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Fast numerical generation and encryption of computer-generated Fresnel holograms

01 Mar 2011-Applied Optics (Optical Society of America)-Vol. 50, Iss: 7

TL;DR: Experimental results reveal that the proposed method can be applied to generate and encrypt holograms with a small number of computations, and can be decoded and reconstructed to the original 3D scene with good fidelity.

AbstractIn the past two decades, generation and encryption of holographic images have been identified as two important areas of investigation in digital holography. The integration of these two technologies has enabled images to be encrypted with more dimensions of freedom on top of simply employing the encryption keys. Despite the moderate success attained to date, and the rapid advancement of computing technology in recent years, the heavy computation load involved in these two processes remains a major bottleneck in the evolution of the digital holography technology. To alleviate this problem, we have proposed a fast and economical solution which is capable of generating, and at the same time encrypting, holograms with numerical means. In our method, the hologram formation mechanism is decomposed into a pair of one-dimensional (1D) processes. In the first stage, a given three-dimensional (3D) scene is partitioned into a stack of uniformed spaced horizontal planes and converted into a set of hologram sublines. Next, the sublines are expanded to a hologram by convolving it with a 1D reference signal. To encrypt the hologram, the reference signal is first convolved with a key function in the form of a maximum length sequence (also known as MLS, or M-sequence). The use of a MLS has two advantages. First, an MLS is spectrally flat so that it will not jeopardize the frequency spectrum of the hologram. Second, the autocorrelation function of an MLS is close to a train of Kronecker delta function. As a result, the encrypted hologram can be decoded by correlating it with the same key that is adopted in the encoding process. Experimental results reveal that the proposed method can be applied to generate and encrypt holograms with a small number of computations. In addition, the encrypted hologram can be decoded and reconstructed to the original 3D scene with good fidelity.

Topics: Digital holography (55%), Holography (54%), Encryption (54%)

Summary (2 min read)

1. Introduction

  • Numerical generation and encryption of holographic data are two important topics in the realm of holography that have instigated numerous research works in the past two decades.
  • Basically, heavy computation is involved in the hologram generation process, and the addition of numerical encryption will further lower the computation efficiency.
  • Second, the authors employed the set of 1D maximum length sequence (M-sequence) [24] as the key signals.
  • The decryption process can be achieved with numerical means with a small amount of arithmetic operations.
  • The vast difference between the reconstructed form of an encrypted hologram that has been decodedwith, andwithout, the right encryption key is illustrated.

A. Fast Generation of Fresnel Holograms

  • Given a set of 3D, self-illuminating object points.
  • Without loss of generality, the authors assumed that the source image has identical size and resolution as the hologram.
  • Compared with the direct implementation in Eq. (1), the alternative hologram generation method based on Eqs. (4) and (5) is significantly faster as it only involves a pair of one-dimensional processes, i.e., the formation of the diffraction pattern Oðx; τÞ in each scan plane and the subsequent generation of the overall diffraction pattern Dðx; yÞ.
  • The authors further noted that Oðx; τÞ can be derived with an economical hardware solution based on field programmable gate array (FPGA) at a throughput of over 100M pixels per second [25].

B. Fast Encryption of Holograms

  • The authors now propose to encrypt the hologram generated with Eq. (5) by convolving the conversion signal RðyÞ with a one-dimensional binary M-sequence signal.
  • In fact, with the prime polynomial, the authors can generate the M-sequence.
  • Subsequently, a new bit value is shifted to the output of the shift register.
  • As the formation of Oðx; yÞ can be realized with the hardware solution, the computation loading on this part of the process is not taken into account.

3. Experimental Evaluation

  • The authors shall evaluate their method with the standard image “Lenna” shown in Fig.
  • The mean square error (MSE) is 3196, and the peak signal to noise ratio (PSNR) is 13:1dB, indicating an extremely low coding fidelity.
  • Next the encrypted hologram is decrypted with the same encryption key S1ðyÞ, and the reconstructed image is shown in Fig. 4(c).
  • The encryption method can be conducted with an encryption key of longer M-sequences.
  • It can be seen that apart from a slight blurriness, together with minor ringing at the boundary and the junction between the upper and lower sections, the reconstructed images at each section are similar to the original one.

