Conventional parallel imaging exploits coil sensitivity profiles to enable image reconstruction from undersampled data acquisition. The extent of undersampling that preserves high quality images is limited in part by the total reduction in signal associated with undersampling, and in part by an additional geometry factor that arises from the position of the coil array with respect to the anatomy being imaged. We propose to further constrain the image reconstruction in order to reduce the geometry factor artifact that limits the use of high acceleration factors. We derive equality constraints from the acquisition model that induce a coupling between the signal intensity at a voxel, and the signal intensity at other neighboring voxels. These additional constraints improve the conditioning of the parallel imaging reconstruction by helping to disambiguate aliasing artifact. Consequently, increased undersampling factors have higher image quality. This in turn enables rapid acquisition of 2D and 3D images. Furthermore, these additional constraints are entirely compatible with alternative sources of information to better condition or regularize the image reconstruction. We propose a SENSE formulation of our augmented image reconstruction equations, derive the equality constraints that reduce the g-factor artifact, and solve for the final image using an Augmented Lagrangian numerical formulation. We also indicate how our formulation can be extended to incorporate image prior models by adding regularized reconstruction or sparsity constraints.