{2{\rm max}(n,m)}}$$
and ${Z_d}$ as the wanted depth of the input image. Note that ${w_p}$ and ${Z_d}$ satisfy the inequalities ${w_p} \ll {D_{\rm{im}}}$ and ${Z_d} \ll {D_{\rm{im}}}$. This will reduce the noise while preserving the correct perspective when applying model rotation. Let us recall that the $n,m$ are dimensions of the input image ${I_{\rm{in}}}$. Matrix with the introduced noise $R_{\rm{im}}^1$ is determined as
$$R_{\rm{im}}^1 = \left[{\begin{array}{*{20}{c}}{{w_p}}&\;\;0&\;\;0\\0&\;\;{{w_p}}&\;\;0\\0&\;\;0&\;\;{{Z_d}}\end{array}} \right] \times U + \frac{{{D_{\rm{im}}}}}{{2||{\rm max}({R_{\rm{im}}}{{)||}_F}}}{R_{\rm{im}}},$$
where $|| \cdot {||_F}$ stands for Frobenius norm. Note that the matrix $R_{\rm{im}}^1$ contains the final information and can be used to calculate single perspective holograms.

C. Point Model Rotation and Perspective

Rotation grants further control of the model, allowing for more straightforward positioning and saving more information, increasing the hologram security capabilities. While model rotation is more useful when dealing with a 3D object because it allows us to see the object from any position, the rotation of the point cloud model gathered from images can also be helpful, as we place it perpendicularly to the point source and rotate it slightly. It is an additional way of increasing security or removing discrete transitions between multiple images.

Let us define three rotation matrices $O_1(\alpha),O_2(\beta),O_3(\gamma)\in\mathbb{R}^{3\times 3}$ given with

$${O_1}(\alpha) = \left[{\begin{array}{*{20}{c}}{\cos \alpha}&\;\;{{-} {\sin} \alpha}&\;\;0\\{\sin \alpha}&\;\;{\cos \alpha}&\;\;0\\0&\;\;0&\;\;1\end{array}} \right],$$
$${O_2}(\beta) = \left[{\begin{array}{*{20}{c}}{\cos \beta}&\;\;0&\;\;{\sin \beta}\\0&\;\;1&\;\;0\\{{-} {\sin} \beta}&\;\;0&\;\;{\cos \beta}\end{array}} \right],$$
$${O_3}(\gamma) = \left[{\begin{array}{*{20}{c}}1&\;\;0&\;\;0\\0&\;\;{\cos \gamma}&\;\;{{-} {\sin} \gamma}\\0&\;\;{\sin \gamma}&\;\;{\cos \gamma}\end{array}} \right],$$
where $\alpha$ controls the yaw of the model, $\beta$ controls the pitch, and $\gamma$ controls the roll. The final 3D point model ${R_{\rm{im}}}(\alpha ,\beta ,\gamma)$ is obtained as
$${R_{\rm{im}}}(\alpha ,\beta ,\gamma) = {O_1}(\alpha) \times {O_2}(\beta) \times {O_3}(\gamma) \times R_{\rm{im}}^1.$$
Changing each yaw, pitch, and roll parameter will produce an object with a different perspective (Fig. 3).
 figure: Fig. 3.

Fig. 3. Examples of perspectives for the box point cloud model with different values for yaw, pitch, and roll.

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D. Holo-Block Calculation and Configuration

Simple reconstruction of a projection or an image is easy to achieve. However, it lacks features compared to saving a 3D model because the different parts of the hologram gain a different perspective. This can be achieved in classic holograms by placing the model close to the hologram plane. In general, the closer the model is to the plane, the greater the change in perspective. However, this approach still has some limitations, such as the maximum degree of rotation, narrow applicability (only on 3D models), and problems in laser light reconstruction. To utilize the hologram to the full extent, we propose a method that divides the whole hologram into holo-blocks, such that every block or segment is calculated with different model rotations or with a different point cloud model altogether (Fig. 4).

 figure: Fig. 4.

Fig. 4. Single perspective and holo-block method examples with (a1), (b1) the whole hologram, (a2), (b2) hologram divided into holo-blocks, and (a3), (b3) FFT of each holo-block. Images labeled with (a) represent a single perspective hologram, while images labeled with (b) represent our method of division and rotation.

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 figure: Fig. 5.

Fig. 5. Example of image imprinting process using the (a) gray-scale hologram with $1000 \times 1000$ resolution, (b) input image with the resolution of $2 \times 2$, and (c) generating the final binary hologram. The low resolution of the input images is used to emphasize the effect.

