Computationally efficient hybrid multi-domain transform fusion for secure image steganography using rate–distortion optimization and Toom–Cook acceleration

1 Introduction

The unprecedented increase in digital communication infrastructures, including cloud computing environments, telemedicine systems, intelligent transportation networks, Internet of Things (IoT) ecosystems and defense-grade communication frameworks, has dramatically increased the amount and complexity of multimedia data (Sanjalawe et al., 2025) that can be transmitted across open networks (Sanjalawe et al., 2025; Sharma et al., 2022; Dhawan et al., 2021; Hamza, 2026). This phenomenon is known as multi-dimensional communications technology. Even though encrypting data is the most effective way to maintain confidentiality, encrypted information indicates the existence of protected communication, which can attract attackers’ attention (El-Den and Raslan, 2025; Alan Anwer Abdulla, 2024). On the other hand, steganography conceals the very existence of such information within unsophisticated digital carriers (images, audio signals) or video streams (Issac et al., 2025; Malik et al., 2025; Nguyen and Le, 2024; Song et al., 2024). With its invisibility feature, steganography is well-suited to digital rights management and protection of sensitive information, such as watermarking for forensic purposes, protecting medical records, and transmitting big data without physical barriers (Alshoura and Alawida, 2025; Sun et al., 2025; Taj et al., 2024). Spatial-domain and transform-domain methods are the two main types of traditional image steganography techniques. Least significant bit (LSB) substitution and other spatial techniques have a high embedding capacity and low computational overhead (Sun et al., 2025; Nguyen and Le, 2024). Direct pixel modification can cause significant changes in the statistical properties of an image, which makes these techniques vulnerable to attacks (Qiu et al., 2022; Sun et al., 2023; Nasr et al., 2024; Gangappa and Satyanarayana, 2025; He et al., 2025) from first- and higher-order steganalysis. Robustness can be enhanced by use of transform-domain techniques, such as discrete cosine transform (DCT) and display wavelet transform (DWT), which incorporate information into frequency coefficients instead of spatial intensities. By utilizing the perceptual frequency spectrum redundancy, these techniques enhance resistance to compression and filtering operations (El-Den and Raslan, 2025; Alshoura and Alawida, 2025). Even with these improvements, one fundamental challenge that remains unresolved is the optimization of imperceptibility, embedding capacity, robustness, statistical invisibility, and computational scalability simultaneously. These objectives are inherently competing. An increase in embedding capacity usually leads to an increase in distortion. The enhancement of robustness often involves the use of stronger coefficient perturbation. Minimal distributional deviation is necessary to improve security against statistical steganalysis (Sun et al., 2025; Qiu et al., 2022; Sun et al., 2023; Zhao et al., 2024). The use of hybrid transforms and deep learning models results in significant scalability limitations for high-resolution images (Issac et al., 2025; Song et al., 2024; Li et al., 2024b). Parameter tuning or architecture modifications are used in most current approaches to address these objectives through empirical means. Very few frameworks have developed the embedding process into a mathematically limited problem of optimization that is based on the theory of distortion (Li et al., 2024a; Pan and Ni, 2024; Menduina et al., 2024).

Despite the speedy evolution of primarily deep-learning-based, model-area, and hybrid steganographic frameworks, a unified and theoretically grounded embedding structure that concurrently optimizes perceptual fidelity, payload potential, statistical security, and computational scalability remains in large part unexplored. Existing schemes primarily based on deep encoder–decoder and generative adversarial network (GAN) models more often than not rely upon empirical balancing among imperceptibility and potential, without explicitly formulating embedding as a constrained rate–distortion minimization problem with analytically enforceable peak signal-to-noise ratio (PSNR)/ structural similarity index measure (SSIM) bounds. Transform-area tactics enhance robustness via wavelet, DCT, or SVD mechanisms, but frequently rely on constant or heuristic embedding rules, lack learning-driven content adaptivity, and offer constrained asymptotic complexity analysis. Similarly, chaos-based and reversible watermarking frameworks focus on possession verification instead of mathematically managed excessive-potential steganographic communication, whilst latest transformer-based steganalysis fashions improve detection instead of embedding design. Furthermore, previous works seldom combine luminance-most effective embedding for chrominance renovation, histogram-conscious statistical constraints to suppress first-order artifacts, and provable sub-quadratic computational acceleration within a single cohesive framework. These barriers inspire the improvement of a mathematically regulated steganographic version that transcends structure-centric or empirically tuned designs. The proposed ML-optimized rate–distortion-managed hybrid steganography is consequently conceived to explicitly version embedding as an optimization problem min D (C, C^#) −βR, put into effect perceptual pleasant constraints, comprise multi-stage DWT–DCT–truncated singular value decomposition (SVD) remodel fusion within the luminance area, and rent distortion-conscious CNN-guided adaptive embedding to decorate statistical invisibility. Moreover, integrating histogram renovation constraints and Toom–Cook-expanded convolution, yielding sub-quadratic complexity O (MN ^1.46), the framework addresses scalability for excessive-decision pictures whilst retaining robustness and high payload efficiency. Collectively, this motivation establishes the need for a theoretically constrained, learning-guided, statistically secure, and computationally efficient hybrid steganographic paradigm able to meet the needs of present-day, high-potential covert communication systems. Systematically dealing with recognizable theoretical, architectural, analytical, and computational limitations in present steganographic strategies, the prevalent approach tries to establish a specific objective surrounding the rate–distortion idea and transform-domain modeling. These goals together guide the improvement of a scalable, learning-driven, and mathematically confined high-potential steganographic framework.

• The first goal is to develop a mathematically grounded embedding version that explicitly minimizes distortion at the same time as maximizing payload while adhering to analytically enforceable perceptual constraints.
• The second goal is to assemble a sturdy transform-area framework integrating multi-stage DWT, block-based DCT, and truncated SVD in the luminance (Y) channel of the YIQ space. This aims to enhance structural stability, chrominance renovation, and robustness in opposition to compression and signal processing attacks, overcoming the constraints of single-area or RGB-based embedding strategies.
• The third objective is to implement a lightweight CNN-based embedding opportunity map trained with a distortion-conscious loss function to gain adaptive, content-based embedding. This immediately addresses the need for learning-driven coefficient choice preference over rule-based or heuristic embedding styles recognized in earlier literature.
• The fourth goal is to incorporate specific first-order statistical renovation constraints into the embedding process, minimizing histogram deviation between service and stego images. This seeks to enhance resistance to current steganalysis strategies that make the most of statistical artifacts, an opening insufficiently addressed in present hybrid and deep approaches.
• The fifth goal is to ensure computational scalability through provable sub-quadratic complexity.

As far as contributions are concerned, the contribution of the proposed framework lies in its unified integration of rate–distortion theory, hybrid multi-decision model embedding, and system learning-driven adaptive management within a computationally efficient architecture. Unlike traditional DWT–DCT or SVD-based totally steganographic schemes that depend upon constant embedding strengths and heuristic coefficient selection, the proposed approach formulates embedding as a restricted optimization problem that explicitly maximizes payload at the same time as bounding perceptual distortion, thereby ensuring high PSNR and SSIM performance. The advent of a distortion-conscious CNN-based, totally embedding chance map enables intelligent, content-adaptive coefficient selection, reworking the embedding system from deterministic to data-driven at the same time as maintaining statistical invisibility. Furthermore, adaptive singular value perturbation regulated by local variance guarantees structurally steady modification, and the incorporation of histogram renovation constraints mitigates first-order statistical artifacts generally exploited in steganalysis. Scalability is improved without compromising robustness by reducing average complexity to sub-quadratic order. This is achieved from a computational perspective by using Toom–Cook enhanced convolution within the DWT analysis level and truncation through SVD. This constitutes an extraordinary and complete development by collaborative conglomeration of formal modeling with hybrid remodel fusion, knowledge-assisted adaptive embedding, analytical renovation, and increasing algorithm speed.

Contributions of the proposed strategy

The primary contribution of the proposed method is the individual applicability of DWT, DCT, SVD, or CNN. These techniques have been investigated previously in steganographic approaches. However, the contribution lies in an optimization-guided combination for adaptive, statistically assured, and high-quality embedding. The major contributions of this approach are combined as follows:

• The proposed method is developed to regulate combining payload capacity and perceptual fidelity with statistical detectability through a constrained rate–distortion optimization structure.
• To control dynamically singular value decomposition depending on local picture complexity and texture properties, a lightweight CNN-driven adaptive embedding strategy is proposed.
• A hybrid transform-domain truncated SVD structure is developed for a combination of multi-resolution presentation with energy compaction and structural durability in a single integrated embedding pipeline.
• A histogram-preserving statistical limitation is introduced for minimization of first-order statistical variance and improving resistance against steganalysis in both classical and machine learning paradigms.
• For processing of high-resolution images, computational scalability is upgraded through acceleration of frequency-domain convolution.

The remainder of the article is organized as follows: Section 2 refers contains the literature review. We present the mathematical formulation in Section 3. The proposed methodology is illustrated in Section 4. The results are presented and discussed in Section 5. Finally, we conclude the manuscript in Section 6.

