Experimental and simulation orthogonal ghost imaging for measurement-domain optical steganography

Abstract

We present an experimentally validated measurement-domain steganography framework based on orthogonal ghost imaging (OGI). In the proposed scheme, a compact single-pixel OGI front-end with orthogonal illumination patterns generates the cover measurements, while a hash-initialized logistic map drives a chaotic keystream that is embedded in the least significant bits (LSBs) of the bucket intensities. This integration into a single optical–digital pipeline avoids holographic components, deep-learning-based reconstruction, and multi-channel architectures, and keeps both the hardware and computational complexity moderate enough for resource-constrained optical platforms. A combination of numerical simulations and single-pixel optical experiments shows that the framework supports imperceptible embedding with stable statistical behaviour. Under the reported operating conditions, the stego ghost images retain PSNR values above 50 dB, SSIM exceeding 0.996, and NPCR close to 99.98%, while intensity histograms, pixel correlations, higher-order moments, and chi-square statistics of the LSB plane remain almost unchanged. These results indicate that chaos-encrypted, bucket-domain LSB modulation can be effectively hidden within the speckle-like statistics and correlation-based reconstruction of OGI, allowing reliable message recovery without visible artifacts while exhibiting statistical consistency with basic steganalysis tests in the OGI domain. Overall, the proposed approach provides a proof-of-concept route to measurement-domain information hiding with hardware-generated covers. Owing to the use of a compact single-pixel architecture and lightweight digital processing, the framework suggests potential relevance to optical scenarios in which system complexity and processing resources are constrained. These considerations are based on architectural simplicity rather than measured power, volume, or embedded runtime metrics. A full system-level implementation and experimental validation on specific embedded or mobile platforms remain beyond the scope of this work and are left for future investigation.

Introduction

Ghost imaging (GI) is an unconventional optical imaging technique that reconstructs an object’s image using second-order intensity correlations rather than direct pixel-wise detection^{1, 2}. Initially demonstrated with entangled photon pairs, GI was later extended to classical thermal light and computational implementations, which substantially broadened its practicality and ease of deployment^3–9. These developments have enabled diverse applications in photonics, including three-dimensional imaging^10,11, X-ray diagnostics^12,13, imaging through turbulence¹⁴, and even quantum-assisted face recognition^15,16. Among the various GI variants, orthogonal ghost imaging (OGI) is particularly attractive because it supports efficient single-pixel reconstruction and is compatible with low-complexity optical setups¹⁷.

The unique randomness and non-locality of GI have recently motivated its use in optical security, including secure key distribution, digital watermarking, and authentication^18–27. A growing line of work integrates GI with steganography, aiming not only to secure data but also to conceal its very existence^28–31. In contrast, conventional spatial- and transform-domain digital steganography operates purely on pre-acquired pixel images and relies on software-generated pseudorandomness, without exploiting the physics of the optical acquisition process^31–36.

Despite these advances, existing GI-based steganographic and security schemes still face several important limitations. First, many approaches rely on deep-learning models or holographic media, which increase hardware and implementation complexity^29,37,38. Second, several recent methods depend on sweeping computational ghost imaging (CGI) strategies that require large measurement counts and are not well suited to lightweight or real-time single-pixel operation³⁰. Third, chaos-based encryption and LSB embedding are typically treated as separate stages^30,39, which complicates the pipeline and can weaken statistical robustness in the stego domain.

Beyond GI-based steganography, recent single-pixel imaging (SPI) studies have explored security from different angles, including adversarial robustness of deep-learning–based reconstruction and encrypted transmission using scrambling and public-key exchange, as well as purely digital chaos-driven image encryption schemes based on DCT compression and nonlinear dynamics^40–42. While these works advance image security and robustness, they focus on encryption or reconstruction stability rather than on measurement-domain steganography with hardware-generated covers. In this work, we adopt a measurement-domain perspective: orthogonal ghost imaging is used as a hardware-assisted cover generator, and the hidden message is embedded directly into the bucket measurements before ghost-image reconstruction, forming a hybrid optical–digital pipeline. To the best of our knowledge, an OGI-based framework for measurement-domain information hiding with bucket-level embedding has not been reported previously.

Beyond its conceptual interest, the proposed framework is motivated by optical communication and imaging scenarios in which front-end hardware complexity is limited. In such contexts, a single-pixel OGI receiver can reduce optical alignment requirements while still enabling bucket-domain embedding prior to reconstruction. The present study does not include a hardware-level evaluation on embedded platforms; consequently, any discussion of deployment on battery-powered, portable, or embedded systems should be regarded as a conceptual outlook rather than an experimentally validated demonstration.

This work presents an experimentally validated OGI-based steganographic framework in which the host signal is generated by a physical optical acquisition process, while message embedding is performed digitally on the resulting bucket measurements within a single optical–digital pipeline. The framework operates in the measurement domain because embedding is applied directly to bucket measurements prior to reconstruction, rather than to reconstructed images. This integration eliminates the need for holography, deep networks, or complex optical components and is specifically designed for lightweight, single-pixel architectures. The main methodological contributions of this study are:

• development of an OGI-based measurement-domain steganographic architecture in which the cover is generated optically and the secret message is embedded directly into the bucket measurements before reconstruction, thereby exploiting the intrinsic randomness and non-local correlations of OGI as a hardware-assisted concealment mechanism;

• integration of hash-initialized logistic-map chaotic encryption with bucket-level LSB embedding in a single, compact pipeline, providing a lightweight chaos layer that is compatible with real-time single-pixel operation; and

• experimental realization of this architecture using a compact single-pixel OGI system with orthogonal illumination patterns, demonstrating a practical hardware configuration that avoids holography and deep-learning-based reconstruction.

At the performance level, extensive simulations and optical experiments show that the proposed framework maintains imperceptible embedding, achieving PSNR above 50 dB, SSIM exceeding 0.996, and NPCR near 99.98%, while preserving histogram and correlation characteristics and offering statistically robust invisibility. To the best of our knowledge, this is the first experimentally validated OGI-based framework that achieves high-fidelity and statistically robust measurement-domain information hiding within a single optical–digital pipeline.

Related work

Digital image steganography has been extensively studied in the spatial, transform, and adaptive domains. Classical surveys outline how simple spatial-domain LSB substitution, while offering high payload and ease of implementation, is highly susceptible to statistical artifacts and modern steganalysis tools^43,44. Transform-domain schemes based on DCT and DWT embedding improve imperceptibility and robustness but at the cost of increased computational complexity and reduced capacity^35,45. More recent surveys emphasize that adaptive and learning-based approaches can enhance security by exploiting image content and data-driven embedding costs^33,34. Nonetheless, all of these methods remain purely digital: they operate on pre-acquired pixel images, rely on software-generated pseudorandom modifications, and ultimately inherit the statistical structure of natural images, which can be exploited by advanced feature-based and deep-learning steganalyzers^35,36.

Ghost imaging (GI) and single-pixel imaging (SPI) have therefore attracted attention as optical platforms for security, where the imaging pipeline itself acts as a hardware-assisted cipher. Ghanbari-Ghalehjoughi et al. proposed a multi-layer CGI-based security framework that combines three-dimensional computational ghost imaging with two single-pixel detectors and multi-channel bucket-data permutation. In their scheme, a grayscale secret image is embedded into a cover object before the ghost-imaging process and later retrieved from noisy CGI reconstructions, demonstrating robustness against eavesdropping in terms of NRMS and a high key-disclosure threshold for successful recovery²⁸. Rajabi-Ghaleh et al. later introduced sweeping computational ghost imaging (SCGI) with integrated LSB steganography and RSA cryptography, where secret images are hidden in SCGI reconstructions and bucket values are additionally encrypted³⁰. These works clearly show that GI can support layered encryption–steganography pipelines, but steganographic embedding is still performed on the final reconstructed images, after CGI/SCGI processing, and relies on relatively high sampling overhead and multi-channel architectures.

