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 [2–8]. 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 [9–13]. 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 [14–23].
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
N−1
ðx
N−1
; y
N−1
; z
N−1
Þ,
the diffraction pattern Dð x; yÞ on a vertical plane
can be generated numerically with the following
equation:
Dðx; yÞ¼
X
N−1
i¼0
a
i
r
i
expðjkr
i
Þ¼
X
N−1
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; yÞRðyÞ: 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
1≤i≤M
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
1≤i≤M
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 1’s, 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
M−1
;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; yÞRðyÞΨ
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 μm×10:58 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