17-07-2012, 11:53 AM
Capture, encryption, compression, and display of digital holograms of three-dimensional objects
[attachment=27796]
ABSTRACT
Digital holography can be used to capture the whole Fresnel field from a reflective or transmissive object. Applications
include imaging and display of three-dimensional (3D) objects, and encryption and pattern recognition
of two-dimensional (2D) and 3D objects. Often, these optical systems employ discrete spatial light modulators
(SLMs) such as liquid-crystal displays. In the 2D case, SLMs can encode the inputs and keys during encryption
and decryption. For 3D processing, the SLM can be used as part of an optical reconstruction technique for 3D
objects, and can also represent the key during encryption and decryption. However, discrete SLMs can represent
only discrete levels of data necessitating a quantisation of continuous valued analog information. To date, many
such optical systems have been proposed in the literature, yet there has been relatively little experimental evaluation
of the practical performance of discrete SLMs in these systems. In this paper, we characterise conventional
phase-modulating liquid-crystal devices and examine their limitations (in terms of phase quantisation, alignment
tolerances, and nonlinear response) for the encryption of 2D and 3D data. Finally, we highlight the practical
importance of a highly controlled discretisation (optimal quantisation) for compression of digital holograms.
Keywords: digital holography, optical encryption, quantization, spatial light modulator, image compression,
three-dimensional image processing
1. INTRODUCTION
Digital holography1–8 is an inherently three-dimensional (3D) technique for the capture of real-world objects.
Many existing 3D imaging and processing techniques are based on the explicit combination of several twodimensional
(2D) perspectives (or light stripes, etc.) through digital image processing. The advantage of recording
a hologram is that multiple 2D perspectives can be optically combined in parallel, and in a constant number
of steps independent of the hologram size. Although holography and its capabilities have been known for many
decades, digital holography has seen renewed interest due to the recent development of megapixel digital sensors
with sufficient spatial resolution and dynamic range. The applications of digital holography could include 3D
television, virtual reality, and medical imaging.
One such application of digital holography is optical encryption9–20 which often produces a complex-valued
encrypted image resulting from a random phase mask positioned in the input, Fresnel, or Fraunhofer domain, or
combination of domains. Digital holography has been applied to the encryption of 2D conventional (real-valued)
images.15–17 Of these, the techniques based on phase-shift interferometry3, 5, 8 (PSI) make good use of detector
resources in that they capture on-axis encrypted digital holograms.16,17 The PSI technique has also been
extended to the encryption of 3D objects.18 The advantage of digital techniques over holographic encryption
methods that use more traditional photorefractive media12, 13 is that the resulting encrypted hologram can be
easily stored, processed, and analysed electronically, and transmitted over conventional communication channels.
Corresponding author: tom.naughton[at]nuim.ie
Often, spatial light modulators (SLMs) such as a liquid-crystal display (LCD) panels are used to encode the
inputs and keys in the encryption and decryption stages. There have been three main approaches to represent
complex-valued data on a SLM. The first is to map the values to be represented on to the values that a SLM can
represent using a minimum Euclidean distance method.21 The second approach referred to as pseudorandom
encoding,22 statistically approximates the desired complex values with those values that are achievable with
a given SLM. Originally developed for phase-only modulators, pseudorandom encoding has been extended to
modulators for which amplitude is a function of phase by transforming the phase statistics to compensate for
the amplitude coupling. The third approach is to encode the complex valued data as a computer generated
hologram23 (CGH) by using a group of pixels to represent a complex datum. Compared to the CGH approach,
the pseudorandom encoding method is a point oriented encoding method and uses the full available spacebandwidth
product of the SLM.
