# Computer-Aided Diagnosis of Alzheimer's Disease via Deep Learning Models and Radiomics Method
## 2 Research Contents and Methods
### Image Registration  
Image registration is the basis of fusion technology, which matches the same area of two images. It is defined by the following formula:  
\[R(x,y)=h[F(f(x,y))] \tag{5}\]  
where \(R\) means reference figure \(R\) and floating figure \(F\). After making the transformation of pixels of figure \(F\) correspond to pixels of figure \(R\), the matching is completed [21]. Correspondence between two images follows: \(f(x,y)\) represents two-dimensional space transformation and \(h(x,y)\) represents one-dimensional gray-scale transformation. Image registration is the crucial step to realize image fusion. In order to highlight the diagnosis results, it is necessary to make the previous image registration as accurate as possible.  
In this paper, affine transformation was used as spatial geometric transformation of registration. Affine transformation of two-dimensional Euclidean space is defined as:  
\[\begin{split} S(X)&=T(X)+C,X=[x,y]^{\prime},\\ T&=[ab,cd]^{\prime},C=[e,f]^{\prime}\end{split} \tag{6}\]  
\(a{\sim}f\) are real numbers. \(x\), \(y\) stands for the coordinate of each pixel. Transform \(S\) is affine transform, which has the characteristic that a finite pixel is mapped to a finite pixel [22]. Two-dimensional affine transformation is linear, including translation, rotation, and scaling. A coordinate point by affine transformation can be expressed as:  
\[\left[\begin{array}{c}x^{\prime}\\ y^{\prime}\end{array}\right]=k\times\left[\begin{array}{cc}\cos\theta&\sin \theta\\ -\sin\theta&\cos\theta\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}\Delta x\\ \Delta y\end{array}\right] \tag{7}\]  
where \(x^{\prime}\), \(y^{\prime}\) stands for the coordinate of pixel after affine, \(k,\theta,\Delta x,\Delta y\) are registration parameters of the two images. This study divides the registration into four steps according to the requirements, as described below. Step 1: read image. Step 2: initial registration (coarse registration). The optimizer selects Powell algorithm, and similarity measure selects the maximum mutual information method. Step 3: improve registration accuracy by changing the optimizer's step size and increasing iterations times. Step 4: use the maximum mutual information as a reference to adjust the step size and iteration times, in order to improve the accuracy. As an optimization strategy of image registration, the Powell algorithm is a multi-parameter local optimization search algorithm that does not need to compute derivatives [23]. Its essence divides the optimization phase into an iterative process, which consists of \(n+1\) times one-dimensional search. First, an extremum point is obtained after searching \(n\) times of different conjugate directions, and the initial value of the search is searched in the direction of the extremum point connection. Then, the latest search direction is replaced by the direction of the extreme points obtained in the previous \(n\) times of search and keep iterating until the function stops decreasing. Maximum mutual information method is a registration similarity measure [24]. Based on the gray level of image, mutual information of two images is taken as the similarity measure, and the optimal transform [23] is obtained by maximizing the similarity measure. The formula is as follows:  
\[I(A,B)=H(A)+H(B)-H(A,B) \tag{8}\]  
\[S{*}=\text{argmax}\,I(S(B),A) \tag{9}\]  
where \(H(A)\) and \(H(B)\) are the entropy of the image which indicates the gray distribution of image. \(S{*}\) is the updated transform result.