# Super-resolved imaging based on spatiotemporal wavefront shaping

Guillaume Noetinger  Institut Langevin, ESPCI Paris, Université PSL, CNRS,
75005 Paris, France  Samuel Métais  Aix Marseille Université, CNRS, Centrale
Marseille, Institut Fresnel, Marseille, France  Geoffroy Lerosey  Greenerwave,
75002 Paris, France  Mathias Fink  Institut Langevin, ESPCI Paris, Université
PSL, CNRS, 75005 Paris, France  Sébastien M. Popoff  Institut Langevin, ESPCI
Paris, Université PSL, CNRS, 75005 Paris, France  Fabrice Lemoult  Institut
Langevin, ESPCI Paris, Université PSL, CNRS, 75005 Paris, France
Corresponding author<EMAIL_ADDRESS>
(September 3, 2024)

###### Abstract

A novel approach to improving the performances of confocal scanning imaging is
proposed. We experimentally demonstrate its feasibility using acoustic waves.
It relies on a new way to encode spatial information using the temporal
dimension. By moving an emitter, used to insonify an object, along a circular
path, we create a temporally modulated wavefield. Due to the cylindrical
symmetry of the problem and its temporal periodicity, the spatiotemporal input
field can be decomposed into harmonics corresponding to different spatial
vortices, or topological charges. Acquiring the back-reflected waves with
receivers which are also rotating, multiple images of the same object with
different Point Spread Functions (PSFs) are obtained. Not only is the
resolution improved compared to a standard confocal configuration, but the
accumulation of information also allows building images beating the
diffraction limit. The topological robustness of the approach promises good
performances in real life conditions.

††preprint: APS/123-QED

Imaging devices exploit waves to retrieve information about an object.
Ultrasound imaging devices or optical microscopes are two widespread
commercially-available technologies relying on different waves, both offering
key insights for medical diagnosis and scientific research. While having
different properties, those approaches rely on the same principles, and their
resolution is limited by the same diffraction effects to a distance of the
order of the wavelength. More specifically, in optical full-field microscopy,
a sample is uniformly illuminated and the waves scattered off an object are
collected with a microscope objective. The finite aperture of the optical
system filters out waves corresponding to high scattering angles and thus the
smaller details [1]. The image of a point, i.e. the Point-Spread Function
(PSF) of the system, is an Airy spot whose first zero is located at a distance
$1.22\frac{\lambda}{2\>N\\!A}$ from the focus (where $\lambda$ is the
operating wavelength and $N\\!A$ the numerical aperture). The Rayleigh
criterion states that two point-like objects closer than this distance cannot
be distinguished [2].

Overcoming this limitation is the domain of super-resolution imaging and many
techniques have already been proposed [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. As
any label-free imaging scheme can be decomposed in two main steps, i.e.
acquiring data and processing the acquired data, those techniques can be
divided in two different classes.

The first strategy consists in shaping the illumination and/or the collection
of waves. The simplest implementation is the well-known confocal microscope
[13, 14], where both the illumination and the detection correspond to a
diffraction-limited volume. In addition to suppressing out-of-focus signals,
it enhances the maximum transmitted spatial frequency by a factor of two [15].
Nevertheless, such improvement is difficult to observe experimentally due to a
poor gain for high spatial frequencies. The increase in lateral resolution is
usually considered to be on the order of roughly 40%. Other approaches were
proposed, all relying on engineering the illumination or the collection
scheme, such as diffractive tomography [16, 17], ptychography [18], structured
illumination [19] or other PSF-engineering [20, 21]. However, all these
techniques remain ultimately limited in resolution by the diffraction.

The second strategy consists in developing algorithms for reconstructing the
image. Indeed, the Rayleigh criterion is somewhat an arbitrary rule. In
particular, it assumes any spatial frequency outside of the transmitted
bandwidth is definitely lost. Nonetheless, strong arguments exist to claim
that the resolution can be limited only by the signal-to-noise ratio [22, 23].
However, the presence of noise in real life experiments makes the problem ill-
posed, forbidding a direct inversion of the mathematical equations. Various
algorithms were developed to regularize the system by compensating for unknown
or noisy information using strong priors [24, 25, 26, 27].

In this letter, we propose an approach combining wavefront shaping and
mathematical deconvolution for improving the resolution. It consists in using
dynamic wavefront shaping to rotate an illumination wavefront, thus using the
temporal domain as an additional way to encode information. We show that it is
equivalent to measuring the image of an object with multiple imaging systems
with different orthogonal PSFs, corresponding to different harmonics of the
received signals. The addition of information provided by those images allows
increasing the effective signal to noise ratio and thus improving the
resolution. We experimentally demonstrate this concept in acoustics in a
confocal-like configuration by reconstructing the image of two small
scatterers. Using a simple deconvolution process we show that two scatterers
can be distinguished below the diffraction limit.

