Lensless imaging is an important and challenging problem. One notable solution to lensless imaging is a single pixel camera which benefits from ideas central to compressive sampling. However, traditional single pixel cameras require many illumination patterns which result in a long acquisition process. Here we present a method for lensless imaging based on compressive ultrafast sensing. Each sensor acquisition is encoded with a different illumination pattern and produces a time series where time is a function of the photon's origin in the scene. Currently available hardware with picosecond time resolution enables time tagging photons as they arrive to an omnidirectional sensor. This allows lensless imaging with significantly fewer patterns compared to regular single pixel imaging. To that end, we develop a framework for designing lensless imaging systems that use ultrafast detectors. We provide an algorithm for ideal sensor placement and an algorithm for optimized active illumination patterns. We show that efficient lensless imaging is possible with ultrafast measurement and compressive sensing. This paves the way for novel imaging architectures and remote sensing in extreme situations where imaging with a lens is not possible.
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