Compressive sensing is considered a huge breakthrough in signal acquisition. It allows recording an image consisting of N(2) pixels using much fewer than N(2) measurements if it can be transformed to a basis where most pixels take on negligibly small values. Standard compressive sensing techniques suffer from the computational overhead needed to reconstruct an image with typical computation times between hours and days and are thus not optimal for applications in physics and spectroscopy. We demonstrate an adaptive compressive sampling technique that performs measurements directly in a sparse basis. It needs much fewer than N(2) measurements without any computational overhead, so the result is available instantly.