3D DWT-DCT and Logistic MAP Based Robust Watermarking for Medical Volume Data

2.1. Logistic Map

A chaotic system has a noise like behavior while it is exactly deterministic, so we can reproduce it if we have its parameters and initial values. These signals are extremely sensitive to the initial conditions. One of the most famous chaotic systems is the Logistic Map, which is a nonlinear return map given by:

Where, x_k((0,1) is the system variable, 0≤μ≤4 is the growth parameter, and k is the number of iterations. The study of the chaotic dynamical system shows that Logistic Map works in a chaotic state when 3.569945≤μ≤4. In this paper, we set μ=4. The chaotic sequences are generated by different initial values, x₀. The Branch image of the Logistic Chaotic is shown in Fig. (1).

We can obtain the scrambled watermarking using the Logistic Map. The scrambled watermarking has the same size with the original one. In the same way, we can restore the original watermarking according to the same initial value based on the Logistic Map.

2.2. The 3D-Discrete Wavelet Transform and Example for Volume Data

The decomposition process of the 3D Discrete Wavelet Transform is shown in Fig. (2). L and H denote the low and high frequency component under low and high frequency filters respectively, which is similar to the 2D Discrete Wavelet Transform.

The medical volume data is decomposed in approximation coefficients (LLL1) which expresses low-frequency characteristic and the detail coefficients which express the high-frequency characteristic. The three-dimensional discrete wavelet transform of medical volume data is shown in Fig. (3). Fig. (3a) is a slice of the medical volume data, Fig. (3b) is the corresponding three-dimensional reconstruction map, Fig. (3c) shows the 3D-DWT of the medical volume data (two layers).

2.3. The 3D-Discrete Cosine Transform

The M×N×P medical volume data’s 3D-DCT is defined by:

$u=0,1,\mathrm{...},M-1;$

$v=0,1,\mathrm{...},N-1;$$w=0,1,\mathrm{...},P-1;$

$c\left(u\right)=\{\begin{array}{c}\sqrt{1/M}\\ \sqrt{2/M}\end{array}\begin{array}{c}u=0\\ u=1,2,\mathrm{...},M-1\end{array}$$c\left(v\right)=\{\begin{array}{c}\sqrt{1/N}\\ \sqrt{2/N}\end{array}\begin{array}{c}v=0\\ v=1,2,\mathrm{...},N-1\end{array}$$c\left(w\right)=\{\begin{array}{c}\sqrt{1/P}\\ \sqrt{2/P}\end{array}\begin{array}{c}w=0\\ w=1,2,\mathrm{...},P-1\end{array}$

where, f(x,y,z) is the value of the medical volume data at the point (x,y,z) and F(u,v,w) is the 3D-DCT coefficient at the point (u,v,w) in frequency domain.

3.1. Extracting the Feature Vector of Medical Volume Data Using 3D DWT-DCT

Applying 3D DWT-DCT, the feature vector of the medical volume data can be obtained. Feature vector of the medical volume data is a group of symbol sequence which represents the visual feature of the medical image. Firstly, 3D-DWT is applied to the original medical volume data to obtain the approximated sub-band LLL1. Then, DCT of the whole LLL1 is computed and the DWT-DCT coefficient is obtained. We choose 8 low-frequency 3D DWT-DCT coefficients (F(1,1,1), F(1,2,1),F(1,1,2),… F(1,2,4)) to compose the coefficient sequence of the volume data, the value of the low-frequency coefficients may change under some attacks. However, the signs of the coefficients remain unchanged, as shown in Table 1. The different attacked images are shown in Fig. (4a-4h). If the coefficient is positive or zero value, it is denoted as “1”; otherwise, it is denoted as “0”. Thus, we can obtain the sign sequence of low-frequency coefficients as shown in Table 1. We can see that the sign sequence remains unchanged after common and geometrical attacks.

Based on the above, we can draw a conclusion that the sign sequence of 3D DWT-DCT low-frequency coefficients can represent the main visual characteristics of the medical volume data and it is robust to common and geometrical attacks.

To prove that the sign sequence of 3D DWT-DCT for volume data is an important visual feature, we choose some different medical volume data which are shown in Fig. (5a-5f) and use 3D DWT-DCT transform. The coefficients and sign sequences of low frequency are different and the NC value between the different sign sequence is very small, and it is less than 0.5. It is shown in Table 2.

3.2. Scrambling the Watermarking

The steps of generating the scrambled watermarking are as follows:

Step1: Generate the chaotic sequence.

The chaotic sequence X(j) was generated by the initial value, x₀, which is regarded as the private key. Each value of X(j) is between 0 and 1. The original watermarking cannot be recovered without the private key.

Step2: Obtain the scrambled watermarking.

First of all, the values of X(j) should be sorted in ascending order then the position of the values will change. According to the position change of the values we can scramble the position space of the watermarking image. In this way, the scrambled watermarking, BW(i,j), can be obtained. The original watermarking and scrambling watermarking are shown in Fig. (6).

3.3. Embedding Watermarking

Step3: Obtain the feature vector of the original volume data using 3D DWT-DCT.

Firstly, 3D-DWT is applied to decompose the original medical volume data F(i,j,k). Then we get the approximation coefficients FA_L. Second, 3D-DCT is applied to the approximation coefficients to obtain the 3D DWT-DCT coefficient matrix DF(i,j,k). After arranging the 3D DWT-DCT coefficients from low to high frequency, the low-frequency sequence can be obtained, which is composed by the top L values. Finally, through the symbolic operation of the low-frequency sequence, the feature vector of the volume data, V(j)={v(j)|v(j)=0,1;1≤j≤L}, can be obtained. Here, the value of L can tune the robustness and capacity of the embedded watermarking. (In this paper, we set L=32 bits).

The process is as follows:

FA_L(i,j,k)=DWT3(F(i,j,k))

DF(i,j,k)=DCT3(FA_L(i,j,k))

V(j)=Sign(DF(i,j,k))

Step4: Obtain the key sequence.

The key sequence Key(i,j) can be computed through the feature vector V(j) and the scrambled watermarking BW(i,j) by HASH function:

Key(i,j)=V(j)⊕BW(i,j)

The key sequence Key(i,j) should be stored for extracting the watermarking later, and then we can get the ownership of the original volume data through registering the key sequence to the third part. The original volume data is of high quality without any effect, which is a zero- watermarking scheme.

3.4. Extracting Watermarking

Step5: Obtain the feature vector of tested volume data using 3D DWT-DCT.

The tested volume data is F’(i,j,k). This procedure of obtaining the feature vector V’(j) of the tested volume data is the same as Step 3 of watermarking embed. The process is as follows:

FA’_L(i,j,k)=DWT3(F’(i,j,k))

DF’(i,j,k)=DCT3(FA’_L(i,j,k))

V’(j)=Sign(DF’(i,j,k))

Step6: Extract the watermarking using the key sequence Key(i,j).

Combining the key sequence Key(i,j) and the feature vector of the tested volume data V’(j), the watermarking can be extracted as follows:

BW’(i,j)=Key(i,j)⊕V’(j)

The extracted watermarking is scrambled. No useful information is obtained after that, and therefore, we must restore it so as to get the accurate information.

3.5. Restoring Watermarking

The chaotic sequence X(j) can be obtained by the same initial value using Logistic Map. By sorting the values of X(j) in ascending order, we can restore the scrambled watermarking according to the position change of X(j). Finally, we can get the restored watermarking W’(i,j). The scrambled watermarking and the restored watermarking are shown in Fig. (7).

4.1. Common Attacks

4.2. Geometrical Attacks