A complementary line of research focuses on computational-imaging encryption and robustness rather than steganography. Wang et al. combined Fourier ghost imaging (FGI) with deep-learning–designed computer-generated holograms (CGHs) to realize a holographic authentication scheme, in which two CGHs encode the sampling path and an authentication-only hidden pattern²⁹. Liang et al. proposed an encrypted SPI transmission scheme based on Hadamard illumination, three-dimensional Arnold scrambling of one-dimensional bucket signals, and elliptic-curve key exchange, thus securing the measurement stream while not concealing the existence of communication⁴¹. The same group also investigated adversarial attacks on deep-learning–based SPI reconstructions and showed that state-of-the-art networks exhibit distinct vulnerabilities to carefully crafted perturbations⁴⁰. These studies primarily address encryption strategies and reconstruction robustness within GI/SPI systems. In contrast, the present work concentrates on embedding information at the level of bucket measurements prior to reconstruction.

Within the orthogonal ghost imaging (OGI) family, Hassanzadeh et al. demonstrated that optimized OGI reconstruction with orthogonal illumination patterns can significantly improve PSNR and computational efficiency in single-pixel imaging, confirming the suitability of OGI for lightweight hardware implementations¹⁷. In follow-up work, they introduced a chaos-enhanced OGI encryption scheme in which logistic-map–generated keystreams are applied to bucket data prior to reconstruction, yielding a compact optical–digital cipher with strong diffusion and confusion but without any steganographic embedding or analysis of histogram/correlation invisibility⁴⁶.

Overall, existing GI- and SPI-based security schemes either (i) implement multi-layer encryption with image-domain steganography on CGI/SCGI reconstructions^28,30, (ii) integrate GI with holographic or deep-learning modules for authentication-oriented encryption²⁹, or (iii) focus on encrypted transmission and adversarial robustness in single-pixel architectures without hiding the presence of the message^40,41,46. These differences are further highlighted in comparison with representative GI/SPI-based schemes in Table 1.

Table 1.

Comparison of the proposed OGI-based measurement-domain optical steganography with representative GI/SPI-based security schemes in terms of core method, experimental setup, and performance Abbreviations: GI = ghost imaging; CGI = computational ghost imaging; SCGI = sweeping computational ghost imaging; SPI = single-pixel imaging; OGI = orthogonal ghost imaging; FGI = Fourier ghost imaging; CGH = computer-generated hologram; DMD = digital micromirror device; SLM = spatial light modulator; PSNR = peak signal-to-noise ratio; SSIM = structural similarity index measure; NPCR = number of pixels change rate; NRMS = normalized root-mean-square error; CC = correlation coefficient; RSA = Rivest-Shamir-Adleman public-key cryptosystem.

Year/Ref.	Core method	Optical/experimental setup	Security/embedding domain	Representative performance/limitations
202028	Multi-channel computational GI with bucket-sequence permutation and CGI-based steganography (no LSB).	Two-channel computational GI with random patterns and a 3D object; up to $\sim 5.4\times {10}^4$ measurements.	Merged multi-channel bucket vector encrypted via permutation; secret image added to the cover at the object/intensity plane before CGI (not pixel-level LSB).	Provides multi-layer security with good NRMS quality, but requires many measurements and lacks detailed cover-statistics analysis.
202430	Sweeping CGI (SCGI) combined with LSB image steganography and RSA cryptography.	Row/column sweeping SCGI for $64\times 64$ binary/grayscale objects; two bucket sequences from row and column scans.	LSB embedding in SCGI reconstructions (pixel domain); RSA applied to bucket data or reconstructed images.	Few-shot acquisition ($m+n=128$) with good NPCR/NRMS/CC security, but stego-image PSNR remains low (13 dB).
202429	FGI encryption with UTN-designed camouflaged CGHs for holographic steganographic authentication.	DMD-based single-pixel FGI encoder plus dual-SLM optical reconstruction of two cascaded CGHs.	Authentication pattern embedded in CGHs; FGI bucket signals used directly as ciphertext (no bucket-level steganography).	Robust encryption/authentication, but relies on deep-learning CGH design and complex holographic hardware.
202441	Encrypted SPI using Hadamard patterns, 3D Arnold scrambling, and elliptic-curve key exchange.	Hadamard single-pixel imaging; 1D bucket data scrambled by a 3D Arnold transform before transmission.	Pure encryption of SPI light-intensity data (plaintext measurements); no steganographic cover or hidden communication channel.	Enhances encryption security but does not address steganographic invisibility or cover statistics.
202546	Dual-layer optical encryption combining orthogonal ghost imaging with Logistic-map chaos.	OGI with orthogonal patterns; bucket data encrypted digitally using a chaotic keystream.	Bucket-level encryption of OGI intensities before reconstruction; no separate hidden message or steganographic cover.	Provides strong diffusion/confusion but is noise-sensitive and does not address physical-layer steganography.
2025 (this work)	OGI-based Measurement-Domain steganography with hash-initialized logistic-map chaos and bucket-level LSB embedding.	Compact single-pixel OGI with orthogonal patterns and a bucket detector; validated in simulation and optical experiment.	Ciphertext embedded into bucket measurements before OGI reconstruction (hardware-generated OGI cover; no DL or holography).	Maintains PSNR$>53$ dB, SSIM$>0.998$, NPCR$\approx 99.98\%$ while preserving histograms/correlations and lightweight complexity.

Problem definition and system model

In this section, we formalize the orthogonal ghost imaging (OGI) optical front-end used to generate the bucket-domain measurements, describe the design of the lightweight chaotic encryption layer, and detail how the secret payload is embedded and extracted in the bucket domain. Finally, we provide a brief theoretical discussion that links this system model to the observed imperceptibility and security characteristics of the proposed framework.

OGI acquisition and reconstruction model

In the proposed framework, the optical front-end relies on orthogonal ghost imaging (OGI) with a single-pixel detector. For each illumination pattern $P_k(x,y)$ projected onto an object with transmission/reflectivity T(x, y), the bucket detector measures a scalar intensity¹⁷

$$\begin{array}{c}\hfill B_k=\int P_k(x,y)T(x,y)dxdy+n_k,\end{array}$$ 1

where $n_k$ models detector and source noise. In the experimental implementation, each pattern is displayed together with its complementary (inverse) pattern, and the final measurement is computed as the difference $B_k=B_k^{(+)}-B_k^{(-)}$. This two-phase acquisition suppresses the DC background and slow ambient-light drift, while preserving the intrinsic fluctuations of the system (shot noise and readout noise).

For reconstruction, the OGI scheme follows the DCT-based orthogonal measurement design reported in our previous work¹⁷. In this implementation, the generic patterns $P_k(x,y)$ are instantiated by a set of 2D-DCT basis patterns $S_{\mathit{uv}}(x,y)$ (ordered in a standard zigzag fashion), with a one-to-one mapping between the scalar index k and the pair (u, v). Let $B_{\mathit{uv}}$ denote the corresponding differential bucket measurements after the two-phase acquisition. The ghost image $\widehat{O}(x,y)$, which provides an estimate of the object transmission T(x, y), is then obtained as an orthogonal expansion over the DCT basis

$$\begin{array}{c}\hfill \widehat{O}(x,y)=\sum _u\sum _v{B}_{\mathit{uv}}{S}_{\mathit{uv}}(x,y),\end{array}$$ 2

which is equivalent to the OGI reconstruction formula derived in Eq. (6) of Ref¹⁷.. No additional temporal windowing or post-reconstruction noise filtering beyond this orthogonal correlation step was applied in the present experiments.