4. Conclusion

  • The authors have proposed a method for fast numerical generation and encryption of a Fresnel hologram.
  • This effectively merges the generation and encryption process into a single entity.
  • First, the amount of complexity involved in the generation of the hologram is significantly smaller than that employing the direct table lookup technique.
  • Experimental results reveal that the visual quality of a reconstructed image from a hologram that is generated and encrypted by their method is extremely poor, if not beyond recognition, if it is obtained without decrypting the hologram with the correct key.
  • The latter can be integrated with existing image encryption schemes, such as [2–8] to foster stronger resistance of optical images towards illegal access and attacks.

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Fast numerical generation and encryption of
computer-generated Fresnel holograms
P. W. M. Tsang,
1,
* T.-C. Poon,
2
and K. W. K. Cheung
1
1
Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
2
The Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA
*Corresponding author: eewmtsan@cityu.edu.hk
Received 23 July 2010; revised 7 November 2010; accepted 10 November 2010;
posted 11 November 2010 (Doc. ID 132144); published 20 December 2010
In the past two decades, generation and encryption of holographic images have been identified as two
important areas of investigation in digital holography. The integration of these two technologies has en-
abled images to be encrypted with more dimensions of freedom on top of simply employing the encryption
keys. Despite the moderate success attained to date, and the rapid advancement of computing technology
in recent years, the heavy computation load involved in these two processes remains a major bottleneck
in the evolution of the digital holography technology. To alleviate this problem, we have proposed a fast
and economical solution which is capable of generating, and at the same time encrypting, holograms with
numerical means. In our method, the hologram formation mechanism is decomposed into a pair of one-
dimensional (1D) processes. In the first stage, a given three-dimensional (3D) scene is partitioned into a
stack of uniformed spaced horizontal planes and converted into a set of hologram sublines. Next, the
sublines are expanded to a hologram by convolving it with a 1D reference signal. To encrypt the holo-
gram, the reference signal is first convolved with a key function in the form of a maximum length se-
quence (also known as MLS, or M-sequence). The use of a MLS has two advantages. First, an MLS is
spectrally flat so that it will not jeopardize the frequency spectrum of the hologram. Second, the auto-
correlation function of an MLS is close to a train of Kronecker delta function. As a result, the encrypted
hologram can be decoded by correlating it with the same key that is adopted in the encoding process.
Experimental results reveal that the proposed method can be applied to generate and encrypt holograms
with a small number of computations. In addition, the encrypted hologram can be decoded and recon-
structed to the original 3D scene with good fidelity. © 2010 Optical Society of America
OCIS codes: 090.0090, 090.1760, 100.4998.
1. Introduction
Numerical generation and encryption (coding) of ho-
lographic data are two important topics in the realm
of holography that have instigated numero us re-
search works in the past two decades. The former
area, commonly known as computer-generated holo-
graphy (CGH), has enabled holograms to be gener-
ated numerically from three dimensional (3D) objects
or synthetic models that do not actually exist in the
real world [1]. The latter one, encryption, allows a ho-
logram to be converted to a form which can only be
reconstructed with prior knowledge, such as the avail-
ability of a key signal, of the encoding process. The in-
tegration of these two technologies, hereafter referred
to as the holographic encryption, has also enabled 3D
optical images to be encrypted with more degrees of
freedom than classical approaches which operate on
the image pixels directly [28]. With the advancement
of the computing technology, the holographic encryp-
tion process can be conducted with numerical means
or a combination of numerical and optical means,
which are more flexible and easier to implement than
the classical optical methods. Along this direction,
successful attempts have been made and published
on the digital holographic processing techniques for
information security [913]. Despite the success
achieved to date on digital holography encryption,
0003-6935/11/070B46-07$15.00/0
© 2011 Optical Society of America
B46 APPLIED OPTICS / Vol. 50, No. 7 / 1 March 2011
Copyright by the Optical Society of America. P. W. M. Tsang, T.-C. Poon, and K. W. K. Cheung, "Fast numerical generation
and encryption of computer-generated Fresnel holograms," Appl. Opt. 50, B46-B52 (2011); doi: 10.1364/ao.50.000b46

