Abstract
Dynamic visual cryptography scheme based on timeaveraged fringes generated by Ronchitype geometric moiré gratings on finite element grids is proposed in this paper. A single cover image is used to encode the secret image and is formed on the surface of a deformable structure. Timeaveraged moiré fringes leak the secret when the structure is oscillated according to a predefined Eigenshape. The envelope functions determining the motion induced blur of the Ronchitype moiré grating depend on the characteristic features of the motion. And though harmonic oscillations do not result into a completely uniform timeaveraged image of the Ronchimoiré grating, initial phase scrambling and phase normalization algorithms are used to encode the secret in the cover image. Theoretical relationships between the amplitude of the Eigenshape, the order of the not completely developed timeaveraged fringe and the pitch of the deformable onedimensional Ronchitype moiré grating are derived.
1. Introduction
Visual cryptography (VC) is a cryptographic technique which allows visual information (pictures, text, etc.) to be encrypted in such a way that the decryption can be performed by the human visual system, without the aid of computers. VC was pioneered by Naor and Shamir in 1994 [1]. They demonstrated a visual secret sharing scheme, where an image was broken up into a number of shares so that only someone with all shares could decrypt the image. Each share was printed on a separate transparency, and decryption was performed by overlaying the shares. When all shares were overlaid, the original image would appear. Since 1994, many advances in visual cryptography have been made. Visual cryptography for color images has been proposed in [2, 3]. Ideal contrast visual cryptography schemes have been introduced in [4]. A general multisecret visual cryptography scheme is presented in [5]; incrementing visual cryptography is described in [6]. A new cheating prevention visual cryptography scheme is discussed in [7].
An image hiding technique, when the secret image leaks in a form of a timeaveraged moiré fringe in an oscillating nondeformable cover image, was first presented in [8]. A stochastic moiré grating is used to embed the secret into a single cover image, and the secret can be visually decoded by the naked eye only when the amplitude of the harmonic oscillations corresponds to an accurately preselected value. The fact that the naked eye cannot interpret the secret from a static cover image makes this image hiding technique similar to VC. Special computational algorithms are required to encode the image, but the decoding is completely visual. The difference from VC is that only a single cover image is used and that it should be oscillated in order to leak the secret. Some additional security measures are implemented as described in [9], where the secret is leaked only when pattern, containing secret information, is oscillated according to a predefined law of motion.
An alternative image hiding scheme is based on deformable moiré gratings, where secret image is leaked from the cover image not when it is oscillated according to a predetermined law of motion – but when it is deformed along the longitudinal coordinate of the stochastic moiré grating. Such implementation requires a special strategy for the formation of the cover image and opens new possibilities for optical control of vibrating structures. A natural extension of such image hiding technique would be a dynamic visual cryptography (DVC) scheme based on harmonic oscillations of the deformable moiré grating according to a preselected Eigenshape of an elastic structure [10]. However, the formation of a moiré grating with a harmonic variation of grayscale levels on a surface of a deformable structure remains a challenging technological problem, especially if microelectromechanical systems (MEMS) are considered. Thus, the main objective of this manuscript is to construct the algorithms for the formation of cover images based on Ronchitype moiré grating for such demanding applications.
For the timeaveraged fringes to form correctly using Ronchitype moiré grating, triangular waveform oscillation function is required. The triangular waveform vibrations are often found in practical applications, such as investigating multilayered structures or image processing. Zigzag scan hysteresis is found after image compression, which, in case of insufficient number of bits’ available, results in macro blocking. Zigzag scan hysteresis also appears in nanosystems, when using nearfield scanning optical microscopes [11]. A layer wise zigzag model for the vibratory response of softcored nonsymmetrical sandwich beams is proposed in [12]. Zigzag effect is used to perfectly bond together adjacent layers of laminate [13]. Thus, it is natural to adapt zigzag/triangular waveform in DVC.