Design of the chaotic encryption layer

Recent optical-encryption studies have introduced advanced high-dimensional chaotic frameworks—such as hyperchaotic systems combined with DNA-based encoding⁴⁷ and lightweight four-dimensional non-degenerate chaotic systems (4D-NDS) for image encryption, exemplified by the compound dynamic diffusion and single-round scheme of Liao et al.^48,49—that offer strong key sensitivity, large key spaces, and enhanced diffusion–permutation behavior. These methods are well suited for standalone optical-encryption pipelines; however, they typically rely on multi-variable state evolution, nonlinear coupling mechanisms, and symbol-level operations that substantially increase computational complexity.

By contrast, CGI operates within a fundamentally lightweight, single-pixel architecture. Prior work has shown that computational and single-pixel ghost imaging require only modest processing overhead for speckle correlation and reconstruction^8,23,24, and recent OGI investigations highlight the efficiency of orthogonal measurement spaces under resource-constrained hardware⁴⁶. Embedding hyperchaotic or 4D-NDS dynamics into such a low-complexity physical pipeline would therefore introduce significant arithmetic cost without delivering proportional security benefits. This is particularly true in our application, where the hidden payload consists of short ASCII sequences rather than full-sized images, and where a dominant part of the security already stems from the physical randomness and nonlocal correlations inherent to OGI itself.

For these reasons, the logistic map is adopted as a lightweight chaotic generator. Its initial conditions are deterministically derived from an MD5 hash of the user key, and a warm-up stage is employed to discard transient dynamics and obtain a more stable statistical regime. The resulting pseudo-random sequence is used to support the embedding of short messages within the measurement-domain steganographic framework proposed in this work.

In this work, the warm-up iterations of the logistic map are used solely to discard transient dynamics and ensure that the sequence is sampled from the fully chaotic regime.

We acknowledge that the one-dimensional logistic map has known limitations in classical cryptographic contexts, including finite-precision effects and reduced resistance under certain attack models. The chaotic layer used in this work is therefore not intended to provide standalone cryptographic security. Its role is limited to generating a pseudo-random sequence that statistically decorrelates the embedded payload within the optical–digital embedding pipeline.

The concealment performance of the proposed framework primarily stems from embedding at the level of bucket measurements in the orthogonal ghost imaging process, where the inherent measurement statistics and correlation structure contribute to masking the payload. The steganographic architecture is independent of the specific chaotic generator employed and remains compatible with alternative pseudo-random sequence generators if required.

Bucket-domain steganographic process

Building on this design rationale, the proposed steganographic framework integrates orthogonal ghost imaging with a logistic-map–driven chaotic encryption layer to achieve lightweight measurement-domain information hiding in the bucket domain. In the embedding phase, the object is illuminated with an orthogonal pattern set and the corresponding bucket measurements ${\{B_k\}}_{k=1}^K$ are acquired by the single-pixel detector; the pre-encrypted message bits are then embedded into these quantized bucket values to form a stego measurement sequence ${\{B_k^{\prime }\}}_{k=1}^K$. In the extraction phase, the authorized receiver uses the same key and pattern set to regenerate the chaotic sequence, read out the embedded bits from the stego buckets, and decrypt the recovered bitstream. A schematic overview of this hybrid optical–digital pipeline is shown in Fig. 1. The ghost image reconstructed from $\{B_k^{\prime }\}$ is mainly used to verify visual imperceptibility and to evaluate PSNR/SSIM, while the hidden message itself resides in the bucket-domain measurements.

Fig. 1.

Schematic of the proposed OGI-based measurement-domain steganography framework. Orthogonal illumination patterns are projected onto the object and a single-pixel detector records the bucket intensity sequence $\{B_k\}$. The secret message is converted to ASCII, encrypted by a logistic-map-generated chaotic keystream, and mapped onto the quantized bucket values to produce stego measurements $\{B_k^{\prime }\}$. The stego bucket sequence is then used by the OGI correlation model to reconstruct a ghost image that is visually indistinguishable from the non-stego reconstruction while securely carrying the hidden message at the physical measurement layer.

Message embedding phase

In this work, the secret payloads are short textual messages encoded using standard printable ASCII characters (codes 32–126). To delimit the payload, we reserve the non-printable value 254 as a dedicated end-of-message symbol, which by construction does not belong to this alphabet and therefore cannot appear inside the message. Under this convention, the appended marker is uniquely identifiable at the decoder and message truncation cannot occur in the use cases considered here.

The bucket-domain embedding phase securely hides the secret message in the OGI measurements through the following steps:

1. Message preprocessing: The input message is converted to ASCII values and represented according to the convention described above, including the appended end-of-message marker.

2. Chaotic key generation: A user-defined key string is hashed using MD5, producing a 128-bit (32-hex-character) value. This 128-bit value is split into two 64-bit halves. After conversion to integers $H_1,H_2\in \{0,\cdots ,2^{64}-1\}$, the logistic-map parameters are computed as

   $$\begin{array}{c}\hfill x_0=\frac{H_1+0.5}{2^{64}}\in (0,1),r=3.57+\frac{H_2}{2^{64}-1}(4-3.57)\in [3.57,4].\end{array}$$ 3

   The pair $(x_0,r)$ uniquely determines the chaotic sequence used for encryption and is reproduced identically whenever the same key string is used.
3. Chaotic sequence and encryption: The logistic map⁵⁰

   $$\begin{array}{c}\hfill x_{n+1}=r{x}_n(1-x_n),\end{array}$$ 4

   is iterated after a warm-up period to produce a discrete chaotic sequence, which is scaled to $[0,255]$. A bitwise XOR between the sequence and the ASCII values generates the encrypted message symbols.

4. Bucket-domain LSB embedding: The encrypted ASCII values are converted into a binary stream. After normalizing and quantizing the bucket measurements $\{B_k\}$ to 8-bit integers $\{b_k\}$, one bit of the encrypted stream is embedded into the least significant bit of each $b_k$ using standard LSB replacement^44,51, yielding the stego bucket sequence $\{b_k^{\prime }\}$. These modified bucket values are then used in the OGI reconstruction model (2) to generate the corresponding stego ghost image.

Message extraction phase

The extraction phase reverses the embedding operations to recover the original message:

1. Stego bucket acquisition: The authorized receiver obtains the stego bucket sequence $\{b_k^{\prime }\}$ corresponding to the OGI illumination patterns $P_k(x,y)$ used during embedding.

2. LSB extraction: Each stego bucket value $b_k^{\prime }$ is quantized (if necessary) and its least significant bit is read out to form the embedded binary stream.

3. Binary-to-ASCII conversion: The bitstream is divided into 8-bit segments to reconstruct the encrypted ASCII sequence.

4. Chaotic key regeneration and decryption: The same user key is hashed to reproduce the logistic map parameters $(x_0,r)$ and regenerate the identical chaotic sequence. Bitwise XOR with the encrypted ASCII values yields the original ASCII message symbols.

5. Message reconstruction: After decryption, the recovered ASCII values are concatenated sequentially until the end-of-message marker (ASCII 254) is encountered, yielding the original plaintext message. It is important to emphasize that the end-of-message marker is interpreted only after successful decryption of the extracted ASCII sequence. The encrypted bitstream itself is never scanned for this marker; therefore, the accidental occurrence of value 254 in the ciphertext cannot cause premature truncation.

The complete stepwise procedure for encryption, embedding, and extraction is presented in Algorithm 1 of the Appendix.