To date, many optical systems of this type have been proposed in the literature, yet there has been relatively
little experimental evaluation of the practical performance of SLMs in encryption systems. Complex-valued data,
when represented on a SLM with a discrete number of levels, leads to quantisation errors. For a double random
phase encoding system,10 Javidi et al.24 and Goudail et al.25 have studied how encrypted image perturbations
affect the decrypted image. Treatments of quantisation in holograms can be found in the literature,26, 27 and
quantisation of real-valued28 and complex-valued29–32 digital holograms, including encrypted digital holograms,
has been studied in the context of data compression. A desirable feature when using a SLM for encryption
and pattern recognition applications is a large space-bandwidth product (number of pixels in the SLM). This
often means a reduction in pixel size, making the optical system less tolerant to misalignment errors of the SLM.
Recently, Unnikrishnan et al.12 have studied the effects of SLM misalignment in a double random phase encoding
system.
In this paper, we consider optical systems that encrypt 2D and 3D objects. Our 3D objects are captured using
in-line phase-shift digital interferometry.8, 33, 34 In addition to amplitude, optical systems offer many degrees
of freedom to encode data such as phase,10, 12, 19, 32, 35, 36 polarization,37 and wavelength.13 In our system,
the encryption is performed using a random phase mask in a Fresnel plane. The random phase mask, acting as
the encryption/decryption key, is displayed on a SLM. The large space bandwidth product of the SLM means
that key sizes in the order of hundreds of thousands or even millions of digits are feasible. The encrypted data
captured by the CCD camera is a complex-valued white-noise-like signal. When decrypting, the complex-valued
data (or some real function of it) is displayed on the first SLM and the decoding mask is displayed on the second
SLM. We use a virtual optics method to simulate the encryption and decryption process. The efficient linear
canonical transform (LCT) algorithm proposed by Hennelly and Sheridan38 could simulate any such optical
system. The display of the encrypted image and decoding mask on SLMs results in quantisation depending on
the number of discrete amplitude or phase levels offered by each SLM.
We characterise conventional phase-modulating liquid-crystal devices and examine their limitations (in terms
of phase quantisation, alignment tolerances, and nonlinear response) in the context of 2D and 3D optical encryption
and decryption. At the decryption side, it is usual for the complex-valued encrypted image to be displayed
on one or more SLMs and propagated through a decryption system. A similar phase-only optical system has been
proposed and demonstrated for the optical reconstruction of 3D objects encoded in digital holograms.30 In this
paper, we consider the effects of phase quantisation and nonlinear SLM response in such an optical reconstruction
system. Also, we highlight the practical importance of highly controlled discretisation for compression of digital
holograms. This will in general be a nonuniform quantisation and it assumes an ideal SLM. We present the
results of one such popular technique (k-means clustering) and compare it to a more computationally efficient
technique that we have implemented: what in communication theory is known as optimal signal quantisation.39
This paper is organised as follows. In Sect. 2 we explain our encryption and decryption mechanism, and
in Sect. 3 explain our method for measuring quantisation and misalignment tolerances. The results of our
investigations into SLM quantisation and SLM misalignment are presented in Sects. 4 and 5, respectively. In
this latter section, too, we characterise the nonlinear response of our phase-modulating SLM and investigate the
optical encryption system’s tolerance to the unwanted amplitude modulation that is coupled with each phase
level. We illustrate the use of nonuniform quantisation for digital hologram compression in Sect. 6, and conclude
in Sect. 7.
d1
CCD
d2
SLM1 SLM2
d1
CCD
d2
SLM1 SLM2
(a) (b)
Figure 1. Schematic of the optical system for encryption and decryption.
2. ENCRYPTION AND DECRYPTION
In this section, we describe the process of encryption and decryption of the data. The following analysis is carried
out in one dimension with extension to two dimensions straightforward. Let f(x) represent the input data to
be encrypted. Let L 1, 1,
1 and L 2, 2,
2 represent LCTs corresponding to two combinations of lenses and free
space propagation distances. The input signal f(·) is transformed using L 1, 1,
1, multiplied by a random phase
mask R and transformed again using L 2, 2,
2.