Figure 1: (a) Acoustic experimental setup.  Left Loudspeakers, arranged in a
circle, emit successively to mimic a single rotating loudspeaker. The pressure
field is recorded in a focal plane by a microphone on a translation stage.
Right The previously recorded field is sent back by a loudspeaker next to the
focal plane’s microphone. This is equivalent to reflection by a point
scatterer. The backscattered field is recorded by microphones arranged on a
circle, which again mimic a rotating receiver by properly time-gating and
summing all the signals.  (b) Temporal signals and spectrums. Backscattered
signals for a scatterer on the rotation axis (top) and off-axis (middle). As
shown by their spectrum (bottom), the on-axis signal is almost monochromatic
while the off-axis one presents peaks at each harmonic $\omega_{0}+n\Omega$.
In the next figure we examine the structure of the fields corresponding to
$n\in[\\![0;3]\\!]$ in space. (black dotted box)

The fundamental idea of the proposal relied on creating a singularity that
would allow discriminating precisely one point from the rest of the object.
This is achieved using rotating wavefronts. Point-like scatterers at different
positions would perceive different time-modulations, depending on their
distance to the rotation axis. Only a scatterer located exactly on the
rotation axis backscatters an unmodulated wave. This modulation can also be
understood using the Doppler effect: in the rotating frame, the object is
rotating and thus Doppler shifts the waves. The further from the rotation
axis, the more the frequency of the signal is shifted. Interestingly, as the
rotation comes periodically to the same position, everything is actually
encoded as a modulation on an initial monochromatic excitation.

For the sake of ease of access to the temporal field, we implement this
concept with acoustic waves in the audible range, but the same framework could
readily be transposed to other wave-based imaging systems, such as optical
microscopes. In order to generate a rotating source, we place a set of 16
loudspeakers and microphones on a 55-cm-radius-circle. Each of them is
connected to a multi-channel Antelope Orion32+ soundboard (see Fig. 1)
controlled through a computer. The speakers emit sequentially a time-gated
monochromatic signals in such a way that it is equivalent to having a rotating
source emitting a monochromatic signal at a frequency
$\omega_{0}=1600\leavevmode\nobreak\ $Hz. (equivalent to a wavelength of
$21.4$ cm). This has the advantage of not requiring any physical rotation. The
rotation angular frequency $\Omega=4\leavevmode\nobreak\ $Hz is chosen so that
the speed of the rotating emitter is low compared to the wave velocity. A
microphone placed on a translation stage records the acoustic field in a plane
(hereinafter called the focal plane) located 1.7 meters above the speakers.

To mimic a point-like scatterer re-emitting in all directions, a microphone
and a loudspeaker are used to record the signal at a target position in the
focal plane and then re-emit the recorded signal (Fig. 1a right). The
microphones arranged on circle then measure the backscattered field. Finally,
the recorded signals are time-gated and summed to mimic a rotating detector
(see SI for experimental details). We first show the transient back-scattered
signals of a scatterer located on and outside the rotation axis in the focal
plane (Fig. 1b). As expected, the spectrum of the signal corresponding to the
scatterer on the axis reveals to be almost monochromatic. On the contrary, the
signal for the off-axis scatterer displays a periodic temporal modulation of
its envelope. In the Fourier space, this corresponds to a frequency comb
centered around $\omega_{0}$ with a spacing between peaks corresponding to the
rotation frequency $\Omega$ (see bottom of Fig. 1b). Similarly, all the
transient signals in this experiment display similar periodic modulation, and
from now on only the data corresponding to the different harmonics
$\omega_{0}+n\Omega$, with $n\in\mathbb{Z}$, are displayed.

In order to investigate the imaging capabilities of our system, a full scan of
the focal plane is performed. The field in the focal plane as well as the
back-scattered signals are recorded. The spatio-temporal signals are then
projected onto the different harmonics. The maps of the complex field in the
focal plane corresponding to the harmonics in the range $[\\![0;3]\\!]$ are
represented in the first part of Fig. 2.

Figure 2:  (a) Field in the focal plane for the harmonics $\omega_{0}+n\Omega$
for $n\in[\\![0;3]\\!]$. (b) Backscattered signals
For each dataset the first row corresponds to the intensity and the bottom row
is the phase (weighted by the amplitude).

The field at harmonic $n=0$, or equivalently at the initial carrier frequency
$\omega_{0}$ (top first column of Fig. 2), reveals a focal spot at the central
position. Its width is approximately $1.3\lambda$. It corresponds to the
diffraction limited focal spot that would be obtained through a thin ring-
aperture of $N\\!A=0.31$ illuminated by a monochromatic wave at $\omega_{0}$.
Reciprocally, it corresponds to the $PSF$ of a full-field microscope with a
thin ring-aperture, with a first zero at a distance
$l_{\textrm{res}}=0.76\frac{\lambda}{2\>N\\!A}=28$ cm from the focal spot (see
SI). The field at the harmonic $n$ is a vortex of topological charge $n$. The
field is concentrated on a ring whose radius increases with $|n|$. As
expected, the central point who is on the rotation axis acts as a singularity.
Similarly to the field at $\omega_{0}$, they are equivalent to the $PSF$ of a
full-field acoustical microscope with a thin ring aperture on which a vortex
plate would have been added.

