Magnetic resonance spectroscopic imaging (MRSI) has been shown to provide valuable information about the biochemistry of the anatomy of interest and thus has been increasingly used in clinical research. However, the long acquisition time associated with multidimensional MRSI is a barrier for translation of this technology to the clinic. A novel approach using the application of compressive sensing, to reduce the acquisition time of MRSI is proposed. Reconstruction of data, simulated to be acquired through compressed sensing is implemented on a computer generated phantom simulating two metabolites of the human brain. The effect of Gaussian noise on this phantom is evaluated. A retrospective analysis of the application of such a reconstruction method for 1H MRSI of previously acquired in vitro brain phantom MRSI data is performed for the first time. On comparison of the reconstruction of the in vitro and computer generated phantoms from undersampled data to that performed from complete k-space; the errors in reconstruction was less than 1%. This indicates that our approach has a significant potential to reduce acquisition times for MRSI studies by 50% which could aid in MRSI being routinely used in the clinic.