Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)The notion of "strength of connectedness" between pixels has been successfully used in image segmentation. We present extensions to these works, which can considerably improve the efficiency of object delineation tasks. A set of pixels is said to be a kappa-connected component with respect to a seed pixel, when the strength of connectedness of any pixel in that set with respect to the seed is higher than or equal to a threshold. We discuss two approaches that define objects based on kappa-connected components with respect to a given seed set: with and without competition among seeds. While the previous approaches either assume no competition with a single threshold for all seeds or eliminate the threshold for seed competition, we show that seeds with different thresholds can improve segmentation in both paradigms. We also propose automatic and user-friendly interactive methods to determining the thresholds. The proposed methods are presented in the framework of the image foresting transform, which naturally leads to efficient and correct graph algorithms. The improvements are demonstrated through several segmentation experiments involving medical images. Copyright (C) 2008 Paulo A. V. Miranda et al.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)FAPESP [05/59808-0, 05/58103-3, 05/56578-4, 03/13424-1]CNPq [302617/2007-8, 472402/2007-2]