# Gear Integrated Error Determination Using the Gaussian Template Convolution-Facet Method
## 3 GTC\(-\)Facet Sub-Pixel Algorithm  
The Facet model, introduced by [PERSON] in 1984 [17], conceptualizes an image as a matrix comprised of numerous pixel units, allowing for its subdivision into several smaller matrices. Each region of these matrices conforms to a specific grayscale distribution, representing a Facet. The grayscale distribution function of each Facet is expressed as a polynomial function of the pixels within its corresponding region. However, directly employing polynomials for coefficient determination exhibits suboptimal resistance to interference, and computing the coefficients of binary cubic polynomials entails a considerable computational load. This article addresses these challenges by transitioning from polynomial fitting of Facet to the utilization of Gaussian template convolution (GTC). The outlined steps are as follows:By Taylor expansion of a unary function at \(x_{0}\), we can obtain the following:  
\[f(x_{0}+\triangle x)=f(x_{0})+\triangle x\cdot f^{\prime}(x_{0})+\frac{1}{2!} \triangle x^{2}\cdot f^{\prime\prime}(x_{0})+o||\triangle x^{2}|| \tag{13}\]  
The mapping of second-order derivatives to two-dimensional and multi-dimensional spaces is the Hessian matrix. In a two-dimensional image, \(f(x,y)\) is a function that represents the grayscale value of the image, and expanding the \(f(x+\triangle x,y+\triangle y)\) at \(f(x_{0},y_{0})\), rewriting it in a matrix form, and rounding off the remaining term \(o||\triangle x^{2}||\) can obtain the following:  
\[f(x+y)\approx f(x_{0}+y_{0})+[\triangle x,\triangle y] \begin{bmatrix}f_{x}(x_ {0},y_{0})\\ f_{y}(x_{0},y_{0})\end{bmatrix} +\frac{1}{2! }[\triangle x,\triangle y] \begin{bmatrix} f_{xx}(x_{0},y_{0})&f_{xy}(x_{0},y_{0})\\ f_{yx}(x_{0},y_{0})&f_{yy}(x_{0},y_{0})\end{bmatrix}\begin{bmatrix}\triangle x \\ \triangle y\end{bmatrix}  \tag{14}\]  
The second matrix in the third term to the right of the above equation, which is the Hessian matrix in two-dimensional space. The Hessian matrix is the second derivative at a point \((x,y)\) in two-dimensional space, written as a partial derivative matrix in the form of  
\[H(x,y)= \begin{bmatrix}f_{xx}(x,y)&f_{xy}(x,y)\\ f_{yx}(x,y)&f_{yy}(x,y)\end{bmatrix} = \begin{bmatrix}\frac{\partial^{2}f(x,y)}{ \partial x^{2}}&\frac{\partial^{2}f(x,y)}{\partial x\partial y}\\ \frac{\partial^{2}f(x,y)}{\partial x\partial y}&\frac{\partial^{2}f(x,y)}{ \partial y^{2}}\end{bmatrix}  \tag{15}\]  
The Hessian matrix simplifies the calculation of the grayscale distribution function by simply finding the elements in the matrix, which is actually the second derivative of the point. Because the data of a two-dimensional image are discrete, the general method for obtaining the second derivative of \(x\) at this point is to convert the derivative of a discrete function into a difference:  
\[f_{xx}(x,y)=f(x,y)-f(x-\triangle x,y)-(f(x-\triangle x,y)-f(x+2\triangle x,y)) \tag{16}\]  
However, this method only considers the information of the three pixels containing itself, and the amount of information is insufficient, so its robustness is poor. According to the linear scale space theory, taking the derivative of a function is equal to the convolution of the derivative of the function and Gaussian function:  
\[\frac{\partial f(x,y)}{\partial x}=f(x,y)*\frac{\partial G(x,y)}{\partial x} \tag{17}\]  
In the equation, * represents convolution, and \(G(x,y)\) is the Gaussian function:  
\[G(x,y)=\frac{1}{2\pi\sigma^{2}}e^{-(x^{2}+y^{2})/(2\sigma^{2})} \tag{18}\]  
So, the elements in the Hessian matrix at \((x_{0},y_{0})\) can be solved by convolving the image grayscale function with the second-order derivative of the Gaussian function:  
\[f_{xx}(x_{0},y_{0})=f(x_{0},y_{0})*\frac{\partial^{2}G(x,y)}{\partial x^{2}} \tag{19}\]  
The Gaussian template is able to include information from all points within the \(n\times n\)-neighborhood range of the point, which makes the results of the second derivative more accurate.