Dynamic visual cryptography is based on hiding of a dichotomous secret image in the regular moiré grating. One pitch of the moiré grating is used to represent the secret image and a slightly different pitch of another moiré grating is used to form the background. The secret is decoded in the form of a pattern of a time-averaged moiré fringe when the cover image is oscillated according to a predefined law of motion. The security of the encoding and the sharpness of the decoded secret are mostly influenced by the selection of the pitches of moiré grating. This paper proposes scheme for the determination of near-optimal pitches of the moiré grating for image hiding in dynamic visual cryptography

Jurate Ragulskiene

Loreta Saunoriene

Sandra Aleksiene

English

Vibroengineering PROCEDIA

 266 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2017, VOL. 13. ISSN 2345-0533  
Near-optimal pitch of a moiré grating for image hiding 
applications in dynamic visual cryptography 
Loreta Saunoriene1, Sandra Aleksiene2, Jurate Ragulskiene3 
Research Group for Mathematical and Numerical Analysis of Dynamical Systems,  
Kaunas University of Technology, Kaunas, Lithuania 
1Corresponding author 
E-mail: 1loreta.saunoriene@ktu.lt, 2sandra.aleksiene@ktu.lt, 3jurate.ragulskiene@ktu.lt 
Received 30 August 2017; accepted 31 August 2017 
DOI https://doi.org/10.21595/vp.2017.19031 
Abstract. Dynamic visual cryptography is based on hiding of a dichotomous secret image in the 
regular moiré grating. One pitch of the moiré grating is used to represent the secret image and a 
slightly different pitch of another moiré grating is used to form the background. The secret is 
decoded in the form of a pattern of a time-averaged moiré fringe when the cover image is oscillated 
according to a predefined law of motion. The security of the encoding and the sharpness of the 
decoded secret are mostly influenced by the selection of the pitches of moiré grating. This paper 
proposes scheme for the determination of near-optimal pitches of the moiré grating for image 
hiding in dynamic visual cryptography. 
Keywords: dynamic visual cryptography, harmonic oscillation, moiré grating. 
1. Introduction 
Visual cryptography (VC) is a special encryption technique used to hide visual information in 
a way that it can be decrypted purely by the human visual system and no computational techniques 
are required. First introduced by Moni Naor and Adi Shamir [1] in 1994, VC is based on the 
division of the original image into several semi-transparent shares. Each share individually does 
not reveal the secret, but if all these shares are overlaid at the right position, a secret message 
appears [2]. Recently many advances have been done in visual cryptography: visual cryptography 
of gray-level and color images was introduced in [3]; probabilistic visual secret sharing scheme 
for grey-scale and color images was presented in [4]; letter-based VC scheme where pixels are 
replaced by letters in the share images was proposed in [5], sharing of multiple secrets in visual 
cryptography was discussed in [6]; security of visual cryptography schemes was analysed in [7-9].  
Geometric moiré is a classical optical experimental technique based on analysis of visual 
patterns formed as a superposition of two regular gratings that geometrically interfere [10, 11]. 
Geometric moiré techniques can be extended to time-average geometric moiré methods. If moiré 
grating is formed on the surface of oscillating structure and time-averaging techniques are applied, 
pattern of time-averaged fringes emerges [12].  
The concept of dynamic visual cryptography was first introduced in [13]. This method is based 
not on static superposition of shares like in VC, but on the time-averaging techniques applied for 
a single encoded image. The secret image is embedded into the moiré grating in such a way, that 
one pitch of the moiré grating is used to form the background of the secret image and a slightly 
different pitch of another moiré grating is used to represent the secret. Phase scrambling 
completely hides the secret and a naked eye cannot interpret the secret from the stationary cover 
image. The secret is decoded in a form of a pattern of time-averaged moiré fringe when the cover 
image is oscillated according to a predefined law of motion. Furthermore, the amplitude of the 
harmonic oscillations must be set to a preselected value in order to leak the secret. The security of 
the encoding and the sharpness of the decoded secret are mostly influenced by the selection of the 
pitches of moiré grating. Pitches of the grating representing the secret image and the background 
cannot differ very significantly – otherwise, encoded secret could be seen by a naked eye in a 
static share. At the same time, the difference between the pitches should ensure the sufficient 
contrast between secret and background areas in time-averaged image. This paper proposes 
NEAR-OPTIMAL PITCH OF A MOIRÉ GRATING FOR IMAGE HIDING APPLICATIONS IN DYNAMIC VISUAL CRYPTOGRAPHY.  
LORETA SAUNORIENE, SANDRA ALEKSIENE, JURATE RAGULSKIENE 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2017, VOL. 13. ISSN 2345-0533 267 
scheme for the determination of near-optimal pitch of the moiré grating for image hiding in 
dynamic visual cryptography. 
2. Optical background 
One-dimensional harmonic moiré grating reads [10, 11]: 
ܨ(ݔ) = 12 +
1
2 cos ൬
2ߨ
ߣ ݔ൰, (1)
where ݔ is the longitudinal coordinate, ߣ is the pitch of the moiré grating. The numerical value 0 
of function ܨ(ݔ) corresponds to black color, 1 – to white color, all values between 0 and 1 
correspond to the appropriate greyscale levels.  
Let us suppose that the moiré grating (Eq. (1)) is formed on the surface of one-dimensional 
non-deformable body and is harmonically oscillated around the state of equilibrium according to 
harmonic function: ܽ ⋅ sin (߱ݐ + ߮), where ܽ is the amplitude of harmonic oscillations, ߱ is the 
frequency and ߮ is the phase. Then grey-scale level of moiré grating at coordinate ݔ  at time 
moment ݐ can be expressed [12]: 
ܨ(ݔ, ݐ) = 12 +
1
2 cos ൬
2ߨ
ߣ (ݔ − ܽsin (߱ݐ + ߮))൰. (2)
Greyscale level of time-averaged geometric moiré can be obtained averaging Eq. (2) in  
time [12]: 
ܨ௧(ݔ) = lim்→ஶ
1
ܶ න ܨ(ݔ, ݐ)݀ݐ
்
଴
= 12 +
1
2 cos ൬
2ߨ
ߣ ݔ൰ ܬ଴ ൬
2ߨ
ߣ ܽ൰, (3)
where ܶ is the exposure time used for the time-averaging, ܬ଴ is the zero order Bessel function of 
the first kind. Note, that oscillation frequency ߱ and phase ߮ have no influence on the formation 
of time-averaged image [11, 12]. Time-averaged image becomes continuously grey at the roots of 
ܬ଴ ቀଶగఒ ܽቁ: 
2ߨ
ߣ ܽ = ݎ௜,     ݅ = 1,2, …, (4)
where ݎ௜ is the ݅th root of the zero order Bessel function of the first kind. 
 