The LCT L , ,
of f(·) is a three parameter transform described as
L , ,
[f](x′) = KZ Z f(x) exp j( x2 − 2 xx′ +
x′2)dx . (1)
The encrypted signal (·) can be written as
(·) = L 2, 2,
2[L 1, 1,
1(f)R] , (2)
and could be decrypted with
f(·) = L− 1,− 1,−
1{R∗L− 2,− 2,−
2[ (·)]} , (3)
where L− ,− ,−
is the inverse transform of L , ,
and indicates complex conjugate. There is a second method
of decryption, suitable for optical implementation, using the following property of LCT
L− ,− ,−
(f) = [L , ,
(f∗)]∗ . (4)
Using this property, we can write Eq. 3 as
f∗(·) = L 1, 1,
1{RL 2, 2,
2[ ∗(·)]} . (5)
If f(·) is real, f∗(·) = f(·). If one takes the conjugate of the encrypted signal and does the LCT operations used
for encryption in reverse order, one gets the conjugate of the input signal. In the present work, L 1, 1,
1 and
L 2, 2,
2 corresponds to free space distances d1 and d2 in which case
1 = 1 =
1 =
d1
, 2 = 2 =
2 =
d2
, (6)
where is the wavelength of light used.
In our experiments, we used digitised photographs as our 2D objects. The phase mask consisted of values
chosen with uniform probability from the range [0, 2) using a pseudo-random number generator. Digital holograms
of 3D objects were captured using an in-line PSI set-up33, 34 based on a Mach-Zehnder interferometer.
Encrypted versions of these holograms can be obtained by positioning the phase mask between the object and the
digital camera. For our experiments, this encryption step was simulated32 after optical capture of the real-world
3D objects.
3. QUANTISATION AND MISALIGNMENT ERRORS DUE TO SLM
Figure 1 shows the optical system used for encryption and decryption. In Fig. 1(a), SLM1 and SLM2 are used
to display the input image and encryption key, respectively. In Fig. 1(b), SLM1 and SLM2 are used to display
the encrypted image and the decryption key, respectively. We perform the encryption and decryption process
by a virtual optics method using the discrete Fresnel transform, which is a special case of the discrete LCT and
for which efficient n log(n) time algorithms exist.38 The wavelength of the coherent source was 532 nm, the
distances d1 and d2 were 80mm, the CCD pixel size was 9 mm and the SLM pixel size was 36 mm. We assumed
a 100% fill factor for both the SLM and CCD.
The error in the decoded image when the SLM is used to represent the encrypted image is studied by
modifying the encrypted image according to
′
i = i + i , (7)
where i represents the value of the ith pixel in the encrypted image, ′
i represents the corresponding value
represented by the SLM, and i is the resulting quantisation error for that pixel. If ai represents the amplitude
and i represents the phase of each quantised complex value ′
i then ai 2 {a1 . . . an} for n predefined amplitude
levels and i 2 { 1 . . . n} for n predefined phase levels. The error r in the decoded image is calculated using a
normalised rms (NRMS) error metric defined as
r =
N
Xi=1
N
Xj
=1
|Id(i, j) − I(i, j)|2
1
2
N
Xi=1
N
Xj
=1
|I(i, j)|2
−
1
2
, (8)
where Id(·) and I(·) are the intensities of the decrypted and original images, respectively.
To study the error in the decoded image when a SLM is used to represent the decryption mask the decryption
mask R′ is modified to conform to
R′
i = exp(ii) (9)
for each pixel i, where i 2 {1 . . . n} for n discrete phase levels. The sensitivity of the position of the mask in
the decryption process is studied by misaligning the decryption mask by a fraction of a pixel in transverse x, y
and longitudinal z directions, according to
R′(x′, y′, z′) = R(x + x, y + y, z + z) , (10)
where x, y, and z are the misalignments (in units of pixels) in the transverse x, y and longitudinal z
directions, respectively.