The images corresponding to the first harmonics of the backscattered signals
are also represented (bottom of Fig. 2). They strongly resemble to the maps in
the focal plane, and notably they all exhibit similar vortices. However, there
is a homothety relationship between the two set of images so that the rings
corresponding to the maximum of intensity at each harmonic have a radius
diminished by the same factor 2. This effect is a direct consequence of the
dynamic configuration used in this experiment. It is similar to the doubling
of the spatial frequency support obtained in structured illumination
microscopy [19] or diffractive tomography [16].

The next step consists in harnessing the new diversity of signals offered by
the dynamic nature of the process to improve the imaging resolution below the
diffraction limit. Our apparatus is equivalent to a confocal microscope in a
monochromatic highly-coherent configuration, except that the illumination
pattern is modulated periodically in time. As a result, the information of the
object is encoded in the different harmonic frequencies. Consequently, we
obtain as many images $\textrm{Im}_{n}(x,y)$ as the number of measured
harmonic frequencies. Each of the images is also the convolution product
between the object and the Point Spread Function $PSF_{n}(x,y)$, which is
specific to the harmonic $n$, and corresponds to a diffraction limited vortex
of vorticity $n$. As two vortices with different topological charges are
orthogonal, each channel carries a different information about the object.
Moreover, thanks to the reduced apparent wavelength previously observed in the
backscattered $PSF$s, the resolution is increased.

Figure 3:  Deconvolution procedure. The images at each harmonic are convolved
with a regularized inverse of the $PSF_{n}$ . All the independent
reconstructions are summed to give a final sharp image.  Figure 4:
Quantification of the resolution limit of the system Two emitters in the focal
plane are imaged in a confocal or dynamic way from full-field resolution to
super-resolution regime. The field of view is approximately $4\lambda$ in X/Y.
The images are shown in normalized intensity and the maximum profile on the
separation axis (X) is shown above each image .
Comparison between the confocal images (top row), the images at $\omega_{0}$
(middle row) and the deconvoluted images (bottom). Close to
$\frac{l_{res}}{2}$, the confocal setup fails to distinguish two diffusers. In
the dynamic images there are hints of the presence of the two diffusers in the
signals: in the $\omega_{0}$ images shown they consists in the observed X/Y
asymettry visible in the secondary lobes. With the deconvolution process the
two point-like objects are made distinguishable further from this limit.

The images $\textrm{Im}_{n}$ corresponding to two point-scatterers separated
by $30$ cm are acquired. The images of the first harmonics are displayed on
the left of Fig. 3 as color coded images of their phase. Each of them shows a
vortex centered on each scatterer. The vortex has a topological charge that
corresponds to the considered harmonic.

For each harmonic, a deconvolution is needed in order to obtain an image of
the two scatterers. This consists in convolving the images by the inverse
$iPSF_{n}$ of the point-spread function $PSF_{n}$ as depicted in Fig. 3. Such
an inverse is calculated from the experimental $PSF_{n}$ obtained with a
single emitter in the spatial Fourier domain as it only involves dot products.
To be robust to the presence of noise, a Tikhonov regularization [28, 29, 30]
is achieved. Each pseudo-inverse operator writes:

$i\widehat{PSF}_{n}=(\widehat{PSF}_{n}^{*}\cdot\widehat{PSF}_{n}+\sigma)^{-1}\cdot\widehat{PSF}_{n}^{*}$
(1)

where $\sigma$ corresponds to the noise-to-signal ratio, $\cdot^{*}$ denotes
the complex conjugate and $\widehat{{\>.\>}}$ the 2D spatial Fourier
transform. Note that in the real space, the inverse $iPSF_{n}$ exhibits a
vorticity with a charge opposite to $PSF_{n}$.

By deconvolution, each harmonic provides its own intensity image of the two-
scatterers. The last step consists in summing all the images from harmonic
$-4$ to $4$ and a final image is built (right of Fig. 3). The signals add up
and the average noise decreases leading to a cleaner final image of the two
scatterers.

