# Dual-ISM: Duality-Based Image Sequence Matching for Similar Image Search
## 2 Related Work
### Low-Dimensional Transformation Techniques  
This section describes a dimension reduction method that reduces high-dimensional to low-dimensional data. The representative dimension reduction algorithms include principal component analysis (PCA), linear discriminant analysis (LDA), and non-negative matrix factorization (NMF).  
The PCA method generates a new main component that is represented by a linear combination of variables through the covariance and correlation matrices of variables. It also refers to a dimension reduction method in which the total variance of the variables can be explained through the main component. The PCA method tends to find a projection matrix, \(W_{opt}\), that maximizes the coefficient of determination of the total scatter matrix [16].  
\[W_{opt}=\underset{w}{argmax}|W^{T}S_{T}W|, \tag{1}\]  
where \(S_{T}\) represents all the scatter matrices and leads to the following equation:  
\[S_{T}=\sum_{i=1}^{M}(x_{i}-\mu)(x_{i}-\mu)^{T}, \tag{2}\]  
where \(\mu\) represents the average shape vector of all the samples in the training set, \(x_{i}\) represents the shape vector of the \(i\)-th sample, and \(M\) represents the total number of training samples.  
The LDA method is a dimension reduction technique that is used in classification problems [17]. The method aims to optimize classifications for given classes by maximizing the ratio of between-class to within-class variances of a given dataset [18]. Here, the key is to separate the class mean in the projected direction while achieving a small variance around the means. Through this method, LDA efficiently reduces multidimensional data into low-dimensional spaces; hence, it is suitable for the graphical representation of datasets [19]. In other words, the LDA method attempts to solve the optimal discrimination projection matrix, \(W_{opt}\), using the following equation [12]:  
\[W_{opt}=\underset{w}{argmax}\frac{\left|W^{T}S_{B}W\right|}{\left|W^{T}S_{W} W\right|}, \tag{3}\]  
where \(S_{b}\) is an interclass scatter matrix and \(S_{W}\) is an intra-class scatter matrix.  
\[S_{b}=\sum_{i=1}^{I}(\mu_{i}-\mu)(\mu_{i}-\mu)^{T}, \tag{4}\]  
\[S_{W}=\sum_{i}\left(x_{i}-\mu_{k_{i}}\right)\left(x_{i}-\mu_{k_{i}}\right)^{T}, \tag{5}\]  
where \(J\) represents the total number of samples in the entire image set, \(\mu_{i}\) is the average shape vector of image class \(i\), and \(n_{i}\) is the number of samples of image class \(i\). Further, \(x_{i}\) denotes a feature vector for a sample, and \(\mu_{k_{i}}\) denotes a vector of an image class to which \(x_{i}\) belongs.  
The NMF [20] is a matrix factorization algorithm that focuses on the analysis of data matrices that are composed of non-negative components. In this case, both the main component and the coefficient must be greater than or equal to zero, and if they are positive, the features can be divided without the superiority or inferiority of the components. The NMF aims to find two non-negative matrices, \(\text{U}=\left[u_{ij}\right]\in\mathbb{R}^{m\times k}\) and \(\text{V}=\left[u_{ij}\right]\in\mathbb{R}^{n\times k}\), that minimize the following objective function [21]:  
\[\text{O}=\|X-\text{UV}^{T}\|_{F}^{2}, \tag{6}\]where each column of \(X\) is a sample vector when \(X\), the data matrix, is \(X=[x_{1},\ x_{2},\ \ldots,\ x_{n}]\in\mathbb{R}^{m\times n}\). [PERSON] et al. [22] proposes a matrix-vector nonnegative tensor factorization (NTF) model, which is an extended method from NMF-based models.