Segmentation of cells in time-lapse bright-field microscopic images is crucial in understanding cell behaviours for oncological research. However, the complex nature of the cells makes it difficult to segment cells accurately. Furthermore, poor contrast, broken cell boundaries and the halo artifact pose additional challenges to this problem. Standard segmentation techniques such as edged-based methods, watershed, or active contours result in poor segmentation. Other existing methods for bright-field images cannot provide good results without localized segmentation steps. In this paper, we present two robust mathematical models to segment bright-field cells automatically for the entire image. These models treat cell image segmentation as a background subtraction problem, which can be formulated as a Principal Component Pursuit (PCP) problem. Our first segmentation model is formulated as a PCP with nonnegative constraints. We exploit the sparse component of the PCP solution for identifying the cell pixels. However, there is no control on the quality of the sparse component and the nonzero entries can scatter all over the image, resulting in a noisy segmentation. The second model is an improvement of the first model by combining PCP with spectral clustering. Seemingly unrelated approaches, we combine the two techniques by incorporating normalized-cut in the PCP as a measure for the quality of the segmentation. These two models have been applied to a set of C2C12 cells obtained from bright-field microscopy. Experimental results demonstrate that the proposed models are effective in segmenting cells from bright-field images.