there are a number of problems which have become
identified as areas of investigation in recent years.
Basically, heav y computation is involved in the holo-
gram generation process, and the addition of numer-
ical encryption will further lower the computation
efficiency. Some of the representative works that pro-
vide a moderate amount of reduction on the computa-
tion load in hologram generation include [1423].
Second, the decoding and reconstruction process is
also computationally demanding as an encrypted ho-
logram has a wider space-bandwidth product than the
source data. Third, it is difficult to find a set of key
signals to encrypt each hologram in a unique manner
(so that each candidate can only be recovered with its
own key), and which guarantee decryption on all the
coded holograms. Fourth, but not the least, the key
signals should not impose excessive disturbance on
the frequency spectrum of a hologram. Otherwise,
the reconstructed image may be contaminated with
noticeable visual artifacts.
In this paper, we propose a method to address the
aforementioned problems. First, we integrated the
hologram generation and encryption mechanism into
a single entity and realized it with a pair of one-
dimensional (1D), finite impulse response (FIR) filter-
ing processes. This results in a substantial reduction
in the computation loading. Second, we employed the
set of 1D maximum length sequence (M-sequence)
[24] as the key signals. The latter are spectrally flat,
which does not impose excessive distortion on the fre-
quency spectrum of the holographic data. According
to [24], the autocorrelation function of each member
in the M-sequence is close to a train of Kronecker delta
function. As a result, the encrypted hologram can be
decoded by simply correlating it with the same key
that is adopted in the encoding process. The decryp-
tion process can be achieved with numerical means
with a small amount of arithmetic operations. Third,
the correlation between different members in the
M-sequence is very low so that each of them can be
employed as a unique encryption key for encoding
each source hologram.
Thepaperisorganizedasfollows.Aftertheintroduc-
tion, the proposed method for fast generation and en-
cryption of the Fresnel hologram is reported. We shall
explain the decomposition of the hologram formation
mechanism into a pair of 1D processes, and how en-
cryption can be integrated without additional compu-
tation overh ead. Experimental evaluation is given in
Section 3. The vast difference between the recon-
structed form of an encrypted hologram that has been
decoded with, and without, the right encryption key is
illustrated. In Section 4, we present a conclusion
summarizing the essential findings.
2. Fast Generation and Encryption of Fresnel
Holograms
A. Fast Generation of Fresnel Holograms
Given a set of 3D, self-illuminating object points P ¼
½p
0
ðx
0
; y
0
; z
0
Þ; p
1
ðx
1
; y
1
; z
1
Þ; ……; p
N1
ðx
N1
; y
N1
; z
N1
Þ,
the diffraction pattern Dð x; yÞ on a vertical plane
can be generated numerically with the following
equation:
Dðx; yÞ¼
X
N1
i¼0
a
i
r
i
expðjkr
i
Þ¼
X
N1
i¼0
a
i
r
i
cosðkr
i
Þþj
a
i
r
i
sinðkr
i
Þ
;
ð1Þ
where a
i
and r
i
represent the intensity of the ith
point and the distance between the object point and
a point ðx; yÞ on the diffraction plane, respectively.
k ¼
2π
λ
is the wavenumber and λ is the wavelength
of the object wave.
The horizontal and vertical extents of the holo-
gram is given by X and Y, respectively. Without loss
of generality, we assumed that the source image has
identical size and resolution as the hologram.
From Eq. (1), it can be seen that the number of
multiplicative operations involved in the hologram
generation process, assuming that all the sine and
cosine terms are precomputed in advance (and there-
fore so as r
i
), is
2 × N × X × Y: ð2Þ
In our proposed method in hologram generation,
the object scene can be partitioned into a vertical
stack of horizontal planes as shown in Fig. 1.
We further assume that the range of depth of the
object points is small and centered at z ¼ z
o
. Suppose
there are NðτÞ object points on the y ¼ τ plane, the
diffraction pattern according to Fresnel diffraction,
contributed by the plane is given by
Dðx; yÞ
τ
X
NðτÞ
i¼0
a
i
r
i
exp
jk
ðx x
i
Þ
2
2z
i
exp
jk
ðy τÞ
2
2z
o
¼ exp
jk
ðy τÞ
2
2z
o
X
NðτÞ
i¼0
a
i
r
i
exp
jk
ðx x
i
Þ
2
2z
i
¼ Rðy τÞOðx; τÞ: ð3Þ
The total object beam is then equal to the super-
position of the diffraction pattern in each scan plane
as
Fig. 1. Partitioning an object scene into a vertical stack (along y
direction) of horizontal planes.
1 March 2011 / Vol. 50, No. 7 / APPLIED OPTICS B47

