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Several important pattern recognition applications are based on feature vector extraction and vector clustering. Directional patterns are commonly represented by rotation-variant vectors {\schmi{\schmi{F}}}_d formed from features uniformly extracted in M directions. It is often desirable that pattern recognition algorithms are invariant under pattern rotation. This paper introduces a distance measure and a K--means-based algorithm, namely, Circular K-means (CK-means) to cluster vectors containing directional information, such as {\schmi{\schmi{F}}}_d, in a circular-shift invariant manner. A circular shift of {\schmi{\schmi{F}}}_d corresponds to pattern rotation, thus, the algorithm is rotation invariant. An efficient Fourier domain representation of the proposed measure is presented to reduce computational complexity. A split and merge approach (SMCK-means), suited to the proposed CK-means technique, is proposed to reduce the possibility of converging at local minima and to estimate the correct number of clusters. Experiments performed for textural images illustrate the superior performance of the proposed algorithm for clustering directional vectors {\schmi{\schmi{F}}}_d, compared to the alternative approach that uses the original K-means and rotation-invariant feature vectors transformed from {\schmi{\schmi{F}}}_d.
Index Terms- Clustering, algorithms, similarity measures.

D. Charalampidis, "A Modified K-Means Algorithm for Circular Invariant Clustering," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 27, no. , pp. 1856-1865, 2005.
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