# Asymmetric Distance in K-Means Clustering Enhances Quality of Cells Raman Imaging
## 4 Results  
### Asymmetric Metric Matrix Benchmark, Compared with Euclidean and Polar Metrics  
We performed K-means clustering analysis on the Raman hyperspectral of cells using the asymmetric metric matrix \(M_{AS}\) to compute the distances between spectra. First, we compared Euclidean and pure polar distances with asymmetric distances, varying the eccentricity parameter _ecc_. A Raman dataset recorded over a HepG2 cell was used for this test. Figure 3 presents the results obtained with the different metrics for some cluster numbers (six, seven, and eight clusters). For all cluster numbers, we observe that Euclidean distance and the asymmetric distance with _ecc_ = 0.50 (first two columns of Figure 3) produce remarkably similar results. This indicates that the KCA algorithm does not perform significantly differently when using these two metrics. The reason becomes clear when observing the isodistance surfaces (Figure 2A,B), with _ecc_ = 1.00 corresponding to Euclidean distance, and _ecc_ = 0.50, which still results in a near-spherical shape. Indeed, with _ecc_ = 0.50 the ratio between the major and minor axes of the prolate ellipsoid is not so pronounced and the isodistance surface is not so far from the sphere of Euclidean metric. The results change significantly when decreasing to _ecc_ = 0.25 and even more to _ecc_ = 0.10. Even with six clusters, the asymmetric metric with _ecc_ = 0.25 highlights subcellular regions (yellow areas in the picture of Figure 3) that are not detected with Euclidean and _ecc_ = 0.50 metrics, while preserving the general structure of other regions (blue, reddish, and green areas). It is important to note that all images in the first row of Figure 3 correspond to clustering with six clusters. But for Euclidean and _ecc_ = 0.50 metrics, we have four different reddish colors (dark red, purple, red, and magenta), while in the case of _ecc_ = 0.25, we have only three reddish colors (purple, red, and magenta) along with one additional yellow cluster. This means that _ecc_ = 0.25 improves segmentation by reducing redundant clusters  
Figure 3: Evaluation of KCA using asymmetric metric distances on a Raman dataset acquired from a HepG2 cell. Different numbers of clusters (six, seven, and eight) and various distance metrics were tested. The first row presents images computed with six clusters using the Euclidean metric (**first column**), asymmetric metrics with eccentricities of 0.50 (**second column**), 0.25 (**third column**), and 0.10 (**fourth column**), as well as the polar metric (**fifth column**). The second and third rows display the corresponding images obtained with seven and eight clusters, respectively. The scale bar (black line in the first image) represents 10 \(\upmu\)m for all images. The color legend on the right is intended to be beneficial to color-blind individuals. (the four reddish tones in the Euclidean and \(ecc\) = 0.50 metrics) and enhancing the detection of biochemical differences due to spectral vector orientation. This capability is even more pronounced at \(ecc\) = 0.10 (again with six clusters): the corresponding image shows well delimited and clean subcellular regions, where blue and reddish areas are better defined within a more extended yellow region (see below in the text for color assignment). The pure polar metric can be regarded as an extreme case of the asymmetric metric, where the ellipsoid becomes highly elongated, approaching a straight-line form. This is just a point of view for comparison with the asymmetric metric. Mathematically, the pure polar distance is a cosine-based metric that considers only angular deviations between spectral vectors. In the last image of the first row (Figure 3), the pure polar result appears excessively noisy: while the reddish and blue areas are well defined, the yellow and green regions are mixed. Moreover, only two reddish colors (red and purple) remain, and a light green cluster is added.