A Novel Hybrid Watermarking Scheme for Medical Images Using Dual-Tree Complex Wavelet and 2D Discrete Cosine Transforms

2.1 Two-dimensional discrete cosine transform (DCT2)

The DCT converts images from spatial to frequency domain for compression like JPEG. For an $M \times N$ image, the 2D DCT formula is [14]:

$\begin{aligned} & T(u, v)=\alpha(u) \alpha(v) \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} f(x, y) \\ & \cos \left(\frac{(2 x+1) u \pi}{2 N}\right) \cos \left(\frac{(2 y+1) v \pi}{2 N}\right)\end{aligned}$               (1)

where, $f(x, y)$ is pixel value, $T(u, v)$ is frequency coefficient, and $\alpha$ factors are, as defined in reference [14]:

$\begin{aligned} & \alpha(u)= \begin{cases}\sqrt{\frac{1}{M}} & \text { for } u=0 \\ \sqrt{\frac{2}{M}} & \text { for } u=1,2, \ldots, M-1\end{cases} \\ & \alpha(v)= \begin{cases}\sqrt{\frac{1}{N}} & \text { for } v=0 \\ \sqrt{\frac{2}{N}} & \text { for } v=1,2, \ldots, N-1\end{cases} \end{aligned}$               (2)

The inverse DCT (2D-IDCT) transforms frequency data back to spatial domain [14]:

$\begin{aligned} & f(x, y)=\sum_{u=0}^{M-1} \sum_{v=0}^{N-1} \alpha(u) \alpha(v) T(x, y) \\ & \cos \left(\frac{(2 x+1) u \pi}{2 N}\right) \cos \left(\frac{(2 y+1) v \pi}{2 N}\right)\end{aligned}$                 (3)

2.2 Dual tree complex wavelet transform

The DTCWT, developed by Kingsbury [10], enhances traditional wavelet transforms by implementing dual parallel filter trees that produce complex coefficients. This approach combines two separates real DWTs with distinct filters (generating real and imaginary components) creating 2^d redundancy for d-dimensional signals [15]. When applied to 2D imagery, the transform yields directionally selective filters at ±15°, ±45°, and ±75° angles, producing two complex low-frequency and six high-frequency subbands per decomposition level [16] (Figure 1). The low-frequency coefficients can be formulated as:

1.jpg

Figure 1. DTCWT subband decomposition: Low-frequency and directional high-frequency components

$y(r, S, i, j)=\Re(y(r, S, i, j))+\jmath \Im(y(r, S, i, j))$, where $S \in\left\{S_1, S_2\right\}$                 (4)

Similarly, the high-frequency components are given by:

$y(R, \phi, i, j)=\Re(y(R, \phi, i, j))+\jmath \Im(y(R, \phi, i, j))$                (5)

where, s, $\Re($.$) \text{and} \mathfrak{I}($.$) \text{denote}$ the real and imaginary parts, respectively. $S_1$ and $S_2$ refer to low-pass subbands obtained from the first and second branches of the decomposition. $R$ denotes the decomposition scale, while $\phi$ specifies the orientation of the sub-band and takes values from the set {-75°, -45°, -15°, +15°, +45°, +75°}. The indices i, j represent the spatial positions within each sub-band and are constrained by: the set {-75°, -45°, -15°, +15°, +45°, +75°}. The indices i and j indicate spatial positions within each sub-band and are constrained as:

$0 \leq i \leq \frac{H}{2^R}-1, \quad 0 \leq j \leq \frac{W}{2^R}-1$                 (6)

This architecture delivers the perfect reconstruction and efficiency of standard DWT while adding shift invariance and directional sensitivity. These properties make DTCWT particularly effective for image processing applications including denoising, segmentation, classification, texture analysis, and digital watermarking, where it provides superior resistance to geometric distortions while maintaining imperceptibility [17].

A comprehensive evaluation of the proposed watermarking scheme was carried out using the publicly accessible COVID-19 chest X-ray database [18]. While CT, MRI, and ultrasound images were obtained from MedPix [19-21]. The evaluation was performed on a Windows 10 system using MATLAB R2023b Update 3, running on an Intel Core i5-6500 CPU (3.2 GHz) with 8GB of RAM. To ensure experimental consistency and mitigate variations arising from dimensional differences, all images were standardized to 512×512 pixels. Figure 4 presents representative examples of these uniformly sized host images from different categories used in the watermarking process. A 32×32 pixel image, representing 1,024 bits of information, was employed as the watermark (Figure 5). This configuration aligns with typical data capacity requirements for securing medical imagery, as documented in the relevant literature [22].