4. PHASE QUANTISATION
In this section we present some results from our simulation study. Figure 2(a) shows the 512×512pixel greyscale
2D image used in our study. We corrupted the 2D image with random phase to make it correspond to the
full Fresnel field immediately in front of a reflective real-world object. The image was encrypted as described
in Sect. 2. The encryption phase mask had 256 discrete levels. First we studied the quantisation effect of the
phase-only SLM displaying the decoding phase mask. The phase of the decrypting phase mask is quantised to
different numbers of discrete levels. Figures 2(b) through (d) show the decrypted image when the phase mask is
quantised to various numbers of levels.
Next we studied the effects due to the finite number of discrete levels of the SLM displaying the encrypted
image. We do not consider the amplitude of the encrypted image and consider only the phase; we assume that
we have access only to a phase-only SLM. (This amplitude removal or amplitude equalisation concept has been
shown to be a useful low-cost technique for the reconstruction of digital holograms using phase-only SLMs.30 )
The phase of the encrypted image was quantised to different numbers of levels ranging from 2 to 256. Several
of these decrypted images are shown in Fig. 3. Plots of NRMS error as a function of phase levels for each of
these two simulation studies are shown in Fig. 4. It was found that the error increases dramatically when the
number of phase levels falls below 10 in each case. In Fig. 4(b) the error never falls below approximately 0.4
even when no phase quantisation is applied. This has been noticed before in relation to digital holograms30 and
corresponds to the fundamental losses introduced by amplitude equalisation.
(a) (b)
© (d)
Figure 2. The decrypted 2D image with (a) no quantisation, and decrypted with (b) 64, © 8, and (d) 2 quantisation
levels in the phase mask.
(a) (b)
© (d)
Figure 3. The decrypted 2D image after the phase-only encrypted image is quantised to (a) 128, (b) 8, © 4, and (d) 2
phase levels.
Normalised rms error
Number of levels in decrypting phase mask
Normalised rms error
Number of levels in encrypted image
(a) (b)
Figure 4. Plots showing (a) NRMS error as a function of number of quantisation levels in the decrypting phase mask,
and (b) NRMS error as a function of number of quantisation levels in the SLM displaying the phase-only information of
the encrypted image. Each plot has a logarithmic scale in its horizontal axis.
[attachment=27796]
ABSTRACT
Digital holography can be used to capture the whole Fresnel field from a reflective or transmissive object. Applications
include imaging and display of three-dimensional (3D) objects, and encryption and pattern recognition
of two-dimensional (2D) and 3D objects. Often, these optical systems employ discrete spatial light modulators
(SLMs) such as liquid-crystal displays. In the 2D case, SLMs can encode the inputs and keys during encryption
and decryption. For 3D processing, the SLM can be used as part of an optical reconstruction technique for 3D
objects, and can also represent the key during encryption and decryption. However, discrete SLMs can represent
only discrete levels of data necessitating a quantisation of continuous valued analog information. To date, many
such optical systems have been proposed in the literature, yet there has been relatively little experimental evaluation
of the practical performance of discrete SLMs in these systems. In this paper, we characterise conventional
phase-modulating liquid-crystal devices and examine their limitations (in terms of phase quantisation, alignment
tolerances, and nonlinear response) for the encryption of 2D and 3D data. Finally, we highlight the practical
importance of a highly controlled discretisation (optimal quantisation) for compression of digital holograms.
Keywords: digital holography, optical encryption, quantization, spatial light modulator, image compression,
three-dimensional image processing
1. INTRODUCTION
Digital holography1–8 is an inherently three-dimensional (3D) technique for the capture of real-world objects.
Many existing 3D imaging and processing techniques are based on the explicit combination of several twodimensional
(2D) perspectives (or light stripes, etc.) through digital image processing. The advantage of recording
a hologram is that multiple 2D perspectives can be optically combined in parallel, and in a constant number
of steps independent of the hologram size. Although holography and its capabilities have been known for many
decades, digital holography has seen renewed interest due to the recent development of megapixel digital sensors
with sufficient spatial resolution and dynamic range. The applications of digital holography could include 3D
television, virtual reality, and medical imaging.