2 Literature review

The current frame of literature demonstrates in Table 1 large development in each deep learning-based research area, steganography, and steganalysis; however, numerous foundational gaps persist. Deep encoder–decoder frameworks incorporating squeeze-and-excitation (SE) attention, inception modules, GAN architectures, and multi-layer coding techniques frequently emphasize empirical upgrades in imperceptibility, robustness, and payload capability, but they commonly lack an express rate–distortion optimization method with analytical aims to maintain PSNR/SSIM constraints or controllable payload–constancy trade-offs. Transform-area processes primarily based on real-time integer wavelet transform (RT–ILWT), IWT–SVD–chaos, or DCT–GAN hybrids enhance robustness and safety, but frequently depend upon constant embedding rules, empirical parameter tuning, and execution-time assessment without formal asymptotic complexity evaluation or provable scalability guarantees. Moreover, present schemes rarely combine multi-degree hybrid rework fusion (e.g., DWT–DCT–SVD), luminance-only embedding for chrominance preservation, histogram-maintaining statistical constraints concentrated on first-order artifacts, and learning-driven content-adaptive embedding inside an unmarried unified framework. In parallel, superior transformer-based steganalysis strategies improve detection accuracy but remain defender-oriented and do not cope with theoretically constrained embedding design. Collectively, those boundaries spotlight a need for a mathematically grounded steganographic structure that unifies express rate–distortion optimization, system learning-guided adaptive embedding, hybrid rework-area modeling, statistical invisibility constraints, and provable computational efficiency, thereby advancing past empirical or structure-centric designs toward theoretically regulated, scalable, and high-capability steady communication structures appropriate for current multimedia applications.

TABLE 1. Comparative analysis of existing steganographic and steganalysis frameworks with identified research gaps.

Sanjalawe et al. (2025) brought a deep mastery-driven multi-layer steganographic framework combining Huffman coding, LSB substitution, and an encoder–decoder structure to decorate payload ability and perceptual quality. Despite this, numerous methodological barriers remain. Their technique seeks to empirically stabilize imperceptibility and embedding rate; however, it does not formulate the embedding system within an express limited rate–distortion optimization framework, nor does it outline a mathematical goal that collectively minimizes distortion D (C, C ^#) at the same time as maximizing payload R R. The absence of a proper optimization formulation limits theoretical guarantees concerning perceptual constancy and ability control. Furthermore, the embedding method operates frequently on the pixel and finds characteristic stages through LSB manipulation and deep characteristic reconstruction, without incorporating hybrid multi-decision transform-area mechanisms along with multi-degree discrete wavelet transform (DWT), block-based discrete cosine transform (DCT), or singular value decomposition (SVD), which might be seen as enhancing robustness in opposition to compression and signal processing attacks. Additionally, no evaluation concerns luminance-selective embedding or chrominance maintenance within perceptually aligned areas along with YIQ or YC_bC_r, thereby overlooking potential gains in structural consistency and constancy. Consequently, while the framework demonstrates promising deep mastering integration, it lacks a mathematically grounded rate–distortion formulation, transform-area robustness evaluation, and a perceptually guided area embedding method, indicating clear scope for a greater theoretically rigorous and multi-area optimized steganographic design.

El-Den and Raslan (2025) proposed a reversible RT, ILWT-based steganographic framework with robust empirical imperceptibility and robustness; numerous theoretical and methodological gaps remain. Their technique no longer formulates embedding as a limited rate–distortion optimization problem, missing a specific goal such as min D (C, C#) −βR or difficult PSNR/SSIM constraints, and alternatively balances ability and imperceptibility empirically. The embedding method is rule-based (constant center bit-planes) without gadget learning-driven, content-adaptive modeling, which the authors perceive as future work. Additionally, no formal asymptotic complexity evaluation is provided, regardless of recounted computational overhead from RT and ILWT integration, and robustness toward present-day ML-based steganalysis is not explicitly optimized via statistical protection constraints. These obstacles justify the need for a hybrid framework incorporating specific rate–distortion optimization, CNN-guided adaptive embedding, provable sub-quadratic computational complexity, and integrated statistical protection mechanisms.

Issac et al. (2025) demonstrated a deep steganographic structure integrating squeeze-and-excitation (SE) attention, Inception modules, and dilated convolutions for stable clinical big-records transmission, demonstrating real-time deployment and best perceptual quality on MRI and OCT datasets. Despite those architectural strengths, the framework stays and provides up-to-date encoder–decoder operation without a specific component of embedding as a limited rate–distortion optimization problem. In particular, no mathematical goal of the form min D (C, C^#) −βR is defined, nor are perceptual first-class metrics, including PSNR and SSIM, incorporated as bounds inside the knowledge objective. The said performance (PSNR ≈39 dB, SSIM ≈0.976) is consequently a function of unconstrained deep steganographic schemes in place of theoretically regulated constancy control. Furthermore, the technique operates immediately in the RGB area and no longer includes luminance-selective embedding, hybrid multi-level DWT–DCT–SVD remodel fusion, or histogram renovation constraints that explicitly deal with first-order statistical artifacts generally exploited in steganalysis. Although computational feasibility is validated on embedded hardware (Jetson TX2), the complexity evaluation is empirical, trading asymptotic bounds or algorithmic acceleration techniques for scalability for ultra-high-decision imagery and massive payload scenarios. In contrast, a mathematically grounded hybrid framework integrating remodel-area modeling, luminance-retaining color-area decomposition, distortion-conscious CNN-guided embedding, histogram-limited statistical security, and Toom–Cook–based sub-quadratic complexity addresses those open gaps, extending deep steganography past architectural innovation in the direction of theoretically limited, statistically robust, and computationally scalable high-ability stable communication.

Malik et al. (2025) proposed a hybrid method comprising DCT-GAN, which integrates LSB-based hiding within DCT coefficients while increasing imperceptibility and robustness. This method employed a deep GAN-based total encoder–decoder method. This mechanism reports an aggressive PSNR/SSIM and bit error rate (BER). However, this empirical layout is predominantly based on an inaccurate rate–distortion optimization version. So, this expressly magnifies payloads below PSNR/SSIM constraints or even on min D (C, C^#) −βR. The method operates inside the DCT and RGB domains without incorporating multi-degree DWT–DCT blending for luminance-satisfactory embedding. This is for chrominance protection, or truncated SVD-based completely singular charge regulation to structurally constrain block-level modifications. Analytical safety is evaluated experimentally instead of through unique histogram-protection or moment-based constraints, and computational scalability is examined via simulation without asymptotic complexity assessment or algorithmic acceleration mechanisms. In contrast, a mathematically defined ML-optimized rate–distortion-controlled hybrid framework addresses these limitations through the approach of integrating luminance-place multi-level DWT–DCT–truncated SVD embedding, distortion-aware CNN-guided adaptivity, histogram-confined statistical safety, and provable sub-quadratic complexity bounds, thereby extending DCT–GAN hybrids within the course of theoretically regulated, statistically robust, and computationally scalable high-capability steganography.

Alshoura and Alawida et al. (2025) present a chaos-enhanced IWT–SVD blind watermarking scheme focused on ownership protection, improving robustness and security through multiple scaling factors and encrypted side information. However, the method does not employ an explicit rate–distortion optimization framework with analytical aims to maintain PSNR/SSIM constraints, and payload–distortion trade-offs are evaluated empirically rather than modeled as min D C , C # − β R .The framework lacks hybrid DWT–DCT fusion, luminance-only embedding, truncated SVD-based complexity control, and learning-driven adaptive embedding strategies, while computational analysis remains empirical without asymptotic guarantees. A mathematically grounded ML-optimized rate–distortion-controlled hybrid approach addresses these gaps by integrating transform-domain fusion, CNN-guided adaptivity, histogram-aware statistical preservation, and sub-quadratic complexity bounds for scalable high-capacity steganography.

Sun et al. (2025) suggest a multi-scale transformer with multi-mission mastering for steganalysis of AMR speech under unknown embedding rates, jointly providing cover–stego classification and embedding-free regression with excessive detection accuracy. However, the framework is defender-oriented and modality-specific, specializing in low-bit-fee speech streams and optimizing detection of overall performance instead of the embedding procedure itself. No express fee–distortion optimization or method of the form min D (C, C^#) −βR is imposed, nor are tough perceptual constraints analytically enforced. Furthermore, the method no longer deals with hybrid rework-area photo embedding, luminance-maintaining color-area modeling, histogram-conscious statistical constraints, or computational scalability through sub-quadratic transforms. In contrast, a mathematically grounded ML-optimized fee–distortion-managed hybrid photo steganography framework explicitly integrates rework fusion, perceptual constraints, statistical preservation, and complexity guarantees, thereby addressing complementary gaps past transformer-based speech steganalysis.

3 Mathematical formulation

This subsection presents the formal mathematical representation for the steganographic problem, which establishes a rigorous basis on which the algorithm proposed later in the article can be built. In the steganographic process, it is mathematically defined as changing from a cover domain to essentially being described as another sub-domain, the “cover” becoming an intermediate layer in the image domain. The embedding process can be analyzed in a systematic manner, rather than being treated as an isolated event. The focus of this section is on the optimization viewpoint in steganography, which involves measuring distortion, capacity, and extraction accuracy to determine the effectiveness or inefficiency of an embedding process.