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Let us define $H(R(\alpha ,\beta ,\gamma),{\vec p_s},{\lambda _s}) \in {{\mathbb R}^{{h_1} \times {h_2}}}$ as the final single perspective gray-scale hologram matrix, calculated for model ${R_{\rm{im}}}(\alpha ,\beta ,\gamma)$ at point source located at ${\vec p_s}$ with light wavelength ${\lambda _s}$. Since the wavelength and the point source position is fixed, we will denote the hologram as $H(R(\alpha ,\beta ,\gamma))$. Vertical and horizontal resolutions are given with ${h_1}$ and ${h_2}$, respectively. Further detail, explanation, and examples for the hologram calculation algorithm can be found in our previous research [15]. We can divide the static hologram matrix into multiple holo-blocks $H_b^{\textit{ij}} \in {{\mathbb R}^{({h_1}/f) \times ({h_2}/k)}}$ for $i = 1, \ldots ,f$ and $k = 1, \ldots ,k$, where $f$ and $k$ are the numbers of vertical and horizontal separate holo-blocks, respectively. Since each holo-block is only a fraction of the entire hologram, the resolution of the final hologram stays the same, and the processing time is affected only by model manipulation. Note, however, that the model rotation has an insignificant impact on the calculation time.

 figure: Fig. 6.

Fig. 6. Single perspective variable-transmittance hologram (VTH) used for calibration. Holograms in the first row are calculated from the identical gray-scale hologram, differing only in the threshold used to calculate the binary hologram. The second row shows the FFT of the matching hologram, while the last row contains the pictures of the optical reconstruction.

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An example of this approach is a simple 3D object, such as a cube. We can manipulate the cube between the holo-blocks, so the total yaw and pitch change is $2\pi$. Using this approach, we can view the object, in this case, a cube, from any direction. Note that $2\pi$ is used to emphasize the advantages of this approach, but the total degree of rotation can be both larger and smaller than $2\pi$. However, each holo-block can also contain a different model. For example, we can separate the hologram into $z$ vertical holo-blocks and calculate that each includes a single number of an ID number with length $z$. Furthermore, each of that holo-blocks can be additionally split into multiple holo-blocks, such that every number has a jaw of a few degrees.

E. Binarization and Image Imprinting

Let us recall that the calculated hologram $H$ is an $h1 \times h2$ gray-scale matrix and that the CtF process can only produce binary holograms. As a final step in the calculation, we need to calculate binary hologram ${H_b}$ from the hologram $H$. While a global constant can be used to achieve this, we can dynamically change the threshold value based on the desired image and imprint another image on the surface of the hologram (Fig. 5).

Note that once the minimum and maximum values of the threshold are empirically determined, this will not compromise the reconstruction quality. A printed variable-transmittance hologram (VTH) is used to determine the minimum and maximum values. Each holo-block in VTH differs in how much light passes through a holo-block, ranging from 95% to 5% with 5% decrements (Fig. 6). Printing the VTH allows us to determine the minimum and maximum threshold value by analyzing the optical reconstruction for each level of transmittance. To have a conclusive and repeatable procedure, we used a gradient-based sharpness function [19], given with

$$F({T_p}) = |\nabla O({T_p}{)|_2},$$
where ${T_p}$ stands for level of transmittance with values $p = 5,10, \ldots ,95\%$, $F({T_p})$ stands for sharpness function, and $O({T_p})$ stands for the image of optical reconstruction with a ${T_p}$ level of transmittance. Before calculating the sharpness of each image, we normalized each input image. Images of optical reconstruction with 80% or higher level of sharpness compared to the sharped image represent usable levels of transmittance (Fig. 7). Note that the value of 80% is an arbitrary value.
 figure: Fig. 7.

Fig. 7. Sharpness function $F({T_p})$ for different values of transmittance. Since the acceptable values have at least 80% of the sharpness of the sharpest image, in this example, holograms with transmittance between 30% and 75% are acceptable.

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If we take a new input gray-scale image $I_{\rm{im}}^s \in {{\mathbb R}^{{c_1} \times {c_2}}}$ and the hologram $H \in {{\mathbb R}^{{h_1} \times {h_2}}}$, then we can define the dimension ratio ${d_r}$ as

$${d_r} = \frac{{{h_1}}}{{{c_1}}} = \frac{{{h_2}}}{{{c_2}}}.$$
Note that the upper equation holds only if the aspect ratio of the input image $I_{\rm{im}}^s$ matches the aspect ratio of the hologram $H$. Otherwise, the input image or the hologram can be cropped so that the proportions match.