We then change the distance between the two scatterers to find when it becomes
impossible to distinguish them. Fig. 4 summarizes the results. For the sake of
generality, classical confocal images, ie. where all the speakers emit
simultaneously and all the microphones receive simultaneously, are also
performed. These images reveal that the two scatterers cannot be resolved when
their distance is less than $22$ cm ($\simeq\lambda=0.79l_{\textrm{res}}$,
corresponding to $30$% of the diffraction limit). The images at harmonic $n=0$
are represented in the central row. They show smaller spatial variations but a
high level of artifacts not allowing to distinguish any objects. Eventually,
when taking into accounts all the available harmonics and their inversion
procedure, it is possible to beat the Rayleigh criterion and reach a
resolution of $12.5\leavevmode\nobreak\ $cm = $0.58\lambda$. It corresponds to
an improvement of 70% compared to the confocal configuration and 10% compared
to structured illumination ($l_{\textrm{res}}/2$). Noise remains the
limitation: the noise-to-signal ratio parameter $\sigma$, estimated at $0.003$
in those experiments, still limits the performance. This is a standard order
of magnitude in microscopy [24], and any enhancement would immediately
translate into an increase of the resolution capabilities. Moreover, further
improvements can be made on the reconstruction using an iterative algorithm
such as Jansson-Van Cittert [31] or a statistical one like Richardson-Lucy
[32] where our solution would be used as an initial guess allowing one to
envision enhanced performances of this approach. In conclusion, we have
proposed to exploit the time domain as a way to improve the resolution of an
imaging device and demonstrated its applicability with an experiment based on
audible acoustics. We showed that projecting a time-periodic pattern on a
sample allows acquiring multiple images of the same sample with orthogonal
$PSF$s. Using a rotating pattern, those $PSF$s correspond to vortices with
different topological charges. The augmentation of the amount of information
allows us to experimentally beat the diffraction limit, even in the presence
of the inherent experimental noise. This new modality of imaging tackles the
issue of diffraction in imaging systems in a novel way. In microscopy, it
could offer an easy way to obtain label-free high or super resolution images.
This strategy could also be combined with non-linear phenomena such as
fluorescence saturation involved in non-linear structured illumination [33].

This work is supported by LABEX WIFI (Laboratory of Excellence within the
French Program “Investments for the Future”) under references ANR-10-LABX-24
and ANR-10-IDEX-0001-02 PSL*, and partially by the Simons
Foundation/Collaboration on Symmetry-Driven Extreme Wave Phenomena.

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# EmbAu: A Novel Technique to Embed Audio Data using
Shuffled Frog Leaping Algorithm ††thanks: Corresponding author: Ankit
Chaudhary<EMAIL_ADDRESS>
Sahil Nokhwal  Dept. Computer Science
University of Memphis
Memphis, TN, USA
<EMAIL_ADDRESS>Saurabh Pahune  Automation analyst
Cardinal Health
Dublin, Ohio, USA
<EMAIL_ADDRESS>Ankit Chaudhary  Dept. Computer Science
University of Missouri
St Louis, MO, USA
<EMAIL_ADDRESS>
###### Abstract

The aim of steganographic algorithms is to identify the appropriate pixel
positions in the host or cover image, where bits of sensitive information can
be concealed for data encryption. Work is being done to improve the capacity
to integrate sensitive information and to maintain the visual appearance of
the steganographic image. Consequently, steganography is a challenging
research area.

In our currently proposed image steganographic technique, we used the Shuffled
Frog Leaping Algorithm (SFLA) to determine the order of pixels by which
sensitive information can be placed in the cover image. To achieve greater
embedding capacity, pixels from the spatial domain of the cover image are
carefully chosen and used for placing the sensitive data. Bolstered via image
steganography, the final image after embedding is resistant to steganalytic
attacks. The SFLA algorithm serves in the optimal pixels selection of any
colored (RGB) cover image for secret bit embedding. Using the fitness
function, the SFLA benefits by reaching a minimum cost value in an acceptable
amount of time. The pixels for embedding are meticulously chosen to minimize
the host image’s distortion upon embedding. Moreover, an effort has been taken
to make the detection of embedded data in the steganographic image a
formidable challenge.

Due to the enormous need for audio data encryption in the current world, we
feel that our suggested method has significant potential in real-world
applications. In this paper, we propose and compare our strategy to existing
steganographic methods.

###### Index Terms:

Steganography, Cipher, Data hiding, Data Security, Confidentiality, Privacy,
Cryptography, SFLA algorithm

##  I Introduction

Increases in both home and office computer use have led to a meteoric rise in
the number of people connected to the internet over the past two decades,
resulting in an increase in privacy concerns, security problems worldwide[1,
2, 3, 4, 5], cybercrime, as criminals attempt to steal high-value assets,
which is now in the form of data. In recent years, there have been a rising
number of eavesdropper attacks and information thefts. Cybersecurity experts
and researchers on the other hand, aim to protect network data transmission
from unauthorized interception by inventing the appropriate method, many of
which are based on cipher algorithms, where data is first encrypted by the
sender, then transmitted and must be decrypted by the receiver in order to get
the confidential message. In this way, access to the data is restricted to
only those who know how to decrypt it. Despite the fact that data encryption
has yielded outstanding results in the past and continues to do so, there is a
significant research interest in enhancing the present encryption methods.

Cipher text is inconspicuous, and thus attracts the attention of unauthorized
users, preventing even authorized recipients from receiving it. A more nuanced
approach would be to disguise the information in a manner that its presence
cannot be detected by unauthorized users. These methods of information hiding
are relatively new areas of research interest and yet have shown considerable
promise. A method of hiding information involves hiding data from one file
into another, the files may or may not have the same format, for instance,
hiding an image in another image or audio data in an image.