One such application of digital holography is optical encryption9–20 which often produces a complex-valued
encrypted image resulting from a random phase mask positioned in the input, Fresnel, or Fraunhofer domain, or
combination of domains. Digital holography has been applied to the encryption of 2D conventional (real-valued)
images.15–17 Of these, the techniques based on phase-shift interferometry3, 5, 8 (PSI) make good use of detector
resources in that they capture on-axis encrypted digital holograms.16,17 The PSI technique has also been
extended to the encryption of 3D objects.18 The advantage of digital techniques over holographic encryption
methods that use more traditional photorefractive media12, 13 is that the resulting encrypted hologram can be
easily stored, processed, and analysed electronically, and transmitted over conventional communication channels.
Corresponding author: tom.naughton[at]nuim.ie
Often, spatial light modulators (SLMs) such as a liquid-crystal display (LCD) panels are used to encode the
inputs and keys in the encryption and decryption stages. There have been three main approaches to represent
complex-valued data on a SLM. The first is to map the values to be represented on to the values that a SLM can
represent using a minimum Euclidean distance method.21 The second approach referred to as pseudorandom
encoding,22 statistically approximates the desired complex values with those values that are achievable with
a given SLM. Originally developed for phase-only modulators, pseudorandom encoding has been extended to
modulators for which amplitude is a function of phase by transforming the phase statistics to compensate for
the amplitude coupling. The third approach is to encode the complex valued data as a computer generated
hologram23 (CGH) by using a group of pixels to represent a complex datum. Compared to the CGH approach,
the pseudorandom encoding method is a point oriented encoding method and uses the full available spacebandwidth
product of the SLM.
To date, many optical systems of this type have been proposed in the literature, yet there has been relatively
little experimental evaluation of the practical performance of SLMs in encryption systems. Complex-valued data,
when represented on a SLM with a discrete number of levels, leads to quantisation errors. For a double random
phase encoding system,10 Javidi et al.24 and Goudail et al.25 have studied how encrypted image perturbations
affect the decrypted image. Treatments of quantisation in holograms can be found in the literature,26, 27 and
quantisation of real-valued28 and complex-valued29–32 digital holograms, including encrypted digital holograms,
has been studied in the context of data compression. A desirable feature when using a SLM for encryption
and pattern recognition applications is a large space-bandwidth product (number of pixels in the SLM). This
often means a reduction in pixel size, making the optical system less tolerant to misalignment errors of the SLM.
Recently, Unnikrishnan et al.12 have studied the effects of SLM misalignment in a double random phase encoding
system.
In this paper, we consider optical systems that encrypt 2D and 3D objects. Our 3D objects are captured using
in-line phase-shift digital interferometry.8, 33, 34 In addition to amplitude, optical systems offer many degrees
of freedom to encode data such as phase,10, 12, 19, 32, 35, 36 polarization,37 and wavelength.13 In our system,
the encryption is performed using a random phase mask in a Fresnel plane. The random phase mask, acting as
the encryption/decryption key, is displayed on a SLM. The large space bandwidth product of the SLM means
that key sizes in the order of hundreds of thousands or even millions of digits are feasible. The encrypted data
captured by the CCD camera is a complex-valued white-noise-like signal. When decrypting, the complex-valued
data (or some real function of it) is displayed on the first SLM and the decoding mask is displayed on the second
SLM. We use a virtual optics method to simulate the encryption and decryption process. The efficient linear
canonical transform (LCT) algorithm proposed by Hennelly and Sheridan38 could simulate any such optical
system. The display of the encrypted image and decoding mask on SLMs results in quantisation depending on
the number of discrete amplitude or phase levels offered by each SLM.