Tricubic interpolation can again be used to scale the image by $0.1\,dr$. Note that the tricubic interpolation is used to fit any input image, which would not be necessary for professional applications. This is done so that every pixel of the input image $I_{\rm{im}}^s$ is used to determine the threshold of a $10 \times 10$ grid of pixels on the hologram. With this, we can achieve 100 shades of gray. Since the hologram resolution ranges from 2400 dpi to 3600 dpi, the image $I_{\rm{im}}^s$ resolution ranges from 240 dpi to 360 dpi, which is the industry standard for printed images.

3. LASER SENSITIVITY CALIBRATION DURING PRODUCTION

Since the CtF process is a new technique for high-resolution CGH printing, to confirm its repeatability and availability, we utilized and tested four different CtF machines working at different resolutions (Table 1). We used two types of films (AGFA Alliance HNm 600BD and AGFA Alliance HND 600BD). Two materials do not differ from each other in any significant way and share the same film sensitivity and minimal element size. Note that every machine and film used in this research can create a high-quality hologram. This means of production is readily available, cheap, and can produce tens or hundreds of unique holograms per minute.

Tables Icon

Table 1. List of Used Printing Machines and Matching Resolutionsa

To evaluate each of the CtF machines, we printed a VTH set (Fig. 6) of binary single perspective holograms calculated for each CtF machine separately. For each machine, multiple laser intensities are tested and measured under an optical microscope to find what intensity best recreates the CGH (Fig. 8). After determining the laser intensity, we used the test set to determine the lowest and highest value of fill percentage that still gives a clear reconstruction. The lowest value is set as the lowest intensity of the image in the image imprinting step, while the highest value corresponds to the highest value in the image (Fig. 6).

 figure: Fig. 8.

Fig. 8. Images of a printed hologram taken with a microscope. Labels of the images correspond to the printing machine No., while letters represent different laser power: (a) min, (b) mid, and (c) max. The left part of the image has no preprocessing, while the right part was denoised, and the threshold was set to have binary output.

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Change in the exposure influences the size of the pixel, meaning that the higher settings will produce a darker hologram with lower transmittance (Fig. 8). Once the noise was removed from images taken by a microscope, we can determine the actual transmittance ${T_m}$ for each laser setting as the ratio of non-zero value pixels ${p_m}$ in the image and the total number of pixels ${p_t}$,

$${T_m} = {p_m}/{p_t}.$$
The average for every exposure setting is calculated for five different positions on the hologram and three different input transmittance levels ${T_t}$, consisting of a total of fifteen measurements. To evaluate each exposure, we calculate the match factor $F$ as
$$F = \frac{{||{T_t} - {T_m}{{||}_2}}}{{||{T_t}{{||}_2}}} \times 100\% .$$
The final results are given in Table 1. Results show that there is no unique solution. Rather, each machine and accompanying raster image processor (RIP) require calibration for best results. Note that this enables us to include the type of mechanical signature of the machine to the final product, discouraging and complicating any forgery attempt.

A. Prepress Differentiation

Utilizing the CGH as a security element is enabled because many parameters change their behavior as a security feature. One crucial step in CGH production is the use of the correct parameters for printing prepress. Using inappropriate prepress will result in a poor-quality hologram, since each machine’s RIP will change the diffraction grid of the hologram. The complete prepress step is elaborated in our previous work [15]. This was important to achieve a repeatable and high-quality manufacturing process. The 3600 dpi is currently the highest printed resolution used for printing holograms on the CtF process.

It is important to note that the identical hologram can be printed on different resolutions. However, for the best results, it is recommended not to utilize interpolation but rather change the physical hologram size to preserve every point. Changing the hologram resolution, and with that its size, will change the size of the reconstructed image as the maximum size ${Y_{{\rm max}}}$ is determined as

$${Y_{{\rm max}}}({H_{{\rm res}}};{\lambda _{{\rm in}}},L) = \frac{{{\lambda _{{\rm in}}}L}}{{\sqrt {0.0508/{H_{{\rm res}}} - {\lambda _{{\rm in}}}}}},$$
where ${H_{{\rm res}}}$ is the resolution in dpi, ${\lambda _{{\rm in}}}$ is the lights wavelength, $L$ is the distance from the hologram to the screen, and the number 0.0508 is the arbitrary constant to convert dpi to dpm. This formula is derived from maxima distance in diffraction grating. Note that approximated change in reconstructed image size is given with
$${Y_{{\rm max}}}(H_{{\rm res}}^\prime ;{\lambda _{{\rm in}}},L) \approx \sqrt {\frac{{H_{{\rm res}}^\prime}}{{{H_{{\rm res}}}}}} {Y_{{\rm max}}}({H_{{\rm res}}};{\lambda _{{\rm in}}},L).$$

4. RESULTS

Progressive binarization allows us to imprint an image on the surface of the hologram. However, increasing the hologram reconstruction quality was the first step in making the CtF printed hologram commercially valuable (Fig. 9).

 figure: Fig. 9.