Steganography as a method of data hiding boasts an incredible storage capacity
[6, 7, 8, 9, 10, 11]. When an image is chosen as the medium of information
hiding, the image into which information is embedded is called the host image
or cover image. After data embedding, the final image is called a
steganographic image [12, 13, 14, 15, 16]. Steganography is performed for
three purposes: 1) The first is to hold or deliver payload (data) [11, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22], and the second is to protect this payload
from being accessed by people other than the intended users, and 3) is to
secure the overall system [23, 24, 6, 25]. Thus, there is much research
interest in methods of embedding data into images, while optimizing for higher
payloads [26, 3, 27] without compromising much on image quality. To ascertain
that a desired standard of quality is maintained, we measure image visual
quality using peak signal-to-noise ratio (PSNR) and Structural Similarity
Index (SSIM) metrics [28, 29]. Also, the storage capacity of an image is
measured in bits-per-pixel. In this research, we present a novel approach for
data embedding in images, in which we seek to determine the ideal sequence of
pixels in which to embed data while maintaining image quality. Here we build
upon the technique proposed in [30] and demonstrate improved results. Our
suggested method gives us control over the convergence speed of data embedding
and thus can be faster. This proposed method is of special interest also due
to it being based on swarm intelligence and thus resulting in optimal pixels
selection for data embedding. The remaining sections in this paper are laid
out as follows: section III explains the related work, followed by explanation
of the SFLA algorithm in section IV, section V provides details about the
proposed technique, and section VI shows the experimental results. In section
VII, the conclusion of our research has been discussed.

Figure 1: Architecture of proposed steganographic method

##  II Motivation

Despite the potential importance of image steganography, it has been largely
overlooked in favor of more well-known security methods like encryption and
watermarking. Limited robustness, stealth, and payload capacity of current
steganographic techniques are three areas where the current literature is
lacking that are being investigated here. Lack of security, a small payload,
poor resilience, and significant computing costs are the salient drawbacks of
existing approaches.

1 Function  _SFLAalgorithm(_$\mathcal{C},\mathcal{A}$_)_:

Input: $\mathcal{C}$ \- Cover image, $\mathcal{A}$ \- audio file (optional)

Output: $\mathcal{Q}$ sequence of optimal pixel locations

2 Begin: $\mathcal{F}\leftarrow$ Generate virtual population of frogs

3 while  _Termination condition not met_ do

4 Arrange the $\mathcal{F}$ frogs in decreasing order of their fitness.

5 Let $X_{g}$ denote the location of the frog with the best global
performance.

6 Form $m$ memeplexes, each containing $n$ frogs.

7 Allocate frog $i$ to memeplex $r=i\mod m$

8

9 for  _each memeplex_ do

10 Create a submemeplex of $q$ chosen at random frogs

11 for  _each submemeplex_ do

12 The submemeplex’s best and worst frogs’ positions will be denoted by
$X_{b}$ and $X_{w}$, respectively.

13 Determine worst frog’s new location ($X_{new1}$) using
$X_{new1}=X_{w}+r(X_{g}-X_{w})$; where $r$ is an arbitrary integer ranging
from 0 to 1 generated uniformly

14

15 if  _Performance of $X_{new1}$ is inferior than $X_{w}$_ then

16 Determine the wrost frog’s new location ($X_{new2}$) using

17 $X_{new2}=X_{w}+r(X_{g}-X_{w})$;

18 if  _Performance of $X_{new2}$ is inferior than $X_{w}$_ then

19 Create a replacement frog $X_{w}$ at random.

20 else

21 Substitute $X_{w}$ for $X_{new2}$

22

23 else

24 Substitute $X_{w}$ for $X_{new1}$

25 Arrange memeplex’s $n$ frogs in decreasing order.

26 Iterate for $N$ iterations

27

28 The memeplexes would be thoroughly merged

29 $\mathcal{Q}\leftarrow$ return the optimal sequence of pixels

Algorithm 1 SFLA algorithm to find a sequence of optimal pixels

##  III Related work

Most existing literature on steganographic techniques is based on the Particle
Swarm Optimization (PSO) algorithm[13]. These works outline PSO-based
techniques that optimally find pixels in which secret information is to be
embedded while retaining image quality. Authors in [31, 32] discuss a well-
performing technique that involves embedding the most significant bit of
secret data in the least significant bit of an image. In [33, 34], use
adaptive neural networks to cut down on noise in the steganographic image.
Authors in [35] proposed a method that utilizes wavelet decomposition. In
[36], a technique is detailed that hides data in images using wavelet
transforms, and uses the PSO algorithm to find optimal pixels for such hiding.
In [37], the proposed approach is a hybrid of optimal chaotic-based encryption
method and noise visibility function which boasts of high payloads. [38] makes
use of the difference between neighboring pixels of the host image, and uses
the modulo function for data hiding. Careful modification of digital image
pixels and hiding of sensitive information in the least significant bits
(LSBs) of image components, the authors of [39, 40], suggested a variety of
effective data concealing techniques. Authors of [41] had previously explored
the random LSBs-based hiding, later built a steganographic approach with
variable least significant bits (VLSB). Using several methods detailed in [42,
43, 41] successfully implemented variable-LSB (VLSB) steganography. For the
data concealing and retrieval processes, they devised a prediction system and
decision structure for selecting the insertable grouping of data bits. Medical
raw data encoding [44], image compression [45], and internet of things
security [46], are also common applications of absolute moment block
truncation coding (AMBTC) based approaches. Confidential data security was
further enhanced by using block-based data-hiding methods, and novel data-
hiding strategies are presented to assure data safety. Some of the promising
block-based methods were proposed in [47, 48].