3.1 Information-theoretic formulation of the steganographic problem

This subsection presents steganography in an information-theoretic context, in which the embedding process is limited by measurable performance criteria (Li et al., 2024a; Fern’andez-Menduina et al., 2025).

Let the RGB carrier image be denoted in Equation 1 asand the grayscale secret image represented in Equation 2 as

C ∈ R M × N × 3 (1)

S ∈ R m × n , m ≤ M , n ≤ N . (2)

The steganographic embedding process produces a stego image depicted by Equation 3 aswhere E · denotes the embedding operator.

C # = E C , S (3)

The design objective is to satisfy the following constraints.

3.2 Imperceptibility constraint

The carrier and stego image difference is quantified by the distortion measure. Deficiencies in distortion remain below a predetermined level, so that embedding does not introduce visible artifacts and thus preserves perceptual fidelity.

Let the distortion between the carrier and stego image be defined by Equation 4 as

D C , C # = 1 M N ∑ i = 1 M ∑ j = 1 N C i , j − C # i , j 2 (4)

Imperceptibility requires representation by Equation 5 as

D C , C # ≤ ϵ . (5)

3.2.1 Robustness constraint

The bit error rate (BER) is the measure of robustness in the extracted secret data, which follows channel distortions such as compression, filtering, or noise. Despite the completion of typical signal processing operations, the embedded information can still be recovered due to this constraint (El-Den and Raslan, 2025; Nasr et al., 2024; Li et al., 2024b).

Let S ^ denote the extracted secret image after channel distortion A ·. Hence, bit error rate (BER) is calculated by Equation 6 as follows:

B E R S , S ^ = 1 m n ∑ i = 1 m ∑ j = 1 n S i , j ≠ S ^ i , j (6)

Robustness is required by Equation 7:

B E R ≤ δ . (7)

3.2.2 Capacity constraint

The aim is to determine the quantity of secret data that can be hidden by contemplating the size of the picture through embedding rates. The proposed algorithm can achieve significant payload capacity (Issac et al., 2025; Zhao et al., 2024; Song et al., 2024) by using the formula without disproportionate distortion.

The embedding rate should be defined by Equation 8 as

R = ∣ S ∣ ∣ C ∣ . (8)

The capacity requirement is defined by Equation 9:

R ≥ γ . (9)

3.3 Constrained rate–distortion optimization model

This subsection presents a steganographic submerging problem, which is basically an optimization problem with a constrained rate–distortion configuration. This algorithm attempts to reduce disfiguration through optimization of embedding strength under qualified constraints rather than selecting a location for embedding randomly. The formal intensification formula offers a conscionable approach to control the trade-off between payload and nonrepresentational quality (Zhu et al., 2024; Zhou et al., 2024; Wu et al., 2025). This formulation is important because empirical parameter adjustment is the base of most steganographic paradigms. The theoretical base of the proposed algorithm is enhanced compared with existing approaches because it provides statistical control on embedding distortion.

Embedding can be formally modeled by Equation 10 as follows:subject to

min ⁡ α , E i , j D C , C # − β R (10)

P S N R C , C # ≥ 50 dB

S S I M C , C # ≥ 0.993

Where embedding strength is calculated by Equation 11 as

R = ∑ i , j E i , j , E i , j ∈ 0 , 1 , α = embedding strength . (11)

This equation differs essentially from heuristic embedding (Li et al., 2024a; Pan and Ni, 2024; Menduina et al., 2024). It administers explicit manage over perceptual constancy restrained optimization.

3.4 Perceptual quality metrics: mathematical analysis

PSNR measures the pixel level deformation among cover and stego pictures. This section derives this statement using the principle that maintaining a distortion limit keeps PSNR values high (Zhang et al., 2023; Zhai et al., 2024). The PSNR is stated in Equation 12 as follows:where L = 255 for 8-bit images.

P S N R = 10 log ⁡ 10 L 2 M S E (12)

Given constraint P S N R ≥ 50, this implies, through Equation 13,

M S E ≤ 255 2 10 50 / 10 . (13)

Thus,

M S E ≤ 0.650 .

This tight bound enforces extremely low distortion (Sharma et al., 2022; Yamni et al., 2024).

Structural Similarity Index (SSIM):

To evaluate perceptual similarity, SSIM takes into account luminance, contrast, and structural information (Li et al., 2024b; Fern’andez-Menduina et al., 2025; Yuan et al., 2025). SSIM is defined by Equation 14 as follows:

S S I M x , y = 2 μ x μ y + C 1 2 σ x y + C 2 μ x 2 + μ y 2 + C 1 σ x 2 + σ y 2 + C 2 . (14)

Maintaining higher SSIM ensures structural and luminance consistency.

These constraints motivate luminance-domain embedding and adaptive coefficient modification.

3.5 Color space transformation and perceptual modeling

This provides an explanation for the conversion of the carrier image from RGB to YIQ color space. Human visual perception is more affected by distortions in chrominance components than by variations in luminance (Wang et al., 2025; Li et al., 2023; Shi et al., 2025). Embedding hidden information in the luminance channel allows the algorithm to exploit perceptual redundancy without introducing visible color effects. In this subsection, the design decision used later in an algorithm is explained as follows: to ensure uniformity of perceptual embedding, we perform embedding only in the luminance channel (Yamni et al., 2024). To preserve chrominance fidelity, the RGB carrier image is converted into YIQ, as given by Equation 15:

Y I Q = T R G B (15)

Where T is defined by Equation 16:

T = 0.299 0.587 0.114 0.596 − 0.274 − 0.322 0.211 − 0.523 0.312 . (16)

Only the luminance channel Y is used for embedding, as given by Equation 17, ensuring

Δ I = 0 , Δ Q = 0 . (17)

This reduces perceptual color distortion (Khokhar et al., 2025).

3.6 Theoretical basis for hybrid transform modeling

3.6.1 Discrete wavelet transform (DWT)

By utilizing DWT, an image can be broken down into subbands with multiple resolutions that represent different frequency components (Yang et al., 2025). The decomposition facilitates embedding in areas in which distortion is less noticeable, and strength is enhanced. DWT decomposes the image into subbands following Equation 18:

L L , L H , H L , H H = D W T Y . (18)

This provides spatial-frequency localization (El-Den and Raslan, 2025).

3.6.2 Discrete cosine transform (DCT)

By concentrating signal energy within certain low-frequency coefficients, DCT produces highly efficient energy compaction (Pan and Ni, 2024) according to Equation 19. This property allows embedding to be performed with minimal spatial distortion. For 8 × 8 block B k:

D k u , v = α u α v ∑ x = 0 7 ∑ y = 0 7 B k x , y cos ⁡ 2 x + 1 u π 16 cos ⁡ 2 y + 1 v π 16 . (19)

DCT compacts energy into low-frequency coefficients (Malik et al., 2025).

3.6.3 Singular value decomposition (SVD)

SVD, as given in Equation 20, describes each picture block as orthogonal matrices with singular values. So, minimum perturbations can affect the results with negligible distortion caused by the singular value property of stability. A combination of this transformation can demonstrate supplementary advantages in relation to recognizability, strength, and stiffness (Taj et al., 2024):

D k = U k Σ k V k T . (20)

Perturbing the controlling singular value confirms minimal structural deformation is attributable to the theory of matrix perturbation described by Equation 21:

IfThen, singular value variation is satisfied by Equation 22:

D k # = D k + Δ , (21)

∣ σ i D k # − σ i D k ∣ ≤ ∥ Δ ∥ 2 . (22)

Thus, minimal modification can preserve structural similarity (Alshoura and Alawida, 2025).

3.7 Hybrid multi-stage transform modeling: mathematical foundations

Integrating several orthogonal transforms inside a consolidated embedding paradigm is motivated by compatible strengths of multi-resolution decomposition, the energy compaction nature of the cosine function, and singular value reliability. Individually, every transform provides fractional robustness collectively, enabling multi-resolution, block-based frequency, and configurational-domain control (Maurya et al., 2023) over embedding deformation. The analytical justification for the proposed hybrid SVD adopted methodology is depicted in this section.

3.7.1 Multi-level discrete wavelet decomposition

Let us consider a luminance domain, denoted by Equation 23 as

Y ∈ R M × N . (23)

A 2-level DWT decomposes Y into hierarchical subbands, as depicted in Equation 24:

L L 1 , L H 1 , H L 1 , H H 1 } = D W T Y , (24)

L L 2 , L H 2 , H L 2 , H H 2 } = D W T L L 1 .

The L L 2 subband denotes approximate low-frequency values, while other subbands assimilate directional high-frequency components.

Wavelet transform can be revealed as convolution with quadrature replicate filters (Equation 25):where h L and h H denote low-pass and high-pass filters, and ↓ 2 indicates dyadic subsampling.

L L = Y * h L T * h L ↓ 2 , L H = Y * h H T * h L ↓ 2 , (25)

Wavelet energy preservation following Equation 26 satisfies

∥ Y ∥ 2 2 = ∥ L L ∥ 2 2 + ∥ L H ∥ 2 2 + ∥ H L ∥ 2 2 + ∥ H H ∥ 2 2 . (26)

This orthonormal disintegration enables embedding within preferred subbands by maintaining global energy uniformity and reliability consistency (Malik et al., 2025; Alshoura and Alawida, 2025; Yamni et al., 2024).

