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Typical digital cameras exhibit a limitation regarding the dynamic range of the scene radiance they can capture. High Dynamic Range (HDR) imaging refers to methods and systems that aim to generate images that exhibit higher dynamic range between the lightest and the darkest parts of the an image. A typical approach for generating HDR images is exposure bracketing where multiple frames, each one with a different exposure setting, are captured and combined to a HDR image of the scene. The large number of images that exposure bracketing requires often leads to motion artefacts that limit the visual quality of the resulting HDR image. In this work, we propose a novel approach in HDR imaging that significantly reduces the necessary number of images. In our proposed system, we employ the notion of random exposure where each pixel of a single frame collects light for a random amount of time. By collecting a small number of such images, the full sequence of low dynamic range images can be reconstructed and subsequently used for HDR generation. The problem is solved by casting the reconstruction of the sequence as a nuclear norm minimization problem following the premises of the recently proposed theory of Matrix Completion. Experimental results suggest that the proposed method is able to reconstruct the sequence from as low as 20% of the images that traditional techniques require with minimal reduction in image quality.