We characterise conventional phase-modulating liquid-crystal devices and examine their limitations (in terms
of phase quantisation, alignment tolerances, and nonlinear response) in the context of 2D and 3D optical encryption
and decryption. At the decryption side, it is usual for the complex-valued encrypted image to be displayed
on one or more SLMs and propagated through a decryption system. A similar phase-only optical system has been
proposed and demonstrated for the optical reconstruction of 3D objects encoded in digital holograms.30 In this
paper, we consider the effects of phase quantisation and nonlinear SLM response in such an optical reconstruction
system. Also, we highlight the practical importance of highly controlled discretisation for compression of digital
holograms. This will in general be a nonuniform quantisation and it assumes an ideal SLM. We present the
results of one such popular technique (k-means clustering) and compare it to a more computationally efficient
technique that we have implemented: what in communication theory is known as optimal signal quantisation.39
This paper is organised as follows. In Sect. 2 we explain our encryption and decryption mechanism, and
in Sect. 3 explain our method for measuring quantisation and misalignment tolerances. The results of our
investigations into SLM quantisation and SLM misalignment are presented in Sects. 4 and 5, respectively. In
this latter section, too, we characterise the nonlinear response of our phase-modulating SLM and investigate the
optical encryption system’s tolerance to the unwanted amplitude modulation that is coupled with each phase
level. We illustrate the use of nonuniform quantisation for digital hologram compression in Sect. 6, and conclude
in Sect. 7.
d1
CCD
d2
SLM1 SLM2
d1
CCD
d2
SLM1 SLM2
(a) (b)
Figure 1. Schematic of the optical system for encryption and decryption.
2. ENCRYPTION AND DECRYPTION
In this section, we describe the process of encryption and decryption of the data. The following analysis is carried
out in one dimension with extension to two dimensions straightforward. Let f(x) represent the input data to
be encrypted. Let L 1, 1,
1 and L 2, 2,
2 represent LCTs corresponding to two combinations of lenses and free
space propagation distances. The input signal f(·) is transformed using L 1, 1,
1, multiplied by a random phase
mask R and transformed again using L 2, 2,
2.
The LCT L , ,
of f(·) is a three parameter transform described as
L , ,
[f](x′) = KZ Z f(x) exp j( x2 − 2 xx′ +
x′2)dx . (1)
The encrypted signal (·) can be written as
(·) = L 2, 2,
2[L 1, 1,
1(f)R] , (2)
and could be decrypted with
f(·) = L− 1,− 1,−
1{R∗L− 2,− 2,−
2[ (·)]} , (3)
where L− ,− ,−
is the inverse transform of L , ,
and indicates complex conjugate. There is a second method
of decryption, suitable for optical implementation, using the following property of LCT
L− ,− ,−
(f) = [L , ,
(f∗)]∗ . (4)
Using this property, we can write Eq. 3 as
f∗(·) = L 1, 1,
1{RL 2, 2,
2[ ∗(·)]} . (5)
If f(·) is real, f∗(·) = f(·). If one takes the conjugate of the encrypted signal and does the LCT operations used
for encryption in reverse order, one gets the conjugate of the input signal. In the present work, L 1, 1,
1 and
L 2, 2,
2 corresponds to free space distances d1 and d2 in which case
1 = 1 =
1 =
d1
, 2 = 2 =
2 =
d2
, (6)
where is the wavelength of light used.
In our experiments, we used digitised photographs as our 2D objects. The phase mask consisted of values
chosen with uniform probability from the range [0, 2) using a pseudo-random number generator. Digital holograms
of 3D objects were captured using an in-line PSI set-up33, 34 based on a Mach-Zehnder interferometer.
Encrypted versions of these holograms can be obtained by positioning the phase mask between the object and the
digital camera. For our experiments, this encryption step was simulated32 after optical capture of the real-world
3D objects.
3. QUANTISATION AND MISALIGNMENT ERRORS DUE TO SLM
Figure 1 shows the optical system used for encryption and decryption. In Fig. 1(a), SLM1 and SLM2 are used
to display the input image and encryption key, respectively. In Fig. 1(b), SLM1 and SLM2 are used to display
the encrypted image and the decryption key, respectively. We perform the encryption and decryption process
by a virtual optics method using the discrete Fresnel transform, which is a special case of the discrete LCT and
for which efficient n log(n) time algorithms exist.38 The wavelength of the coherent source was 532 nm, the
distances d1 and d2 were 80mm, the CCD pixel size was 9 mm and the SLM pixel size was 36 mm. We assumed
a 100% fill factor for both the SLM and CCD.
