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We propose a flexible light field camera architecture that is at the convergence of optics, sensor electronics, and applied mathematics. Through the co-design of a sensor that comprises tailored, Angle Sensitive Pixels and advanced reconstruction algorithms, we show that—contrary to light field cameras today—our system can use the same measurements captured in a single sensor image to recover either a high-resolution 2D image, a low-resolution 4D light field using fast, linear processing, or a high-resolution light field using sparsity-constrained optimization.