To evaluate the performance of the proposed watermarking method, two primary aspects are considered: imperceptibility and robustness against various attacks. The quality of both the watermarked and restored images is quantitatively assessed using two standard image quality metrics: Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM). These metrics provide reliable indications of perceptual similarity and distortion levels and are computed according to the formulations described in equations [23]:

${PSNR}({Imo}, {Imw})=10 log _{10}\left(\frac{M x^2}{M S E({Imo}, {Imw})}\right)$                   (10)

${MSE}({Imo}, {Imw})=\frac{1}{M^2} \sum_{i, j=1}^M\left({Im} o_{i, j}-{Im} w_{i, j}\right)^2$            (11)

${SSIM}({Imo}, {Imw})=\frac{2 \mu_{ {Imo }} \mu_{ {Imw }}+v_1}{\mu_{ {Imo }}^2+\mu_{ {Imw }}^2+v_1} \times \frac{2 \sigma_{ {ImoImw }}+v_2}{\sigma_{ {Imo }}^2+\sigma_{ {Imw }}^2+v_2}$                 (12)

4.jpg

Figure 4. Sample cover medical images

where,

• Imo, Imw: denote the original and watermarked images.
• M: represent the dimensions of the image.
• Mx: max pixel intensity.
•  $\mu_{ {Imo }}$, $\mu_{ {Imw }}$: represent the mean values of the cover and watermarked images respectively.
• $\sigma_{ {Imo }}^2$, $\sigma_{I m w}^2$: indicate the variances of the cover and watermarked images respectively.
• $\sigma_{ {Imo \text {Imw} }}$: denotes the covariance between the original and the watermarked images.
• $v_1, v_2$: are small positive values significantly less than 1.

SSIM ranges from [0-1], with 1 indicating perfect similarity.

Watermark robustness is assessed using normalized cross-correlation (NC) [24] and Bit Correct Rate (BCR) [25].

$\text{N C}\left(W_{\text {Orig }}, \text W_\text{Ext}\right)=\frac{\sum_{\text i=1}^\text n \sum_{\text j=1}^\text n \text W_{\text {Orig }_{\text i,\text j}} \times \text W_{\text {Ext}_\text {i,j}}}{\sqrt{\sum_{\text i=1}^n \sum_{\text j=1}^n {\text W_{\text {Orig }_\text{i,j}}{}_\text{i,j}^2}} \sqrt{\sum_\text {i=1}^\text n \sum_\text {j=1}^\text n {\text W_{\text {Ext}_\text {i, j}}}_\text{i,j}^2}}$                (13)

where, ${W_{{orig}}}_{i, j}$ and ${W_{E x t}}_{i, j}$ represent the pixel intensities at position (i, j) in the original and extracted watermarks, respectively. NC value of 1 indicates perfect recovery of the watermark. Lower NC values reflect reduced robustness of the watermark against image processing operations or intentional attacks.

The Bit Correct Ratio (BCR) in Eq. (14) measures the discrepancy between the extracted watermark bits and the originally embedded ones, where $\oplus$ denotes the (XOR) operation. A BCR of 100% indicates a perfect extraction, meaning no bit errors occurred during the retrieval of the watermark.

$B C R=\frac{1}{L} \sum_{k=0}^{L-1} \overline{W_{ {Orig }}(k) \oplus W_{ {Ext }}(k)} \times 100 \%$                 (14)

5.jpg

Figure 5. watermark image

4.1 Gain factor determination

The parameter α regulates the trade-off between imperceptibility and robustness: a smaller α reduces embedding distortion but weakens watermark resilience, whereas a larger α strengthens robustness at the expense of image quality as shown in Figure 6. To identify the optimal operating point, we evaluated α over the range [0.1-1.0] in increments of 0.1 and measured the average PSNR, SSIM, NC, and BCR across the dataset. Two constraints were imposed: (i) imperceptibility thresholds of PSNR ≥44dB and SSIM ≥0.97 to ensure preservation of diagnostic quality, and (ii) robustness thresholds of NC ≥0.95 and BCR ≥95% under JPEG compression (Q=50) and Gaussian noise (σ²=0.01). The results showed that only the interval 0.4-0.5 simultaneously satisfied both conditions. Lower values of α (<0.3) failed to ensure sufficient robustness, while higher values (>0.6) caused visible quality degradation. Therefore, α =0.4 was selected as the default embedding strength, since it provides the best compromise between imperceptibility and robustness in medical image watermarking.

