## 3 Technology Background
### Fun
In both scenarios, \(\textit{pixels}_{\text{block}}\) has been set to 32, which is the maximum number of pixels that can be stored in the local memory inside ARTICo\({}^{3}\) accelerators. An experimental evaluation (see Table 4) shows that the impact of \(\textit{pixels}_{\text{block}}\) on the extraction effectiveness is negligible when this parameter is large enough (i.e., when \(\textit{pixels}_{\text{block}}\geq 16\)). The distance between extracted and real endmembers has been measured using the spectral angle, which can be calculated using Equation (1):

\[\textit{Spectral Angle}=\arccos\frac{e_{\text{real}}\bullet e_{\text{extracted}}}{\|e_{\text{real}}\| \cdot\|e_{\text{extracted}}\|} \tag{1}\]

where \(e_{\text{real}}\) represents known endmembers (a pixel with all its spectral bands) and \(e_{\text{extracted}}\) represents endmembers obtained with the algorithm. Perfect matches between real and extracted endmemberswould render a spectral angle of 0\({}^{\rm o}\), whereas for completely unrelated pixels, the spectral angle would tend to 90\({}^{\rm o}\). It is important to note that, for real hyperspectral images, if the algorithm finds \(P\) endmembers and the image is known to have only \(P_{\rm real}\), with \(P_{\rm real}<P\), the spectral angle is computed as the minimum value from all possible combinations (i.e., the extracted \(P_{\rm real}\) endmembers that are closer to the real ones). As it can be seen, the reduction-based endmember extraction provides results that are close to the ones obtained using the original FUN algorithm, which in turn are not far from the actual endmembers present in the image. A comparison with state-of-the-art alternatives (Table 3) shows that deviations in the spectral angle are acceptable either because it is still below the value obtained with some of the most widely used unmixing algorithms (e.g., VCA or NMF), or because (even with comparable results) the algorithm is too complex to enable a runtime adaptive hardware implementation as the one presented in this work (e.g., DgS-NMF). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Image Characteristics**} & \multicolumn{2}{c}{**Mean Spectral Angle (\({}^{\rm o}\))**} \\ \hline
**SNR (dB)** & **Endmembers** & **Abundances** & **FUN** & **Reduction-Based FUN** \\ \hline  & & Gauss Spheric & 1.5115 & 2.6709 \\  & 4 & Legendre Polynomial & 1.5433 & 1.4739 \\  & & Dirichlet & 3.7219 & 2.0508 \\
20 & 8 & \begin{tabular}{c} Gauss Spheric \\ Legendre Polynomial \\ Dirichlet \\ \end{tabular} & 2.8381 & 2.4991 \\  & 12 & Legendre Polynomial & 3.6176 & 3.8983 \\  & 12 & \
\begin{tabular}{c} Dirichlet \\ Gauss Spheric \\ Legendre Polynomial \\ Dirichlet \\ \end{tabular} & 2.5348 & 2.7655 \\  & 13 &
\begin{tabular}{c} Gauss Spheric \\ Legendre Polynomial \\ Dirichlet \\ \end{tabular} & 3.7033 & 5.4387 \\ \hline  & & Gauss Spheric & 0.0798 & 0.2163 \\  & 4 & Legendre Polynomial & 0.3327 & 0.1521 \\  & & Dirichlet & 0.1254 & 0.4491 \\  & & Gauss Spheric & 0.4679 & 0.3806 \\
40 & 8 & Legendre Polynomial & 0.6072 & 0.8048 \\  & & Dirichlet & 0.3019 & 1.2459 \\  & 12 &
\begin{tabular}{c} Gauss Spheric \\ Legendre Polynomial \\ Dirichlet \\ \end{tabular} & 0.4252 & 1.1471 \\  & 13 & Legendre Polynomial & 0.3992 & 0.3853 \\  & & Dirichlet & 0.3895 & 0.8428 \\ \hline  & & Gauss Spheric & 0.0658 & 0.0726 \\  & 4 & Legendre Polynomial & 0.0367 & 0.2022 \\  & & Dirichlet & 0.0682 & 0.0895 \\  & & Gauss Spheric & 0.0459 & 0.0936 \\
60 & 8 & Legendre Polynomial & 0.0737 & 0.0975 \\  & & Dirichlet & 0.1463 & 0.1672 \\  & & Gauss Spheric & 0.0876 & 0.0860 \\  & 12 & Legendre Polynomial & 0.1705 & 0.1953 \\  & & Dirichlet & 0.1384 & 0.2409 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Endmember extraction accuracy (synthetic hyperspectral images with known endmembers).