This paper is organized as follows. Optical background of the problem is discussed in section 2; the construction of a deformable moiré grating is presented in section 3; the DVC scheme based on deformable gratings is illustrated in section 4; concluding remarks are given in section 5.
2. Optical relationships
Let us consider two different onedimensional geometrical moiré gratings – a harmonic moiré grating:
and a Ronchitype moiré grating:
where $x$ is the longitudinal coordinate; $\lambda $ is the pitch of the grating; the numerical value 0 corresponds to the black color; 1 – to the white color.
Firstly, let us assume that these gratings are formed on a surface of nondeformable structure which does oscillate around the state of equilibrium according harmonic law of motion:
where $a$ is the amplitude of oscillations, $\omega $ is the angular frequency and ${f}_{i}$ is the phase of harmonic oscillations. Moiré gratings are blurred due to these oscillations; the motion induced blur in the timeaveraged image reads [8]:
where $T$ is the exposure time and ${J}_{0}$ is the zero order Bessel function of the first kind. The decay of the contrast of the timeaveraged harmonic moiré grating at increasing amplitudes of harmonic oscillation is nonmonotonous. Timeaveraged moiré fringes are formed at such amplitudes which correspond to the roots of ${J}_{0}$:
where ${r}_{k}$ is the $k$th root of ${J}_{0}$. However, the relationship (4) does not hold for the Ronchitype moiré grating – timeaveraged fringes do not form at any amplitude of harmonic oscillations [9] (Fig. 1). Ronchitype moiré gratings generate timeaveraged fringes only if the waveform of the oscillation is triangular [9] – this phenomenon could be exploited as an additional factor of encoding security in DVC applications.
Fig. 1Oscillation of the inelastic onedimensional moiré grating (λ=0.03) produces timeaveraged fringes. Time averaged image is shown on the left; the RMSE errors from the equilibrium and graph of sinc or Bessel function – at the right part of the figure. Timeaveraged fringes do form correctly if harmonic cover image is oscillated according harmonic law (a). If image is oscillated according triangular waveform function (b), fringes does also form, but at sinc function’s roots
a)
b)
To prove and adapt this phenomenon on dynamic visual cryptography it could be split into two parts. Firstly, provided idea can be simplified by using harmonic moiré grating instead of Ronchitype moiré grating, oscillated according triangular waveform function. If amplitudes, where timeaveraged moiré fringes forms, could be verified on the simplified version, hypothesis that Ronchitype moiré grating also forms at the same amplitudes could be proved experimentally.
Thus, a cover image formed on a surface of a deformable structure would not oscillate according to the law described in (3). Let’s define triangular waveform function used in oscillations with period $2\pi $ and value range from 1 to 1:
Let the deformation from the state of equilibrium at the point $x$ at time moment $t$ be equal to $u\left(x,t\right)$. Then the explicit deformation of the moiré grating reads:
if only $x$ can be explicitly expressed from the equality:
in the following form:
Let us describe an oscillation around the state of equilibrium with function $u\left(x,t\right)$:
It is the simple form of triangular waveform function, where $a\left(x\right)$ is the Eigenshape of inplane oscillations and time $t\in \left[1,1\right]$. The field of amplitudes $a\left(x\right)$ can be linearized around the point ${x}_{0}$:
where ${a}_{0}={a(x}_{0})$; ${\dot{a}}_{0}={\left.\frac{\mathrm{d}a\left(x\right)}{\mathrm{d}x}\right}_{x={x}_{0}}$. Then equalities 8, 9 and 10 yield:
Finally, the grayscale level of the moiré grating at coordinate $x$ and time moment $t$ reads:
2.1. Nondeformable moiré grating
Let us assume that $a\left(x\right)=A$ ($A$ is a constant) and oscillations are done using triangular waveform function. This means that the deflection, which describes the oscillation of a nondeformable body around the state of equilibrium, equals $u\left(x,t\right)=At$ [8]. Then the grayscale level of the moiré grating at time step $t$ reads:
Timeaveraging techniques can be used to register the image of the grating [8, 14]:
where $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(x\right)=\frac{\mathrm{sin}\left(x\right)}{x}$ is cardinal sine function. Note that the distribution of the grayscale level in the timeaveraged image does not depend on characteristics of triangular waveform function (9) and this is why the simple form could be used in provided calculations.