6a.jpg

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6b.jpg

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Figure 6. PSNR values of the watermarked images in relation to the gain factor α (a). of the retrieved watermark in relation to the gain factor α following JPEG lossy compression with quality factor Q=50 (b)

The proposed approach is evaluated through visual and numerical analyses to assess imperceptibility and robustness. For privacy preservation, the watermark must remain undetectable. Established thresholds define this requirement: a peak signal-to-noise ratio (PSNR) of ≥29dB indicates high image quality, while values <25dB suggest visible degradation [26]. Similarly, a structural similarity index (SSIM) of ≥0.90 ensures perceptual invisibility [27]. As indicated in Figures 7(a) and 7(b), our watermarking method achieves strong imperceptibility, demonstrated by an average PSNR of 44.90dB -ranging from 44.84 to 44.93dB- across 115 test images. This high PSNR, combined with an SSIM exceeding 0.9827 -ranging from 0.9783 to 0.9867-, confirms that the method introduces modifications indistinguishable to human observers. Figure 8 exemplifies this imperceptibility across diverse medical imaging modalities, showing watermarked images (top row) that maintain excellent visual quality with PSNR values consistently above 44dB and SSIM values exceeding 0.98, while the corresponding extracted watermarks (bottom row) demonstrate perfect recovery without any attack scenarios. The near-ideal SSIM values further ensure the preservation of critical diagnostic details-a necessity in medical imaging, where even minor alterations risk clinical misinterpretation. In comparative evaluations, as indicated in Table 1, our method demonstrates superior imperceptibility performance across diverse medical imaging modalities. For X-ray images using the COVID-19 Radiography dataset, our approach achieves a PSNR of 44.94 dB, outperforming Saïd et al. [27] (38.74 dB), Fares et al. [4] (44.98 dB), and Hebbache et al. [1] (44.23 dB). Across CT, MRI, and ultrasound images from the MedPix dataset, our method consistently delivers competitive performance with PSNR values of 44.55 dB, 44.53 dB, and 44.54 dB respectively, significantly surpassing Saïd et al. [27] (39.85 dB and 39.38 dB for CT and MRI) while maintaining comparable quality to Hebbache et al. [1] (44.23dB across all modalities).

7a.jpg

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7b.jpg

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Figure 7. Image quality metrics across 115 test images at gain factor α=0.4: (a) PSNR values and (b) SSIM values

8.jpg

Figure 8. Watermarked images and their extracted watermarks without any attack

Table 1. Comparing the imperceptibility of our proposed method with related methods

4.3 Computational complexity and runtime analysis

The computational cost of the proposed DTCWT-DCT2 watermarking scheme arises mainly from the application of the Dual-Tree Complex Wavelet Transform (DTCWT) and the two-dimensional Discrete Cosine Transform (DCT2). For an input image of size M×M, a single-level DTCWT requires O(M2) operations due to the filtering and downsampling stages, while the DCT2 applied to the two low-frequency subbands (each of size 2M×2M) requires O(M2logM) operations using fast DCT algorithms. The embedding process involves vectorization and differential modification of coefficients, which is linear in the watermark length L, i.e., O(L). Similarly, the extraction stage mirrors the embedding with comparable complexity. Consequently, the overall complexity of the scheme is dominated by the transform operations and can be expressed as O(M2logM).

To assess practical runtime performance, the method was implemented in MATLAB R2023b on a desktop equipped with an Intel Core i5-6500 CPU (3.2GHz) and 8GB RAM. For medical images of size 512×512 pixels, the average embedding time was 0.1336 seconds, while watermark extraction required 0.0615 seconds (Table 2). Memory consumption remained modest, as only a limited number of transform subbands and coefficient vectors are stored at each stage.

Table 2. Computational time for embedding and extraction of the proposed watermarking scheme

Input Images	Embedding Time (Sec.)	Extraction Time (Sec.)	Total Time (Sec.)
X-ray image	0.1337	0.0607	0.1944
MRI image	0.1332	0.0614	0.1946
CT scan image	0.1316	0.0635	0.1952
Ultrasound image	0.1358	0.0601	0.1959
The average	0.1336	0.0615	0.1950

These results demonstrate that the proposed watermarking system is computationally efficient and suitable for near real-time telemedicine applications. In practical deployments, further optimization through compiled languages (e.g., C/C++) or parallelization on GPU hardware would enable faster processing, making the scheme highly applicable for secure and timely medical image transmission.

