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Magnetic resonance spectroscopic imaging requires a great deal of time to gather the data necessary to achieve satisfactory resolution. When the image has a limited region of support (ROS), it is possible to reconstruct the image from a subset of k-space samples. Therefore, the authors desire to choose the best possible combination of a small number of k-space samples to guarantee the quality of the reconstructed image. Sequential forward selection (SFS) is appealing as an optimization method because the previously selected sample can be observed while the next sample is selected. However, when the number of selected k-space samples is less than the number of unknowns at the beginning of the selection process, the optimality criterion is undefined and the resulting SFS algorithm cannot be used. Here, the authors present a modified form of the criterion that overcomes this problem and develop an SFS algorithm for the new criterion. Then the authors develop an efficient computational strategy for this algorithm as well as for the standard SFS algorithm. The combined algorithm efficiently selects a reduced set of k-space samples from which the ROS can be reconstructed with minimal noise amplification.