Hyperspectral data processing typically demands enormous computational resources in terms of storage, computation and I/O throughputs, especially when real-time processing is desired. In this paper, we investigate a low-complexity scheme for hyperspectral data compression and reconstruction. In this scheme, compressed hyperspectral data are acquired directly by a device similar to the single-pixel camera based on the principle of compressive sensing. To decode the compressed data, we propose a numerical procedure to directly compute the unmixed abundance fractions of given endmembers, completely bypassing high-complexity tasks involving the hyperspectral data cube itself. The reconstruction model is to minimize the total variational of the abundance fractions subject to a pre-processed fidelity equation with a significantly reduced size, and other side constraints. An augmented Lagrangian type algorithm is developed to solve this model. We conduct extensive numerical experiments to demonstrate the feasibility and efficiency of the proposed approach, using both synthetic data and hardware-measured data. Experimental and computational evidence obtained from this study indicates that the proposed scheme has a high potential in real-world applications.