# Fusion of Bilateral 2 DPCA Information for Image Reconstruction and Recognition
## 2 Related Works  
Besides the unilateral and bilateral 2 DPCA-based algorithms mentioned above, there are many other advances of 2 DPCA, as discussed below. To extract more structure information from images, [PERSON] et al. [24] and [PERSON] et al. [25; 26] made improvements on the traditional 2 DPCA by utilizing it on subimagesof the original image. The subimages are generated by sliding a window on the original image or by partitioning the original image into blocks. By this way, they can extract more structure information from images. However, the procedures of these algorithms are complex and their performances are largely determined by how the image is partitioned. Traditional 2 DPCA utilizes \(\ell_{2}\)-norm as the distance metric, which aggravates the effect of outliers on the objective function. On the other hand, the features extracted by 2 DPCA are dense, which means that 2 DPCA cannot automatically eliminate irrelevant features. To introduce robustness and sparsity into traditional 2 DPCA, several algorithms have been proposed. [PERSON] et al. [27] and [PERSON] et al. [28] replaced \(\ell_{2}\)-norm with \(\ell_{1}\)-norm on the objective function of 2 DPCA to improve its robustness, and proposed the greedy 2 DPCA-L1 and the non-greedy 2 DPCA-L1. [PERSON] et al. [29] further replaced \(\ell_{1}\)-norm with transformed \(\ell_{1}\)-norm, which is expected to be more robust, and proposed the 2 DPCA-\(T\ell_{1}\). [PERSON] and [PERSON] [30] imposed \(\ell_{1}\)-norm both on the objective function and the constraint of 2 DPCA for simultaneously robust and sparse modeling, and proposed the 2 DPCA1-S. Considering that replacing \(\ell_{2}\)-norm with \(\ell_{1}\)-norm brings many benefits to the related algorithms, it is worthwhile to replace the two norms by an arbitrary norm, i.e., \(\ell_{p}\)-norm, to exploit more possibilities. Based on this idea, [PERSON] [31] imposed \(\ell_{p}\)-norm both on the objective function and the constraint of 2 DPCA, and proposed the generalized 2 DPCA with \(\ell_{p}\)-norm (G2 DPCA-Lp), further enhancing its robustness and sparseness. Rotational invariance is another desired property of 2 DPCA-based algorithm which can be introduced by \(\ell_{2,1}\)-norm or \(F\)-norm. \(\ell_{2,1}\)-norm is a rotational invariant \(\ell_{1}\)-norm, hence the name \(R_{1}\)-norm [32; 33]. [PERSON] et al. [34] introduced \(R_{1}\)-norm into 2 DPCA and proposed the \(R_{1}\)-2 DPCA. [PERSON] et al. [35] replaced square \(F\)-norm with \(F\)-norm in 2 DPCA and proposed the \(F\)-2 DPCA. [PERSON] et al. [36] minimized the ratio of reconstruction error to the projected variances that are measured with \(F\)-norm, and proposed the Angle-2 DPCA. [PERSON] and [PERSON] [37] minimized the product of reconstruction error and projected variance that are measured with \(F\)-norm and proposed the Area-2 DPCA. [PERSON] et al. [38] maximized the ratio of projected variance to original data that are measured with \(F\)-norm and proposed the Cos-2 DPCA. [PERSON] et al. [39] maximized the ratio of projected variance to original data that are measured with \(R_{1}\)-norm and considered the optimal mean in data centralization, proposing the robust optimal mean cosine angle 2 DPCA (ROMCA-2 DPCA). [PERSON] et al. [40] introduced 2D outliers-robust PCA (ORPCA) by relaxing the orthogonal constraints and penalizing the regression coefficient. Inspired by extending \(\ell_{2}\)-norm and \(\ell_{1}\)-norm to \(\ell_{p}\)-norm, the \(\ell_{2,1}\)-norm and \(F\)-norm were extended to arbitrary powers correspondingly. By extending \(\ell_{2,1}\)-norm to \(\ell_{2,p}\)-norm, [PERSON] et al. [41] proposed the \(\ell_{2,p}\)-2 DPCA, [PERSON] et al. [42] proposed the generalized centered 2 DPCA (GC-2 DPCA). By extending \(F\)-norm to \(F_{p}\)-norm, [PERSON] et al. [43] proposed the \(F_{p}\)-2 DPCA. Following the two-stage idea of RC2 DPCA [23], [PERSON] et al. [44] proposed the bilateral Angle-2 DPCA (BA2 DPCA), [PERSON] et al. [45] proposed the \(\ell_{2,p}\)-norm sequential bilateral 2 DPCA (\(\ell_{2,p}\)-SB-2 DPCA).