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This paper presents a new framework for blind source separation (BSS) of non-negative source signals. The proposed framework, referred herein to as convex analysis of mixtures of non-negative sources (CAMNS), is deterministic requiring no source independence assumption, the entrenched premise in many existing (usually statistical) BSS frameworks. The development is based on a special assumption called local dominance. It is a good assumption for source signals exhibiting sparsity or high contrast, and thus is considered realistic to many real-world problems such as multichannel biomedical imaging. Under local dominance and several standard assumptions, we apply convex analysis to establish a new BSS criterion, which states that the source signals can be perfectly identified (in a blind fashion) by finding the extreme points of an observation-constructed polyhedral set. Methods for fulfilling the CAMNS criterion are also derived, using either linear programming or simplex geometry. Simulation results on several data sets are presented to demonstrate the efficacy of the proposed method over several other reported BSS methods.