3.7.1.1 Selection L L 2 for rationale

Embedding within excessively high-frequency bands may degrade robustness under compression and filtering. Conversely, embedding in a pure spatial domain increases detectability. The second-level approximation band L L 2 provides balanced robustness and imperceptibility due to reduced spatial sensitivity and controlled frequency representation (Wang et al., 2024).

3.7.2 Block-based discrete cosine transform (DCT)

Each non-overlapping 8 × 8 block B k in L L 2 undergoes DCT, as given by Equation 27:

D k u , v = α u α v ∑ x = 0 7 ∑ y = 0 7 B k x , y cos ⁡ 2 x + 1 u π 16 cos ⁡ 2 y + 1 v π 16 . (27)

The energy compaction property is defined by Equation 28 as follows:

∑ u , v D k u , v 2 = ∑ x , y B k x , y 2 . (28)

DCT exhibits strong decorrelation capability, concentrating energy into low-frequency coefficients. This property ensures that small perturbations in selected coefficients distribute minimally within spatial reconstruction.

The combination of DWT and DCT yields

• Multi-resolution robustness (DWT)
• Local block energy control (DCT)
• Reduced inter-coefficient correlation.

3.7.3 Singular value decomposition and structural stability

For each DCT block stated in Equation 29:

D k = U k Σ k V k T . (29)

Here,

• U k and V k are orthogonal matrices,
• Σ k = diag σ k 1 , σ k 2 , … , σ k 8.

Singular values represent the intrinsic geometric structure of the block.

3.7.4 Perturbation stability theorem

Let the perturbed matrix be defined by Equation 30:

D k # = D k + Δ k . (30)

Weyl’s inequality shown in Equation 31 states

∣ σ i D k # − σ i D k ∣ ≤ ∥ Δ k ∥ 2 . (31)

Thus, modifying only the dominant singular value σ k 1 with a small perturbation ensures bounded structural distortion.

3.7.5 Truncated SVD for complexity reduction

Full SVD complexity for n × n block is given as Equation 32:

O n 3 . (32)

For 8 × 8 blocks, although constant-sized, full SVD across all blocks yields cumulative cost:

O M N 64 · 8 3 = O M N .

However, singular vectors are not modified in embedding. Only the dominant singular value is required.

Thus, truncated SVD approximations are calculated using Equation 33:

D k ≈ U k 1 σ k 1 V k 1 T . (33)

This reduces computation to dominant eigenvalue extraction using power iteration, as shown by Equation 34:

σ k 1 = max ⁡ ∥ x ∥ = 1 ∥ D k x ∥ . (34)

Power method complexity is depicted by Equation 35:

O n 2 . (35)

Therefore, truncated SVD provides structural robustness with reduced computational cost.

3.8 Adaptive embedding optimization

Adaptive embedding optimization enhances content-specific embedding methods by adjusting the strength of embedded content to match local image traits. The main point of contention is that smooth areas are more susceptible to distortion, while textured areas can tolerate more severe modifications. The algorithm’s dynamic modification of embedding strength reduces perceptual distortion, while also maximizing embedding capacity. Embedding is performed via singular value modification (Dhawan et al., 2021), as shown by Equation 36:

σ k 1 # = σ k 1 + α i , j · S ′ i , j , (36)

Where

S ′ i , j = S i , j 255 .

The reconstructed block can be calculated by Equation 37:

D k # = U k Σ k # V k T . (37)

3.8.1 Adaptive strength function

The adaptive strength function determines embedding intensity based on local variance. Regions with higher variance correspond to complex textures where embedding modifications are less noticeable. This mechanism implements the perceptual masking principle, ensuring that embedding operations remain visually imperceptible (Dhawan et al., 2021). Embedding strength is not constant. It is defined by Equation 38 as follows:

α i , j = α max · σ l o c a l i , j σ g l o b a l . (38)

Let local variance be given by Equation 39:

σ l o c a l 2 i , j = 1 w 2 ∑ p , q ∈ Ω Y p , q − μ Ω 2 . (39)

High-texture regions exhibit larger variance (Equation 40),

α i , j ↑ if texture high . (40)

This adaptive modulation satisfies perceptual masking principle, as shown in Equation 41:

∂ D ∂ α ≤ 0 in textured regions . (41)

3.8.2 Embedding map via machine learning

Using local image patches, a lightweight CNN is introduced to learn about embedding probability maps. Without defining them manually, CNN predicts areas that are suitable for embedding. The ability of the algorithm through its learning to capture convolute patterns of texture helps to enhance statistical invisibility. The embedding probability map can be defined by Equation 42:where P i j depicts a local patch centered at i , j .

P e m b i , j = f θ P i j , (42)

Decision rules can be depicted using Equation 43:

E i , j = 1 if P e m b i , j > τ 0 otherwise . (43)

Loss function during training is calculated by Equation 44:

L = L C E + γ · M S E Y , Y # . (44)

This strengthens deformation-aware learning (He et al., 2025; Song et al., 2024).

3.8.3 Constrained Lagrangian optimization embedding

To determine the embedding strength, theoretical deduction illustrates the use of Lagrangian optimization paradigm. The derivation proves that marginal distortion is equal to the marginal embedding charge gain, resulting in foremost embedding and a principled method for balancing payload and picture quality. This technique uses the following equation (Sun et al., 2025; Qiu et al., 2022; Sun et al., 2023; Song et al., 2024).

The restrained objective is defined by Equation 45:It is reformulated using Equation 46:

min ⁡ D − β R (45)

J α , E = ∑ i , j Y i , j − Y # i , j 2 − β ∑ i , j E i , j . (46)

It takes the derivative in Equation 47:

∂ J ∂ α = 2 Y − Y # ∂ Y # ∂ α − β ∂ R ∂ α . (47)

Optimal embedding strength is satisfied using Equation 48:

2 Y − Y # ∂ Y # ∂ α = β . (48)

Thus, embedding will increase till marginal distortion equals marginal charge gain (Sharma et al., 2022). This formal derivation distinguishes the proposed method from heuristic embedding.

3.9 Statistical invisibility constraint

By inspecting statistical irregularities in pixel-level distributions, current strategies for steganalysis perceive covert information. A limitation is covered in this section for restrictions on histogram variations among cover and stego pictures, preserving the equal amount.

First-order histogram preservation is presented in Equation 49:

min ∥ H Y − H Y # ∥ 2 2 . (49)

Let us say the bin probability of the histogram is given by Equation 50:

p k = n k M N . (50)

Embedding discomposure shifts bin counts calculated by Equation 51:

p k # = p k + Δ k . (51)

Constraint requires the following:

∑ k Δ k 2 → 0 .

Thus, analytical divergence is minimized by adjusting embedding strength.

3.10 Computational scalability and Toom–Cook accelerated convolution

3.10.1 Computational bottleneck in hybrid transform steganography

Though hybrid frequency domain embedding enhances robustness and imperceptibility, it also introduces significant computational overhead. In multi-dimensional DWT, convolution operations influence processing cost. For an image Y ∈ R M × N, standard separable 2D convolution with kernel size k × k requires the following (Equation 52):

O M N k 2 (52)

For ideal wavelet filters where k is proportionally equivalent to signal length in multi-spectral decomposition, convolution cost approaches are expressed as in Equation 53:for large-scale perpetration or non-optimized filtering.

O M N 2 (53)

The quadratic development becomes computationally preventive when it is extended to high-resolution pictures such as 4K medical images. Thus, it reduces convolution complexity without compromising transform accuracy, which is essential for scalable steganographic deployment. The proposed method addresses these limitations by incorporating Toom–Cook polynomial multiplication, a divide-and-conquer convolution acceleration strategy which minimizes convergent complexity to sub-quadratic order.

3.10.2 Convolution polynomial multiplication

Consider 1D convolution as given in Equation 54:

c n = ∑ i = 0 k − 1 a i b n − i (54)

This is interpreted as polynomial multiplication, as shown in Equations 55, 56:

A x = ∑ i = 0 k − 1 a i x i (55)

B x = ∑ j = 0 k − 1 b j x j (56)

Then:

C x = A x B x (57)

Coefficient extraction of C x, as shown in Equation 57, yields a convolution result.

Thus, complexity defined in Equation 58 generated due to convolution minimizes to polynomial multiplication complexity. Standard polynomial multiplication requires the following:

O n 2 multiplications . (58)

3.10.3 Toom–Cook multiplication: divide-and-conquer strategy

Here, Karatsuba multiplication is generalized by Toom–Cook multiplication, where the polynomial is sliced into three parts.

Let us consider the polynomial of degree n to be partitioned by Equations 59, 60: where m = n 3.

A x = A 0 + A 1 x m + A 2 x 2 m (59)

B x = B 0 + B 1 x m + B 2 x 2 m (60)

Multiplication following Equation 61 becomes

C x = A x B x (61)

Instead of nine naive multiplications, Toom-3 evaluates polynomials shown in Equation 62 at selected points:

x ∈ 0 , 1 , − 1 , 2 , ∞ (62)

This should be computed for five evaluation points using Equation 63:

P i = A x i B x i (63)

Thus, only five recursive multiplications of size n / 3 are required instead of nine. Reconstruction (interpolation) combines evaluated products to obtain coefficients of C x.

