4.1. Sampling Theorem Based on the FrT
We obtain the sampling theorem by using the FrT and the new processing operations described in the previous section. The sampling theorem is very important for the modern optical implementation of information processing systems that are composed of cameras, spatial light modulators and other optical and optoelectronic devices. The correct sampling at the spatial domain and in the FrD is a critical aspect for the information processing systems based on the free-space optical propagation because the characteristics of optical waves propagation can be preserved or modified when these samplings are defined correctly or incorrectly, respectively.
In order to simplify the mathematical process of obtaining the sampling theorem based on the FrT, we introduce a new Dirac comb function for parameters
,
z and period
T, which is given by:
The FrT for the parameters
and
z in the previous equation is:
The previous equation shows that the FrT for parameters
and
z of the proposed Dirac comb function for parameters
,
z and period
T corresponds to the other Dirac comb function for parameters
,
and period
. We describe the sampling theorem based on the FrT for a function
whose FrT
for parameters
and
z has finite support
for coordinate
u at the FrD. The following expression
denotes the shifted replicas of
:
The FrT for parameters
and
of Equation (
19) is:
The function
of the previous equation represents a sampled version of
with a sampling period that is defined by
. The new convolution operation of
Section 3.2 for parameters
and
z is utilized to perform the filtering of the function
using a low-pass filter in the FrD. This filtering allows us to reconstruct
from
, as follows:
where
in order to avoid aliasing for the reconstruction of
, and the low-pass filter in the FrD is performed with a rectangle function whose support over the coordinate
u is
, which corresponds to the finite support of the FrT
. This last equation denotes the sampling theorem based on the FrT, and this reconstruction equation of
allows us to obtain the values of
for every
x from the sampled version of
when the sampling period is
.
4.2. The Double Random Phase Encoding (DRPE) in the FrD
Double random phase encoding (DRPE) is a very important technique used to encrypt images with a high level of security by using an optical system [
6,
7,
8]. DRPE uses two random phase masks (RPMs) to convert (encrypt) an original image into a noisy image (encrypted image). Initially, the DRPE was implemented in the FD using a holographic 4
f-processor [
1]. Later, the optical DRPE was extended from the FD to the FrD [
12], the fractional Fourier domain [
13] and the Collins domain [
14] in order to improve the security of the encryption system. The DRPE can also be optically implemented with a nonlinear joint transform correlator (JTC) architecture in different optical processing domains (such as FD and FrD) with the purpose of increasing the quality of the decrypted image and obtaining better security over the encrypted image and practical setups with no hard alignment requirements [
3,
4,
5,
15,
16,
17,
18,
19,
20].
In this section, we propose a modified version of the DRPE using the new processing operations based on FrT. This modified version of the DRPE in the FrD introduces new security keys that are given by the parameters of shift and distance of the proposed generalized shift operation in order to increase the security of the encrypted image. The original image to encrypt is a real-valued grayscale image
with values in the interval [0, 1]. We use two RPMs for the encryption-decryption system, which are given by:
where
x and
u are the coordinates in the spatial domain and the FrD, respectively, and
and
are normalized positive functions that are randomly generated, statistically independent and uniformly distributed in the interval [0, 1] [
6,
15,
16].
We define the encrypted image by using the original image
that is encoded in amplitude, the two RPMs
and
, and the proposed generalized shift and convolution operations. The first step of the encryption system is:
where
and
. In this first step of the encryption process, the original image to needs to be encrypted
is encoded in the amplitude and multiplied by the RPM
on the input plane of the encryption system. Then, this product of
is shifted to
and modulated by a pure linear phase term by means of the generalized shift operation
. We perform the convolution for parameters
and
in the first FrD for the parameters
and
between the shifted and modulated product
and the distribution
, which is shifted to
and modulated by another pure linear phase. This convolution in the first FrD for the parameters
and
is computed by the product between the Fresnel propagation (FrT) of the distribution
at distance
when this distribution is illuminated with a plane wave of wavelength
, and the distribution given by the multiplication of the RPM
, a pure linear phase term and a quadratic phase term. The data distribution
is placed on the input plane of the encryption system, and its FrT plane, which is the first FrD for the parameters
and
, is located at distance
with respect to the input plane of the system.
Equation (
23) allows us to place two data distributions centred at different locations on the input plane of the encryption system by using the generalized shift operation; this feature is not possible for the proper working of the DRPE in the FrD presented in [
12]. The application of this generalized shift operations on the input plane of the encryption system allow us to ensure the complete overlapping of the data distributions
and
in the FrD for the parameters
and
. This feature is very important to implement the DRPE technique because the encrypted image will be a noisy image when this feature is fulfilled [
3,
6,
7,
8,
12,
15,
16,
17,
18]. Therefore, the correct implementation of the DRPE in the FrD is achieved due to proposed generalized shift and convolution operations based on the FrT. The proposed new processing operations based on the FrT can be used to describe and analyse the nonlinear encryption systems based on a JTC in the FrD in a compact form [
3,
19,
20].
