We propose a novel design for light field image acquisition based on compressive sensing principles. By placing a randomly coded mask at the aperture of a camera, incoherent measurements of the light passing through different parts of the lens are encoded in the captured images. Each captured image is a random linear combination of different angular views of a scene. The encoded images are then used to recover the original light field image via a novel Bayesian reconstruction algorithm. Using the principles of compressive sensing, we show that light field images with a large number of angular views can be recovered from only a few acquisitions. Moreover, the proposed acquisition and recovery method provides light field images with high spatial resolution and signal-to-noise-ratio, and therefore is not affected by limitations common to existing light field camera designs. We present a prototype camera design based on the proposed framework by modifying a regular digital camera. Finally, we demonstrate the effectiveness of the proposed system using experimental results with both synthetic and real images.