# Asymmetric Distance in K-Means Clustering Enhances Quality of Cells Raman Imaging
## 1 Introduction  
Raman spectroscopy is a powerful, non-invasive analytical technique that provides detailed molecular information about biological samples, making it particularly valuable in biomedical research. By measuring the inelastic scattering of monochromatic light, Raman spectroscopy generates spectral fingerprints that reflect the biochemical composition of cells, tissues, and biofluids. This capability has led to its widespread application in, among others, cellular biology [1, 2], disease diagnosis [3, 4], and drug development [5, 6]. However, the complexity and high dimensionality of Raman spectral data pose significant challenges in data interpretation. Spectra often contain overlapping peaks, autofluorescence, and noise background, along with an intrinsic variability due to the heterogeneity of biological samples. Primarily, raw data need to be preprocessed with background reduction (such as polynomial subtraction or similar techniques) and somehow normalized in intensity to reduce, as much as possible, the presence of non-Raman signals and signal differences due to different optical parameters (laser power, integration time, variations in focusing on the sample). However, after the most efficient preprocessing and in thepresence of clean Raman signals, the extraction of meaningful insights from Raman imaging datasets of complex biological samples requires the use of computational approaches, such as machine learning and, more recently, deep learning techniques. Among machine learning approaches, K-means clustering analysis (KCA) can still be considered an effective unsupervised method for analyzing Raman hyperspectral images. K-means clustering is a widely used partitioning algorithm that groups data points into clusters based on spectral similarity. By minimizing intra-cluster variance, it allows for the identification of distinct biochemical regions within a Raman image, facilitating the differentiation of cellular components, disease states, or metabolic processes. Compared to other clustering techniques such as hierarchical clustering or Gaussian mixture models, K-means offers computational efficiency and scalability, making it suitable for large Raman imaging datasets. However, the choice of distance metric in K-means clustering has crucial importance in achieving a meaningful segmentation of Raman data. Different metrics, such as Euclidean distance or polar distance (cosine function), can significantly impact the clustering outcome by influencing how spectral similarities and differences are measured. Selecting an appropriate metric ensures that biologically relevant spectral variations are effectively captured, leading to improved cluster separability and the more accurate identification of biochemical structures within the sample. In the last two decades, several efforts have been made to introduce non-Euclidean metrics in the framework of clustering analysis, and limiting the view to angular-based metrics, several works have been proposed, like polar coordinates for circular clustering [7] and for density peaks clustering [8], polar transforms with k-means segmentation [9], and clustering using polar self-organizing maps [10]. Also, efforts for exploring and exploiting asymmetries have been proposed, like asymmetric self-organizing maps [11], unsupervised anisotropic clustering [12], the asymmetric K-means algorithm [13], and K-means clustering on asymmetric data [14]. However, in these latter works on asymmetric clustering, all the proposed methods develop a non-metric use of data asymmetry by introducing Gaussian kernels or dissimilarity measures. In this study, we develop an asymmetric metric distance to improve clustering performance, and an introduced ad hoc parameter called eccentricity will move from Euclidean to polar distance through different asymmetry degrees. By applying K-means clustering to the Raman imaging of cell samples, we explore the optimization of two clustering parameters, the eccentricity and the number of clusters.