Low sensitivity MR techniques such as magnetic resonance spectroscopic imaging (MRSI) greatly benefit from the gain in signal-to-noise provided by ultra-high field MR. High-resolution and whole-slab brain MRSI remains however very challenging due to lengthy acquisition, low signal, lipid contamination and field inhomogeneity. In this study, we propose an acquisition-reconstruction scheme that combines 1H free-induction-decay (FID)-MRSI sequence, short TR acquisition, compressed sensing acceleration and low-rank modeling with total-generalized-variation constraint to achieve metabolite imaging in two and three dimensions at 7 Tesla. The resulting images and volumes reveal highly detailed distributions that are specific to each metabolite and follow the underlying brain anatomy. The MRSI method was validated in a high-resolution phantom containing fine metabolite structures, and in five healthy volunteers. This new application of compressed sensing acceleration paves the way for high-resolution MRSI in clinical setting with acquisition times of 5 min for 2D MRSI at 2.5 mm and of 20 min for 3D MRSI at 3.3 mm isotropic.