4.4 Assessment of watermarking resilience against attacks

We evaluated the watermarking scheme's robustness across multiple attack categories, with performance metrics detailed in Tables 3 and 4. The scheme achieved perfect resilience (NC=1.0) against histogram equalization, sharpening (0.2), average filtering (3×1), median filtering (3×3), Wiener filtering (3×3), gamma correction (0.5 and 1.5), Gaussian LPF (3×3 with variances 0.05 and 0.05), and speckle noise (variance 0.01). This robustness to intensity transformations and smoothing operations is critical for medical imaging applications where such enhancements are routine.

Table 3. NC values and extracted watermarks of the proposed DTCWT-DCT2 based watermarking scheme under different attacks

Attacks	BCR (%)	NC Values	Extracted Watermark
Histogram Equalization	100	1	t1.png
Gaussian noise (0, 0.01)	97.66	0.9793	t2.png
Gaussian noise (0, 0.005)	98.83	0.9932	t3.png
Sharpening (0.2)	100	1	t4.png
Average filtering (3×1)	100	1	t5.png
Average filtering (3×3)	98.05	0.9831	t6.png
Cropping (25%)	50.78	0.5320	t7.png
Cropping (50%)	54.30	0.5841	t8.png
Rotation (0.25°)	62.11	0.9829	t9.png
Rotation (0.5°)	98.05	0.6689	t10.png
JPEG Compression (Q=30)	92.97	0.9388	t11.png
JPEG Compression (Q=40)	99.22	0.9932	t12.png
JPEG Compression (Q=50)	100	1	t13.png
Salt & Pepper Noise (2 %)	94.53	0.9511	t14.png
Scaling (50 %)	98.44	0.9864	t15.png
Gaussian LPF (3×3), var=0.05	100	1	t16.png
Gaussian LPF (3×1), var=0.05	100	1	t17.png
Gamma correction (0.5)	100	1	t18.png
Gamma correction (1.5)	100	1	t19.png
Speckle noise (var=0.01)	100	1	t20.png
Speckle noise (var=0.02)	98.83	0.9898	t21.png
Median filter (3×3)	100	1	t22.png
Wiener filter (3×3)	100	1	t23.png

Table 4. The extracted watermark after various attacks

Surrounding Crop (35%)	Surrounding Crop (40%)	Surrounding Crop (30%)	JPEG Compression (Q=60)	Gaussian Noise (Var =0.01)	Gamma Correction (2)
BCR=88.6719% NC=0.8975	BCR=84.7656% NC=0.8651	BCR=89.0625% NC=0.9008	BCR=98.0469% NC=0.9828	BCR=95.7031% NC=0.9622	BCR=99.2188% NC=0.9932
t41.png	t42.png	t43.png	t44.png	t45.png	t46.png
t47.png	t48.png	t49.png	t50.png	t51.png	t52.png

The watermark demonstrated excellent noise tolerance across different distortion types. Gaussian noise yielded NC values of 0.9932 (variance 0.005) and 0.9793 (variance 0.01), while salt & pepper noise (2% density) achieved NC=0.9511. Speckle noise with variance 0.02 maintained high performance with NC=0.9898, indicating strong resistance to noise-based attacks commonly encountered during medical image acquisition and transmission. The slight degradation in average filtering performance when kernel size increases from 3×1 (NC=1.0) to 3×3 (NC=0.9831) demonstrates sensitivity to more aggressive smoothing operations.

Compression performance validation showed strong resilience under JPEG standards across all quality factors tested. JPEG compression achieved NC=0.9388 at Q=30, NC=0.9932 at Q=40, and perfect reconstruction (NC=1.0) at Q=50. Table 4 demonstrates additional compression robustness with JPEG compression at Q=60 yielding BCR=98.0469% and NC=0.9828. Scaling to 50% maintained excellent performance with NC=0.9864, confirming the watermark's stability under resolution changes.

However, the scheme exhibits significant weaknesses against geometric transformations. Rotation attacks demonstrate severe performance degradation with increasing angles, where NC=0.9829 for 0.25° rotation drops dramatically to 0.6689 for 0.5° rotation. This steep decline indicates high sensitivity to angular displacement, likely due to the spatial domain embedding approach disrupting coefficient relationships during geometric transformation.

