This paper presents a new approach to sampling two-dimensional data, e.g. images, in which samples are taken on a cutset with respect to a graphical image model having one node for each pixel and undirected edges between neighboring pixels.  The cutsets considered in this paper are, primarily, Manhattan grids consisting of every Nth row and column of the image.  Such sampling is motivated both by applications in which there are physical constraints, e.g. a ship taking water samples, and by the fact that dense sampling along lines may permit image edges to be better reconstructed than conventional lattice-based sampling at the same density.  The focus of the paper is on reconstruction methods to be used with cutset sampling.  These are mostly minimum MSE estimation methods based on Markov random field image models and belief propagation, or simple image autocorrelation models.  Attention is also paid to new approaches to lossless and lossy image compression in which the first step is to losslessly encode the pixels on a cutset.