Fig. 9. Reconstruction with different levels of computational method and prepress refinement: (a) CGH calculated at a resolution of 2540 dpi with no machine parameters compensation, prepress, or noise introduction, (b) CGH calculated at the resolution of 2540 dpi with machine compensation with no progressive binarization, and (c) CGH calculated at the resolution of 3600 dpi with machine compensation, noise introduction, and progressive binarization. Note that the difference in the reconstruction size (c) and (b) comes from increasing the resolution.

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The final product, the printed security element, is combining three levels of security:

  • 1. Computer object manipulation and holo-block composition in order to achieve a large number of combinations,
  • 2. Progressive binarization, used to imprint an image onto the hologram surface, and
  • 3. Machine signature, as described in the previous section.

Combining the three levels starts with the calculation of the CGH composed of $N$ holo-blocks. Every holo-block contains information for a different object perspective or a different object altogether. Holo-block dimensions of $1\,\,{\rm mm}\, \times \,1\,\,{\rm mm}$ have proven to be enough to create a seamless illusion when looking through the hologram and are large enough to reconstruct only one holo-block using a laser. The proposed CGH contains $N = 1369$ holograms and has the final size of $3.7\,\,{\rm cm}\, \times \,3.7\,\,{\rm cm}$. Since the CGH has gray-scale values after calculation, we use the progressive binarization and imprint an image onto the surface of the CGH. If the correct values for the lowest and the highest transmittance (Fig. 6) are used, reconstruction quality will not degrade when compared to other means of binarization (Fig. 10).

 figure: Fig. 10.

Fig. 10. Example of progressive binarization used for image imprinting. (a1), (b1) Input CGH composed of holo-blocks, (a2), (b2) input image, and (a3), (b3) final security element. Two-positions on the (a3) and (b3) are enlarged for better visualization of the rasterization effect.

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An image imprinted on the CGH by the means of the progressive binarization creates a clear image that is not connected to the point model saved in the CGH (Fig. 11).

 figure: Fig. 11.

Fig. 11. Examples for two different images and possible reconstruction based on what part of the hologram is reconstructed. Note that in this example a low number of holo-blocks is used (25 in total per hologram) for clearer visualization. In practice, the number of holo-blocks is a couple of orders of magnitude greater.

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Expanding the point cloud model calculations to be able to use 2D images and advancing from single perspective hologram calculations opened a new area of security element printing while preserving the low cost and high throughput. Note that once calibrated, CtF machines will produce the same quality prints and will not require frequent maintenance. Final printed and reconstructed results can be seen in Figs. 1012 and in video format (Visualization 1, [20]). Reconstruction of the model saved in the CGH can be obtained by using a coherent light laser [Figs. 12(a1) and 12(b1)—12(b3)], which enables precise measurements using an optical setup. However, it can also be obtained with a point white light source, enabling a fast security check by using a simple light source such as a smart-phone flashlight [Figs. 12(c1)–12(c3)].

 figure: Fig. 12.

Fig. 12. Example of the final printed and reconstructed hologram. Figure (a1) shows an example of a finished ID containing a total of four holograms. Figures (a2), (a3), (b1), (b2), and (b3) show a monochromatic reconstruction of a single holo-block or a single perspective hologram. Figures (c1), (c2), and (c3) show reconstructions of three separate holo-blocks using a white point light source. Images (b1), (b2), and (b3) contain point cloud objects obtained from images. Note that figure (b2) contains an 11 digit number that can be used to save each person’s own ID number, emphasizing the capability of this method to make every hologram unique.

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5. CONCLUSION

When combined, the advantages of the proposed computer method and the acquired information of the CtF machine yield a fast production method for unique CGH that have broad applications in security. The final product’s security elements are not just based on a hologram diffraction grating. Instead, the synergy of the three proposed security elements strengthens the security while preserving the ability to be printed for a low price on standard commercial machines.

While the manufacturing process is based on a well-known CtF technique, the security basis of this approach lies in the latest advantages of computer science.

In future research, we will expand the existing list of machines, increasing the maximum resolution to 6400 dpi or higher, measure the film sensitivity to UV exposure, and experiment with different methods for lamination and their effect on the optical reconstruction. Expanding upon the successfully implemented CGH production on the CtF, we plan to test other production methods, such as the flexo and computer-to-plate process, using parameters and procedures presented in this paper and the previous research.

