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Human visual system (HVS) can perceive constant color under varying illumination conditions while digital images record information of both reflectance (physical color) of objects and illumination. Retinex theory, formulated by Edwin H. Land, aimed to simulate and explain this feature of HVS. However, to recover the reflectance from a given image is in general an ill-posed problem. In this paper, we establish an L1-based variational model for Retinex theory that can be solved by a fast computational approach based on Bregman iteration. Compared with previous works, our L1-Retinex method is more accurate for recovering the reflectance, which is illustrated by examples and statistics. In medical images such as magnetic resonance imaging (MRI), intensity inhomogeneity is often encountered due to bias fields. This is a similar formulation to Retinex theory while the MRI has some specific properties. We then modify the L1-Retinex method and develop a new algorithm for MRI data. We demonstrate the performance of our method by comparison with previous work on simulated and real data.