Hard C-means (HCM; k-means) is one of the most widely used partitive clustering techniques. However, HCM is strongly affected by noise objects and cannot represent cluster overlap. To reduce the influence of noise objects, objects distant from cluster centers are rejected in some noise rejection approaches including general noise rejection (GNR) and cluster-wise noise rejection (CNR). Generalized rough C-means (GRCM) can deal with positive, negative, and boundary belonging of object to clusters by reference to rough set theory. GRCM realizes cluster overlap by the linear function threshold-based object-cluster assignment. In this study, as a unified approach for GNR and CNR in HCM, we propose linear function threshold-based C-means (LiFTCM) by relaxing GRCM. We show that the linear function threshold-based assignment in LiFTCM includes GNR, CNR, and their combinations as well as rough assignment of GRCM. The classification boundary is visualized so that the characteristics of LiFTCM in various parameter settings are clarified. Numerical experiments demonstrate that the combinations of rough clustering or the combinations of GNR and CNR realized by LiFTCM yield satisfactory results.
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