3.10.4 Recurrence relation and complexity derivation

Let T n given in Equation 64 denote multiplication cost.

3.10.4.1 Recurrence

T n = 5 T n 3 + O n (64)

The linear term corresponds to evaluation and interpolation steps.

Applying the Master Theorem, we get

T n = O n log ⁡ 3 5 .

Because

log ⁡ 3 5 = log ⁡ 5 log ⁡ 3 ≈ 1.4649 .

T n = O n 1.46

This is strictly sub-quadratic, as

1.46 < 2 .

3.10.5 Extension to 2D convolution

Wavelet filtering requires 2D convolution, as given in Equation 65, implemented separably:

Y * h = Y * h x * h y . (65)

If 1D convolution complexity is as follows,then 2D separable filtering complexity becomes

O n 1.46 ,

O M N 1.46 .

For square image M = N:Compare this to the standard:in naive 2D convolution.

O N 2.46

O N 3

Thus, computational reduction ratio asymptotically follows Equation 66 and becomes

N 2.46 N 3 = N − 0.54 . (66)

For large N, savings increase significantly.

3.10.6 Multi-level DWT complexity with Toom–cook

For 2-level DWT:

Level 1 operates on the full image given in Equation 67:

O M N 1.46 (67)

Level 2 operates on quarter-size approximation following Equation 68:

O M N 4 1.46 (68)

Total complexity is calculated by Equation 69:

T D W T = O M N 1.46 1 + 1 4 1.46 (69)

4 1.46 ≈ 8.93

T D W T ≈ O M N 1.46 1 + 0.112

T D W T ≈ 1.112 O M N 1.46

Dominant order remains are as follows:

O M N 1.46

3.10.7 Computational acceleration of DWT convolution

Let M = N. The proposed hybrid steganography algorithm 1.1 and algorithm 1.2 operates in sub-quadratic time relative to standard convolution-based transform implementations.

Standard convolution-based DWT given by Equation 70 :

O N 3 (70)

The proposed Toom–Cook accelerated DWT is given as Equation 71:

O N 2.46 (71)

2.46 < 3 ,

It follows using Equation 72 that

lim ⁡ N → ∞ N 2.46 N 3 = lim ⁡ N → ∞ N − 0.54 = 0 . (72)

Thus, asymptotically defined by Equation 73,

O N 2.46 = O N 3 . (73)

Thus, the proposed strategy conceptually decreases the convolutional complexity of the DWT operation through fragmenting and controlling polynomial multiplication. However, the complexity of the total workflow remains dominated by DCT, CNN inference, and block-based SVD operations.

3.10.8 Theoretical implication for real-time steganography

Though the acceleration in the DWT phase decreases convolutional overhead conceptually, practically, the overall complexity of the complete embedded pipeline remains almost linear in connection with picture size due to fixed-size transformation and block-based operations.

• Suitable for large multimedia datasets, the proposed algorithm achieves a reduction in convolutional complexity
• It is deployable in near real-time systems
• It is feasible for GPU acceleration
• It is scalable for high-capacity embedding

This computational guarantee distinguishes the framework from empirical deep-learning-only or full-SVD-based methods lacking asymptotic analysis (Issac et al., 2025; Li et al., 2024b).

We present a theoretically grounded steganographic framework that merges rate–distortion constrained optimization, hybrid transform-domain modeling, adaptive machine learning control, and computational acceleration into a unified coherent architecture. The proposed design utilizes DWT–DCT–truncated SVD fusion and incorporates complementary characteristics such as multi-resolution frequency representation, energy compaction, and structural stability for distortion-minimized embedding. A strategy of machine learning-based adaptive embedding regulates the strength of embedding based on local image complexity, while a statistical constraint that preserves histogram strengthens resistance to modern steganalysis. To maintain practical scalability, the framework employs Toom–Cook convolution and reduces dominant computational complexity to sub-quadratic orders.

4 Proposed method: ML-optimized rate–distortion controlled hybrid steganography

The proposed ML-optimized rate–distortion managed hybrid steganography framework demonstrates the benefit of full-size development over traditional remodel-area embedding strategies via integration of multi-stage DWT, block-based DCT, truncated SVD, and system learning-guided adaptive embedding inside a unified mathematical optimization model. By formulating the embedding method as a confined rate–distortion minimization problem, the approach explicitly balances payload ability and perceptual fidelity, ensuring PSNR values remain above 50 dB (≈51.40 dB), and SSIM values approximately around 0.9816 whilst retaining structural and chromatic consistency via luminance-area embedding in the YIQ color space. The adaptive singular rate perturbation strategy, regulated through a manner of nearest variance and complexity measures, minimizes distortion in smooth regions and strategically enhances embedding strength in high-texture regions, thereby enhancing imperceptibility and robustness. The integration of a lightweight CNN-based embedding opportunity map similarly improves statistical safety through a manner of mastering content-aware embedding distributions under a distortion-penalized loss function, lowering susceptibility to trendy steganalysis strategies. Moreover, histogram analysis maintains constraint-diminishing first-order analytical deviations, thereby resisting detection. Additionally, from a computational overview, adoption of Toom–Cook convolution in DWT calculation and SVD substantially diminishes the time trade-off to quadratic order O M N 1.46. Thus, it makes the strategy scalable for highly decisive pictures without affecting robustness. Collectively, the hybrid redesign fusion, adaptive attending to know-driven embedding, formal optimization modeling, and computational acceleration install the proposed set of policies as a balanced, secure, and inexperienced steganographic solution suitable for high-capacity potential covert verbal exchange applications.

4.1 Problem formulation

Let the RGB carrier image be defined using Equation 1

C ∈ R M × N × 3

and the secret grayscale image by Equation 2

S ∈ R m × n

with m ≤ M , n ≤ N .

The objective is to construct a stego image C # such that

• The imperceptibility constraint uses Equation 3

P S N R C , C # ≥ 40 dB S S I M C , C # ≥ 0.9875

• 2. There is maximum embedding capacity
• 3. There is minimum computational complexity
• 4. There is robustness against statistical steganalysis.

This is formulated by Equation 10 as a constrained rate–distortion optimization problem:where E i , j ∈ 0 , 1 denotes the embedding map and α the embedding strength.

min ⁡ α , E D C , C # − β R

D = ∑ i , j Y i , j − Y # i , j 2

R = ∑ i , j E i , j

4.2 Algorithms

Algorithm 1.1 Proposed CNN-Based Toom–Cook Accelerated DWT–DCT–Truncated SVD Image Embedding Algorithm.

Algorithm 1.2 Secret Image Extraction Algorithm.

4.3 Formal computational complexity analysis

Let the image size be M × N.

1. YIQ Transformation

O M N

2. 2-Level DWT (Toom–Cook Accelerated)

Each level:

O M N

With Toom–Cook convolution:

O M N 1.46

3. Block DCT

Number of blocks:

M N 64

Per block complexity:

O 64 ⁡ log ⁡ 8

Total:

O M N ⁡ log ⁡ 8 ) ≈ O ( M N

4. Truncated SVD

Per block:

O 64 2

Total:

O M N

5. CNN Embedding Map

If kernel size k × k:

O M N k 2

For lightweight CNN (small k):

O M N

4.4. Total Complexity

T M , N = O M N 1.46 + O M N

The dominant term is O M N 1.46, which is lower than conventional O M N 2 convolution-based implementations.

4.4.1. Computational Efficiency

For sufficiently large N,

M N 1.46 < M N 2

Because 1.46 < 2, the proposed Toom–Cook accelerated framework achieves sub-quadratic complexity relative to standard convolution.

5 Result analysis and discussion

5.1 Dataset description

For evaluation of the performance of the proposed method against benchmarked datasets that are publicly available and widely used in JPEG format from Kotak for research of steganography including color and gray scale images, such as Auditorium, Fox, House, Iguana, Peeper, Seabeach, Strawberry, and Sunset given in Figures 1a–h is taken into consideration, which is provides multiple characteristics like edges, texture, and smooth regions. All the images below are stored in standard JPEG format for real compression environments, which ensures live applicability. The datasets contain an equilibrium mixture of high- and low-textured images. This diversification is needed to validate the adaptability of embedding; the strategy depends on picture complexity.

5.2 Resolution and sample size

The original cover pictures shown in Figures 2a–h were collected in RGB color format, which was subsequently standardized into a uniform spatial resolution of 512 × 512 pixels with 8-bit depth per color domain (24-bit color pictures), as shown in Table 2. Secret information to be hidden as Signature_AB as shown in Figure 2i. This ensures consistency during experimental evaluations. The picture dataset consists of pictures with diversifying texture densities, luminance allocations, and structural contents. This helps the comprehensive assessment of imperceptibility, capacity, robustness, and resistance against modern steganalysis.

• The total number of test images was 8 (benchmark set) (http://sipi.usc.edu/database/; Eastman Kodak Company, 2026; http://istock.com/beach).
• The color space was as follows: transformation of RGB into a YIQ color space, where embedding is incorporated only in the luminance (Y) channel.
• The bit depth was 8 bits per channel.

TABLE 2. Features of selected dataset pictures.

Test pictures were resized to maintain uniformity of resolution: 512 × 512 pixels.

Each experiment was carried out over all pictures, and mean results were noted as evidence for validation of robustness and statistical consistency.