Dðx; yÞ¼
X
τ
Dðx; yÞ
τ
¼
X
τ
Oðx; τÞRðy τÞ: ð4Þ
Equation (4) is equivalent to a convolution process
which can be represented by
Dðx; yÞ¼Oðx; yÞRðyÞ: ð5Þ
For clarity of explanation we refer to RðyÞ, which is
employed to convert the sublines into a Fresnel ho-
logram, as the conversion signal. Compared w ith
the direct implementation in Eq. (1), the alternative
hologram generation method based on Eqs. (4) and
(5) is significantly faster as it only involves a pair of
one-dimensional processes, i.e., the formation of the
diffraction pattern Oðx; τÞ in each scan plane and the
subsequent generation of the overall diffraction pat-
tern Dðx; yÞ. We further noted that Oðx; τÞ can be de-
rived with an economical hardware solution based on
field programmable gate array (FPGA) at a through-
put of over 100 M pixels per second [25].
Subsequently, a hologram can be generated from
the diffraction pattern Dðx; yÞ by adding a planar
or a spherical reference beam.
B. Fast Encryption of Holograms
We now propose to encrypt the hologram generated
with Eq. (5) by convolving the conversion signal RðyÞ
with a one-dimensional binary M-sequence signal.
Mathematically, the latter can be expressed as the
primitive (prime) polynomial [24]
SðyÞ¼1 þ g
1
y
1
þ g
2
y
2
þ þ g
M
y
M
; ð6Þ
where all mathematical operations are performed in
modulo-2, in other words, þ denotes the modulo-2 ad-
dition, and the value of SðyÞ and g
i
j
1iM
is either 1 or
0. For a given M in Eq. (6), there are 2
M
1 binary
bits in the M-sequence.
And there exist certain settings for the coefficients
g
i
j
1iM
where one or more M-sequences can be gen-
erated. The M-sequence can be generated by a linear
feedback shift register (LFSR) as shown in Fig. 2,
where a cascade series of M stages of a single binary
memory is shown.
Let Ψ
k;M
denote the kth output for a sequence of
length 2
M
1. The output signals at the selected bin-
ary memory (commonly referred to as taps) are com-
bined with the exclusive or (XOR) operation (which
is basically the modulo-2 addition), and the result is
feedback to the input of the LFSR.
For example with M ¼ 6, two prime polynominals
can be formed from Eq. (6) with
g
i
¼
1 ði ¼ 5; 6Þ
0 otherwise
; ð7aÞ
for the first polynominal, and
g
i
¼
1 ði ¼ 1; 6Þ
0 otherwise
; ð7bÞ
for the second polynominal.
In other words with M ¼ 6, a pair of M-sequences
can be generated with a sequence length of 2
6
1.
In fact, with the prime polynomial, we can generate
the M-sequence. For instance, with g
i
¼ 1 for i ¼ 5,6;
g
i
¼ 0 otherwise given above, and assuming all the
binary memories have been loaded with 1s, we take
the fifth and sixth binary memory outputs and XOR
them to obtain a new bit for the input of the shift reg-
ister. This bit, as well as the rest of the contents in the
shift register, are then shifted right at the command of
a clock. Subsequently, a new bit value is shifted to the
output of the shift register. This process is repeated
through the control of the clock to generate 2
6
1
pseudorandom bits before repeating the same
sequence.
We now have the M-sequence, denoted as Ψ
M
ðyÞ¼
ðΨ
1;M
; Ψ
2;M
; Ψ
2
M1
;M
Þ, where the argument y repre-
sents that we are applying the sequence along the y
direction.
The diffraction pattern encrypted with the M-
sequence Ψ
M
ðyÞ is then given by
D
E
ðx; yÞ¼Oðx; yÞRðyÞΨ
M
ðyÞ: ð8Þ
Taking the Fourier transform on both sides of
Eq. (8) along the vertical direction, we have,
~
D
E
ðx; e
jω
Þ¼
~
Oðx; e
jω
Þ
~
Rðe
jω
Þ
~
Ψ
M
ðe
jω
Þ¼
~
Oðx; e
jω
Þ
~
Eðe
jω
Þ;
ð9Þ
where
~
Eðe
jω
Þ¼
~
Rðe
jω
Þ
~
Ψ
M
ðe
jω
Þ.
From Eq. (9), it can be seen that the generation and
encryption of the hologram can be merged into a sin-
gle process and conducted in the frequency domain.
In other words, no extra computation load is required
in the encryption process. Subsequently, the diffrac-
tion pattern can be obtained by inverse Fourier
transforming
~
D
E
ðx; e
jω
Þ as
D
E
ðx; yÞ¼FT
1
f
~
D
E
ðx; e
jω
Þg: ð10Þ
Fig. 2. M-stage LFSR for the realization of M-sequence.
B48 APPLIED OPTICS / Vol. 50, No. 7 / 1 March 2011

