# Tamper and Clone-Resistant Authentication Scheme for Medical Image Systems
## 4 A Proposed New Secured Unclonable Medical Watermarking Scheme
### The Jacobian Model for Generating Watermarks  
The Jacobian matrix is a matrix defined from a vector function F and a given point \((x_{1},\ldots,x_{n})\in R^{n}\). It is the matrix of partial derivatives of the first order of a vector function.  
Let \(F\) be a function defined from \(R^{n}\) to \(R^{m}\), by its \(m\) component functions with real values, as follows:  
\[F:\left(\begin{array}{c}x_{1}\\ \vdots\\ x_{n}\end{array}\right)\rightarrow\left(\begin{array}{c}f_{1}(x_{1},\ \cdots,\ x_{n})\\ \vdots\\ f_{m}(x_{1},\ \cdots,\ x_{n})\end{array}\right) \tag{5}\]  
The partial derivatives of these one-point functions \(M\), if they exist, can be arranged in a matrix with \(m\) rows and \(n\) columns, called the Jacobian matrix of \(F\):  
\[J_{F}(M)=\left(\begin{array}{ccc}\frac{\partial f_{1}}{\partial x_{1}}& \cdots&\frac{\partial f_{1}}{\partial x_{n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{m}}{\partial x_{1}}&\cdots&\frac{\partial f_{m}}{\partial x _{n}}\end{array}\right) \tag{6}\]  
Watermark Generation Using the Jacobian Model  
Pertinent features namely skewness, entropy, and median are extracted from the original image. The patient name is extracted from the header of the DICOM image and the corresponding initials (the first letter of the given name and family name) are transformed into a binary matrix of size \(16\times 16\). A matrix of size \(16\times 16\) called _add_mat_ is then generated from the original image by a cumulative subtraction process. All previous information is used to build a meaningful watermark WM based on the Jacobian model.  
We suggest 16 functions with 16 parameters to generate a \(16\times 16\) matrix that can be exploited to build the watermark. We build all the functions using the binary matrix of the patient name, the three pertinent features (skewness, entropy, and median) extracted from the host image and the matrix _add_mat_ extracted from the host image. The proposed Jacobian matrix model is based on a vector \(Y\) of 16 functions : \(Y_{j}:R^{16}\to R\), \(I=1,2,\ldots,16\).  
These functions \(Y_{1},Y,\dots,\ Y_{16}\) are defined by:  
\[Y_{i}(x_{1},x_{2},\dots x_{16})=\sum\
olimits_{j=1}^{16}\frac{add\_mat(i,j)}{f\_ val_{i}}\frac{x_{j}^{2}}{2}\,\ j=1,\dots,\ 16 \tag{7}\]  
And,  
\[f\_val_{i}=\left\{\begin{array}{l}\text{skewness value if }i=1,\dots,\ 5\\ \text{entropy value if }i=6,\dots,\ 10\\ \text{median value if }i=11,\dots,\ 16\end{array}\right. \tag{8}\]  
The Jacobian matrix \(J\) of \(Y\) at (\(x_{1}\), \(x_{2}\), \(\dots,x_{16}\) ) is a \(16\times 16\) matrix defined as follows:  
\[I_{Y}(x_{1}\dots x_{16})=\left(\begin{array}{cccc}\frac{add\_mat(1,1)}{f\_ val_{1}}x_{1}&\cdots&\frac{add\_mat(1,16)}{f\_val_{1}}x_{16}\\ \vdots&\ddots&\vdots\\ \frac{add\_mat(5,1)}{f\_val_{1}}x_{1}&\cdots&\frac{add\_mat(5,16)}{f\_val_{1}} x_{16}\\ \frac{add\_mat(6,1)}{f\_val_{2}}x_{1}&\ddots&\frac{add\_mat(6,16)}{f\_val_{1}} x_{16}\\ \vdots&\ddots&\vdots\\ \frac{add\_mat(10,1)}{f\_val_{2}}x_{1}&\cdots&\frac{add\_mat(10,16)}{f\_val_{1}} x_{16}\\ \frac{add\_mat(11,1)}{f\_val_{3}}x_{1}&\cdots&\frac{add\_mat(11,16)}{f\_val_{3}} x_{16}\\ \vdots&\ddots&\vdots\\ \frac{add\_mat(16,1)}{f\_val_{3}}x_{1}&\cdots&\frac{add\_mat(16,16)}{f\_val_{3}} x_{16}\end{array}\right) \tag{9}\]  
This \(16\times 16\) Jacobian matrix is an image matrix used as a watermark intended for embedding in the original image. Examples of watermarks generated with the Jacobian model are presented in Figure 13.  
Figure 13: Original images and corresponding watermarks examples.