1 Function  _EmbeddingData(_$\mathcal{C},\mathcal{K},\mathcal{A}$_)_:

Input: $\mathcal{C}$ \- Cover image to hide the audio data, $\mathcal{K}$ \-
Secret key used for encrypting audio data, $\mathcal{A}$ \- audio file

Output: $\mathcal{S}$ steganographic image

2

3 $\mathcal{B}\leftarrow$ Read audio data and convert it into bits

4 $\mathcal{B}\leftarrow$ Encrypt audio bits using AES algorithm

5 $\mathcal{F}\leftarrow$ Generate virtual population of frogs

6 $\mathcal{Q}\leftarrow$ gets optimal sequence of pixels to hide data using
the SFLA algorithm 1

7 $\mathcal{S}\leftarrow$ Embed the encrypted audio bits into the cover image
and return steganographic image

Algorithm 2 Proposed algorithm for embedding data

1 Function  _RetrievingData(_$\mathcal{S},\mathcal{K}$_)_:

Input: $\mathcal{S}$ \- Steganographic image to retrieve the audio data,
$\mathcal{K}$ \- Secret key used for decrypting the audio data

Output: $\mathcal{A}$ audio file

2 $\mathcal{F}\leftarrow$ Generate virtual population of frogs

3

4 $\mathcal{Q}\leftarrow$ gets optimal sequence of pixels to hide data using
the SFLA algorithm 1

5

6 $\mathcal{A}\leftarrow$ Retrieve encrypted audio data

7 $\mathcal{A}\leftarrow$ Decrypt audio bits using AES algorithm

8 $\mathcal{A}\leftarrow$ Make a audio file and return

9

Algorithm 3 Proposed algorithm for retrieving data

##  IV The SFLA algorithm

Based on the flocking behavior of frogs, a swarm-based, memetic algorithm SFLA
was designed that performs a local and global search to find an optimal or
near optimal solution. The SFLA algorithm makes use of both stochastic and
deterministic techniques. Because of its deterministic approach, the SFLA
algorithm is able to make good use of the information provided by the local
search values to direct the heuristic exploration, much as is done in the PSO
algorithm. The search process is adaptable and reliable because of the random
variables. Initially, a considerable number of random frogs $\mathcal{F}$ are
released into the marsh to swarm everywhere. The population is split up into a
few separate parallel groups, known as memeplexes, and each of these groups is
allowed to grow independently so that they may explore the space in a wide
range of ways. The frogs undergo a memetic evolution as a result of being
influenced by the concepts of other frogs while they are participating in each
memeplex. The effectiveness of every meme is improved via the process of
memetic evolution, which also increases a frog’s overall effectiveness toward
achieving a desired result. Frogs that have superior memes (ideas) are allowed
to contribute greater amounts to the production of new ideas than frogs that
have poor ideas. This is done to guarantee that the influenced procedure
remains competitive. A competitive edge may be gained by superior ideas via
the use of a triangle probability distribution in the selection of frogs.
Frogs may update their memes throughout the evolution by taking the most
effective ideas from the whole population or just the memeplex. Updates across
the memotype(s) that occur incrementally correlate to the size of a leaping
stride, and the appearance of a new meme represents the frog’s new location.
Each frog is given the opportunity to improve itself before being released
back into the memeplex. There is no delay in using the acquired knowledge to
again improve the condition. This method differs from a genetic algorithm in
that it does not need the whole population to be altered prior to new
findings. Instead, this method provides quick access to newly discovered
information. Here, the graphic depiction of frogs is utilized as a comparison,
but in terms of the dissemination of ideas, the parallel may be used to groups
of scientists or scholars enhancing an idea or experts incrementally improving
a model. When the meta-heuristic evolutionary cycles reach a specific
threshold, the memeplexes are compelled to interbreed, and novel memeplexes
get generated as a result of a shuffling procedure. After coming into contact
with frogs from various parts of the such swamp, the memes benefit from this
mixing and matching. The process of looking for a potential solution is
accelerated by the shuffling of frogs, which also assures that there is no
local prejudice in the progress of any special relevance.

##  V Proposed Technique

The proposed technique has been divided into two sections, i.e., embedding
audio data into the cover image; and retrieving the already hidden data from
the steganographic image.

###  V-A Embedding audio data using the SFLA algorithm

Our proposed technique aims to locate the sequence of the optimal pixels for
embedding audio data so that distortion in the cover image is minimized. The
steps for our proposed technique are

  1. 1.

First read the confidential audio file that needs to be stored in the cover
image in its decimal format

  2. 2.