A breakdown of the number of complex multiplica-
tion operations involved in Eqs. (10) (formulated
based on Eqs. (8) and (9)) is given in Table 1. As the
formation of Oðx; yÞ can be realized with the hard-
ware solution, the computation loading on this part
of the process is not taken into account.
According to Table 1, the amount of complex multi-
plication operation involved in the generation of the
encrypted hologram D
E
ðx; yÞ is given by
ðX × Y log
2
YÞþðX × YÞþðX × Y log
2
YÞ
¼ðX × YÞð1 þ 2Y log
2
YÞ: ð11Þ
However, one complex multiplication is composed
of four scalar multiplication operations. Hence in
Eq. (11 ), the total number of scalar multiplication op-
erations is equivalent to
4ðX × YÞð1 þ 2Y log
2
YÞ: ð12Þ
Comparing with Eq. (2), the computation advan-
tage (CA) of the proposed method over the direct
hologram generation process based on Eq. ( 1) is given
by
CA ¼
2 × N × X × Y
4 × X × Y½1 þ 2 log
2
Y
¼
N
2½1 þ 2 log
2
Y
: ð13Þ
Suppose the source image is of identical size and
resolution as the hologram, and all the pixels in
the source image are object points (i.e., N ¼ X × Y),
we have
CA ¼
XY
2½1 þ 2 log
2
Y
: ð14Þ
For a small hologram the size of 1024 × 1024 pixels,
our proposed method is over 2 × 10
4
times faster than
the direct hologram generation method based on
Eq. (1).
The decryption process can be conducted in the fre-
quency space by inverse filtering
~
D
E
ðx; e
jω
Þ as given
by
Dðx; yÞ¼FT
1
~
D
E
ðx; e
jω
Þ
~
Eðe
jω
Þ
: ð15Þ
Equation (15) indicates that the encrypted holo-
gram can be perfectly reverted to its original form
if it is decrypted with the same key.
3. Experimental Evaluation
We shall evaluate our method with the standard im-
age Lenna shown in Fig. 3. The image is positioned
at a distance of 0:5 m from the hologram an d the op-
tical setting is given in Table 2.
The image Lenna is converted into a hologram
without encryption according to Eq. (5), and the nu-
merical reconstruction is shown in Fig. 4(a). It can be
seen that apart from a slight blurriness of the image
and minor ringing patterns near the boundary which
is caused by the finite size of the hologram, the result
is close to the original image.
Next, we apply our proposed method to generate,
and concurrently encrypt the hologram with an M-
sequence derived from a six-taps LFSR representing
the polynominal S
1
ðyÞ¼1 þ y
5
þ y
6
[see Eq. (6) with
M ¼ 6 and the g
i
given in Eq. (7a) for ð1 i 6Þ]. For
simplicity of explanation, we refer the polynominals
representing the M-sequence as the encryption
keys.
The image reconstructed from the encrypted holo-
gram is shown in Fig. 4(b). The result is barely visible
and the original content is completely lost. The mean
square error (MSE) is 3196, and the peak signal to
noise ratio (PSNR) is 13:1 dB, indicating an extre-
mely low coding fidelity.
Next the encrypted hologram is decrypted with the
same encryption key S
1
ðyÞ, and the reconstructed im-
age is shown in Fig. 4(c). It can be seen that the re-
constructed image is identical to the reconstru cted
image derived from the unencrypted hologram
shown in Fig. 4(a). The MSE is 0 which is in accor-
dance with the theoretical deduction in Eq. (15).
Fig. 3. Test image Lenna positioned at 0.5 m from the hologram.
Table 1. Breakdown of Number of Complex Multiplications for Each
Step in Eq. (10)
Operation Number of Complex Multiplications
Oðx; yÞ
~
Oðx; e
jω
Þ X × Y log
2
Y
~
D
E
ðx; e
jω
Þ¼
~
Oðx; e
jω
Þ
~
Eðe
jω
Þ X × Y
~
D
E
ðx; e
jω
Þ D
E
ðx; yÞ X × Y log
2
Y
Table 2. Optical Setting for Generating the Hologram
Wavelength 650 nm
Pixel size of hologram 10:58 μ10:58 μm
Image/hologram dimension X ¼ Y ¼ 1024
z
o
of Eq. (3) 0:5 m
1 March 2011 / Vol. 50, No. 7 / APPLIED OPTICS B49

