Hyperspectral data processing typically demands enormous computational resources in terms of storage, computation, and input/output throughputs, particularly when real-time processing is desired. In this paper, a proof-of-concept study is conducted on compressive sensing (CS) and unmixing for hyperspectral imaging. Specifically, 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 CS. To decode the compressed data, we propose a numerical procedure to compute directly the unmixed abundance fractions of given end members, completely bypassing high-complexity tasks involving the hyperspectral data cube itself. The reconstruction model is to minimize the total variation of the abundance fractions subject to a preprocessed 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 evidences obtained from this paper indicate that the proposed scheme has a high potential in real-world applications.