## 2 Materials and Methods
### IRS-RFL Algorithm
#### 2.2.2 LMaFit Fingerprint Database Reconstruction Algorithm  
When building a fingerprint database, the number of fingerprint reference points is too large, which will lead to a relatively large workload in building the fingerprint database. Therefore, in order to reduce the workload of building a fingerprint database, in this study, we used the LMaFit algorithm to build a complete fingerprint database while only collecting fingerprint data from a partial fingerprint database. The algorithm first samples the fingerprint database \(\mathbf{F}^{\prime}\), and the number of sampled elements is significantly smaller than the total number of elements \(N\times(M\times W)\) in the fingerprint database \(\mathbf{F}^{\prime}\). Let \(\Gamma\) be the set of subscripts of the sampled elements. The reconstructed fingerprint database \(\mathbf{X}^{\prime}\), which is the reconstructed fingerprint database \(\mathbf{F}^{\prime}\), is obtained following reconstruction using the LMaFit algorithm in the case of the known fingerprint data corresponding to \(\Gamma\). Define \(G_{\Gamma}\) as the projection mapping of the fingerprint database \(\mathbf{F}^{\prime}\) onto \(\Gamma\). \[G_{\Gamma}\big{(}\mathbf{F}^{\prime}\big{)}_{uv}= \begin{cases}\mathbf{F}^{\prime}_{ uv},(v,u)\in\Gamma\\ 0,(v,u)\
otin\Gamma\end{cases}  \tag{20}\]  
According to the matrix reconstruction theory, the reconstructed fingerprint database \(\mathbf{X}^{\prime}\) satisfies Equation (21). \[ \begin{array}{ll}\min&rank\big{(}\mathbf{X}^{\prime}\big{)}\\ \textit{s.t.}&G_{\Gamma}\big{(}\mathbf{X}^{\prime}\big{)}=G_{\Gamma}(\mathbf{F}^{ \prime})\end{array}  \tag{21}\]  
where \(\min\) denotes the minimization, \(\textit{s.t. }\) denotes the constraint, and \(rank\big{(}\mathbf{X}^{\prime}\big{)}\) denotes the rank of \(\mathbf{X}^{\prime}\). Equation (21) indicates that if there is a unique low-rank matrix \(\mathbf{X}^{\prime}\) satisfying the above conditions, this matrix is the complete estimation matrix of matrix \(\mathbf{F}^{\prime}\). Considering that this problem is computationally extremely intensive, the LMaFit algorithm was used to transform the rank minimization problem in Equation (21) into the following problem:  
\[ \begin{array}{ll}\min&1/2&\left\|\mathbf{U}\mathbf{R}^{T}-\mathbf{1}\right\|_{F-norm}^{2 }\\ \textit{s.t.}&G_{\Gamma}(\mathbf{I})=G_{\Gamma}(\mathbf{F}^{\prime})\end{array}  \tag{22}\]Let the rank of matrix \(\mathbf{F}^{\prime}\) be \(r\). We introduce the intermediate matrix \(\mathbf{\mathrm{I}}\) to facilitate the calculation and introduce two matrices of dimensions \(N\times r\) and \((M\times W)\times r\) of \(\mathbf{U}\) and \(\mathbf{R}\) as the matrix to be estimated, that is, \(\mathbf{\mathrm{X}}^{\prime}=\mathbf{U}\mathbf{R}^{T}\). Equation (22) is nonconvex and can be transformed into the following Lagrange form for computational convenience:  
\[ \begin{array}{l}L(\mathbf{U},\mathbf{R},\mathbf{\mathrm{I}},\mathbf{\mathrm{\Omega}})\\ =1/2\big{\|}\mathbf{U}\mathbf{R}^{T}-\mathbf{\mathrm{I}}\big{\|}_{F-norm}^{2}-\big{\langle} \mathbf{\mathrm{\Omega}},G_{\Gamma}(\mathbf{\mathrm{I}}-\mathbf{F}^{\prime})\big{\rangle} \\ =1/2\big{\|}\mathbf{U}\mathbf{R}^{T}-\mathbf{\mathrm{I}}\big{\|}_{F-norm}^{2}-trace\big{(} \mathbf{\mathrm{\Omega}}^{T}G_{\Gamma}(\mathbf{\mathrm{I}}-\mathbf{F}^{\prime})\big{)}, \end{array}  \tag{23}\]  
where \(\langle\mathbf{\mathrm{\Omega}},G_{\Gamma}\rangle=\sum_{a,b}\mathbf{\mathrm{\Omega}}_ {a,b}(G_{\Gamma})_{a,b}\) is the inner product of matrices, and the matrix \(\mathbf{\mathrm{\Omega}}\) is a Lagrange multiplier that satisfies \(\mathbf{\mathrm{\Omega}}=G_{\Gamma}(\mathbf{\mathrm{\Omega}})\). As \(\|\mathbf{\mathrm{P}}\|_{F-norm}^{2}=trace\big{(}\mathbf{\mathrm{P}}^{T}\mathbf{\mathrm{P}} \big{)}\), Equation (23) can be transformed into Equation (24) as follows:  
\[