## 2 Materials and Methods
### IRS-RFL Algorithm
#### 2.2.2 LMaFit Fingerprint Database Reconstruction Algorithm
\begin{array}{l}L(\mathbf{U},\mathbf{R},\mathbf{\mathrm{I}},\mathbf{\mathrm{\Omega}})\\ =\frac{1}{2}trace\big{(}\big{(}\mathbf{U}^{T}\mathbf{R}-\mathbf{\mathrm{I}}^{T}\big{)} \big{(}\mathbf{U}\mathbf{R}^{T}-\mathbf{\mathrm{I}}\big{)}\big{)}\\ -trace\big{(}\mathbf{\mathrm{\Omega}}^{T}G_{\Gamma}(\mathbf{\mathrm{I}}-\mathbf{F}^{\prime })\big{)}\end{array}  \tag{24}\]  
The above equation is differentiated, and the differential equation is set to \(0\), which leads to the optimization condition of Equation (22). This can be expressed as Equation (25). \[ \begin{array}{l}\left\{\begin{array}{l}\frac{\partial L}{\partial\mathbf{ \mathrm{I}}}=\big{(}\mathbf{U}\mathbf{R}^{T}-\mathbf{\mathrm{I}}\big{)}\mathbf{R}=0\\ \frac{\partial L}{\partial\mathbf{\mathrm{I}}}=\mathbf{U}^{T}\big{(}\mathbf{U}\mathbf{R}^{T}- \mathbf{\mathrm{I}}\big{)}=0\\ \frac{\partial L}{\partial\mathbf{\mathrm{I}}}=-G_{\Gamma}(\mathbf{\mathrm{I}}-\mathbf{F}^ {\prime})=0\\ \frac{\partial L}{\partial\mathbf{\mathrm{I}}}=\mathbf{U}^{T}\big{(}\mathbf{\mathrm{I}}- \mathbf{U}\mathbf{R}^{T}\big{)}-\mathbf{\mathrm{\Omega}}=0\end{array}\right.\end{array}  \tag{25}\]  
Because \(\mathbf{\mathrm{\Omega}}_{uv}=0,(u,v)\
otin\Gamma\) exists, the final version of Equation (25) can be written as \(G_{\Gamma^{c}}(\mathbf{\mathrm{I}}-\mathbf{U}\mathbf{R}^{T})=0\), where \(\Gamma^{c}\) complements \(\Gamma\). To solve Equation (25) using the immovable-point iteration method, the first equation in Equation (25) is transformed as follows:  
\[\mathbf{U}\mathbf{R}^{T}\mathbf{R}=\mathbf{\mathrm{I}}\mathbf{R} \tag{26}\]  
This equation can be solved as follows:  
\[\mathbf{U}=\mathbf{\mathrm{I}}\mathbf{R}\Big{(}\mathbf{R}^{T}\mathbf{R}\Big{)}^{+}=\mathbf{\mathrm{I}} \big{(}\mathbf{R}^{+}\big{)}^{T} \tag{27}\]  
where \(\mathbf{R}^{+}\) is the Moore-Penrose pseudo-inverse of \(\mathbf{R}\). Matrix \(\mathbf{U}\) can be updated iteratively according to (27), and similarly, the matrices \(\mathbf{\mathrm{I}}\) and \(\mathbf{R}\) can be updated in the iteration using the remaining conditions of Equation (25). In the convergence calculation process, the approach of dynamically selecting the step size for \(\omega\) at each iteration was adopted as follows. Here, \(\mathbf{E}_{i-1}(\omega)\) denotes the error matrix for the last step size equal to the \(\omega\) iteration, and \(\mathbf{E}_{i}(\omega)\) is the error matrix for the current step size equal to \(\omega\). Consequently, the error rate was calculated using Equation (28). \[\sigma(\omega)=\|\mathbf{E}_{i}(\omega)\|_{F-norm}/\|\mathbf{E}_{i-1}(\omega)\|_the value of \(\omega\) could be increased moderately when \(\sigma(\omega)>\sigma_{1}\). The stopping criterion of the algorithm is expressed as Equation (29).  
\[\frac{\left\|G_{\Gamma}\left(\mathbf{X}^{\prime}{step}-\mathbf{F}^{\prime} \right)\right\|_{F-norm}}{\left\|G_{\Gamma}(\mathbf{F}^{\prime})\right\|_{F- norm}}\leq error, \tag{29}\]  
where \(step\) denotes the number of iterations and \(error\) denotes the minimum allowable error. Thus, the steps of the LMaFit fingerprint database reconstruction algorithm are as Algorithm 1. Input: Fingerprint database \(\mathbf{F}^{\prime}\) set of subscripts of the sampled elements \(\Gamma\) sampled fingerprint database \(G_{\Gamma}(\mathbf{F}^{\prime})\), iteration step \(\omega\), maximum number of iterations \(step(\varsigma=1,\ldots,step)\), allowed minimum error \(error\), initialization: \(\mathbf{U}\), \(\mathbf{R}\), \(\mathbf{I}\)Repeat:  
1. Update the error matrix \(\mathbf{E}\) using equation \(\mathbf{E}_{\varsigma+1}=G_{\Gamma}\left(\mathbf{F}^{\prime}-\mathbf{U}_{ \varsigma}{R_{\varsigma}}^{T}\right)\). 2. Update the intermediate matrix \(\mathbf{I}\) using equation \(\mathbf{I}_{\varsigma+1}=\mathbf{U}_{\varsigma}{R_{\varsigma}}+\omega{ \mathcal{E}}_{\varsigma+1}\). 3. Update the matrix \(\mathbf{U}\) using equation \(\mathbf{U}_{\varsigma+1}=\omega\mathbf{I}_{\varsigma+1}{R_{\varsigma}}\left( {R_{\varsigma}}^{T}{R_{\varsigma}}\right)^{-1}+(1-\omega)\mathbf{U}_{\varsigma}\). 4.