The second step of the encryption system is to perform the FrT for parameters
and
over the data distribution
:
The previous equation represents a Fresnel propagation (FrT) of the distribution
from the first FrD for parameters
and
to the distance
with the same wavelength
. The resulting data distribution of Equation (
24) is located in the second FrD for parameters
and
. Finally, the encrypted image is defined by:
The encrypted image
is defined in the second FrD for parameters
and
, and it corresponds to image
that is shifted to
and modulated by a pure linear phase term. The six security keys of the encryption system are given by the RPM
, the wavelength
and the distances
,
,
and
. The security keys for the encryption system of this subsection, apart from the RPM key, are added because we used the new processing operations based on the FrT. The proposed encryption system introduces two new security keys given by the distances
and
with respect to the previous DRPE in the FrD [
12]. When the shifts
,
and
are equal to zero, the proposed encryption of this paper is very similar to the DRPE in the FrD presented in [
12].
Figure 1 shows the optical encryption scheme based on the FrT and the new processing operations proposed in this work.
The correct output of the decryption system, which is the correct decrypted image, is obtained from the encrypted image
and the six security key RPMs (
,
,
,
,
and
). The first step of the decryption system is to apply the generalized shift operation with parameters
,
and
to the encrypted image
where
,
and
. In order to retrieve the data distribution
of Equation (
24) without performing shift and modulation by using a pure linear phase term from
, the correct three security keys used in the first step of decryption system have to be set exactly as
,
and
. The second step of the decryption system is to apply the FrT for parameters
and
to the data distribution
In this second step of the decryption system, we use two security keys that are given by
and
. If the following security keys are set by
,
,
and
, the result of Equation (
27) that is given by
would be very different to the
of Equation (
23), and thus, the decryption process will be unsuccessful because the decrypted image will have a noisy distribution. For the case when
,
,
and
, the result of Equation (
27) is:
where
K is a complex-valued constant. The result of Equation (
28) is a shifted and modulated version of the data distribution
, which is defined in Equation (
23). Now, if
,
,
and
, Equation (
28) is reduced to
. The third step of the decryption system is to compute the following product:
where
and
. Equation (
29) is the proposed correlation in the FrD for the parameters
and
between the data distributions of
and
. In this third step of the decryption system, we use the security keys corresponding to
and
. If the following keys are set by
,
,
,
,
and
, Equation (
29) will never be reduced to the data distribution
. Finally, when the six security keys are correct in the decryption system, i.e.,
,
,
,
,
and
, the decrypted image is obtained as follows:
The decrypted image
will be a replica of the original image
when the six security keys in the decryption system are correct. If one or more of the six keys are wrong, the decrypted image will have a noisy data distribution. Therefore, the correct retrieval of the original image at the output of the decryption system is only possible when all of the values of the six security keys used in the decryption system are the same values values that were used in the encryption system. The optical schematic diagram of
Figure 1 can be used to implement the proposed decryption system by using the complex conjugate of
on the input plane of this optical schematic [
12]. The design, description and analysis of the proposed encryption and decryption systems were performed by using the new processing operations proposed in this work. The use of these new processing operations produces a simple and compact characterization and implementation of the behaviour of the proposed modified version of the DRPE in the FrD of this subsection.
Figure 2 shows the computational results for the proposed encryption and decryption systems based on the generalized shift, convolution and correlation operations in the FrD. The images in
Figure 2a,b represent the original image that needs to be encrypted
and the random code image
of the RPM
, respectively. The random code image
of the RPM
is very similar in appearance to the image depicted in
Figure 2b, but the values of random image
are different from the values of random image
. The amplitude and phase of the encrypted image
for the keys
nm,
mm,
mm,
mm and
mm, are shown in
Figure 2c and
Figure 2d, respectively. The right decrypted image
is presented in
Figure 2e, where the correct values of the six security keys are provided to the decryption system. If the wrong value of the security key, such as
, or
, is introduced in the decryption system with the other correct five security keys, the resulting decrypted images will be the noisy images that are depicted in
Figure 2f and
Figure 2g, respectively.
4.3. Possible Case Studies Based on the New Processing Operators of This Work
In our research group, we numerically simulate new phase elements for photovoltaic cells and solar collectors with the purpose of focusing the thermal energy irradiated from the sun. This focusing of the sun’s rays can be performed with new phase elements based on the generalized shift operation proposed in this work. We are designing new phase elements that allow the sun’s rays to be focused on the same point or region, regardless of their orientation when they enter Earth’s atmosphere in order to eliminate the heavy mechatronic solar tracking systems.
The FrT is very important for optical processing systems because this optical transform allows us to implement centred, compact and lens-less optical systems for near-field optics [
3,
19,
20]. Therefore, the new processing operators in this work can describe and analyse these optical processing system in the near-field regime. For instance, the JTC is an optical architecture that can be used for holography, target recognition, object distance detection, video surveillance, authentication, encryption, decryption and DNA sequence alignment, among other applications. The JTC in the FrD allows us to implement compact and lens-less optical systems when the generalized shift operation of this work is applied to each data distribution of the input plane of the JTC because all of the resulting data distributions on the output plane of the JTC in the FrD, which are contained in the joint Fresnel power distribution (JFPD) [
3], are centred at the same place. The different terms of the JFPD can be easily described and analysed by the proposed convolution and correlation operations in this work. This description and analysis of the JTC in the FrD will be simple and compact, which would allow researchers to suitably study of the behaviour for the optical processing systems based on the JTC architecture in the FrD.
Finally, we will expect that several optical processing systems in the FrD for applications in holography, imaging systems, microscopy, diffractive optical elements, spectroscopy, beam focusing, surface modification, optical tweezers, security systems and image filtering systems, among others applications, will be designed, implemented, analysed and improved by using the new processing operators proposed in this work.