Theoretical rationale for imperceptibility and security

According to the OGI reconstruction model in (2), the reconstructed image $\widehat{O}(x,y)$ is a linear functional of the bucket measurements ${\{B_k\}}_{k=1}^{\mathit{MN}}$. After 8-bit quantization, each bucket value is represented as an integer $b_k\in \{0,\cdots ,255\}$.

Let $M\times N$ denote the spatial resolution of the reconstructed ghost image, so that the total number of bucket coefficients used in reconstruction is MN. Suppose that $N_{\text{modified}}$ least significant bits (LSBs) are altered during embedding. We define the LSB modification ratio as

$$\begin{array}{c}\hfill \alpha =\frac{N_{\text{modified}}}{\mathit{MN}}.\end{array}$$ 5

Flipping the LSB of an 8-bit integer changes its value by at most $\pm 1$. Denoting the embedding perturbation by ${\delta }_k\in \{-1,0,+1\}$, we have

$$\begin{array}{c}\hfill |I(i,j)-\widehat{I}(i,j)|=|{\delta }_k|\le 1.\end{array}$$ 6

Since only a fraction $\alpha$ of the coefficients are modified, the mean squared embedding distortion (MSE) satisfies

$$\begin{array}{c}\hfill \text{MSE}=\frac{1}{\mathit{MN}}\sum _{k=1}^{\mathit{MN}}{\delta }_k^2\le \frac{N_{\text{modified}}}{\mathit{MN}}=\alpha .\end{array}$$ 7

Therefore, the peak signal-to-noise ratio (PSNR), defined as⁵²:

$$\begin{array}{c}\hfill \text{PSNR}=10{log}_{10}\left(\frac{{255}^2}{\text{MSE}}\right),\end{array}$$ 8

admits the lower bound

$$\begin{array}{c}\hfill \text{PSNR}\ge 10{log}_{10}\left(\frac{{255}^2}{\alpha }\right).\end{array}$$ 9

Equation (9) explicitly shows that the PSNR increases monotonically as the LSB modification ratio $\alpha$ decreases. For the payload sizes considered in this work, $\alpha$ remains well below unity, which theoretically explains the consistently high PSNR values observed in the simulation and experimental sections.

Experimental setup

In this work, the term “physical-layer” denotes that steganographic operations are applied directly to the optically acquired bucket-intensity sequence, before any conventional digital image reconstruction or post-processing. The OGI system projects an orthogonal set of illumination patterns onto the object and records the corresponding bucket intensities with a single-pixel detector. Thus, the host measurements are generated by the physical propagation of light, detector fluctuations, and illumination variability, rather than from a pre-existing digital image.

As illustrated in Fig. 2, the proposed optical steganography framework was implemented using a compact single-pixel OGI system. A Casio XJ-A146 digital projector sequentially displayed $64\times 64$ orthogonal patterns with 256 grayscale levels onto a white A4 sheet, forming a reflective target area of approximately $2\times 2$ cm$_{}^2$. The distance between the projector and the reflective screen was fixed at 72 cm, and each pattern was projected for 0.15 s, resulting in a total of 8192 measurements for full image reconstruction. The reflected light from the target was collected by a Thorlabs DCC1545M camera, which operated as a single-pixel (bucket) detector by integrating the total reflected intensity without spatial resolution. The bucket detector is implemented using a Thorlabs DCC1545M USB 2.0 CMOS camera (MT9M001 monochrome sensor, $1280\times 1024$ pixels, maximum full-frame rate 25 fps, pixel clock 5–40 MHz). According to the sensor datasheet, the peak quantum efficiency is approximately 50% around 550 nm, with a temporal read noise of about 10 e$_{}^{-}$ RMS and a typical dark current of $\sim 30$ e$_{}^{-}$/s at 25$_{}^{\circ }$C. The acquired intensity sequence was processed using the OGI correlation algorithm to reconstruct the ghost image of the target scene, as detailed in our prior work on improving PSNR and efficiency in OGI¹⁷.

Fig. 2.

Experimental single-pixel OGI setup used for physical-layer steganography. A Casio XJ-A146 digital projector sequentially displays $64\times 64$ orthogonal patterns onto a $2\times 2$ cm$_{}^2$ reflective area on a white A4 sheet, while the reflected light is collected by a Thorlabs DCC1545M camera operating as a single-pixel (bucket) detector. The resulting bucket-intensity sequence is used as the host signal for chaos-encrypted LSB embedding before OGI reconstruction, yielding a stego ghost image that remains visually indistinguishable from the non-stego reconstruction.

In the experiments, each illumination pattern and its complementary (inverse) pattern were displayed sequentially on the projector, and the corresponding camera frames were combined to obtain the differential bucket sequence, as formalized in Sec. 3. The resulting bucket stream was then normalized and quantized to 8-bit integers, which served as the host sequence for LSB embedding of the encrypted message bits⁵¹. After embedding, the modified bucket values were fed into the OGI reconstruction algorithm in Eq. (2) to generate the stego ghost image.

Evaluation metrics overview

To objectively evaluate the performance of the proposed optical steganography framework, we employ a set of well-established quantitative metrics that are widely used in image-based data hiding and optical security studies. These metrics allow for a comprehensive assessment of visual fidelity, embedding imperceptibility, statistical consistency, and robustness against steganalysis. A brief description of the adopted criteria is provided below.

• Structural Similarity Index (SSIM): SSIM evaluates the structural resemblance between two images, considering luminance, contrast, and local patterns, making it more consistent with human perception than pixel-wise error alone⁵³:

  $$\begin{array}{c}\hfill \text{SSIM}(x,y)=\frac{(2{\mu }_x{\mu }_y+C_1)(2{\sigma }_{\mathit{xy}}+C_2)}{({\mu }_x^2+{\mu }_y^2+C_1)({\sigma }_x^2+{\sigma }_y^2+C_2)},\end{array}$$ 10

  where ${\mu }_x$ and ${\mu }_y$ are local means, ${\sigma }_x^2$ and ${\sigma }_y^2$ are local variances, and ${\sigma }_{\mathit{xy}}$ is the local covariance. Constants $C_1={(K_{1}L)}^2$ and $C_2={(K_{2}L)}^2$ ensure numerical stability.

• Payload Ratio (Payload Percentage): In this paper, the payload percentage is defined as the number of embedded message bits divided by the total LSB capacity of the image, i.e., $M\times N$ for an $M\times N$ 8-bit grayscale image. For a given payload percentage, the message length is scaled proportionally to $M\times N$, so that the same payload ratio corresponds to the same fraction of LSB positions regardless of image resolution.

