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The presence of clumps in biological cell images may degrade the performance of automated disease detection methods using them. We present techniques to split the clumps based on dominant point detection from contours. The selection of optimal dominant points and split lines is achieved using some logical rules based on geometry of the clumps. The scheme first discards some of the dominant points by processing three successive dominant points and labels the remaining points as split points. The split points are then joined by finding pairs of optimal split points that achieve a good split. Pairs of split points that are opposite to each other are joined first and then the remaining split points are joined by using appropriate heuristics. The performance of the scheme is evaluated using several blood smear images and the results show that the method is capable of handling complex clumps.