To demonstrate the effect of employing a wrong
key to decrypt the encrypted hologram, the latter
is decrypted with another encryption key S
2
ðyÞ given
by [see Eq. (6) with M ¼ 6 and the g
i
given in Eq. (7b)
for ð1 i 6Þ]
S
2
ðyÞ¼1 þ y
1
þ y
6
;
and the reconstructed image is shown in Fig. 4(d).It
can be seen that the original image is beyond recog-
nition. The MSE is 1519, and the PSNR is 16:3 dB,
indicating an extremely low coding fidelity.
The encryption method can be conducted with an
encryption key of longer M-sequences. To demon-
strate this, the proposed method is applied to gener-
ate and encrypt the hologram with an M-sequence
derived from a eight-taps LFSR representing the
polynominal S
3
ðyÞ¼1 þ y
1
þ y
6
þ y
8
. The eight-taps
LFSR generates an M-sequence of length 256, which
is four times the one generated with a six-taps LFSR.
Subsequently, the encrypted hologram is decrypted
with the same key S
3
ðyÞ and the result is shown
in Fig. 4(e). It can be seen that the reconstructed im-
age is identical to the one corresponding to the unen-
crypted hologram shown in Fig. 4(a). The MSE is 0
demonstrating that the quality of the reconstructed
image is not affected by the length of the M-sequence.
Next, we apply our proposed method to generate
and encrypt a hologram generated from the image
peppers as shown in Fig. 5(a). The horizontal and
vertical extent of the test image is 512 × 512, and the
image is evenly divided into an upper and a lower por-
tion located at a distance of z
1
and z
2
from the holo-
gram as shown in Fig. 5(b) . A hologram is first
generated without encryption with z
1
¼ 0:502 m,
z
2
¼ 0:498 m, and z
o
¼ 0:50 m according to the setting
given in Table 1. The reconstructed images at dis-
tances of z
1
and z
2
are shown in Figs. 6(a) and 6(b),
respectively.
It can be seen that apart from a slight blurriness,
together with minor ringing at the boundary and the
junction between the upper and lower sections, the
reconstructed images at each section are similar to
the original one.
We then apply the propose d method to generate
and encode the hologram for the source image in
Fig. 5(a). Subsequently, the encoded hologram is de-
coded, and the reconstructed images at z ¼ 0:502 m
and z ¼ 0:498 m are shown in Figs. 6(c) and 6(d), re-
spectively. For both cases, the reconstructed images
are identical to the ones obtained with the unen-
crypted holograms and the MSE is 0.
A similar test is performed on the source image in
Fig. 5(a) with z
1
¼ 0:49 m, z
2
¼ 0:51 m, and z
o
¼
0:50 m. The reconstructed images of the unencrypted
hologram at these two depth planes are shown in
Figs. 6(e) and 6(f). It can be seen that when z
1
and
z
2
are further away from z
0
, the reconstructed images
are slightly inferior than those shown in Figs. 6(c) and
6(d). The reason is that in applying the proposed
method in generating the hologram [i.e., Eq. (3)], it
has been assumed that the range of depth of the object
points is within a close neighborhood of z
o
. As a result,
the quality of the reconstructed images will decrease
as the deviation from the assumption escalates.
An encoded hologram based on the key S
3
ðyÞ is
then generated with the proposed method. The en-
crypted hologram is decoded, and the reconstructed
Fig. 4. (a) Reconstructed image from a hologram generated from the test image in Fig. 3. (b) Reconstructed image from a hologram
generated from the test image in Fig. 3 and encrypted with the M-sequence generated by S
1
ðyÞ. PSNR ¼ 13:1 dB. (c) Reconstructed image
from a hologram generated from the test image in Fig. 1 and that has been encrypted and decrypted with the same encryption key, S
1
ðyÞ.
The MSE, as compared with the reconstructed image of the unencrypted hologram, is 0. (d) Reconstructed image from a hologram gen-
erated from the test image in Fig. 1 and that has been encrypted with the encryption key S
1
ðyÞ and decrypted with the key S
2
ðyÞ.
PSNR ¼ 16:3 dB. (e) Reconstructed image from a hologram generated from the test image in Fig. 3 and that has been encrypted and
decrypted with the same encryption key, S
3
ðyÞ. The key is derived from an eight-taps M-sequence. The MSE, as compared with the re-
constructed image of the unencrypted hologram in Fig. 4(a),is0.
Fig. 5. (a) Test image evenly divided into an upper and lower sec-
tion, each locating at a different depth from the hologram as shown
in Fig. 5(b). (b) Depth of each section of the image in Fig. 5(a) to the
hologram.
B50 APPLIED OPTICS / Vol. 50, No. 7 / 1 March 2011

