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Computing a consensus object from a set of given objects is a core problem in machine learning and pattern recognition. One popular approach is to formulate it as an optimization problem using the generalized median. Previous methods like the Prototype and Distance-Preserving Embedding methods transform objects into a vector space, solve the generalized median problem in this space, and inversely transform back into the original space. Both of these methods have been successfully applied to a wide range of object domains, where the generalized median problem has inherent high computational complexity (typically $\mathcal{NP}$-hard) and therefore approximate solutions are required. Previously, explicit embedding methods were used in the computation, which often do not reflect the spatial relationship between objects exactly. In this work we introduce a kernel-based generalized median framework that is applicable to both positive definite and indefinite kernels. This framework computes the relationship between objects and its generalized median in kernel space, without the need of an explicit embedding. We show that the spatial relationship between objects is more accurately represented in kernel space than in an explicit vector space using easy-to-compute kernels, and demonstrate superior performance of generalized median computation on datasets of three different domains. A software toolbox resulting from our work is made publicly available to encourage other researchers to explore the generalized median computation and applications.