Sparse representation of 3D images is considered within the context of data reduction. The goal is to produce high quality approximations of 3D images using fewer elementary components than the number of intensity points in the 3D array. This is achieved by means of a highly redundant dictionary and a dedicated pursuit strategy especially designed for low memory requirements. The benefit of the proposed framework is illustrated in the first instance by demonstrating the gain in dimensionality reduction obtained when approximating true color images as very thin 3D arrays, instead of performing an independent channel by channel approximation. The full power of the approach is further exemplified by producing high quality approximations of hyper-spectral images with a reduction of up to 371 times the number of data points in the representation.