5.3 Parameter settings

The proposed method incorporates frequency domain embedding with adaptive maximization. The parameters are chosen to maintain the balance of the triad: imperceptibility, robustness, and security.

5.3.1 DWT configuration

• Wavelet type: Haar wavelet
• Segregation level: 2D-DWT
• Embedding channel: Level-2 approximation of LL2 subband of the luminance (Y) component.

5.3.2 DCT configuration

• Block size: 8 × 8
• Transformation: 2D DCT applied to selected DWT subbands
• Selection of coefficients: the mid-frequency domain is selected to avoid perceptible distortion and ensure robustness.

5.3.3 Truncated SVD parameters

• Disintegration applied to DCT blocks
• Truncated rank:

e m p i r i c a l s e l e c t i o n r = 20

The reason behind this truncation is to minimize redundancy, stabilize the embedding process, and improve resistance against noise, which thereby enhances the compression paradigm.

The CNN module in the proposed strategy is employed exclusively to create maps for adaptive embedding probability depending on local picture texture and complexity properties. It is not incorporated as the preliminary steganalysis detector. Evaluation of steganalysis resistance is independently executed by employing RS analysis, chi-square attack, SPA attack, and support vector machine (SVM)-based classifiers.

5.3.3.1 CNN structure for adaptive embedding guidance

A lightweight CNN-dependent method is employed to evaluate resistance against learning-based steg attacks.

Input: 256 × 256 image patches

• Convolutional layers: three layers
• Conv (32 filters, 3 × 3) + ReLU
• Conv (64 filters, 3 × 3) + ReLU
• Conv (128 filters, 3 × 3) + ReLU
• Pooling: Max pooling (2 × 2)
• Fully connected layers: 2 layers
• Output: Binary classification (cover vs. stego)
• Loss function: Binary Cross-Entropy
• Optimizer: Adam (learning rate = 0.001)

From selected picture datasets, benchmark pairs of cover and stego pictures were generated by a lightweight CNN, which was trained using benchmark datasets. Then every picture was subdivided into non-overlapping 256 × 256 blocks for training. This label is created according to local embedding suitability. Here, highly textured regions were allocated higher embedding probabilities and seamless regions with lower probabilities. The dataset was subdivided into 80% for training and 20% for testing samples. The network was trained for 20 cycles using a batch size of 32 with the Adam optimizer (learning rate = 0.001). Binary Cross-Entropy loss was incorporated to optimize prediction of embedding probability and minimize distortion-sensitive embed decisions.

5.3.3.2 Parameters for embedding strength

Depending on standardized picture complexity, embedding strength was adaptively administered using Equation 74.

α = α min + α max − α min · C n o r m (74)

• α min = 0.05 (for smooth regions).
• α max = 0.25 (for high-texture regions).
• C n o r m ∈ 0 , 1: measurement of normalized complexity.

The key characteristics are that low embedding in smooth regions preserves values of PSNR and SSIM, while robustness and capacity improve in high embedded textured areas.

5.4 Imperceptibility analysis

The imperceptibility assessment illustrates that the proposed method conserves the perceptible quality of stego pictures with invariably high PSNR values (∼51.40 dB) and SSIM (∼0.9816). The minimum mean squared error (MSE) and extraordinary low BER further validate that embedding introduces only insignificant distortion. Visual inspection suggests, as seen in Figures 3a–h, no major artifacts, even in smooth areas, indicating efficient management of embedding strength. This is accomplished through the adaptive, complexity-conscious embedding mechanism, which methodically modifies textured areas while securing homogeneous regions. Overall, the results validate that the approach achieves excessive payload capability without compromising perceptual quality, a key necessity for a secure steganographic paradigm.

Table 3 presents the quantitative imperceptibility performance of the proposed method, demonstrating high PSNR (>49 dB), strong structural similarity (SSIM >0.97), and negligible bit error rates.

TABLE 3. Quantitative imperceptibility performance analysis.

5.5 Payload capacity evaluation

The proposed method actively calibrates embedding capacity based on picture complexity; this is indicated by results of semantics density (Table 4). Payload allocation varies according to local picture complexity. The proposed strategy maintains consistency from medium- to high-capacity embedding across all the benchmark pictures (Table 5), while preserving perceptual accuracy. This means that the embedding strategy takes into account the complexity and effectiveness of the method. This also enables a balance trade-off between imperceptibility and payload, as shown in Figures 4, 5.

TABLE 4. Payload capacity (bits per pixel - BPP).

TABLE 5. Rate–distortion values (BPP vs. PSNR).

5.6 Rate–distortion optimization performance

The rate–distortion performance of the proposed strategy is evaluated by analyzing the relationship between embedding rate (BPP) and distortion (measured in PSNR). The experimental evidence shown in Table 5 highlights the consistency of the proposed method, which maintains a high PSNR between 49.92 and 53.76 dB across diversified payload capabilities, indicating effective control over embedding-stimulated distortion. Remarkably, even with increasing embedding rates of bpp ≈0.21, the PSNR achieves values above 50.37 dB. This reflects the strength of the constrained optimization strategy in sustaining perceptual commitment. Moreover, the proposed approach exhibits a more stable receiver operating characteristic (ROC), compared to conventional methods, while increasing payload does not significantly degrade image quality. This suggests that embedding energy is optimally regulated and in step with nearby picture characteristics.

In most cases, this behavior is attributed to the combination of rate–distortion formulation, CNN-guided adaptive embedding, and transform-area coefficient selection; together, these ensure that distortion is minimized even as embedding capacity is maximized, as depicted in Figure 6. This improves the balance between the allocation of higher embedding to texture-rich areas and the restriction of any modifications in the smooth regions of images. This can be done by enhancing the adaptive modulation of embedding strength depending on local variance. Consequently, the proposed method describes outstanding performance relative to traditional and hybrid embedding strategies while maintaining an impactful balance between payload capacity, perceptual quality, and rate–distortion optimization. The proposed model successfully implements high-capability embedding and imperceptible strego communication by validating empirically and theoretically the optimization of the rate–distortion mechanism shown in Figure 6.

The adaptive embedding strength variation as a function of normalization of picture complexity can be illustrated by the strength-distortion curve given in Figure 7. From this, it is clear that embedding capacity monotonically increases with the increase of complexity. This reflects the principle of perceptual masking. Critically, low-complexity smooth regions are assigned lower embedding to minimize visual distortion compared to high-complexity, textured areas. This increases the strength of the embedding capacity without compromising visual quality and strength. Optimal embedding efficacy is ensured by creating an adaptive environment while validating the effectiveness of CNN-dependent complexity.

5.7 Robustness against image processing attacks

Robustness evaluation shows that the proposed adaptive stego strategy maintains an extraction accuracy above 96% (Table 6) under different steganalysis attacks like compression and distortion by noise. This is a direct consequence of the image complexity-guided adaptive nature of the embed method. Some images with high texture, like Seabeach.jpg, reveal superiority in terms of robustness measures due to powerful embedding in characteristically complex regions; by comparison, stability is maintained in smoother pictures such as Auditorium.jpg through conservation of embedding.

TABLE 6. Extraction accuracy (%) under different attacks.

Under attacks of geometric such as cropping and scaling, the method preserves sustainable performance above 92% for recovery which indicate effectiveness of spatial redundancy and payload allocation distribution. The lower bound value of bit error rate 0.00051 (average) ensures that propagation of disruption does not significantly intrude within extraction errors.

The proposed methodology accomplished an equilibrium trade-off in terms of robustness, imperceptibility, and security in combination with poor steganalysis detection depicted by ROC-AUC ≈0.0, having a minimum true positive rate (TPR) at an increasing false positive rate (FPR) simultaneously in traditional schemes. This highlights contribution of the strategy leveraging perceptive adaptive submerging with persistency payload distribution. This is highly suitable for assured and covert picture communication in conflicting environments.

The experimental results of bit error rate (BER) analysis shown in Table 7 confirms that the proposed method achieves a lower BER under different signal processing attacks like Gaussian noise and compression specifically. Gaussian noise attacks show values lower than 0.3, validating stability for embedding bits in frequency appearance regions. Cropping introduces the highest level of BER due to the removal of the entire embedding region, although the error remains under control due to the distribution of embedding. Throughout the evaluation, it was clear that the highly textured pictures, such as Seabeach.jpg, consistently achieve the lowest BER across different attacks, strengthening the novelty of the embedded principle depending on picture complexity. This suggests that the method not only enhances imperceptibility but also enhances robustness selectively where channel dimension is naturally advanced, which is shown to be a key contribution to the embedding strategy. The proposed strategy does not apply a definite geometric synchronization process, such as an affine transformation or key point-based positioning. Thereby, robustness is accomplished preliminarily through dispensing of frequency-domain embedding and context-dependent redundancy allocation rather than accurate geometric restoration against cropping and rescaling.

TABLE 7. Bit error rate (BER) under image processing attacks.

5.8 Steganalysis resistance

Examination was carried out to test steganalysis attacks against analytical and knowledge bases for evaluation of the security of the proposed steganographic framework. The analysis incorporates classical first-order and higher-order analytical detectors. Modern machine knowledge-based classifiers are also incorporated for this purpose, as shown in Table 8. The utilization of CNN in this method is a more distinctive approach than the external steganalysis evaluation models. The stated statistical steganalysis results are procured independently using analytical and machine learning-based detectors.