• LSB Modification Percentage: This metric quantifies the ratio of altered least significant bits to the total number of LSBs in the image. A lower modification percentage implies higher imperceptibility and improved resistance to LSB-based steganalysis⁵⁴:

  $$\begin{array}{c}\hfill \text{LSB Modification Percentage}=\frac{N_{\text{modified}}}{N_{\text{total}}}\times 100,\end{array}$$ 11

  where $N_{\text{modified}}$ is the number of modified LSBs and $N_{\text{total}}$ is the total LSB count.
• Number of Pixel Change Rate (NPCR): NPCR assesses the diffusion strength of the steganographic and encryption processes. It measures the percentage of pixels that change between two stego images generated from slightly different inputs, such as a one-bit variation in the message or key^55,56:

  $$\begin{array}{c}\hfill NPCR=\frac{{\sum }_{i,j}D(i,j)}{M\times N}\times 100\%,\end{array}$$ 12

  $$\begin{array}{c}\hfill D(i,j)=\left\{\begin{array}{cc}1,\hfill & \text{if}{C}_1(i,j)\ne C_2(i,j),\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}\right)\end{array}$$ 13

  A value above 99% is generally considered strong evidence of effective diffusion.
• Higher-Order Statistics (HOS): HOS provide a global statistical perspective of pixel intensity distributions and are often used to assess the subtle effects of data embedding. These include the mean $(\mu )$, variance $({\sigma }^2)$, skewness $(\gamma )$, and kurtosis $(\kappa )$:

  $$\begin{array}{cc}\hfill \mu & =\frac{1}{\mathit{MN}}\sum _{i=1}^M\sum _{j=1}^{N}I(i,j),\hfill \end{array}$$ 14

  $$\begin{array}{cc}\hfill {\sigma }^2& =\frac{1}{\mathit{MN}}\sum _{i=1}^M\sum _{j=1}^N{(I(i,j)-\mu )}^2,\hfill \end{array}$$ 15

  $$\begin{array}{cc}\hfill \gamma & =\frac{1}{MN·{\sigma }^3}\sum _{i=1}^M\sum _{j=1}^N{(I(i,j)-\mu )}^3,\hfill \end{array}$$ 16

  $$\begin{array}{cc}\hfill \kappa & =\frac{1}{MN·{\sigma }^4}\sum _{i=1}^M\sum _{j=1}^N{(I(i,j)-\mu )}^4.\hfill \end{array}$$ 17

  Stability of these descriptors indicates robustness against statistical steganalysis³⁶.
• Chi-Square Test: The chi-square test evaluates the uniformity of the LSB plane in cover and stego images⁴⁴. It is computed as

  $$\begin{array}{c}\hfill {\chi }^2=\sum _{i=1}^k\frac{{(O_i-E_i)}^2}{E_i},\end{array}$$ 18

  where $O_i$ and $E_i$ are the observed and expected frequencies of the i-th category ($k=2$ for binary LSBs). A high p-value indicates that no significant statistical anomalies have been introduced by the embedding process. In this work, the chi-square statistic was computed using the overall LSB distribution of the entire image by counting the total number of zeros and ones, rather than using a block-wise or local grouping method.

Overall, these complementary metrics capture both visual and statistical aspects of image quality and security. Using this set of indicators ensures a balanced evaluation of the proposed framework in terms of imperceptibility, robustness, and resistance to basic statistical steganalysis tests that are applicable to OGI-based measurement-domain bucket covers. It is worth noting that rich-model steganalyzers such as the Spatial Rich Model (SRM) and its selection-channel-aware variant maxSRM, originally developed and tuned for natural images in the spatial and JPEG domains^57,58, rely on local prediction residuals and high-order co-occurrence statistics. These feature spaces were primarily developed for natural image statistics and may not directly capture the speckle-driven and correlation-based characteristics of OGI measurements. Furthermore, classical pixel-domain steganalysis algorithms such as RS analysis, pixel–pair correlation detectors, wavelet-moment-based methods, and LSB-matching detectors^44,56,59,60 operate on fully reconstructed spatial images (or transform-domain coefficients) rather than on pre-reconstruction bucket intensities. Since the embedding in our framework is performed directly in the bucket-intensity domain, a principled use of these detectors would require redesigning their feature-extraction stage and retraining them on OGI-specific data. Accordingly, this study evaluates statistical consistency using (HOS, ${\chi }^2$, etc.) that remain directly interpretable for OGI-based measurement-domain covers, and leave a full rich-model steganalysis of OGI stego as separate future work.

Simulation and experimental results

This section presents a systematic evaluation of the proposed framework through both simulation and experimental analyses. The analysis focuses on three primary aspects: message recoverability, visual imperceptibility, and practical applicability in optical settings. All evaluations are supported by quantitative measurements and visual inspection.

Simulation results

The initial assessment was performed using synthetically reconstructed ghost images derived from six widely used grayscale test images (house, lake, mandrill, pirate, and cameraman), each with a resolution of $100\times 100$ pixels. A fixed-length 1250-character secret message was embedded into each image using the logistic-map-based steganographic scheme described in the previous sections.

Figure 3 illustrates a representative example for the cameraman image. The left column shows the ghost images before and after embedding, the middle column displays the corresponding intensity histograms, and the right column presents pixel correlation plots. The visual similarity and negligible differences in histograms and correlation structures demonstrate the high imperceptibility of the proposed method.

Fig. 3.

Comparative analysis between the original cover image and the stego-image after embedding. The left column depicts the reconstructed ghost images, the middle column shows the corresponding intensity histograms, and the right column presents pixel correlation plots. Minimal variations confirm that the embedding process preserves visual fidelity and statistical properties.

Quantitative results are summarized in Table 2. Across the five standard $100\times 100$ grayscale test images, these results correspond to an average PSNR of $50.47\pm 0.20$ dB and an average SSIM of $0.9980\pm 0.0013$, with an LSB modification ratio of $50.07\pm 0.45\%$. The small standard deviations indicate that both visual distortion and bit-plane perturbations remain highly stable across different scenes under identical embedding conditions. At the maximum payload, corresponding to embedding a 1249-character secret message plus a one-character end-of-message marker (1250 characters in total, i.e., 10 000 bits) into each $100\times 100$ image, all stego-images maintain PSNR values around 50 dB and SSIM scores in the range 0.996–0.999, confirming that the visual distortion remains negligible even at full LSB utilization. In this full-capacity regime, the LSB modification rate is close to 50% for all test images, reflecting the fact that the chaos-encrypted bitstream is statistically balanced and uniformly distributed over the available LSB positions. As discussed in our definition of the payload ratio, this configuration corresponds to a 100% payload level, where every LSB position carries an embedded bit and, on average, only about half of them flip, in agreement with the values reported in Table 2. In all cases, the embedded messages were extracted without errors.

Table 2.

Quantitative evaluation of stego-images generated from six $100\times 100$ grayscale test images after embedding a 1250-character message.

Image	Thumbnail	PSNR (dB)	SSIM	LSB Change (%)	Message Match
House		50.78	0.9973	49.51	$✓$
Lake		50.41	0.9986	50.54	$✓$
Mandrill		50.48	0.9991	49.73	$✓$
Pirate		50.24	0.9990	50.11	$✓$
Cameraman		50.45	0.9961	50.45	$✓$
Mean	–	50.47	0.9980	50.07	–
Std	–	0.20	0.0011	0.39	–

To further investigate the trade-off between payload size and image quality, we varied the length of the embedded message from 5% to 100% of the available LSB capacity for the $100\times 100$ cameraman image. Table 3 and Fig. 4 summarize the results. As expected, the PSNR decreases gradually from 64.24 dB at 5% payload to 51.12 dB at 100% payload, while the SSIM remains above 0.996 across the entire range. To interpret these values in terms of human visual perception, we relate them to commonly used criteria in the image steganography and watermarking literature. Several recent works report that stego images with PSNR values above approximately 40 dB can generally be regarded as visually imperceptible or of excellent quality from the viewpoint of human observers^61–63. In our experiments, all tested payloads from 5% up to the full 100% LSB capacity satisfy this condition, which is consistent with the strong masking effect of the speckle-like OGI statistics. The LSB modification percentage increases with payload, as expected, but all embedded messages were recovered successfully.

Table 3.

Performance analysis for the cameraman image under varying payload ratios. PSNR, SSIM, LSB modification, and message recovery are reported to assess the impact of payload size.

Payload (%)	PSNR (dB)	SSIM	LSB Change (%)	Message Match
5	64.24	0.9998	2.45	$✓$
10	61.12	0.9995	5.03	$✓$
20	58.14	0.9992	9.97	$✓$
50	54.11	0.9982	25.25	$✓$
80	52.10	0.9973	40.10	$✓$
100	51.12	0.9964	50.22	$✓$

Fig. 4.