Gray timeaveraged moiré fringes do form when $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(x\right)=0$ – what happens at amplitudes $\frac{2\pi}{\lambda}{A}_{k}={r}_{k}$ (${r}_{k}=\pi k$ are roots of $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(x\right)$; $k=\mathrm{1,2},\dots $) and are illustrated in Fig. 1(b).
2.2. Deformable moiré grating; linear deformation field
Next assume that $a\left(x\right)=Ax$. The principal difference from nondeformable moiré gratings (described in Section 2.1), is that the moiré grating will deform proportionally to the coordinate $x$, when the body is oscillated in time. However, harmonic moiré grating can still be formed on the surface of the onedimensional body in the state of equilibrium.
Linearization around ${x}_{0}$ yields: $a\left(x\right)=A{x}_{0}+A\left(x{x}_{0}\right)$; ${a}_{0}=A{x}_{0}$; ${\dot{a}}_{0}=A$. Thus, Eq. (12) now reads:
Note that a singularity occurs at $A=1$ in Eq. (15). Thus, it is assumed that $0<A\ll 1$. Finally, the timeaveraged image reads:
Fig. 2Deformable onedimensional moiré grating produces timeaveraged fringes when oscillated according to the triangular waveform. One period (t∈1, 1) of oscillations is shown in the top left image; onedimensional timeaveraged image at A*=0.05 is shown at the bottom on the left; the formation of timeaveraged fringes then different amplitudes is illustrated on the right; A=0.001, 0.1
In this case, timeaveraged moiré fringes form at $\frac{2\pi}{\lambda}Ax={r}_{k}=\pi k$; $k=\mathrm{1,2},\dots $. Figure 2 shows the oscillating moiré grating in the left upper image. The onedimensional moiré grating is motionlessly fixed on the left side and the right side of the grating deforms at an amplitude ${A}^{\mathrm{*}}=0.05$. The vertical dashed line marks the equilibrium state of the onedimensional deformable structure; the pitch of the moiré grating is $\lambda =0.015$. The right part of Fig. 2 shows the timeaveraged images of the onedimensional moiré grating at different amplitudes $A$. The left bottom part of Fig. 2 illustrates timeaveraged moiré fringes at ${A}^{\mathrm{*}}=0.05$.
2.3. Deformable moiré grating; nonlinear deformation field
The main objective of this paper is to develop an image hiding scheme based on deformable moiré gratings on finite element grids under the assumption that the deformation field $a\left(x\right)$ is a nonlinear function. Without losing the generality we set ${x}_{0}=0$. Let us denote $\stackrel{}{a}\left(x\right)={a}_{0}+{\dot{a}}_{0}x$. Now, Eq. (12) reads:
If ${\dot{a}}_{0}\ll 1$ and ${t}^{2}\le 1$ then ${a}_{0}{\dot{a}}_{0}{t}^{2}\ll 1$ and:
Note that ${\int}_{1}^{1}\mathrm{sin}\left(\frac{2\pi}{\lambda}\stackrel{}{a}\left(x\right)t\right)dt=0$ due to the evenness of the sine function. Then, the timeaveraged image reads:
Thus, time averaged moiré fringes form at $\frac{2\pi}{\lambda}\stackrel{}{a}\left(x\right)={r}_{k}=\pi k$; $k=\mathrm{1,2},\dots $, which corresponds well to the results produced in Sections 2.1 and 2.2. The goal is to transform the whole image into a timeaveraged moiré fringe (only by varying the pitch $\lambda \left(x\right))$. Thus, the distribution of the pitch reads:
In previous relationship $\lambda $ still depends on the linearized field of amplitudes $\stackrel{}{a}\left(x\right)$. The conjecture that $\stackrel{}{a}\left(x\right)$ can be replaced with $a\left(x\right)$ will be tested and validated by using computational tools. Let us assume that a onedimensional elastic structure oscillates according to the following law:
The above stated presumption implies that a timeaveraged moiré fringe must form in the whole domain of $x$ when the stationary moiré grating has the variable pitch in respect of $x$:
The parameter $k$ is fixed to 1, because the contrast around the first timeaveraged moiré fringe is the highest (the first root of $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}$${r}_{1}=\pi $). Now, instead of applying the oscillations of the moiré grating according to Eq. (21) we set the oscillation process to:
where the parameter $b$ is varied from 0 to 0.2 (Fig. 3). It can be clearly seen that the timeaveraged moiré fringe does form at $b=0.1$. Thus, the conjecture stating that that the linearized field $\stackrel{}{a}\left(x\right)$ can be replaced by $a\left(x\right)$ in Eq. (20) does hold.