Cropping attacks cause the most severe performance degradation across all tested levels. NC values drop substantially to 0.5320 for 25% cropping and 0.5841 for 50% cropping, with corresponding BCR values of 50.78% and 54.30% respectively. Table 4 provides comprehensive evidence showing surrounding crop attacks at 35% (BCR=88.6719%, NC=0.8975), 40% (BCR=84.7656%, NC=0.8651), and 50% (BCR=89.0625%, NC=0.9008) all result in severely compromised watermark integrity. The extracted watermarks display visible corruption patterns and noise artifacts, demonstrating that while partial watermark information survives, complete reconstruction becomes impossible due to direct coefficient loss in cropped regions. The visual comparison in Table 4 clearly illustrates the progressive degradation of watermark quality as cropping severity increases, with the most dramatic deterioration observed in the 40% cropping scenario despite maintaining a relatively high BCR of 84.7656%.

The geometric attack vulnerabilities highlight two primary error sources that limit the scheme's robustness. First, spatial coefficient displacement during rotation disrupts the embedding pattern, causing misalignment between watermark extraction coordinates and actual coefficient locations. Second, direct coefficient loss during cropping removes portions of the embedded watermark data, making complete recovery impossible. The consistently poor performance across all cropping percentages indicates this vulnerability is inherent to the current embedding strategy rather than parameter-dependent. Potential improvements to address these limitations include implementing rotation-invariant transforms such as log-polar mapping, deploying redundant embedding across multiple image regions to ensure partial recovery capability, and incorporating error correction coding mechanisms to reconstruct watermark data from surviving coefficients.

As summarized in Tables 5 and 6 compare the proposed DTCWT-DCT2 scheme with recent watermarking methods [1, 4, 13, 27] across diverse attacks and imaging modalities (CT, MRI, Ultrasound). The results show that our approach consistently achieves superior robustness. For example, it attains NC=1.0 under histogram equalization, sharpening, filtering, and gamma correction, where competing methods drop significantly (e.g., NC=0.7223 [1] for histogram equalization and 0.7693 [4] for sharpening). Against Gaussian and salt & pepper noise, our scheme maintains NC above 0.95, outperforming [4, 13]. While geometric attacks such as cropping and small rotations remain challenging (NC≈0.58-0.63), performance under scaling is notably higher (NC=0.9864 vs. 0.7157 [1]). The strong results across modalities highlight the generalization ability of the method. This robustness is attributed to the complementary strengths of DTCWT (shift invariance and directional selectivity) and DCT (energy compaction), which together balance imperceptibility and resilience.

Table 5. Comparison of NC values under various attacks with the schemes proposed by Hebbache et al. [1], Fares et al. [4], Khaldi et al. [13] and Saïd et al. [27]

Table 6. Comparison of NC values under various attacks with the schemes proposed by Hebbache et al. [1]

4.5 Comparative performance discussion

While the numerical results presented in Tables 2-4 confirm that the proposed scheme achieves higher PSNR, SSIM, and NC values compared to existing methods, it is important to analyze the mechanisms that drive these improvements. The superior robustness of the proposed DTCWT-DCT2 approach can be attributed to the following factors:

4.5.1 Synergistic use of DTCWT and DCT2

• The DTCWT provides approximate shift invariance and strong directional selectivity, which increases resilience against geometric attacks such as scaling and filtering.
• The DCT2 offers strong energy compaction, allowing the watermark to be embedded in stable mid-frequency coefficients that are less sensitive to noise and compression.
• Together, these transforms provide complementary strengths, outperforming methods that rely on a single transform (e.g., pure DCT or DWT).

4.5.2 Differential embedding strategy

By embedding watermark bits based on the difference between paired coefficients, the method achieves stronger resistance to common intensity operations (e.g., histogram equalization, gamma correction), as relative relationships are preserved even under global modifications.

4.5.3 Optimized embedding in high-energy frequency regions

Selecting embedding positions adaptively in high-energy mid-frequency bands balances invisibility and robustness. This explains why the proposed method consistently maintains PSNR >44dB and SSIM >0.95, while also withstanding compression and noise attacks better than competing methods.

4.5.4 Blind extraction capability

Unlike some prior methods that require the original image for watermark retrieval, our approach enables blind extraction. This not only increases practicality but also reduces error accumulation when the watermarked image undergoes multiple processing steps.

In summary, the superior performance of the proposed scheme is not only reflected in quantitative metrics but is also rooted in the combined theoretical advantages of the hybrid transform framework, the robustness of the differential embedding mechanism, and the optimized selection of embedding positions.