Figures (8)
Citations
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Journal ArticleDOI
TL;DR: The proposed algorithm can compress and encrypt image signal, especially can encrypt multiple images once, and is of high security and good compression performance.
Abstract: Based on hyper-chaotic system and discrete fractional random transform, an image compression-encryption algorithm is designed. The original image is first transformed into a spectrum by the discrete cosine transform and the resulting spectrum is compressed according to the method of spectrum cutting. The random matrix of the discrete fractional random transform is controlled by a chaotic sequence originated from the high dimensional hyper-chaotic system. Then the compressed spectrum is encrypted by the discrete fractional random transform. The order of DFrRT and the parameters of the hyper-chaotic system are the main keys of this image compression and encryption algorithm. The proposed algorithm can compress and encrypt image signal, especially can encrypt multiple images once. To achieve the compression of multiple images, the images are transformed into spectra by the discrete cosine transform, and then the spectra are incised and spliced into a composite spectrum by Zigzag scanning. Simulation results demonstrate that the proposed image compression and encryption algorithm is of high security and good compression performance.

55 citations


Journal ArticleDOI
TL;DR: A novel asymmetric color information cryptosystem based on an optical coherent superposition method and phase-truncated gyrator transform (GT) and two phase keys provide asymmetric keys that offer a high-level robustness against existing attacks is proposed.
Abstract: A novel asymmetric color information cryptosystem based on an optical coherent superposition method and phase-truncated gyrator transform (GT) is proposed. In this proposal, an original color image is converted into three independent channels, i.e., red, green, and blue. Each channel is separated into a random phase masks (RPM) and a key phase mask (KPM) using a coherent superposition method. The KPM is a modulation of the RPM by the color channel and used as decryption key. The same RPM, which is independent of plaintext, can be chosen for different images of the same size; however, KPMs, which are related to the original color images, are different. The RPM and the KPM are independently gyrator transformed. Then two gyrator spectra are, respectively, phase truncated to obtain two encoded images and amplitude truncated to generate two asymmetric phase keys. The KPM and two phase keys provide asymmetric keys. The transformation angles of the GT give additional keys for each channel and thus offer a high-level robustness against existing attacks. The proposed optical design is free from axial movement. Numerical simulations are demonstrated to verify the flexibility and effectiveness of the proposed method.

45 citations


Journal ArticleDOI
Jian Liu1, Hongzhen Jin1, Lihong Ma1, Yong Li1, Weimin Jin1 
TL;DR: A novel technique of optical color image encryption and decryption based on computer generated hologram (CGH) and chaotic theory is proposed, which has the feasibility and its robustness against occlusion and noise attacks.
Abstract: A novel technique of optical color image encryption and decryption based on computer generated hologram (CGH) and chaotic theory is proposed. The tri-color separated images of an image to be encrypted are encoded with three random phase arrays constructed by a chaotic sequence of the deterministic non-linear system, respectively. Then Burch's encoding method using the modified off-axis reference beam is adopted to fabricate the CGH as the encryption image. A clear original color image can be reconstructed as long as the correct initial value of chaotic sequence and the correct system parameters are given. The initial value of chaotic function with a very small change will lead to the generation of an entirely different chaotic sequences. As a result, the random phase array changes dramatically and the original image cannot be recovered rightly. Serving as the secret keys, the initial values of chaotic sequence and system parameters reduce the amount of the key data. And the digital encryption image is also more favorable to be stored and transmitted. The feasibility and its robustness against occlusion and noise attacks are verified by numerical simulations.

32 citations


Journal ArticleDOI
TL;DR: A study of the collision property of the optical image encryption technique based on interference is presented and an efficient way to avoid this potential risk is provided by adding an extra unit in the original structure, which can be called a double-factor verification system.
Abstract: From the perspective of cryptography, collision is an undesirable situation that occurs when two or more distinct inputs into a security system produce undistinguishable outputs. In this paper we present a study of the collision property of the optical image encryption technique based on interference (Zhang et al. Optics Letters; 2008:33(21)). For an arbitrary secret image (output) to be encrypted, we can find various distinct pairs of phase-only masks (inputs), which yield almost the same outputs by use of a modified phase retrieval algorithm. Meanwhile, we also provide an efficient way to avoid this potential risk by adding an extra unit in the original structure, which can thus be called a double-factor verification system. A series of computer simulations were also carried out to demonstrate our concern and strategy.