Fig. 3Timeaveraged image of the onedimensional grating (the variation of the pitch is determined according to Eq. (22)); the variation of the amplitude b is determined by Eq. (23)
3. Ronchitype moiré gratings on finite element grids
The main objective of this paper is to develop an image hiding scheme based on deformable Ronchitype moiré gratings on finite element grids. In other words, cover image should only have two colors black or white. In section 2 it was proved, that timeaveraged fringes forms then harmonic moiré grating is oscillated according triangular waveform function. This raises a hypothesis that the same should hold true with Ronchitype moiré gratings Eq. (2).
Fig. 4Timeaveraged fringes do form correctly if Ronchitype moiré grating is oscillated according triangular waveform function. Fringes form, at sinc function’s roots rk=πk
Hypothesis can be confirmed experimentally the same way as with harmonic moiré gratings. From Fig. 4 it can be seen that time averaged fringes do form correctly at ${r}_{k}=\pi k$ if Ronchitype moiré grating is oscillated according triangular waveform function.
4. Dynamic visual cryptography based on deformable moiré gratings on Finite Element grids
The timeaveraged moiré fringes will be formed using previously mentioned nonlinear deformation fields. This process does results only in 1D moiré gratings, thus to extend it to 2D model, the field of deformations should be sliced horizontally (or vertically). The deformation field must be determined by FEM computations. This process is illustrated in Fig. 5.
The 2D deformation field is shown in Fig. 5(a), there the 10th Eigenshape of a plate is selected. White zones stand for no oscillation and dark zones for maximum deformations from the equilibrium. The resolution of Fig. 5(a) is 500×500 pixels, so all further calculations are related to this size. Because, provided model forms only onedimensional timeaveraged moiré, an array of 500 onedimensional moiré gratings must be formed and concatenated vertically. This results in 2D stationary moiré grating shown in Fig. 5(b). Grating pitch is constructed using Eq. (22), there the linearized deformation field $\stackrel{}{a}\left(x\right)$ is replaced by $ka\left(x\right)+b$. This transformation of the numerical values of the Eigenshapes $a\left(x\right)$ is necessary in order to avoid singularities at the points where the amplitudes $a\left(x\right)$ become equal to 0. $k$, $b$ are positive constants greater than 0, and all further computations is set to $k=0.0045$ and $b=0.0055$. Thus, if the initial range of the Eigenmode is $[1,1]$, after this transformation the working range of amplitudes becomes $[0.001,0.01]$.
Now, we must note that initial phase of all 500 horizontal 1D gratings is the same and equals to 0. This results in clean pattern and especially – interpretable Eigenshape function. To avoid this the stochastic initial phase scrambling algorithm [8] should be used to complicate the pattern (Fig. 5(c)) (during this process the pitch in every single onedimensional grating is not altered).
After this preparation, every onedimensional grating is timeaveraged according to the $x$axis, using triangular function. These inplane unidirectional oscillations result in completely gray image (except the left and right boundaries where the image becomes slightly uneven) shown in Fig. 5(d).
