2.4 Summary
3.1.2 Selective Signal Removal (SSR)
3.1.2.4 Identification of Optimal Block Size
Block size B is an essential parameter for the heuristic function h, as it defines the number of pixels in each super-pixel. For example, B = 4 means that the heuristic function considers the super-pixel of size 4×4. While up-sampling from IB to IB′ , the whole block is allocated the corresponding boolean form, as is IB. Thus in the final image, I′ is obtained by applying Hadamard product between the original image I and IB′ , i.e., I′ = I ⊙IB′ . The block size manifests as the coarseness of the removed image signals. Figure 3.5 shows how block size affects the appearance of the final imageI′.
In order to find the best suitable block size, we have defined the Block Corre- lation (Bcor) as the ratio of the total number of non-zero pixels in the difference image of cover and stego processed by the SSR scheme and without the SSR scheme. Let X and Y denote a cover and CMD-stego images, respectively, and let X′ and Y′ indicate a cover and CMD-stego image after processed by theSSR scheme. Let us denote the difference between the cover and theCMD-stego image processed by the SSR scheme asDSSR =Y′−X′ and the difference between the cover and CMD-stego asD=Y −X. ThenBcor is given by eq. (3.3).
Bcor = no. of non-zero pixels in DSSR
no. of non-zero pixels in D (3.3) A high value of Bcor implies that the image I′ processed by SSR has many pix- els that are different in cover and stego, which is obvious since many irrelevant super-pixels are removed (which are not affected by the embedding). In order to
Block correlation vs. Block size
4 x 4 8 x 8 16 x 16 32 x 32 64 x 64 128 x 128
Block Size 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Block Correlation
2 x 2
Figure 3.6: Illustration of the effect on Block Correlation (Bcor) with different block size.
select the most affected region of the image by the embedding, a high block cor- relation is preferable. In order to justify this claim,Bcor is compared for different block sizes over the entire dataset (10,000 images of BossBase [85]) with CMD-S- UNIWARD [36]. Figure 3.6shows the variation of Bcor over different block sizes.
The bar graph in Figure 3.6 shows an initial rise fromB = 2 to B = 4 and then falls monotonously up to B = 128. This observation is intuitive due to the obvi- ous geometrical constraints on large block sizes and its inability to represent finer distributions of embedding locations. For example, in the experiments, a block size of 128 makes it inappropriate to choose 8 out of 16 boxes (image dimension
= 512×512) such that the correlation is high. This is because there are only sixteen 128×128 blocks in the image, and picking 8 out of the 16 such that they map the embedding pixels perfectly is unlikely. However, if the block size is very small, then theDCTcoefficients may fail to capture the texture variation. In that case, the edges and corners in the image have a high sum of squares DCTvalues even though they may not belong to a texture-rich region. This can be observed for B = 2, where the Bcor is 0.782606 as compared to an average correlation of 0.797211 forB = 4. SRMcaptures the four-dimensional inter-pixel dependencies of the image. Therefore, the SRM features consider every four consecutive pixels
Table 3.1: Comparison of the classification error (in %) of SRM and proposed SSR- SRM scheme for different block sizes on the different payloads. The SSR-SRM has lower classification error for the block-size 16×16. The lower classification errors are shown in boldface.
Payload (bpp)
SRM SSR- SRM 4×4
SSR- SRM 8×8
SSR- SRM 16×16
SSR- SRM 32×32
SSR- SRM 64×64 0.5 20.79 19.42 16.99 16.51 16.69 17.14 0.4 25.46 24.22 21.02 20.52 20.88 21.89 0.3 30.65 29.18 25.77 25.53 25.74 27.04 0.2 35.34 34.86 32.66 31.24 32.47 34.12
and their relation to each other. Thus, a smaller block size may fail to model the inter-pixel dependency used in SRM, whereas the correlation with embedded pixels (non zero entries ofD) is lower for large block sizes. Intuitively, a relatively smaller block can model±1 pixels of Dmore accurately. A larger block preserves the inter-pixel dependencies (that are modeled as the co-occurrence matrix in the SRM), which means that the amount of spatial distortion introduced in the image is relatively less for a large block size as compared to a smaller block. This trade-off implies that intermediate block size should be chosen for the assignment algorithm. In order to find an optimum block size, the proposed SSR technique is tested on the CMD-S-UNIWARD at embedding rates of 0.2, 0.3, 0.4, and 0.5 bpp. The classification test error of the scheme is shown in Table 3.1. Block size of 16×16 dominates all other block sizes across all the embedding rates and may be the optimal parameter for our proposed scheme. The relation between the block correlation and the classification accuracies for different block-sizes are shown in Figure 3.7. From Figure 3.7, it can be observed that the highest test accuracy is achieved for the block size 16×16, which has a block correlation of
∼0.74. This result is consistent with our hypothesis that an intermediate block size provides a balance between block correlation with embedding locations and the effective population of SRM features.
Block Correlation and Classification accuracy vs. Block size
4 x 4 8 x 8 16 x 16 32 x 32 64 x 64
Block Size ( pixels ) 0
10 20 30 40 50 60 70 80 90
Percentage (%)
Block correlation Classification accuracy
Figure 3.7: Classification accuracy (Green) and block correlation (Blue) with different block size.