# Parameter Optimization for Low-Rank Matrix Recovery in Hyperspectral Imaging
## 2 Methods and Materials  
### Methods  
The first LRMR model was proposed in [18] as a robust principle component analysis approach. It was further developed in [13] for hyperspectral image restoration by combining nonlinear optimization in the inner iteration loop in order to solve the restoration model. In this work, we apply optimization with respect to the initial parameters in terms of an outer iteration loop.  
In the following, we present the LRMR model. Given the real matrix \(D\) of size \(m\times n\) containing the observed data and assuming corruption by the sparse error matrix \(S\) and random Gaussian noise modeled by the matrix \(N\), the goal is to recover the low-rank matrix \(L\) with \(D=L+S+N\), which are all real-number matrices of the same size: \(m\times n\). The minimization problem  
\[\min_{L,S}\|D-L-S\|_{F}^{2}\text{ s.t. }\text{rank}(L)\leq r,\text{card}(S)\leq p \tag{1}\]  
is solved, with \(r\) denoting the upper bound for the rank of \(L\) and \(p\) for the cardinality of \(S\), which is related to the estimation of noise corruption. The norm \(\|\cdot\|_{F}\) denotes the Frobenius-norm. The formulation (1) of LRMR can be found in [13; 14]. Redundancies between the wavebands yield the low-rank property required for LRMR. LRMR modeling is then applied together with the GoDec algorithm [14] in order to solve the subproblems exploiting the low-rank property of HSI. The subproblems are created by taking subcubes centered at a pixel in the spatial dimension. Thus, if the whole data cube is of size \(m\times n\times w\), where \(w\) denotes the number of spectral reflectance bands, then the subcubes are of size \(b\times b\times w\), with \(b<m\) and \(b<n\). We define the entries of the subcube's spectral band by the spectral band. The entries of each subcube band-by-band are then organized in lexicographical order to obtain two-dimensional matrices of size \(b^{2}\times w\). The subcubes  
Figure 1: An example of a hyperspectral data cube (courtesy of NASA/JPL-Caltech).  
are then processed iteratively, providing local image restoration. We denote further the iteration stepsize by \(s\). More details on the specific LRMR model can be found in [13] and on GoDec in [14].  
The main focus of this work is the detailed investigation of the LRMR method with respect to the starting values for its main variable parameters, including rank \(r\) and blocksize of the subcubes \(b\), estimation parameter for the percentage of noise corruption \(p\), and stepsize \(s\) of the inner iteration. In the following, we denote by \(r\), \(p\), \(b\), and \(s\) the initial settings. We apply nonlinear optimization in order to determine the best parameter values for the parameter settings with respect to the peak signal-to-noise ratio (PSNR). We choose the Nelder-Mead simplex algorithm as presented in [15] for the nonlinear optimization. The PSNR is computed by  
\[\text{PSNR}=10*\log_{10}\frac{\max(C)^{2}*m*n*w}{\|C-\tilde{C}\|^{2}}, \tag{2}\]  
where \(C\) and \(\tilde{C}\) denote the original and denoised data cube, respectively. The sizes of the spatial and spectral dimensions are denoted by \(m,n\), and \(w\). The performance of LRMR is analyzed in terms of computational efficiency and memory with regard to different parameter choices. The computational time taken by the algorithm in MATLAB and Python is compared in order to study the performance of LRMR and differences in the implementation between the two programming languages. The goal is to investigate possible difficulties with parameter optimization.