Funding

AKD d.o.o. (Ministry of the Interior of the Republic of Croatia); Hrvatska Zaklada za Znanost (HrZZ IP-2019-04-6418).

Acknowledgment

Material and equipment support came from the AKD d.o.o. (Ministry of the Interior of the Republic of Croatia) project “Analysis of the possible application of printed computer-generated holograms as security elements on identification documents.” The equipment was supplied by the Croatian Science Foundation project “Laser synthesis of nanoparticles and applications” (HrZZ IP-2019-04-6418), PI: dr. sc. N. Krstulović.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are available in Visualization 1, Ref. [20].

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7. T. Zhao, J. Liu, Q. Gao, P. He, Y. Han, and Y. Wang, “Accelerating computation of CGH using symmetric compressed look-up-table in color holographic display,” Opt. Express 26, 16063–16073 (2018). [CrossRef]  

8. Y. Pan, X. Xu, S. Solanki, X. Liang, R. B. A. Tanjung, C. Tan, and T.-C. Chong, “Fast CGH computation using S-LUT on GPU,” Opt. Express 17, 18543–18555 (2009). [CrossRef]  

9. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18, 19504–19509 (2010). [CrossRef]  

10. F. Depasse, M. A. Paesler, D. Courjon, and J. M. Vigoureux, “Huygens–Fresnel principle in the near field,” Opt. Lett. 20, 234–236 (1995). [CrossRef]  

11. X. Zhang, X. Liu, and X. Chen, “Computer-generated holograms for 3D objects using the Fresnel zone plate,” Proc. SPIE 5636, 109–115 (2005). [CrossRef]  

12. A. J. Lee and D. P. Casasent, “Computer generated hologram recording using a laser printer,” Appl. Opt. 26, 136–138 (1987). [CrossRef]  

13. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969). [CrossRef]  

14. J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018). [CrossRef]  

15. V. Cviljušac, A. L. Brkić, A. Divjak, and D. Modrić, “Utilizing standard high-resolution graphic computer-to-film process for computer-generated hologram printing,” Appl. Opt. 58, G143–G148 (2019). [CrossRef]  

16. P. Tsang, T.-C. Poon, W.-K. Cheung, and J.-P. Liu, “Computer generation of binary Fresnel holography,” Appl. Opt. 50, B88–B95 (2011). [CrossRef]  

17. C. Martinez, F. Laulagnet, and O. Lemonnier, “Gray tone image watermarking with complementary computer generated holography,” Opt. Express 21, 15438–15451 (2013). [CrossRef]  

18. F. Lekien and J. Marsden, “Tricubic interpolation in three dimensions,” Int. J. Numer. Methods Eng. 63, 455–471 (2005). [CrossRef]  

19. M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).

20. “Visualization of the results (CGH in security printing),” figshare (2021) [CrossRef]  .

References

  • View by:

