This paper proposes a novel compressive hyperspectral (HS) imaging approach that allows for high-resolution HS images to be captured in a single image. The proposed architecture comprises three key components: spatial-spectral encoded optical camera design, over-complete HS dictionary learning and sparse-constraint computational reconstruction. Our spatial-spectral encoded sampling scheme provides a higher degree of randomness in the measured projections than previous compressive HS imaging approaches; and a robust nonlinear sparse reconstruction method is employed to recover the HS images from the coded projection with higher performance. To exploit the sparsity constraint on the nature HS images for computational reconstruction, an over-complete HS dictionary is learned to represent the HS images in a sparser way than previous representations. We validate the proposed approach on both synthetic and real captured data, and show successful recovery of HS images for both indoor and outdoor scenes. In addition, we demonstrate other applications for the over-complete HS dictionary and sparse coding techniques, including 3D HS images compression and denoising.