The error in the decoded image when the SLM is used to represent the encrypted image is studied by
modifying the encrypted image according to
′
i = i + i , (7)
where i represents the value of the ith pixel in the encrypted image, ′
i represents the corresponding value
represented by the SLM, and i is the resulting quantisation error for that pixel. If ai represents the amplitude
and i represents the phase of each quantised complex value ′
i then ai 2 {a1 . . . an} for n predefined amplitude
levels and i 2 { 1 . . . n} for n predefined phase levels. The error r in the decoded image is calculated using a
normalised rms (NRMS) error metric defined as
r =
N
Xi=1
N
Xj
=1
|Id(i, j) − I(i, j)|2
1
2
N
Xi=1
N
Xj
=1
|I(i, j)|2
−
1
2
, (8)
where Id(·) and I(·) are the intensities of the decrypted and original images, respectively.
To study the error in the decoded image when a SLM is used to represent the decryption mask the decryption
mask R′ is modified to conform to
R′
i = exp(ii) (9)
for each pixel i, where i 2 {1 . . . n} for n discrete phase levels. The sensitivity of the position of the mask in
the decryption process is studied by misaligning the decryption mask by a fraction of a pixel in transverse x, y
and longitudinal z directions, according to
R′(x′, y′, z′) = R(x + x, y + y, z + z) , (10)
where x, y, and z are the misalignments (in units of pixels) in the transverse x, y and longitudinal z
directions, respectively.
4. PHASE QUANTISATION
In this section we present some results from our simulation study. Figure 2(a) shows the 512×512pixel greyscale
2D image used in our study. We corrupted the 2D image with random phase to make it correspond to the
full Fresnel field immediately in front of a reflective real-world object. The image was encrypted as described
in Sect. 2. The encryption phase mask had 256 discrete levels. First we studied the quantisation effect of the
phase-only SLM displaying the decoding phase mask. The phase of the decrypting phase mask is quantised to
different numbers of discrete levels. Figures 2(b) through (d) show the decrypted image when the phase mask is
quantised to various numbers of levels.
Next we studied the effects due to the finite number of discrete levels of the SLM displaying the encrypted
image. We do not consider the amplitude of the encrypted image and consider only the phase; we assume that
we have access only to a phase-only SLM. (This amplitude removal or amplitude equalisation concept has been
shown to be a useful low-cost technique for the reconstruction of digital holograms using phase-only SLMs.30 )
The phase of the encrypted image was quantised to different numbers of levels ranging from 2 to 256. Several
of these decrypted images are shown in Fig. 3. Plots of NRMS error as a function of phase levels for each of
these two simulation studies are shown in Fig. 4. It was found that the error increases dramatically when the
number of phase levels falls below 10 in each case. In Fig. 4(b) the error never falls below approximately 0.4
even when no phase quantisation is applied. This has been noticed before in relation to digital holograms30 and
corresponds to the fundamental losses introduced by amplitude equalisation.
(a) (b)
© (d)
Figure 2. The decrypted 2D image with (a) no quantisation, and decrypted with (b) 64, © 8, and (d) 2 quantisation
levels in the phase mask.
(a) (b)
© (d)
Figure 3. The decrypted 2D image after the phase-only encrypted image is quantised to (a) 128, (b) 8, © 4, and (d) 2
phase levels.
Normalised rms error
Number of levels in decrypting phase mask
Normalised rms error
Number of levels in encrypted image
(a) (b)
Figure 4. Plots showing (a) NRMS error as a function of number of quantisation levels in the decrypting phase mask,
and (b) NRMS error as a function of number of quantisation levels in the SLM displaying the phase-only information of
the encrypted image. Each plot has a logarithmic scale in its horizontal axis.