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We optimize over the set of corrected Laplacians (CL) associated with a weighted graph to improve the average case normalized cut (NCut) of a graph. Unlike edge-relaxation SDPs, optimizing over the set CL naturally exploits the matrix sparsity by operating solely on the diagonal. This structure is critical to image segmentation applications because the number of vertices is generally proportional to the number of pixels in the image. CL optimization provides a guiding principle for improving the combinatorial solution over the spectral relaxation, which is important because small improvements in the cut cost often result in significant improvements in the perceptual relevance of the segmentation. We develop an optimization procedure to accommodate prior information in the form of statistical shape models, resulting in a segmentation method that produces foreground regions which are consistent with a parameterized family of shapes. We validate our technique with ground truth on MRI medical images, providing a quantitative comparison against results produced by current spectral relaxation approaches to graph partitioning.