  1. J. Vacca, Holograms & Holography: Design, Techniques, & Commercial Applications (Charles River Media, 2001).
  2. M. Sutkowski, M. Kujawinska, and M. T. Stadnik, “Holovideo based on digitally stored holograms,” Proc. SPIE 4659, 309–318 (2002).
    [Crossref]
  3. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39, 6595–6601 (2000).
    [Crossref]
  4. A. Symeonidou, D. Blinder, and P. Schelkens, “Colour computer-generated holography for point clouds utilizing the phong illumination model,” Opt. Express 26, 10282–10298 (2018).
    [Crossref]
  5. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005), pp. 1–164.
  6. V. Cviljušac, A. Divjak, and D. Modrić, “Computer generated holograms of 3D Points cloud,” Teh. Vjesnik - Tech. Gaz. 25, 1020–1027 (2018).
    [Crossref]
  7. T. Zhao, J. Liu, Q. Gao, P. He, Y. Han, and Y. Wang, “Accelerating computation of CGH using symmetric compressed look-up-table in color holographic display,” Opt. Express 26, 16063–16073 (2018).
    [Crossref]
  8. Y. Pan, X. Xu, S. Solanki, X. Liang, R. B. A. Tanjung, C. Tan, and T.-C. Chong, “Fast CGH computation using S-LUT on GPU,” Opt. Express 17, 18543–18555 (2009).
    [Crossref]
  9. T. Shimobaba, H. Nakayama, N. Masuda, and T. Ito, “Rapid calculation algorithm of Fresnel computer-generated-hologram using look-up table and wavefront-recording plane methods for three-dimensional display,” Opt. Express 18, 19504–19509 (2010).
    [Crossref]
  10. F. Depasse, M. A. Paesler, D. Courjon, and J. M. Vigoureux, “Huygens–Fresnel principle in the near field,” Opt. Lett. 20, 234–236 (1995).
    [Crossref]
  11. X. Zhang, X. Liu, and X. Chen, “Computer-generated holograms for 3D objects using the Fresnel zone plate,” Proc. SPIE 5636, 109–115 (2005).
    [Crossref]
  12. A. J. Lee and D. P. Casasent, “Computer generated hologram recording using a laser printer,” Appl. Opt. 26, 136–138 (1987).
    [Crossref]
  13. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
    [Crossref]
  14. J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
    [Crossref]
  15. V. Cviljušac, A. L. Brkić, A. Divjak, and D. Modrić, “Utilizing standard high-resolution graphic computer-to-film process for computer-generated hologram printing,” Appl. Opt. 58, G143–G148 (2019).
    [Crossref]
  16. P. Tsang, T.-C. Poon, W.-K. Cheung, and J.-P. Liu, “Computer generation of binary Fresnel holography,” Appl. Opt. 50, B88–B95 (2011).
    [Crossref]
  17. C. Martinez, F. Laulagnet, and O. Lemonnier, “Gray tone image watermarking with complementary computer generated holography,” Opt. Express 21, 15438–15451 (2013).
    [Crossref]
  18. F. Lekien and J. Marsden, “Tricubic interpolation in three dimensions,” Int. J. Numer. Methods Eng. 63, 455–471 (2005).
    [Crossref]
  19. M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).
  20. “Visualization of the results (CGH in security printing),” figshare (2021).
    [Crossref]

2019 (1)

2018 (4)

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

A. Symeonidou, D. Blinder, and P. Schelkens, “Colour computer-generated holography for point clouds utilizing the phong illumination model,” Opt. Express 26, 10282–10298 (2018).
[Crossref]

V. Cviljušac, A. Divjak, and D. Modrić, “Computer generated holograms of 3D Points cloud,” Teh. Vjesnik - Tech. Gaz. 25, 1020–1027 (2018).
[Crossref]

T. Zhao, J. Liu, Q. Gao, P. He, Y. Han, and Y. Wang, “Accelerating computation of CGH using symmetric compressed look-up-table in color holographic display,” Opt. Express 26, 16063–16073 (2018).
[Crossref]

2013 (1)

2011 (2)

M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).

P. Tsang, T.-C. Poon, W.-K. Cheung, and J.-P. Liu, “Computer generation of binary Fresnel holography,” Appl. Opt. 50, B88–B95 (2011).
[Crossref]

2010 (1)

2009 (1)

2005 (2)

X. Zhang, X. Liu, and X. Chen, “Computer-generated holograms for 3D objects using the Fresnel zone plate,” Proc. SPIE 5636, 109–115 (2005).
[Crossref]

F. Lekien and J. Marsden, “Tricubic interpolation in three dimensions,” Int. J. Numer. Methods Eng. 63, 455–471 (2005).
[Crossref]

2002 (1)

M. Sutkowski, M. Kujawinska, and M. T. Stadnik, “Holovideo based on digitally stored holograms,” Proc. SPIE 4659, 309–318 (2002).
[Crossref]

2000 (1)

1995 (1)

1987 (1)

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Blinder, D.

Brkic, A. L.

Casasent, D. P.

Chen, X.

X. Zhang, X. Liu, and X. Chen, “Computer-generated holograms for 3D objects using the Fresnel zone plate,” Proc. SPIE 5636, 109–115 (2005).
[Crossref]

Chen, Y.

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

Cheung, W.-K.

Chong, T.-C.

Courjon, D.

Cviljušac, V.

Depasse, F.

Divjak, A.

Gao, Q.

Han, Y.

He, P.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Huang, Y.

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

Ito, T.

Javidi, B.

Jiang, X.

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005), pp. 1–164.

Kujawinska, M.

M. Sutkowski, M. Kujawinska, and M. T. Stadnik, “Holovideo based on digitally stored holograms,” Proc. SPIE 4659, 309–318 (2002).
[Crossref]

Laulagnet, F.

Lee, A. J.

Lekien, F.

F. Lekien and J. Marsden, “Tricubic interpolation in three dimensions,” Int. J. Numer. Methods Eng. 63, 455–471 (2005).
[Crossref]

Lemonnier, O.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Liang, X.

Liu, J.

Liu, J.-P.

Liu, X.

