In this paper, we describe a variational segmentation method to segment an image in two regions based on the piecewise constant case of the Fuzzy Region Competition method. The proposed model introduces a local weighting into the refered model to improve the detection of objects composed by large intensity variations or some kind of texture. Furthermore, the proposed model is computationally efficient compared to other unsupervised local techniques derived from Fuzzy Region Competition. The experiments showed that the proposed model is very robust in relation to noise and provides better results than the piecewise constant case of Fuzzy Region Competition when dealing with texturized images.