Variation of PSNR and SSIM with embedding payload for the cameraman image. PSNR decreases with increasing payload, while SSIM remains consistently high, confirming strong imperceptibility.

From a statistical point of view, the main benefit of the chaos-based encryption layer is that it whitens the embedded bitstream. Plaintext messages (e.g., ASCII text or binary payloads with structure) generally exhibit highly non-uniform and message-dependent bit patterns, so directly embedding them may introduce detectable biases in the histogram of the stego LSB plane. In contrast, the logistic-map/XOR operation produces a pseudo-random keystream whose bits are approximately balanced, with $Pr\{1\}\approx Pr\{0\}\approx 0.5$. As a consequence, for each candidate bucket position the probability that its LSB is flipped is close to 50% and essentially independent of the underlying plaintext. This drives the LSB statistics of the stego OGI measurements towards a uniform distribution and removes simple message-dependent patterns, thereby suppressing straightforward histogram- and ${\chi }^2$-based steganalysis, while, as shown by the PSNR/SSIM results, having negligible impact on perceptual image quality.

Experimental validation

To demonstrate the practical feasibility of the proposed framework, the standard cameraman test image was projected onto a white A4 sheet to form the physical scene of a single-pixel OGI setup. The scene was sequentially illuminated with orthogonal patterns, and the reflected light was collected by a bucket detector to reconstruct the ghost image. The reconstructed image then served as the cover medium for embedding the pre-encrypted secret message using the logistic-map-driven LSB approach.

Figure 5 presents a comparison of the experimentally reconstructed ghost image before and after embedding. The corresponding intensity histograms and pixel correlation plots reveal that the embedding process introduces negligible visual or statistical distortion. These experimental results align well with the simulation outcomes, confirming that the proposed scheme maintains high imperceptibility under real optical conditions. Moreover, the method’s ability to preserve visual quality in a low-complexity optical setup highlights its potential for lightweight physical-layer optical communication scenarios.

Fig. 5.

Experimental comparison of ghost images before and after embedding. Left: original and stego ghost images. Middle: corresponding intensity histograms. Right: pixel correlation plots. High visual and statistical similarity confirm imperceptible embedding in the real optical experiment.

Computational complexity analysis

The proposed framework is designed for lightweight and scalable implementation. The OGI reconstruction exhibits a computational complexity of $\mathcal{O}(N·m^2)$, where $m\times m$ is the image size and N is the number of orthogonal illumination patterns⁴⁶. This structure allows efficient reconstruction, especially when accelerated by matrix operations or parallel processing.

The logistic-map-based chaotic encryption adds negligible overhead with a complexity of $\mathcal{O}(N)$, as each iteration involves only one multiplication and one subtraction. Following our previous OGI-related study⁴⁶, we adopt the logistic map as a lightweight keystream generator due to its implementation simplicity. In the present framework, it is used solely to generate a pseudo-random sequence that supports short-message embedding within the measurement-domain steganographic pipeline, rather than as a standalone cryptographic component.

Moreover, the additional digital steps required by the steganographic layer—XOR-based encryption, LSB embedding in the bucket domain, and the symmetric extraction and decryption operations at the receiver—are likewise dominated by simple arithmetic and bitwise operations with linear complexity in the number of processed buckets and payload bits.

While execution times are hardware-dependent and not explicitly measured, the low-order arithmetic confirms that the method is suitable for resource-constrained optical systems in scenarios where moderate computational overhead is acceptable.

It should be noted that the computational analysis presented in this work is intended to characterize the algorithmic complexity of the proposed framework rather than to provide optimized runtime benchmarks. The dominant execution time in OGI-based systems is governed by the optical acquisition process (pattern display and detector integration), whereas introduce comparatively low algorithmic overhead relative to the optical acquisition stage. As such, the reported complexity reflects structural algorithmic cost rather than hardware-specific timing performance.

From a hardware perspective, these observations indicate that the proposed framework suggests potential feasibility for compact computing platforms with moderate computational and power budgets. However, a detailed hardware-specific characterization of execution time, power consumption, and energy efficiency on concrete embedded platforms (e.g., Raspberry Pi or low-power FPGA boards), as well as projector/detector power budgeting for battery-powered operation and portable form factors, is beyond the scope of this proof-of-concept study and is left for future work.

Security analysis

A secure steganographic system must guarantee both the confidentiality of the hidden message and the invisibility of its presence. To evaluate the security of the proposed optical framework, a multi-dimensional assessment was conducted. The analysis considers visual imperceptibility, resistance to statistical steganalysis, key sensitivity and diffusion, and comparison with prior CGI-based approaches. All evaluations are supported by both simulation and experimental data to demonstrate the method’s robustness under practical optical conditions.

Imperceptibility analysis

Imperceptibility requires that the embedded message remain visually undetectable to human observers. Quantitative results in Table 2 show that, even at the maximum payload of a 1249-character message plus the end-of-message marker, all stego-images retain PSNR values around 50 dB and SSIM scores between 0.996 and 0.999. Together with the payload-variation study on the cameraman image in Table 3, where PSNR stays above roughly 50 dB and SSIM above 0.996 for payloads ranging from 5% to 100% of the available LSB capacity, these figures place the reconstructed stego OGI images comfortably within the visually lossless regime typically reported in steganography and watermarking literature.

Although the LSB modification rate approaches 50% at full-capacity embedding, the combination of speckle-like OGI statistics and a statistically balanced encrypted bitstream ensures that the induced perturbations are noise-like and spatially well spread. As a result, low-order statistics such as histograms and pixel correlations remain almost unchanged, and no visible artifacts are apparent in the reconstructed images. These observations confirm that the proposed embedding strategy is effectively imperceptible under the considered operating conditions and is therefore suitable for physical-layer optical steganography.

Statistical resistance

Robust steganographic systems must also preserve the global and local statistical properties of the host image. As discussed in the simulation results and illustrated in Figs. 3 and 5, the intensity histograms and spatial correlation patterns of the cover and stego images remain almost unchanged after embedding. This preservation of low-order statistics limits the effectiveness of classical histogram- and correlation-based steganalysis techniques^44,64.

The controlled embedding mechanism minimizes detectable perturbations, and the close agreement between simulated and experimental measurements indicates that the proposed framework introduces only negligible statistical artifacts in practical optical settings. Taken together, these observations show that the scheme exhibits strong resilience to classical statistical attacks, which is critical for covert optical communication.

Key sensitivity and diffusion analysis

High key sensitivity is desirable so that small changes in the encryption key or hidden message (e.g., a single-bit difference in the input string) lead to decorrelated stego-images. In the proposed design, the user key is first processed by an MD5 hash to derive the initial conditions of the logistic map; cryptographic hash functions are designed to exhibit an avalanche effect, whereby small changes in the input cause large, pseudo-random changes in the output⁶⁵. In this framework, the hash function is used solely as a deterministic numerical seeding mechanism for the chaotic generator and is not employed for authentication or standalone cryptographic protection; any modern hash function (e.g., SHA-256) could be used interchangeably in this role without affecting the key sensitivity, embedding performance, or statistical results reported in this work. Combined with the fully chaotic regime of the logistic map, this ensures that neighbouring keys generate distinct chaotic sequences and, consequently, visually unrelated stego reconstructions. In our experiments, key and message sensitivity are assessed at the image level using the NPCR metric. For the cameraman image, the proposed framework achieves an NPCR of 99.98%, indicating that small input variations propagate across the output and disrupt statistical dependencies, thereby providing strong protection against differential analysis. A detailed bit-wise sensitivity profiling of the hash–chaos core is beyond the scope of this work and will be addressed in future studies focused specifically on cryptographic analysis.