X. Zhang, X. Liu, and X. Chen, “Computer-generated holograms for 3D objects using the Fresnel zone plate,” Proc. SPIE 5636, 109–115 (2005).
[Crossref]

Marsden, J.

F. Lekien and J. Marsden, “Tricubic interpolation in three dimensions,” Int. J. Numer. Methods Eng. 63, 455–471 (2005).
[Crossref]

Martinez, C.

Masuda, N.

Mattheij, R. M. M.

M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).

Maubach, J. J.

M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).

Modric, D.

Nakayama, H.

Paesler, M. A.

Pan, Y.

Poon, T.-C.

Rudnaya, M. E.

M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).

Schelkens, P.

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005), pp. 1–164.

Shimobaba, T.

Solanki, S.

Stadnik, M. T.

M. Sutkowski, M. Kujawinska, and M. T. Stadnik, “Holovideo based on digitally stored holograms,” Proc. SPIE 4659, 309–318 (2002).
[Crossref]

Su, J.

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

Sutkowski, M.

M. Sutkowski, M. Kujawinska, and M. T. Stadnik, “Holovideo based on digitally stored holograms,” Proc. SPIE 4659, 309–318 (2002).
[Crossref]

Symeonidou, A.

Tajahuerce, E.

Tan, C.

Tanjung, R. B. A.

ter Hg, M.

M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).

Tsang, P.

Vacca, J.

J. Vacca, Holograms & Holography: Design, Techniques, & Commercial Applications (Charles River Media, 2001).

Vigoureux, J. M.

Wang, Y.

Xu, X.

Yan, X.

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

Zhang, T.

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

Zhang, X.

X. Zhang, X. Liu, and X. Chen, “Computer-generated holograms for 3D objects using the Fresnel zone plate,” Proc. SPIE 5636, 109–115 (2005).
[Crossref]

Zhao, T.

Appl. Opt. (4)

Appl. Sci. (1)

J. Su, X. Yan, Y. Huang, X. Jiang, Y. Chen, and T. Zhang, “Progress in the synthetic holographic stereogram printing technique,” Appl. Sci. 8, 851 (2018).
[Crossref]

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Int. J. Numer. Methods Eng. (1)

F. Lekien and J. Marsden, “Tricubic interpolation in three dimensions,” Int. J. Numer. Methods Eng. 63, 455–471 (2005).
[Crossref]

J. Manuf. Sci. Eng. (1)

M. E. Rudnaya, R. M. M. Mattheij, J. J. Maubach, and M. ter Hg, “Gradient-based sharpness function,” J. Manuf. Sci. Eng. 1126, 301–306 (2011).

Opt. Express (5)

Opt. Lett. (1)

Proc. SPIE (2)

X. Zhang, X. Liu, and X. Chen, “Computer-generated holograms for 3D objects using the Fresnel zone plate,” Proc. SPIE 5636, 109–115 (2005).
[Crossref]

M. Sutkowski, M. Kujawinska, and M. T. Stadnik, “Holovideo based on digitally stored holograms,” Proc. SPIE 4659, 309–318 (2002).
[Crossref]

Teh. Vjesnik - Tech. Gaz. (1)

V. Cviljušac, A. Divjak, and D. Modrić, “Computer generated holograms of 3D Points cloud,” Teh. Vjesnik - Tech. Gaz. 25, 1020–1027 (2018).
[Crossref]

Other (3)

“Visualization of the results (CGH in security printing),” figshare (2021).
[Crossref]

J. Vacca, Holograms & Holography: Design, Techniques, & Commercial Applications (Charles River Media, 2001).

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, 2005), pp. 1–164.

Supplementary Material (1)

NameDescription
Visualization 1       For a clearer understanding, we recorded the results of our scientific work. The video shows the optical reconstruction of printed computer-generated holograms with application in security printing.

Data Availability

Data underlying the results presented in this paper are available in Visualization 1, Ref. [20].

20. “Visualization of the results (CGH in security printing),” figshare (2021) [CrossRef]  .