Encrypt the decimal values obtained from the audio file using the AES
encryption algorithm which outputs 128-bit data

  3. 3.

In order to save this encrypted data in the cover image, these encrypted
128-bits data would be transformed into a series of binary bits

  4. 4.

The cover image’s pixels are all transformed to their 8-bit equivalents

  5. 5.

A custom objective function is developed based on the multi-objective
scenario. It is a metric used to measure the quality of a steganographic image
compared to the original signal. PSNR is calculated as the ratio of the
maximum possible power of a signal to the power of the noise that affects the
fidelity of its representation. The PSNR value is expressed in decibels (dB)
and it provides a quantitative measure of the amount of distortion present in
the steganographic signal. The higher the PSNR value, the better the quality
of the steganographic image.

The Structural Similarity Index (SSIM) is a metric used to measure the
similarity between two images. It is a structural similarity measure that
takes into account the structural information and luminance of the images
being compared. It attempts to model the human visual system by measuring the
changes in structural information, luminance, and contrast between the two
images. SSIM is calculated by comparing the luminance, contrast, and
structural information of two images. It outputs a value between -1 and 1,
where a value of 1 indicates that the two images are identical, and a value of
-1 indicates that the two images are completely dissimilar.

Mean Squared Error (MSE) is a commonly used metric to measure the average
squared difference between the estimated values and the true values of two
images. It measures the quality of a steganographic image. The calculation is
done by taking the average of the squared differences between the
corresponding pixels or samples in the cover and steganographic images.

$MSE=\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}\left[\mathcal{X}(i,j)-\mathcal{Y}(i,j)\right]^{2}$

where $(i,j)$ represents the $i^{th}$ and $j^{th}$ pixel of the cover
$\mathcal{X}(i,j)$ and steganographic $\mathcal{Y}(i,j)$ image, and $|m|=|n|$
as the total pixels of the cover and steganographic image are equal.

The equation for PSNR can be formulated as

$PSNR=10\times\log\left(\dfrac{MAX^{2}_{C}}{MSE}\right)$

$PSNR=10\times\log\left(MAX^{2}_{C}\right)-10\times\log\left(MSE\right)$

Where $MAX_{C}$ represents the maximum pixel value of the cover image.
Therefore, a general fitness function can be formulated as

$Z(\mathcal{X},\mathcal{Y})=\alpha\times
SSIM(\mathcal{X},\mathcal{Y})+(1-\alpha)\times\dfrac{PSNR(\mathcal{X},\mathcal{Y})}{100}$

Here, $\mathcal{X}$ and $\mathcal{Y}$ represent two images that are being
compared. In our work, $\mathcal{X}$ represents a steganographic image, and
$\mathcal{Y}$ represents a cover image. As the cover image $\mathcal{Y}$ is
fixed during the entire process of embedding, the fitness function can be
formulated as

$Z(\mathcal{S},\mathcal{C})=\alpha\times
SSIM(\mathcal{S},\mathcal{C})+(1-\alpha)\times\dfrac{PSNR(\mathcal{S},\mathcal{C})}{100}$
(1)

Here the value of $\alpha$ is taken as 0.5.

Here the value of SSIM gives the universal image quality index, which is
described as follows

$Q=\dfrac{4\sigma_{uv}\hat{u}\hat{v}}{(\hat{u}^{2}+\hat{v}^{2})(\sigma_{u}^{2}+\sigma_{v}^{2})}$

Here, the steganographic image’s window size is represented by $u$ and $v$,
and its average is represented by $\hat{u}$ and $\hat{v}$. The variance of $u$
and $v$ is represented by $\sigma^{u}_{2}$ and $\sigma^{v}_{2}$, and their
covariance is represented by $\sigma_{uv}$.

The metric takes Human Visual System (HVS) into account. The HVS’s model
distortions comprise of three different elements: luminance distortion, loss
of correction, and contrast distortion. They are mathematically modeled as
follows:

$Q=\dfrac{2\sigma_{uv}}{\sigma_{u}\sigma_{v}}\times\dfrac{2\hat{u}\hat{v}}{\hat{u}^{2}+\hat{v}^{2}}\times\dfrac{\sigma_{u}\sigma_{v}}{(\sigma_{u}^{2}+\sigma_{v}^{2})}$

The first element exhibits some linear correlation and has an inclusive
dynamic range of -1 to 1. The dynamic range of the second element is between 0
and 1 inclusive, and it displays the brightness $u$ and $v$. The range of the
third element, which represents how comparable the images’ contrasts are, is
between 0 and 1. Consequently, the dynamic range of $Q$ is between -1 and 1.
When the denominator turn in the QUI approaches zero, then results become
unstable. To avoid this problem, we use a generalized metric called SSIM and
it tells how similar two images are using the following equation

$SSIM(u,v)=\dfrac{(2\hat{u}\hat{v}+c_{1})(2\sigma_{uv}+c_{2})}{(\hat{u}^{2}+\hat{v}^{2}+1)(\sigma_{u}^{2}+\sigma_{v}^{2}+c_{2})}$

Where $u$ and $v$ indicate the window size for the original image and
steganographic image, respectively, and $u$ and $v$ denote the mean of $u$ and
$v$. $\sigma^{u}_{2}$ and $\sigma^{v}_{2}$ denote the variances of $u$ and
$v$, whereas $\sigma_{u}v$ indicates their covariance.