TABLE 8. Steganalysis detection performance.

The receiver operating characteristic (ROC) curve was obtained by changing the decision threshold of the steganalysis separator throughout the testing datasets. The experimental result achieved near 0.5 for the area under the curve (AUC), which strongly indicates random differentiation competency between cover and stego pictures. This behavior suggests that the embedding paradigm efficiently minimizes comprehensible statistical artifacts.

A balanced dataset was constructed for steganalysis evaluation, consisting of cover and stego pictures comprising 500 sample pictures, including 250 each of cover and stego obtained from benchmark datasets having versatile texture complexity. This dataset was segregated into 80% as training and 20% as testing subsets. Along with classical steganalysis strategies like RS, chi-square, and sample pair analysis (SPA), ML-based steganalysis is also employed using an SVM classifier for analytical and texture-dependent feature presentations, which are extracted from frequency channel coefficients and histogram allocation.

The proposed method was additionally tested against lightweight CNN-guided steganalysis, which was trained on the same partition for assessment of robustness against modern steganalysis attacks. The experimental calculation shown in Table 9 indicated maintenance of random classification behavior for detection. Observation indicates that adaptive embedding strategy and histogram preservation restrain predominantly decreasing features divisibility even under deep feature extraction.

TABLE 9. Comparative steganalysis evaluation.

Experimental evaluation shows that the proposed method exhibits almost random detectability over both analytical and knowledge-based attacks. This indicates strong steganalysis resistance toward detection of secrecy. Moreover, RS and Chi² are unable to identify hidden artifacts due to nonuniformity and complex adaptive distribution of payload, thereby avoiding incorporating consistent statistical anomaly. AUC ≈0.5 for machine knowledge-based training of steganalysis indicates poor performance, which suggests that embedded characteristics do not form discrimination in the pattern of the feature space. This is due to the initial incorporation of frequency channel embedding in hybrid DWT–DCT with adaptive control, which functionally preserves both first-order and higher statistics of the cover picture.

Unlike conventional fixed or pixel domain embedding, the proposed method dynamically accommodates embedding strength depending on local picture complexity, thereby maintaining robustness by decreasing detectable distortions. This could signify novelty and experimental consolidation under an adversarial situation by dual optimization of both security and detectability.

5.9 Evaluation metrics for steganalysis resistance

To assess the quantitative detectability of the proposed framework, primary metrics are investigated: Detection Accuracy, ROC Curve, and False positive rate (FPR).

5.9.1 Normalized histogram similarity analysis

Moreover, evaluations were estimated for normalization of similarity between cover and stego images to provide a more meticulous statistical interpretation of histogram conservation. Normalized measurements provide a scale-independent quantification of statistical similarity for which Kullback–Leibler (KL) divergence (Equation 75), chi-square distance (Equation 76), histogram intersection (Equation 77), and EMD (Equation 78) are utilized, unlike raw histogram deviation values. Lowering those values indicates a reduction in statistical distortion, while higher values of histogram intersection represent stronger preservation, as depicted in Table 10.

TABLE 10. Normalized histogram similarity analysis.

KL Divergence

D K L P ∣ ∣ Q = ∑ i P i log ⁡ P i Q i (75)

Chi-Square Distance

χ 2 P , Q = ∑ i P i + Q i 2 P i + Q i (76)

Histogram Intersection

H I P , Q = ∑ i min ⁡ P i , Q i (77)

Earth Mover’s Distance

E M D P , Q = inf ⁡ γ ∈ Γ P , Q ∑ i , j γ i , j d i , j (78)

where P , Q denote normalized cover and stego histograms.

The results metrics in Table 10 prove that the proposed method introduces only minimal first-order statistical distortion. Lower KL divergence, chi-square distance, and EMD metrics together with high histogram intersection scores signify that the histogram-preserving embedding paradigm effectively maintains statistical consistency between cover and stego pictures, thereby improving resistance against histogram-guided steganalysis.

5.9.2 Detection accuracy

Using Equation 79, a steganalysis framework can correctly classify cover and stego images for detection accuracy:

A c c u r a c y = T P + T N T P + T N + F P + F N × 100 (79)

where

• T P: True Positives (accurate detection of stego pictures)
• T N: True Negatives (accurate identification of cover pictures)
• F P: False Positives
• F N: False Negatives

In a secure steganographic system, detection accuracy should ideally be close to 50%, indicating random guessing behavior.

5.9.3 Receiver operating characteristic (ROC) curve

The ROC curve demonstrates a trade-off between true positive rate T P R and false positive rate F P R across diverse categorization thresholds in Equation 80:

T P R = T P T P + F N , F P R = F P F P + T N (80)

A curve near the diagonal line signifies poor distinctness required for steganography.

The area under the curve (AUC) calculates overall detectability:

• A U C ≈ 0.5 → Secure (random detection).
• A U C → 1 → Easily detectable.

Based on your data:

• F P R = [0.0, 0.125, 1.0, 1.0]
• T P R = [0.0, 0.0, 0.0, 1.0]

This yields AUC ≈0.5, confirming strong resistance.

5.9.4 False positive rate F P R

FPR estimates in Equation 81, the probability of erroneously classifying a cover picture as stego:

F P R = F P F P + T N (81)

Lower FPR signifies fewer negative alarms.

• For robustness, F P remains immensely unstable throughout thresholds, prohibiting reliable detection.

Observed F P mutationexhibits unpredictable classifier behavior, emphasizing security.

0.0 → 1.0

Figure 8 shows detection accuracy of ∼50% along with non-stable F P distribution combines near the diagonal of the ROC curve, indicating that the proposed method efficiently conceals analytical footprints. Unlike conventional embedding strategies, this adaptive complexity-guided scheme unsettles learnable patterns, making each statistical and ML-dependent steganalysis counterproductive. This substantiates evidence that the algorithm fulfils practical minimum perceptibility under evaluation of steganalysis conditions, which is a critical contribution alongside its higher imperceptibility and robustness.

5.10 Histogram preservation analysis

Deviations in histogram analysis (Table 11) indicate the perseverance of statistical dissemination of pixel intensity efficiently for all pictures. Remarkably, Seabeach and Strawberry again exhibit a minimal number of deviations. This signifies that the highly consistent pictures shown in Figures 9a–h depict a negligible amount of distortion after incorporation of the adaptive embedding strategy. Though small higher deviations were observed for smooth pictures like Auditorium, overall, dissemination remains visually consistent, thereby maintaining reliable analytical detection quality.

TABLE 11. Histogram deviation analysis.

This finding confirms that the set of rules effectively keeps the histogram consistent, which is essential for resisting first-order statistical steganalysis assaults. The inclusion of C h i 2 and RS analysis reinforces the safety of the proposed framework.

The histogram deviation provides, in Table 11, essential insight into the statistical constancy of the stego pictures produced by using the proposed framework. In general, a decrease in histogram deviation suggests higher preservation of the authentic pixel depth distribution, which, without delay, contributes to resistance against first-order statistical steganalysis attacks, including chi-square and RS analysis. From the table, it is determined that Seabeach (13,998) and Strawberry (142,088) show the bottom histogram deviation, in Table 8, indicating that the embedding procedure introduces minimum statistical distortion in particularly textured pictures. This behavior harmonizes with the proposed complexity-adaptive embedding arrangement, wherein increased payload is allotted to rich texture zones, seemingly covering embedding artifacts

Moderate deviations are observed for pictures like House (372,602), Fox (426,834), and Iguana (657,132), which suggests a proportionate trade-off between hiding capability and analytical preservation. In contrast, specifically higher divergence in smoother pictures like Peeper (756,036) and Auditorium (794,801) can be ascribed to the limited availability of better frequency areas, making even little modifications more statistically noticeable. Despite several differences, the deviations remained controlled, which is a reasonable property. Unlike conventional strategies, which addressed regular histogram changes, the proposed method avoids anticipated patterns, thereby lowering the efficiency of statistical detectors.

The overall attainment confirms successful compatibility of the proposed method with histograms while preserving picture features. This achieves concurrent subtlety and steganalysis resistance.

5.11 Visual quality analysis for embedding map visualization

The embedding map generated is shown in Figures 11a–h through the CNN, giving a spatial illustration of adaptive embedding power throughout the image, wherein better values imply areas decided on for insertion of more potent facts. Visual inspection of the embedding maps shows that the community assigns better weights to textured and edge-dense areas, while suppressing embedding in clean and homogeneous areas. This behavior confirms that the CNN successfully learns content-conscious characteristic representations, permitting selective embedding that minimizes perceptual distortion. In high-complexity regions, more potent embedding is carried out without introducing visible artifacts, while in flat regions, the embedding electricity is decreased to maintain visible fine and statistical consistency. Moreover, the embedding map exhibits non-uniformity and asymmetrical patterns that prevent the evolution of detectable analytical signatures. This significantly enhances robustness against traditional steganalysis like RS or chi² and knowledge base detection because the embedding situation does not observe anticipated or static rules.

Overall, the visual artifacts validate the central novelty of the proposed paradigm of a CNN-adopted adaptive embedding strategy that intelligently manages the equilibrium triad for imperceptibility, robustness, and safety through aligning embedding power with intrinsic picture characteristics.