In contrast to purely digital chaos-based image-encryption schemes such as the Fibonacci-interleaved diffusion with non-degenerate discrete chaos proposed by Xie et al.⁶⁶, which primarily aim at maximizing key sensitivity, NPCR, and resistance to differential or known-plaintext attacks in the pixel domain, our framework targets Measurement-Domain steganography on OGI bucket measurements while maintaining both the higher-order statistics and the visual quality of the reconstructed ghost image. Nevertheless, the use of NPCR, histogram, and key-sensitivity analyses in this work follows the same cryptanalytic rationale, showing that small perturbations in the plaintext or key induce near-ideal randomness in the embedded ciphertext while leaving the cover statistics essentially unchanged.

Resilience against statistical detection

To assess higher-order statistical stealthiness, we computed global moment-based features (mean, variance, skewness, and kurtosis) together with a chi-square goodness-of-fit test on the LSB plane. As summarized in Table 4, the higher-order statistics of the original and stego cameraman images differ only marginally, indicating that the overall intensity distribution and histogram shape are largely preserved after embedding.

Table 4.

Statistical features before and after embedding for the cameraman image.

Metric	Before Embedding	After Embedding
Mean	118.3134	121.111
Variance	3618.8973	3801.7509
Skewness	−0.78675	−0.78975
Kurtosis	2.1988	2.2026
Chi-square value	0.6400	3.24
Chi-square p-value	0.4237	0.071861

For the LSB histogram, the chi-square statistics of the cover and stego images are ${\chi }^2=0.64$ ($p=0.4237$) and ${\chi }^2=3.24$ ($p=0.071861$), respectively. At the conventional significance level of $5\%$, both p-values are greater than 0.05, so the null hypothesis of an unbiased LSB distribution cannot be rejected in either case. In other words, the embedding does not introduce a statistically detectable bias into the LSB plane. This behaviour is consistent with the classical analysis of Westfeld⁶⁷, who showed that for low embedding rates the chi-square test typically produces non-significant p-values fluctuating between 0.05 and 0.95, making it unreliable for detecting weak LSB embedding.

Additionally, we extracted texture-sensitive descriptors such as SPAM and LBP before and after embedding, following their standard definitions in^68,69. The empirical distributions of these features exhibit only subtle changes at the embedding strengths considered here, which is in line with prior steganalytic observations that low-rate LSB matching causes only weak, hard-to-detect perturbations in natural textured images⁵⁹. Overall, these results suggest that straightforward statistical and texture-based detectors have limited power against the proposed bucket-domain embedding strategy.

Practical robustness and physical-layer security

In practical OGI-based systems, environmental factors such as slow ambient-illumination drift or mild optical misalignment primarily influence the signal-to-noise ratio of the acquired bucket sequence and, consequently, the visual fidelity of the reconstructed ghost image. In the proposed framework, however, the hidden message is embedded directly into the time-ordered bucket measurements prior to any image reconstruction. The extraction process relies only on the known correspondence between the illumination patterns and the acquired measurements under controlled laboratory conditions. From a conceptual standpoint, this design suggests that moderate degradation in reconstruction quality does not necessarily preclude correct message extraction, provided that the acquisition order and pattern set remain unchanged.

It is important to emphasize that the above observation reflects a qualitative analysis based on the structure of the OGI measurement process, rather than a quantitative experimental validation under adverse conditions. In the present study, experiments were conducted under stable and well-controlled optical settings, and no systematic evaluation was performed under strong background illumination, severe optical misalignment, or photon-starved regimes. As such, the robustness discussed here should be interpreted as structural robustness associated with the nonlocal pattern–measurement coupling in OGI, rather than demonstrated robustness against all forms of environmental perturbation.

In addition to environmental robustness considerations, it is necessary to clarify the adversarial model under which the proposed measurement-domain steganographic framework is evaluated. In this work, the security discussion is formulated under a passive measurement-domain observer model, in which the adversary has access only to the transmitted stego bucket sequence $\{b_k^{\prime }\}$. The original pre-embedding bucket measurements $\{b_k\}$ are not assumed to be publicly available, since embedding is performed immediately after optical acquisition within the same optical–digital pipeline.

Under this assumption, detectability must be considered directly in the bucket domain. The bucket measurements in OGI arise from orthogonal pattern projections combined with detector noise and illumination variability, resulting in a stochastic measurement sequence. Furthermore, the embedded payload is decorrelated via a pseudo-random whitening stage prior to LSB insertion. Consequently, the embedding operation replaces the least significant bit of each selected bucket value with a statistically decorrelated bitstream.

We emphasize that the present framework is concealment-oriented and does not claim resistance against strong insider or known-cover attack models, in which the exact pre-embedding bucket sequence of the same acquisition is available to an adversary. Such scenarios would require additional countermeasures beyond basic LSB replacement and are outside the scope of the current study.

A further characteristic of the proposed framework is that the host measurements are generated directly in the optical domain rather than synthesized from a fixed digital cover image. As a result, repeated acquisitions using the same user key and message can produce different realizations of the bucket sequence due to physical variability in illumination and detection, while still allowing correct payload extraction under identical experimental conditions. From a security perspective, this physical diversity may complicate statistical modeling of the cover sequence by an adversary, as it hinders the construction of a single, deterministic reference distribution. This interpretation is consistent with the stable statistical indicators (e.g., histogram similarity, ${\chi }^2$, HOS, and NPCR) reported in Section 6.

Overall, by generating the host (cover) sequence through a hardware OGI front-end and performing message embedding digitally at the level of bucket measurements prior to image reconstruction, the proposed framework implements a hardware-assisted measurement-domain steganographic architecture. In this formulation, measurement-domain refers to the fact that embedding is applied directly to acquired bucket data before image formation, rather than to reconstructed spatial images. The combination of nonlocal OGI measurements with a lightweight chaotic modulation layer contributes to concealment performance under the controlled conditions examined in this study. A systematic investigation of robustness under challenging environmental conditions, such as strong background light or photon-limited operation, remains an important direction for future research.

Comparison with previous works

Several recent works have combined computational ghost imaging (CGI) and related single-pixel imaging (SPI) architectures with cryptographic primitives and, in some cases, steganography or watermarking to enhance optical security. Representative examples include CGI-based ghost cryptography and steganography with conventional public-key encryption⁷⁰, multi-channel CGI with multi-layer encryption and steganographic embedding²⁸, sweeping CGI (SCGI) with LSB-based image hiding and RSA³⁰, color ghost cryptography and steganography assisted by MDGANs⁷¹, Fourier ghost imaging combined with computational holography and steganographic authentication²⁹, as well as SPI-based encrypted transmission⁴¹ and non-GI optical image-security schemes based on Gyrator transforms and transform-domain embedding⁷². CGI-based blind watermarking in the wavelet domain has also been explored for robust logo embedding rather than full steganographic transmission²⁶. A qualitative comparison of these schemes with the proposed framework, in terms of imaging modality, embedding domain, hardware complexity, and use of learning-based components, is summarized in Table 1.

From a methodological perspective, most CGI/SPI-based security schemes operate either in the reconstructed image domain or in transform/hologram domains, often requiring multi-channel acquisition, Fourier optics, holographic reconstruction, or deep neural networks to achieve acceptable visual quality. By contrast, the proposed framework relies on an orthogonal ghost imaging (OGI) front-end with a single-pixel detector and performs steganographic embedding directly in the bucket domain using LSB modulation combined with a lightweight logistic-map-based encryption layer. This design eliminates the need for complex optical components and data-driven reconstruction models, while still providing a physically motivated source of randomness and diversity at the measurement level.