30 citations


Journal ArticleDOI
Dezhao Kong, Liangcai Cao1, Xueju Shen, Hao Zhang1, Guofan Jin1 
TL;DR: An encryption method based on interleaved computer-generated holograms (CGHs) displayed by a spatial light modulator (SLM) is demonstrated and may avoid the inherent silhouette problem and alleviate the precise alignment requirements of interference encryption.
Abstract: An encryption method based on interleaved computer-generated holograms (CGHs) displayed by a spatial light modulator (SLM) is demonstrated. Arbitrary decrypted complex optical wave fields are reconstructed in the rear focal plane of two phase-only holograms, generated from original image using a vector decomposition algorithm. Two CGHs are encoded into one hologram by interleaving the column of pixels, which optically combines the optical wave fields of two neighboring phase-only modulated pixels. The designed image encryption system may avoid the inherent silhouette problem and alleviate the precise alignment requirements of interference encryption. Video encryption and real-time dynamic decryption is demonstrated using one SLM.

23 citations


References
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"Fast numerical generation and encry..." refers background or methods in this paper

  • ...Second, we employed the set of 1D maximum length sequence (M-sequence) [24] as the key signals....

    [...]

  • ...According to [24], the autocorrelation function of each member in theM-sequence is close toa train ofKroneckerdelta function....

    [...]

  • ...Mathematically, the latter can be expressed as the primitive (prime) polynomial [24]...

    [...]


Journal ArticleDOI
TL;DR: An information security method that uses a digital holographic technique that provides secure storage and data transmission and can be electrically decrypted by use of the digital hologram of the key.
Abstract: An information security method that uses a digital holographic technique is presented. An encrypted image is stored as a digital hologram. The decryption key is also stored as a digital hologram. The encrypted image can be electrically decrypted by use of the digital hologram of the key. This security technique provides secure storage and data transmission. Experimental results are presented to demonstrate the proposed method.

432 citations


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TL;DR: A method for optical encryption of three-dimensional (3D) information by use of digital holography using a phase-shifting interferometer and an intensity-recording device.
Abstract: A method for optical encryption of three-dimensional (3D) information by use of digital holography is presented. A phase-shifting interferometer records the phase and amplitude information generated by a 3D object at a plane located in the Fresnel diffraction region with an intensity-recording device. Encryption is performed optically by use of the Fresnel diffraction pattern of a random phase code. Images of the 3D object with different perspectives and focused at different planes can be generated digital or optically after decryption with the proper key. Experimental results are presented.

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TL;DR: A novel method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture.
Abstract: A novel method for simulating field propagation is presented. The method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture. This method can be implemented by using the fast Fourier transformation twice and a spectrum interpolation. It features computation time that is comparable with that of standard calculation methods for diffraction or propagation between parallel planes. To demonstrate the method, numerical results as well as a general formulation are reported for a single-axis rotation.

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Journal ArticleDOI
TL;DR: A novel approach to dramatically reduce the size of the conventional LUT, still keeping its advantage of fast computational speed, is proposed, which is called here a novel LUT (N-LUT) method.
Abstract: Several approaches for increasing the speed in computation of the digital holograms of three-dimensional objects have been presented with applications to real-time display of holographic images. Among them, a look-up table (LUT) approach, in which the precalculated principal fringe patterns for all possible image points of the object are provided, has gained a large speed increase in generation of computer-generated holograms. But the greatest drawback of this method is the enormous memory size of the LUT. A novel approach to dramatically reduce the size of the conventional LUT, still keeping its advantage of fast computational speed, is proposed, which is called here a novel LUT (N-LUT) method. A three-dimensional object can be treated as a set of image planes discretely sliced in the z direction, in which each image plane having a fixed depth is approximated as some collection of self-luminous object points of light. In the proposed method, only the fringe patterns of the center points on each image plane are precalculated, called principal fringe patterns (PFPs) and stored in the LUT. Then, the fringe patterns for other object points on each image plane can be obtained by simply shifting this precalculated PFP according to the displaced values from the center to those points and adding them together. Some experimental results reveal that the computational speed and the required memory size of the proposed approach are found to be 69.5 times faster than that of the ray-tracing method and 744 times smaller than that of the conventional LUT method, respectively.

233 citations