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Figures (12)

Fig. 1.
Fig. 1. Example of point cloud model creation from an input image with (a) an input gray-scale image, (b) semi-stochastic point model calculation visualization, and (c) the fixed threshold calculation visualization. For both examples, we used ${N_d} = 2000$ .
Fig. 2.
Fig. 2. Fast Fourier transform (FFT) reconstruction of holograms (a) without the noise (whole domain), (b) cropped reconstruction without the noise, and (c) reconstruction with the implemented noise. Note that the noise added to the cloud point model is exaggerated for better representation. The repeating line artifact is drastically lowered compared to the FFT reconstruction without the noise.
Fig. 3.
Fig. 3. Examples of perspectives for the box point cloud model with different values for yaw, pitch, and roll.
Fig. 4.
Fig. 4. Single perspective and holo-block method examples with (a1), (b1) the whole hologram, (a2), (b2) hologram divided into holo-blocks, and (a3), (b3) FFT of each holo-block. Images labeled with (a) represent a single perspective hologram, while images labeled with (b) represent our method of division and rotation.
Fig. 5.
Fig. 5. Example of image imprinting process using the (a) gray-scale hologram with $1000 \times 1000$ resolution, (b) input image with the resolution of $2 \times 2$ , and (c) generating the final binary hologram. The low resolution of the input images is used to emphasize the effect.
Fig. 6.
Fig. 6. Single perspective variable-transmittance hologram (VTH) used for calibration. Holograms in the first row are calculated from the identical gray-scale hologram, differing only in the threshold used to calculate the binary hologram. The second row shows the FFT of the matching hologram, while the last row contains the pictures of the optical reconstruction.
Fig. 7.
Fig. 7. Sharpness function $F({T_p})$ for different values of transmittance. Since the acceptable values have at least 80% of the sharpness of the sharpest image, in this example, holograms with transmittance between 30% and 75% are acceptable.
Fig. 8.
Fig. 8. Images of a printed hologram taken with a microscope. Labels of the images correspond to the printing machine No., while letters represent different laser power: (a) min, (b) mid, and (c) max. The left part of the image has no preprocessing, while the right part was denoised, and the threshold was set to have binary output.
Fig. 9.
Fig. 9. Reconstruction with different levels of computational method and prepress refinement: (a) CGH calculated at a resolution of 2540 dpi with no machine parameters compensation, prepress, or noise introduction, (b) CGH calculated at the resolution of 2540 dpi with machine compensation with no progressive binarization, and (c) CGH calculated at the resolution of 3600 dpi with machine compensation, noise introduction, and progressive binarization. Note that the difference in the reconstruction size (c) and (b) comes from increasing the resolution.
Fig. 10.
Fig. 10. Example of progressive binarization used for image imprinting. (a1), (b1) Input CGH composed of holo-blocks, (a2), (b2) input image, and (a3), (b3) final security element. Two-positions on the (a3) and (b3) are enlarged for better visualization of the rasterization effect.
Fig. 11.
Fig. 11. Examples for two different images and possible reconstruction based on what part of the hologram is reconstructed. Note that in this example a low number of holo-blocks is used (25 in total per hologram) for clearer visualization. In practice, the number of holo-blocks is a couple of orders of magnitude greater.
Fig. 12.
Fig. 12. Example of the final printed and reconstructed hologram. Figure (a1) shows an example of a finished ID containing a total of four holograms. Figures (a2), (a3), (b1), (b2), and (b3) show a monochromatic reconstruction of a single holo-block or a single perspective hologram. Figures (c1), (c2), and (c3) show reconstructions of three separate holo-blocks using a white point light source. Images (b1), (b2), and (b3) contain point cloud objects obtained from images. Note that figure (b2) contains an 11 digit number that can be used to save each person’s own ID number, emphasizing the capability of this method to make every hologram unique.

Tables (1)

Tables Icon

Table 1. List of Used Printing Machines and Matching Resolutions a

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

R = [ x 1 x 2 x N d y 1 y 2 y N d z 1 z 2 z N d ] .
p ( i , j ) = 1 I i n ( i , j ) ,
S = N d i = 1 , j = 1 n , m p ( i , j )
w p = D i m 2 m a x ( n , m )
R i m 1 = [ w p 0 0 0 w p 0 0 0 Z d ] × U + D i m 2 | | m a x ( R i m ) | | F R i m ,
O 1 ( α ) = [ cos α sin α 0 sin α cos α 0 0 0 1 ] ,
O 2 ( β ) = [ cos β 0 sin β 0 1 0 sin β 0 cos β ] ,
O 3 ( γ ) = [ 1 0 0 0 cos γ sin γ 0 sin γ cos γ ] ,
R i m ( α , β , γ ) = O 1 ( α ) × O 2 ( β ) × O 3 ( γ ) × R i m 1 .
F ( T p ) = | O ( T p ) | 2 ,
d r = h 1 c 1 = h 2 c 2 .
T m = p m / p t .
F = | | T t T m | | 2 | | T t | | 2 × 100 % .
Y m a x ( H r e s ; λ i n , L ) = λ i n L 0.0508 / H r e s λ i n ,
Y m a x ( H r e s ; λ i n , L ) H r e s H r e s Y m a x ( H r e s ; λ i n , L ) .

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