In terms of image quality and statistical security, existing CGI/SPI-based schemes typically report PSNR and SSIM values for the hidden or recovered images that are considerably lower than those achieved in the present work, often in the approximate range of $10\text{--}40\text{dB}$ under nontrivial payloads or attack conditions^26,30,71,72. In our OGI-based framework, the reconstructed stego images consistently attain PSNR above 50 dB and SSIM above 0.998 in the considered grayscale scenes. This behaviour is closely linked to the use of orthogonal patterns and two-phase differential acquisition. In our OGI setup, the reconstruction operator is well-conditioned, so LSB-level perturbations in the bucket measurements translate into small, noise-like fluctuations in the image domain. Since the encrypted LSB sequence produced by the logistic map is approximately white and of bounded amplitude, the resulting perturbations are effectively masked by the intrinsic OGI reconstruction noise, which explains why the PSNR between the cover and stego OGI images remains in a $>50$ dB regime under the tested sampling factors. Note that, unlike most existing CGI/SPI-based schemes where PSNR is measured between the ground-truth scene or host image and the recovered image after multiple optical and digital operations, our reported $>50$ dB values refer to the difference between the OGI cover and stego images under the same reconstruction pipeline.

We emphasize, however, that these figures are obtained under heterogeneous experimental configurations, with substantial differences across the literature in image size and content (grayscale natural scenes versus color images or binary logos), payload type and rate (pure encryption, watermarking, or full-image steganography), and optical architectures (multi-channel CGI, color CGI with MDGANs, Fourier ghost imaging, SPI-only encryption, etc.). As a consequence, a strict one-to-one numerical ranking across all methods would be of limited scientific value and potentially misleading.

For this reason, rather than enforcing a consolidated benchmark table with partially incompatible metrics, we focus on a statistically robust evaluation of the proposed OGI-based framework under well-controlled conditions. In the revised manuscript, we report average PSNR and SSIM (together with standard deviations) over multiple test images, perform a detailed payload–capacity analysis from low rates up to full LSB utilization, and complement these fidelity measures with security-oriented metrics such as NPCR and a discussion of LSB-level statistical whitening induced by the chaos-based encryption layer. Within this evaluation, the reconstructed stego images consistently achieve PSNR above 50 dB and SSIM above 0.998 for the considered grayscale OGI scenes, while the image-domain NPCR reaches $99.98\%$, indicating strong diffusion and key sensitivity. Taken together with the qualitative comparison in Table 1, these results highlight that the proposed OGI-based scheme offers a practically attractive trade-off between hardware simplicity, physical-layer operation, high-fidelity steganography, and statistical security compared with existing GI/SPI-based optical security methods.

Noise robustness and practical considerations

The experiments in this work were conducted in a controlled indoor single-pixel OGI setup without a systematic sweep over externally injected optical noise levels. In such conditions, the dominant noise sources are projector intensity fluctuations, detector readout and dark noise, and residual ambient light, all of which are integrated into the bucket measurements and partially averaged out by the correlation-based reconstruction. Since the steganographic embedding is applied after optical acquisition and digitization, directly on the bucket-intensity sequence, moderate optical noise predominantly affects the PSNR/SSIM of the reconstructed ghost image rather than corrupting the embedded bitstream, as long as the pattern-measurement correspondence and digital sampling remain reliable.

It is well established that ghost imaging and single-pixel imaging inherently exhibit a degree of resilience to random optical noise, because the image is recovered from statistical correlations rather than direct pixel-wise measurements. Differential and adaptive correlation strategies have been shown to mitigate the impact of Gaussian and Poisson noise on bucket measurements, improving image quality in noisy environments⁷³. Likewise, compressive and correlation-based single-pixel imaging in scattering or ambient-light-dominated media has demonstrated that uncorrelated background fluctuations can be substantially suppressed, enabling image retrieval even through diffusive or biological samples⁷⁴. Our previous OGI study also showed that the same two-phase differential illumination scheme used here cancels quasi-static background offsets and enhances SNR under laboratory conditions¹⁷. Together, these works suggest that the proposed OGI-based steganographic framework can inherit similar noise-robust behavior. A dedicated quantitative analysis of PSNR/SSIM degradation and message-extraction performance under controlled Gaussian/Poisson noise and realistic outdoor conditions is therefore identified as an important and practically relevant direction for future work.

Conclusion

This work presents an experimentally validated OGI-based measurement-domain steganography framework that integrates a hash-initialized logistic map for chaotic encryption with bucket-level LSB embedding within a unified optical and digital pipeline. By relying on a compact single-pixel OGI front end and orthogonal illumination patterns, the scheme avoids holographic components, deep learning reconstruction, and multichannel architectures, and thus remains compatible with optical platforms operating under moderate computational constraints.

Through a combination of numerical simulations and single-pixel optical experiments, the proposed framework was shown to provide imperceptible embedding and statistically consistent concealment performance. In the reported configurations, the stego ghost images consistently maintained PSNR above 50 dB, SSIM exceeding 0.996, and NPCR close to 99.98%, while preserving intensity histograms, pixel correlations, higher-order moments, and the chi-square behaviour of the LSB plane. These observations indicate that chaos-encrypted bucket-domain LSB modulation can be effectively masked by the speckle-like statistics and correlation-based reconstruction of OGI, enabling reliable message recovery without visually perceivable artifacts.

At the same time, the present study has several limitations that suggest clear directions for future work. First, the experimental validation is restricted to grayscale still images acquired in a controlled indoor environment with moderate optical SNR. A systematic robustness study under controlled Gaussian and Poisson noise, photon-starved acquisition, and strong background-light fluctuations, together with countermeasures such as adaptive or coded illumination, multi-frame bucket accumulation, and error-correcting codes in the bucket domain, remains an important extension. Second, the current implementation operates on single-frame grayscale data. Extending the framework to color and video will require appropriate multi-channel OGI acquisition strategies and channel-specific keys for RGB (or spectrally multiplexed) operation, as well as temporally structured pattern groups and key schedules that exploit spatio–temporal redundancy across consecutive frames.

Finally, the cryptographic core is deliberately confined to a one-dimensional logistic map combined with single-bit LSB embedding. Future work will investigate higher-dimensional chaotic maps and alternative embedding strategies, such as multi-bit or transform-domain embedding in the bucket sequence, and will subject the framework to systematic evaluation using modern rich-model and deep-learning-based steganalyzers and GI-specific attacks on larger, more diverse OGI datasets. Constructing such datasets and mapping the receiver-operating characteristics of SRM/maxSRM and CNN-based detectors as functions of measurement rate, embedding strength, and illumination design would provide a more complete picture of detectability under powerful data-driven adversaries and complement the measurement-domain analysis reported in this paper.

Taken together, these results indicate that OGI-based measurement-domain steganography can be realized within a compact single-pixel framework and modest digital post-processing. While this proof-of-concept study does not address system-level constraints such as power consumption, volume, or real-time embedded execution, it establishes a foundation for future investigations in resource-constrained optical settings.

Appendix: Embedding, extraction, and parameter overview

Author contributions

K.H. conceived the main idea, designed the study, conducted the methodology and investigation, performed the analyses, and drafted the manuscript. S.A.-K. supervised the research, contributed to validation, and assisted in reviewing and editing the manuscript. R.K. contributed to validation and supported the review and editing process. S.A.M. assisted with formal analysis, software, and resources. All authors discussed the results, provided feedback, and approved the final manuscript.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.