  6. 6.

_Population initialization:_ Initial population generation of frogs is based
on random initialization and then dividing the population evenly by the
required number of memeplexes. Generate random $\mathcal{F}$ frogs
$\mathcal{U}(1)$, $\mathcal{U}(2)$, $\mathcal{U}(3)$,$\mathellipsis$,
$\mathcal{U}(F)$ those lies in solution space $\Omega\in\mathbb{R}^{d}$, where
$\mathcal{D}$ represents the total number of determinants. Therefore, a vector
of values $\mathcal{U}(i)$ = $\mathcal{U}(2)$,
$\mathcal{U}(3)$,$\mathellipsis$, $\mathcal{U}(F)$ is described for the
$i^{th}$, decision variable stands in for the $i^{th}$ frog. Determine the
$\mathcal{f}(i)$ value to every frog in the set $\mathcal{U}(i)$.

  7. 7.

_Initialize memeplexes:_ The number of frogs $\mathcal{F}$, is divided into
memeplexes, i.e., $m$ and each memeplex has $\eta$ frogs, where $\eta$ is an
integer. Hence, the equation depicting the aggregate number of samples
$\mathcal{F}$ in the swamp can be formulated as

$\mathcal{F}=m\times\eta$

Here $\mathcal{F}$ depicts the total number of frogs in the initial
population, $m$ represents the total number of memeplexes, and $\eta$
corresponds to the number of frogs within every memeplex.

  8. 8.

_Rank frogs:_ Each frog’s fitness is determined and used to arrange the
population in memeplexes in descending order and in a round-robin manner.

  9. 9.

_Perform Search:_ A local search is performed in each memeplex based on a
memetic algorithm. After a few iterations, a few frogs jump from one memeplex
to another, then the fitness of each frog is recalculated, after which they
rejoin the memeplex based on their fitness. Thus, a new population is
constructed. These steps are repeated until terminating conditions are met or
an optimal solution is found.

  10. 10.

Using the SFLA approach, determine the best set of pixels to mask the data in
the host image.

  11. 11.

Utilize the 1-LSB approach to embed the encrypted data bits in the cover image
with minimal image distortion

###  V-B Retrieving the embedded audio data from steganographic image

  1. 1.

Resize the steganographic color image of 3 channels (m x n x 3) into m’ x n’
image (2-dimensional)

  2. 2.

A custom objective function based on SSIM and PSNR values is developed as per
equation 1

  3. 3.

_Population and memeplexes initialization:_ The initial population generation
of frogs is done as per as the section V-A

  4. 4.

After configuring all settings, the SFLA algorithm is executed to minimize the
objective function, yielding an optimal sequence of pixels for retrieving
audio data

  5. 5.

Retrieve the encrypted data based on the sequence obtained from the previous
step

  6. 6.

Decrypt the data into its binary representation using the AES algorithm

  7. 7.

Make an audio file using the decrypted binary bits

Figure 2: Before

Figure 3: After

Figure 4: Before and after steganography - Fashion

Figure 5: Before

Figure 6: After

Figure 7: Before and after steganography - Cosmic

Figure 8: Before

Figure 9: After

Figure 10: Before and after steganography - F16

Figure 11: Before

Figure 12: After

Figure 13: Before and after steganography - Stars

Figure 14: Before

Figure 15: After

Figure 16: Before and after steganography - Ship Figure 17: Comparing results
of different SOTA steganographic methods

##  VI Experimental Results

Extensive experiments have been conducted to evaluate the proposed method’s
performance and the proposed method has shown state-of-the-art results in
hiding an audio secret data in an image. We have compared our results with
state-of-the-art steganographic methods. The results figure shows the
comparison of Normal LSB, [32], [49], [30] and our proposed method.

##  VII Conclusion

In this work, we propose a unique method for concealing audio data inside
otherwise unnoticeable image files, one that achieves outstanding results with
minimal degradation of the original images. In order to obtain the best
possible sequence of pixel locations to hide the secret audio data, the SFLA
algorithm has played a crucial role. By mimicking frog leaping behavior, the
SFLA algorithm search makes use of nature’s evolutionary mechanisms to quickly
arrive at the optimal solution. In keeping with the practice of most pixel
replacement techniques, we have ensured that even after embedding audio in the
host image, the overall size is not increased. When compared to other state-
of-the-art approaches that aim to conceal binary audio data in an image, the
proposed method excels in both the speed with which the optimal/sub-optimal
solution is reached and the low degree of distortion introduced to the host
image.

## Acknowledgment

We want to thank Dr. Nirman Kumar<EMAIL_ADDRESS>for insightful
discussions.

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