# Active Sonar Target Classification Method Based on Fisher's Dictionary Learning
## 3 Active Sonar Target Classification Method Based on [PERSON]'s Dictionary Learning
### Fisher Discriminant Model and Solution Method  
#### 3.1.1 Fisher Discriminant Model  
\(A=[A_{1},A_{2},\ldots A_{k}]\) denotes a set of training samples, where \(A_{k}\) is the subset of the training samples from class \(k\), where \(D=[D_{1},D_{2},\ldots D_{k}]\) denotes the matrix of the dictionary; \(D_{k}\) is the dictionary matrix corresponding to the training signal of class \(k\). We can write \(X=[X_{1},X_{2},\ldots X_{k}]\), where \(X\) is the sparse coefficient matrix of the training signal\(A\) on dictionary \(D\), and \(X_{i}\) is the sparse representation matrix \(A_{i}\) over \(D\). This method can be expressed as follows:  
\[J_{(D,X)}=\text{argmin}_{(D,X)}\{r(A,D,X)+\lambda_{1}\
orm{X}_{1}+\lambda_{2}f(X)\} \tag{1}\]  
where \(r(A,D,X)\) is the limited-data fidelity item, \(\
orm{X}_{1}\) is the sparse penalty, \(f(X)\) is the sparse coefficient limit term, \(\lambda_{1}\) is the regularization parameter, \(\lambda_{2}\) is the adjustment parameter, and \(\lambda_{1}>0\), \(\lambda_{2}>0\).  
(1) The limited-data fidelity item \(r(A,D,X)\)  
The dictionary items of similar signals have a greater degree of relevance to the information of this type of signal, and a smaller degree of relevance to other types of signal information. We write a limited-data fidelity item \(r(A,D,X)\) to restrict the dictionary.  
We denote the sparse representation matrix as \(X_{i}=[X_{i}^{1},\dots,X_{i}^{j},\dots,X_{i}^{k}]\) over \(D\), where \(X_{i}^{k}\) is the sparse coefficient of the training signal \(A_{i}\) with class \(i\) over the \(k\)-th sub-dictionary \(D_{k}\). The model of the algorithm is given as  
\[r(A_{i},D,X_{i})=\
orm{A_{i}-DX_{i}}_{F}^{2}+\
orm{A_{i}-D_{i}X_{i}}_{F}^{2}+ \sum_{j=1\atop j\
eq i}^{r}\
orm{D_{j}X_{i}^{j}}_{F}^{2} \tag{2}\]  
where \(\
orm{A_{i}-DX_{i}}_{F}^{2}\) is the penalty of the training signal \(A_{i}\) over the entire dictionary \(D\), \(\
orm{A_{i}-D_{i}X_{i}}_{F}^{2}\) is the penalty of the training signal \(A_{i}\) over the \(i\)-th sub-dictionary \(D_{i}\), and \(\
orm{D_{j}X_{i}^{j}}_{F}^{2}\) is the penalty of the training signal \(A_{i}\) over the \(j\)-th sub-dictionary \(D_{j}\). In order to make the sub-dictionary corresponding to each type of signal more recognizable to the corresponding type of signal, \(\
orm{A_{i}-D_{i}X_{i}^{i}}_{F}^{2}\) and \(\
orm{D_{j}X_{i}^{j}}_{F}^{2}\) are assigned smaller values.  
(2) Sparse coefficient limit term: \(f(X)\)  
In order to improve the classification and recognition of the dictionary, the sparse coefficient within-class divergence \(S_{w}(X)\) is minimized and the inter-class divergence \(S_{b}(X)\) is maximized by [PERSON]'s criterion, and the sparse coefficient is restricted. The model can be expressed as follows:  
\[S_{w}(X)=\sum_{i=1}^{K}\sum_{x_{k}\in X_{i}}{(x_{k}-m_{i})(x_{k}-m_{i})}^{T} \tag{3}\]  
\[S_{b}(X)=\sum_{i=1}^{K}n_{i}(m_{i}-m)(m_{i}-m)^{T} \tag{4}\]  
where \(m_{i}\) and \(m\) are the mean vectors of the sparse coefficients \(X_{i}\) and \(X\), respectively, and \(n_{i}\) is the number of samples of class \(A_{i}\). In order to determine the optimal solution, we intuitively define this item in the form of a convex optimization function:  
\[f(X)=tr(S_{w}(X))-tr(S_{b}(X))+\eta\
orm{X}_{F}^{2} \tag{5}\]  
To this end, we propose the model \(J_{(D,X)}\) of FDDL, as follows:  
\[J_{(D,X)}=\text{argmin}_{(D,X)}=\{\sum_{i=1}^{K}r(A_{i},D,X)+\lambda_{1}\
orm{ X}_{1}+\eta\
orm{X}_{F}^{2}+\lambda_{2}[tr(S_{w}(X))-tr(S_{b}(X))]\} \tag{6}\]