# Asymmetric Distance in K-Means Clustering Enhances Quality of Cells Raman Imaging
## 4 Results
### Concluding Remarks on Asymmetric Metric Benchmark  
In summary, seven clusters are needed for Euclidean and \(ecc\) = 0.50 asymmetric metrics to detect the presence of mitochondria signals, but at the expense of redundant segmentation in the lipidic regions, with four clusters assigned to lipids. On the other hand, \(ecc\) = 0.25 and even more \(ecc\) = 0.10 asymmetric metrics properly address mitochondria signals already with six clusters, and only a slight redundancy is present in the lipidic region with three clusters. Passing to seven clusters for \(ecc\) = 0.25 and 0.10 is not helpful and only increases the redundancy in the lipidic segmentation. The polar metric with six and seven clusters reveals the presence of mitochondria and works fine in the segmentation of lipidic signals, assigning only two clusters to lipids. But redundancy is produced in the other regions: in the case of seven clusters, nearly overlapping average spectra are assigned in both nuclear and outer membrane regions, while in the case of six clusters, this redundancy is present in the outer membrane. However, the main disadvantage of polar clustering is the excessive noise of the produced images, with mixed clusters and not well-defined subcellular regions. The segmentation accuracy is assessed, in the case of six clusters, by projecting the clusters onto the first two principal components (PCs) computed by Principal Component Analysis (PCA). Unfortunately, there is no ground truth to compare with, and many internal metrics used for segmentation accuracy in the absence of ground truth, like the Dunn Index, the Silhouette method, the Davies-Bouldin Index, and the Calinski-Harabasz Index, are based on intra-cluster and inter-cluster characteristic distances. The effectiveness of these internal metrics with Euclidean distance reflects their fundamental reliance on symmetric distance metrics. In our case, an asymmetric distance metric is employed for clustering, which could impair the reliability of such indices (Dunn, DBI, CHI, and the Silhouette method). Instead, the projection of clusters on PCs (Figure 5) provides a visual inspection of how the clustering behaves under different distance metrics. Since the PC1 axis is much more elongated than it appears in the figure (note the different scales for PC1 and PC2 scores), the Euclidean metric (top panel) fails to identify the yellow cluster dominated by mitochondrial signal, and merges it with the blue cluster dominated by nuclear signals. Since the Euclidean distance is isotropic and the PC1 axis dominates in terms of variance, clustering segmentation is primarily driven along the PC1 direction in the case of Euclidean distance. Conversely, the polar metric (bottom panel) is sensitive only to directional information (cosine similarities) and not to spectral vector magnitudes. It correctly addresses lipidic signals using only two clusters (red and purple symbols), but leads to significant overlap among mitochondria, cytosol, and membrane regions (yellow, green, and light green clusters). The proposed asymmetric distance offers a compromise between the Euclidean and polar cases by tuning the eccentricity parameter \(ecc\). In the figure, clustering with \(ecc\) = 0.10 is shown (middle panel). Mitochondrial (yellow), nuclear (blue), and membrane (green) regions are clearly separated, while lipidic signals are assigned to three clusters (red, magenta, and purple symbols), which are less redundant than the four clusters of the Euclidean case. Hence, PCA projection suggests that the asymmetric distance improves the separation of relevant biochemical regions without producing excessive redundancy compared to the Euclidean and polar metrics. Finally, the test performed with eight clusters (last row of Figure 3) produces redundancy with all metrics, but to different extents. Compared to the seven-cluster attempt, Euclidean and asymmetric metrics introduce a new redundancy in the outer membrane, where an unnecessary division into two regions (green and light green) can be observed.