Fig. 1. Time-averaged moiré at ߣ = 0.75: a) one-dimensional time-averaged moiré in case of harmonic 
oscillations at increasing amplitudes ܽ; b) standard deviation of time-averaged moiré 
NEAR-OPTIMAL PITCH OF A MOIRÉ GRATING FOR IMAGE HIDING APPLICATIONS IN DYNAMIC VISUAL CRYPTOGRAPHY.  
LORETA SAUNORIENE, SANDRA ALEKSIENE, JURATE RAGULSKIENE 
268 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2017, VOL. 13. ISSN 2345-0533  
It can be noted, that standard deviation of time-averaged image, described by Eq. (3), is 
calculated as follows [14]: 
ܵ൫ܨ௧(ݔ)൯ =
ቚܬ଴ ቀ2ߨߣ ܽቁቚ
√8 . 
(5)
Fig. 1(a) provides one-dimensional time-averaged moiré image at increasing amplitudes. It is 
clear that time-averaged fringes form at amplitudes corresponding to the Eq. (4). Fig. 1(b) shows 
the variation of the standard deviation of the time-averaged moiré at increasing amplitudes. Value 
of the standard deviation within the time-averaged image depends on a blur level in time-averaged 
moiré grating: sharp structure of the grating ensures sufficiently high standard deviation. Note that 
standard deviation is equal to zero at the centers of time-averaged fringes. 
3. Estimation of near-optimal pitch of a moiré grating 
The main idea of hiding a secret in the dynamic visual cryptography is to encode this secret 
image into moiré grating: one pitch of the moiré grating is used to form the background of the 
secret image and a slightly different pitch of another moiré grating is used to represent the secret. 
Let the pitch of the grating in the area of secret information be denoted ߣ௦, the pitch of the grating 
in the background – ߣ௕ . Pitches of the grating ߣ௦  and ߣ௕  cannot differ very significantly |ߣ௦ − ߣ௕| ≤ ߝ , otherwise the encoded secret could be seen by a naked eye in a static share. 
Simultaneously, the difference between ߣ௦ and ߣ௕ should ensure the sufficient contrast between 
secret and background areas in time-averaged image. 
The secret information is leaked if only the cover image is oscillated harmonically. Secret 
appears as a uniformly grey area in time-averaged image while background contains slightly 
blurred pattern of the cover image.  
Let the standard deviations of the secret and background areas in time-averaged image be 
denoted as ߪ௦ and ߪ௕ respectively. Values of ߪ௦ and ߪ௕ are calculated as [14]: 
ߪ௦ =
ฬܬ଴ ൬2ߨߣ௦ ܽ൰ฬ
√8   ,    ߪ௕ =
ฬܬ଴ ൬2ߨߣ௕ ܽ൰ฬ
√8 . 
(6)
Sufficient contrast between the leaked secret and the background in the time-averaged cover 
image is obtained if only |ߪ௦ − ߪ௕| ≥ ߜ. The graphical representation of standard deviations ߪ௦ 
and ߪ௕ is proposed in Fig. 2. 
 
