# Pattern Recognition in Dried Milk Droplets Using Lacunarity and Deep Learning
## 3 Results
### Discrimination Between Milk Types Using Lacunarity Analysis  
Figure 3 presents the decrease in lacunarity (\(\lambda\)) as a function of the natural logarithm of the normalized box size (\(\ln(\epsilon)\)) for dried drops of whole and lactose-free milk with varying NaCl concentrations. In Figure 3a, lactose-free milk deposits without NaCl show a maximum lacunarity value of \(\lambda_{max}=6.29\) at \(\ln(\epsilon)=-4.60\), whereas whole milk without NaCl reaches a considerably higher maximum lacunarity of \(\lambda_{max}=15.21\) at the same \(\ln(\epsilon)\). No significant differences in \(\lambda\) are observed between the two milk types as the box size decreases. In Figure 3b, at a NaCl concentration of \(\phi_{N}=2\) wt%, lactose-free milk deposits exhibit a maximum lacunarity value of \(\lambda_{max}=1.97\) at \(\ln(\epsilon)=-4.60\), while whole milk deposits reach a slightly higher maximum of \(\lambda_{max}=2.13\) at the same value of \(\ln(\epsilon)\). The overall trend remains consistent with no substantial variations between the two milk types. Figure 3c, for NaCl concentrations of \(\phi_{N}=4\) wt%, shows significant differences  
Figure 2: Morphological patterns of dried milk droplets at different NaCl concentrations. **(a)** Whole milk and **(b)** lactose-free milk, with NaCl concentrations of \(\phi_{N}=0\) wt%, 2 wt%, and 4 wt%. Panel I shows the full deposits formed under controlled conditions (\(T=25\)\({}^{\circ}\)C, RH = 30%, \(V=3\)\(\mu\)L). Panel II displays magnified views of the central regions, indicated by blue squares in Panel I. Scale bars correspond to 0.5 mm. between the two types of milk. Lactose-free milk deposits reach a maximum lacunarity of \(\lambda_{max}=1.93\) at \(\ln(\epsilon)=-4.60\), while whole milk deposits achieve a higher maximum value of \(\lambda_{max}=2.33\) at the same point. We determined the optimal box sizes for the lacunarity analysis by calculating the relative differences \(|\lambda_{D}-\lambda_{E}|/\lambda_{E}\), where \(\lambda_{D}\) and \(\lambda_{E}\) represent the mean lacunarity values for whole milk and lactose-free milk, respectively. As shown in Figure 4a, the smallest box sizes display the most significant differences between the lacunarity values, with \(\ln(\epsilon)=-4.6\) (\(l=2\) pixels, \(\epsilon=0.01\)), corresponding to the largest observed discrepancy. Based on these findings, we utilized the aforementioned lacunarity values to perform discriminant analysis using Receiver Operating Characteristic (ROC) curves. Panel I in Figure 4b demonstrates the typical behavior of excellent performance between Sensitivity and 1-Specificity for deposits at \(\phi_{N}=0\) wt%. Panels II and III in Figure 4b show that the ROC curve's shape for \(\phi_{N}=2\) wt% and 4 wt% indicates very good differentiation between elements of the two groups, respectively. The area under the ROC curve (AUC) quantifies a classification method's efficiency in distinguishing between two groups. Specifically, the ROC curve represents the probability of correctly classifying a given deposit. In this study, the AUC was employed to assess the accuracy of the classification between whole milk and lactose-free milk deposits. Figure 5  
Figure 3: Lacunarity of milk deposits. Lacunarity \(\lambda\) as a function of natural logarithm of the normalized box size \(\ln(\epsilon)\) at \(\phi_{N}=0\) wt% (**a**), 2 wt% (**b**), and \(\phi_{N}=4\) wt% (**c**). Black and red colors represent whole milk and lactose-free milk, respectively. presents the accuracy of the lacunarity analysis applied to dry milk drops with varying NaCl concentrations. The highest classification performance was observed at \(\phi_{N}=0\) wt%, where the accuracy reached 1.0, while the values decreased to 0.92 and 0.91 at \(\phi_{N}=2\) wt% and \(\phi_{N}=4\) wt%, respectively. Figure 4: Relative difference of lacunarity and Receiver Operating Characteristic (ROC).