The texture vs. smooth vicinity shown in Figures 10a–h has an embedding visibility that definitely shows that the proposed technique allocates better embedding energy to textured areas whilst maintaining smooth regions with minimum modification. In edge-concentrated and excessive-frequency environments, the embedding is more concentrated, leveraging natural interference-like traits to mask hidden information effectively. Conversely, in polished areas, the embedding is considerably decreased to avoid visual artifacts and statistical distortion. This selective conduct guarantees excessive imperceptibility and structural preservation, as contemplated in steady PSNR and SSIM values. Overall, the visualization confirms the effectiveness of the adaptive, complexity-guided embedding method in balancing payload capability and visible quality.

5.12 Computational complexity and runtime

Experiments were performed on a system with an Intel Core i7 processor, 16 GB RAM, and NVIDIA RTX-series GPU support using a Python hybrid implementation environment. Runtime measurements were procured under identical execution conditions using repeated experimental trials. The average execution time lasted for over 20 independent runs to minimize transient system-level fluctuations.

The proposed method accomplishes effective processing with an average runtime of 2.08 s, as shown in Table 12. Images with higher structural complexity require higher processing time due to enhancements in frequency-domain operations and embedding decisions, as depicted in the runtime scalability graph Figure 12. Compared to the high-textured images like Seabeach, these manifest significantly lower runtime because the embedding is less constrained and more uninterrupted. Regardless of incorporating several stages of DWT, DCT, SVD, and adaptive control, the paradigm maintains an empirical computational efficacy that accelerates efficiently to reduce overhead. This aims to maintain the suitability of the framework for real-time or close-to-real-time stable photograph communication applications in addition to strengthening its practical relevance.

TABLE 12. Encoding vs. decoding time analysis.

Experimental results obtained describe variations in runtime execution due to differences in textual complexity, edge density, distribution of local entropy, and adaptive embed even though all benchmark pictures possess identical spatial dimensions. The computational overhead increases due to highly textual pictures, which require increased analysis of frequency channel coefficients, decision of adaptive embedding, and estimation of lightweight CNN-guided complexity. Those pictures that are structurally smooth permit a quicker embedding and extraction procedure due to reduced adaptive modulation.

To investigate the impact of Toom–Cook acceleration within the DWT phase as shown in Table 13, a comparative estimation of the runtime between traditional convolutional DWT AND Toom–Cook accelerated DWT filter operation was conducted for different picture resolutions.

TABLE 13. Runtime comparison with and without Toom–Cook acceleration.

The comparative analysis of runtime depicted in Table 14 shows that the advantage of Toom–Cook acceleration increases significantly for high-resolution pictures. While the normal DWT filter already performs effectively for small-sized pictures, the acceleration in the polynomial multiplication paradigm diminishes convolutional overhead during multi-level decomposition for large-scale image processing. This supports the scalability of the proposed strategy’s ability to perform on the frequency channel for large multimedia objects.

TABLE 14. Runtime comparison of conventional DWT and Toom–Cook accelerated DWT.

Implementations of DWT in execution generally use small, fixed wavelet filtering, while incorporating Toom–Cook for polynomial multiplication was investigated for acceleration in a scalable high-definition frequency domain embedding strategy. The experimental runtime analysis in Figure 12 describes reducing computational overhead in multi-resolution filter operations precisely for large picture dimensions. Moreover, total complexity also depends on DCT, truncated SVD, and a lightweight CNN-based adaptive embedding phase.

All runtime evaluations were performed using the same implementation environment, identical parameter configuration, and fixed image resolution to ensure fair computational comparison.

5.13 Comparative performance study

Benchmark picture data were used as described in Section 5.1 for experimental evaluation of the proposed method, whereas comparative values for existing methods were collected from their respective published studies. The comparative evaluation in Table 15 exhibits a clean trade-off between embedding capability and imperceptibility throughout current steganographic methods. LSB steganography, like pixel domain methods, has a high capability of embedding, often exceeding 0.5 bpp, but these methods suffer from low security measures and are vulnerable to modern statistical or machine learning-based approaches. In contrast, frequency domain methodologies like DWT, DCT, and SVD demonstrate significant improvement in terms of imperceptibility and robustness. This is achieved due to embedding of secret information in frequency or structural constituents. However, this will reduce embedding payload capability. The visual quality of the output image gets enhanced by incorporating deep knowledge-based adaptive embedding approaches; it also offers flexibility, but this mechanism generally falls short of explicit rate–distortion refinement. Hybrid methods attempted to achieve stability of these competing objectives, although these methods depend on heuristic specification tuning, which cannot provide theoretical guarantees. The proposed methods achieve more productive impartiality through a combination of hybrid frequency-domain embedding with knowledge of CNN adaptivity beneath a validated rate–distortion refinement paradigm, which enables dynamic payload efficacy. This approach also maintains a significant amount of higher cognitive quality (PSNR >50 dB) and improved analytical security.

TABLE 15. Qualitative comparative analysis with existing steganographic methods reported in literature.

The comparative evaluation provided in Table 14 is presented as a qualitative literature assessment in place of a strict experimental benchmark under equal implementation conditions. Because current techniques are evaluated based on their respective research, the use of extraordinary datasets, payload capacities, attack models, photograph resolutions, assessment protocols, and direct numerical assessment must be interpreted cautiously. The goal of this assessment is to assess the proposed framework in relation to the consultant transform-domain, hybrid, and deep learning-based steganographic methods in terms of well-known embedding characteristics, perceptual quality, adaptivity, and robustness trends.

A comparative assessment using the same datasets, payload constraints, and attack criteria for all competing techniques might require extensive reimplementation and retraining of more than one recent algorithm, which remains a crucial course for future investigation.

5.14 Ablation study

The ablation study shown in Table 16 illustrates that each characteristic of the proposed work influences overall performance significantly. Moreover, removing the SVD module leads to a notable decline in both indiscernibility and robustness, while excluding the histogram constraint enhances susceptibility to statistical steganalysis. To achieve this, the CNN component plays a crucial role in security against knowledge-based attacks.

TABLE 16. Ablation study of proposed framework.

5.15 Statistical significance analysis

The statistical importance analysis shown in Table 17 confirms that the enhancements executed via means of the proposed version are not due to random variation. The extraordinarily low p-values obtained from each t-test and Wilcoxon test validate the reliability and consistency of the overall performance gains, thereby strengthening the credibility and novelty of the proposed approach.

TABLE 17. Statistical significance analysis.

6 Conclusion

The above proposed methodology represents an analytically grounded and computationally effective steganographic strategy which consolidated rate–distortion optimization, hybrid frequency domain embedding, and knowledge-guided adaptivity within a single unified coherent architecture. The proposed method also accomplishes an equilibrium between inappreciability, embedded capacity, robustness, and security by incorporating multi-level DWT–DCT truncated by SVD with illumination-dependent domain embedding in the YIQ color space. Moreover, incorporation of CNN-defined adaptive embedding and histogram conservation constraints successfully minimizes perceptible artifacts, which results in close-to-random steganalysis detection achievement of AUC ≈0.5 even as retaining enough perceptual quality (PSNR >50 dB, SSIM ≈0.9816). Furthermore, adaptation of Toom–Cook accelerated convolution aims to maintain quasilinear analytical complexity, making it adaptable for high-resolution images and real-time implementation. Extensive experimental evaluations across several benchmark datasets validate the superiority of the proposed method over existing spatial, converted, and deep knowledge-based techniques in terms of both theoretical precision and practical performance. In terms of future scope, despite its strong performance, several instructions remain for further enhancement and generalization of the proposed framework. The major contribution of the proposed strategy lies in establishing a coordinated strategy for optimizing a framework that adaptively balances embedding capacity, perceptual fidelity, statistical security, and computational scalability and not merely in combining existing transforms. Future research can amplify the method toward the path of cross-region steganography, which includes video, hyperspectral, and clinical imaging modalities, wherein temporal and multi-channel correlations can be jointly exploited. The integration of advanced deep architectures that incorporate transformers or diffusion-based methods completely defines methods that can also similarly enhance embedding adaptivity and resistance toward growing AI-driven steganalysis strategies. Geometric normalization mechanisms and feature-guided alignment techniques for further improving robustness against severe cropping, rotation, and scaling attacks can be assessed in a future study. Additionally, exploring traditional strategies and game-theoretic optimization may help make the system more robust under adaptive attack scenarios. From a computational perspective, implementation on parallel hardware (GPU/FPGA) and optimization for resource-constrained devices would possibly permit real-time deployment in resource-constrained environments. Finally, extending the theoretical formula to include higher-order statistical constraints and perceptual models aligned with human visible gadget traits may also enhance both invisibility and security, paving the way for next-generation smart and provably secure steganographic systems.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frsip.2026.1864603/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Mohamed Amine Tahiri, Universite Sidi Mohamed Ben Abdellah Laboratoire d’Ingenierie Systemes et Applications, Morocco

Reviewed by: Ahmed Bencherqui, Sidi Mohamed Ben Abdellah University, Morocco

Senthilraja B, Rajalakshmi Engineering College, India

Footnotes

http://istock.com/beach

http://sipi.usc.edu/database/