Fig. 5Triangular waveform oscillations according to the 10th Eigenmode of a Ronchitype moiré, produce a gray twodimensional image; part (a) shows the Eigenshape; part (b) illustrates the stationary moiré grating (the pitch of the grating varies in the interval λ=0.002, 0.02; λ(x)=2a(x); part (c) shows the cover image produced from the moiré grating; part (d) illustrates the timeaveraged image when the cover image is oscillated according the 10th Eigenmode
a)
b)
c)
d)
We have experimentally proved that Ronchitype moiré, when oscillated according to triangular waveform function, as well works with nonlinear amplitude functions. The last step is to encode secret image into the cover image, which is done by modifying the phase regularization algorithm introduced in [8] (Fig. 6). The variation of the amplitude $a\left(x\right)$ is depicted in Fig. 6(a). The corresponding grayscale level of the onedimensional moiré grating is illustrated in Fig. 6(b). Let us place the block of “secret” information in the middle of the grating, meaning that timeaveraged moiré fringe should form everywhere, except in the middle.
Now, the white zones (the left and rightthird of Fig. 5(b) and the middlethird of Fig. 5(d)) are taken into the composite grating illustrated in Fig. 6(e). Direct copying results into a noncontinuous grating with phase jumps at the joining points, which are eliminated with phaseregulation algorithm (Fig. 6(f)). Note that the variation of the pitch is not altered in this process. Lastly, timeaveraging of Fig. 6(f) results into Fig. 6(g) as it is oscillated by the law defined by Eq. (8) and the field of amplitudes $a\left(x\right)$ is determined by Fig. 6(a). Timeaveraged moiré fringes do form in the middlethird of the timeaveraged image; the leftthird and the rightthird of the image does clearly stand out from the gray background.
Fig. 6A schematic diagram illustrating the encoding of the secret in a onedimensional moiré grating. Part (a) shows the field of amplitudes a(x); part (b) illustrates the corresponding moiré grating. Part (c) shows the field of amplitudes used in the regions occupied by the secret; part (d) illustrates the corresponding moiré grating. The composite moiré grating uses the left and the right thirds from part (b) and the middle third from part (d). All discontinuities in part (e) are eliminated by the phase regularization algorithm (part(f)). The timeaveraged image of (f) is shown in parts (g, h).
Provided image hiding scheme effective embeds dichotomous images into the cover images. To assure the functionality of such image hiding scheme based on dynamic visual cryptography, lets embed secret dichotomous image Fig. 7(a) into the cover image Fig. 7(b). The cover image generated according to the 12th Eigenmode, where initial phase is stochastic in all horizontal onedimensional gratings and phase regularization algorithms are used to hide the secret. The encoding result is uninterpretable to naked eyes.
Moreover, decoding process dependency on the initial Eigenshape of the structure, introduces one more security feature: secret information is inaccessible without using correct Eigenshape to oscillate the cover image. Thus, the Eigenmode itself can be considered as a key for the visual decoding procedure. Such dependence is shown in Fig. 8 where visual decoding is executed using different Eigenmodes. The contrast enhancement procedures [15] are used to highlight moiré fringes in timeaveraged images.
Fig. 7The secret image is shown in part (a); the cover image with the embedded secret according to the 12th Eigenmode is shown in part (b).
a)
b)
Fig. 8The Eigenmode serves as the key for visual decryption of the cover image. The first row shows different Eigenshapes; the row column – timeaveraged images; the third row – contrast enhanced timeaveraged images
5. Conclusions
Dynamic visual cryptography scheme based on Rochi gratings and triangular waveforms is presented in this paper. An image encoding scheme in deformable onedimensional moiré gratings oscillating according to a predefined Eigenmode is developed and implemented for the construction of twodimensional digital dichotomous secret images. The secret is leaked from the cover image when it is oscillated according to a predefined Eigenmode. The efficiency of the proposed scheme is illustrated by computational examples.
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