Fig. 2. Variation of standard deviations of time-averaged moiré. Thick solid line stands for standard 
deviation of the secret ߪ௦ at ߣ௦ = 0.45; thin solid and thin dashed lines represent  
standard deviations of the background ߪ௕ଵ and ߪ௕ଶ at ߣ௕ଵ = 0.7, ߣ௕ଶ = 0.35 accordingly 
NEAR-OPTIMAL PITCH OF A MOIRÉ GRATING FOR IMAGE HIDING APPLICATIONS IN DYNAMIC VISUAL CRYPTOGRAPHY.  
LORETA SAUNORIENE, SANDRA ALEKSIENE, JURATE RAGULSKIENE 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2017, VOL. 13. ISSN 2345-0533 269 
As it is mentioned in Eq. (4), the area of the secret information in time-averaged image 
becomes uniformly grey if only the amplitude of oscillations ܽ is equal to ܽ = ఒೞଶగ ݎ௜, ݅ = 1,2, … . 
It is clear that standard deviation of uniformly grey area is equal to 0, thus ߪ௦ = 0 if only the 
amplitude of oscillations is equal to ఒೞଶగ ݎଵ . Therefore, the contrast between the secret and the 
background depends on the value of ߪ௕  only (note that distance AC in Fig. 2 is equal to  
|ߪ௕ଵ − ߪ௦| = ߪ௕ଵ and distance AB is equal to |ߪ௕ଶ − ߪ௦| = ߪ௕ଶ at the point where ଶగఒೞ ܽ = ݎଵ). 
Let us assume that the contrast of the leaked secret in time-averaged cover image is sufficient, 
if standard deviation ߪ௕ is equal or exceeds threshold ߜ : 
ฬܬ଴ ൬2ߨߣ௕ ܽ൰ฬ
√8 ≥ ߜ, 
(7)
whereas amplitude of oscillation is preset to ܽ = ఒଶగ ݎଵ, Eq. (7) yields: 
ฬܬ଴ ൬ߣ௦ߣ௕ ݎଵ൰ฬ
√8 ≥ ߜ. 
(8)
The inequality Eq. (8) is visualized in Fig. 3 when ߣ௦ is fixed. Striped intervals on the graph 
show the intervals of ߣ௕ for which the inequality ቚܬ଴ ቀఒೞఒ್ ݎଵቁቚ √8ൗ ≥ ߜ holds true. Zeroes of the 
function ܬ଴ ቀఒೞఒ್ ݎଵቁ are located at ߣ௕ = ߣ௦
௥భ
௥೔
, ݅ = 1,2, ….  
Function ܬ଴ ቀఒೞఒ್ ݎଵቁ can be approximated by its tangent in the root surrounding if the threshold 
ߜ is relatively small. It is known, that ܬ଴ᇱ (ݔ) = −ܬଵ(ݔ), where ܬଵ(ݔ) is first order Bessel function 
of the first kind. Therefore, the slope of tangent line at point ߣ௕ = ߣ௦ is ݇ଵ = ܬ଴′(ݎଵ) = ௥భఒೞ ܬଵ(ݎଵ). 
Analogously the slope at point ߣ௕ = ߣ௦ ௥భ௥మ reads ݇ଶ = ܬ଴′(ݎଶ) =
௥మమ
௥భఒೞ ܬଵ(ݎଶ). Now Δ௜, ݅ = 1,2, … 
(Fig. 3) can be approximated as Δ௜ = ఋ௞೔. 
 
