## 2 Materials and Methods  
### Low-Rank Matrix Approximation  
Thermal variance during the short acquisition intervals provides an indication of thermal heterogeneity, which can be detected by different matrix factorization (MF) methods that are widely used in noninvasive evaluation [33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47], such as principal component analysis (PCA) in thermography, named as PCT [33; 34], and non-negative matrix factorization (NMF) [36; 37; 38; 39]. High dimensionality of thermography in various applications is still an open research challenge in the field. PCT decomposes the vectorized thermal images that are stacked in a matrix, called a heat matrix. It sorts the thermal base vectors based on their variance. Some methods, such as fixed eigenvector analysis [36], incremental PCT (IPCT) [37], and candid covariance-free incremental principal component thermography (CCIPCT) [40], attempt to overcome the computational load by a fixed set of previously generated eigenvectors and a covariance-free approach, respectively. SparsePCT [41; 42] (also known as sparse dictionary matrix decomposition [38]) adds regularization terms  
Figure 2: Scheme of the presented thermographic system to be used prior to mammography for the initial diagnosis of breast cancer along with clinical breast examination (CBE). to the PCT to not only restrict the domain of solution, but to also make it nonlinear, which strengthens the important basis of improving the detection of defective patterns. NMF [35; 39] adds constraints to the basis and coefficients and converts MF to a clustering problem, which helps in thermography to better evaluate the thermal abnormalities with no overlapping bases [43; 44]. Semi-NMF and sparse NMF are used for infrared, noninvasive evaluation and extraction of the thermal patterns as well as for preliminary breast cancer screening, along with all the aforementioned methods for comparison [45; 46; 47]. Imaging features extracted from thermal images, called thermomics [47], are used to provide diagnostic solutions along with deep learning-based thermomics [48]. Deep learning approaches are also used to diagnose breast cancer using thermographic imaging [49; 50]. Input data \(\mathbf{X}\) is the heat matrix obtained by stacking the vectorized thermal images \(\mathbf{X}\in\mathbb{R}^{MN\times\tau}\). PCT decomposes the input heat matrix \(\mathbf{X}\) to eigenvalue and eigenvector matrices, which are also known as coefficients and bases. The bases matrix represents the highest variance of thermal patterns, sorted in descending variance order corresponding to the coefficients. SparsePCT [41; 42] adds penalty terms to the original PCT to limit the domain of the solutions, which leads to nonlinear behavior and a sparse outcome. NMF modifies the PCA's decomposition using non-negative constraints, whereas the bases and coefficients in PCA are allowed to be negative. \(\mathbf{X}\) can be presented as a linear relation of \(\tau\) bases \(\mathbf{B}=\left\{\beta_{1},\beta_{2},\ldots,\beta_{p}\right\},\ \mathbf{B}\in \mathbb{R}^{+}_{MN\times\tau}\) and A coefficient matrix \(\mathbf{A}\in\mathbb{R}^{+}_{\tau\times\tau}\), and \(\mathbf{X}=\mathbf{B}\mathbf{A}\)_s.t._\(\mathbf{A}\geq 0,\ \mathbf{B}\geq 0\). The \(\ell_{1}\) norm penalty term, along with a regularization parameter, is added to NMF for compensating the bases' distinctiveness and improving their presentation, which is shown by the following equation:  
\[\text{C}_{\text{Sparse}}\text{NMF}=\frac{1}{2}\|\mathbf{X}-\mathbf{B}\mathbf{A }\|_{\text{F}}^{2}+\lambda\|\mathbf{B}\|_{1} \tag{1}\]  
where \(\|\mathbf{B}\|_{p}\) represents the \(\ell_{p}\) norm of B, defined by \(\ell_{p}=\sum\limits_{\text{d},m}\|\mathbf{B}_{\text{d},m}\|_{\text{P}}^{1/ \text{P}}\), which is similar to the \(\ell_{1}\) penalty term [51; 52] calculating \(\mathbf{B}\) for convex \(\mathbf{A}\).