Fig. 3. Visualization of the inequality ቚܬ଴ ቀఒೞఒ್ ݎଵቁቚ √8ൗ ≥ ߜ in the interval (0.2ߣ௦; 1.5ߣ௦). Thick solid line 
represents the variation of ቚܬ଴ ቀఒೞఒ್ ݎଵቁቚ √8ൗ ; tangents at points ߣ௦ and ߣ௦
௥భ
௥మ are displayed by thin lines 
Finally, the approximate solution of inequality in Eq. (8) if ߜ is relatively small can be written 
NEAR-OPTIMAL PITCH OF A MOIRÉ GRATING FOR IMAGE HIDING APPLICATIONS IN DYNAMIC VISUAL CRYPTOGRAPHY.  
LORETA SAUNORIENE, SANDRA ALEKSIENE, JURATE RAGULSKIENE 
270 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2017, VOL. 13. ISSN 2345-0533  
as: 
ۖە
۔
ۖۓ ߣ௕ ≥ ߣ௦ ൬1 +
ߜ
ݎଵܬଵ(ݎଵ)൰,                                        
ߣ௦ ൬1 −
ߜ
ݎଵܬଵ(ݎଵ)൰ ≥ ߣ௕ ≥ ߣ௦
ݎଵ
ݎଶ ൬1 +
ߜ
ݎଶܬଵ(ݎଶ)൰ .                                                              
 (9)
Computational solutions of inequality in Eq. (8) are presented in Fig. 4: all ߣ௕ located in the 
uniformly grey and striped grey areas fit the inequality at the predefined values of ߜ. Black lines 
show the boundaries of the approximate solution in Eq. (9).  
 
Fig. 4. Solutions of the inequality ቚܬ଴ ቀఒೞఒ್ ݎଵቁቚ √8ൗ ≥ ߜ: grey color indicates areas where  
ܬ଴ ቀఒೞఒ್ ݎଵቁ √8ൗ ≥ ߜ, striped grey areas – where ܬ଴ ቀ
ఒೞ
ఒ್ ݎଵቁ √8ൗ ≤ −ߜ,  
black lines corresponds to the boundaries of the approximate solution in Eq. (9) 
If we want to obtain sufficient predetermined contrast between secret and background areas 
(ߪ௕ = ߜ), it would be optimal to choose such values ߣ௕ that lye on the contours of the grey and 
striped grey areas in Fig. 3, or, if ߜ is small, the approximate optimal solution is: 
ߣ௕ = ߣ௦ ൬1 +
ߜ
ݎଵܬଵ(ݎଵ)൰    or   ߣ௕ = ߣ௦ ൬1 −
ߜ
ݎଵܬଵ(ݎଵ)൰. (10)
Note, that the difference between values ߣ௦ and ߣ௕ still should be small enough in order to 
ensure the safety of the visual encoding scheme. 
4. Conclusions 
The secure encryption and successful decryption of the secret information in dynamic visual 
cryptography is generally based on the correct determination of the moiré grating parameters. This 
paper proposes the graphical scheme as well as the approximate solutions for the preselection of 
the near optimal pitches of moiré grating. The pitch standing for the secret information should 
ensure uniformly grey secret area in the time-averaged image (standard deviation of the area is 
equal to zero). The pitch of the background should guarantee high enough standard deviation in 
the time-averaged background. Such near-optimal pair of the pitches of moiré grating provides 
sufficient contrast of the decoded image and ensures that no secret information is visible in the 
NEAR-OPTIMAL PITCH OF A MOIRÉ GRATING FOR IMAGE HIDING APPLICATIONS IN DYNAMIC VISUAL CRYPTOGRAPHY.  
LORETA SAUNORIENE, SANDRA ALEKSIENE, JURATE RAGULSKIENE 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. SEP 2017, VOL. 13. ISSN 2345-0533 271 
static cover image. 
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Near-optimal pitch of a moiré grating for image hiding applications in dynamic visual cryptography

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Near-optimal pitch of a moiré grating for image hiding applications in dynamic visual cryptography

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