Shannon's sampling theorem provides a link between the continuous and the discrete realms stating that bandlimited signals are uniquely determined by its values on a discrete set. This theorem is realized in practice using so called analog-to-digital converters (ADCs). Unlike Shannon's sampling theorem, the ADCs are limited in dynamic range. Whenever a signal exceeds some preset threshold, the ADC saturates, resulting in aliasing due to clipping. The goal of this paper is to analyze an alternative approach that does not suffer from these problems. Our work is based on recent developments in ADC design, which allow for ADCs that reset rather than to saturate, thus producing modulo samples. An open problem that remains is: Given such modulo samples of a bandlimited function as well as the dynamic range of the ADC, how can the original signal be recovered and what are the sufficient conditions that guarantee perfect recovery? In this paper, we prove such sufficiency conditions and complement them with a stable recovery algorithm. Our results not limited to certain amplitude ranges, in fact even the same circuit architecture allows for the recovery of arbitrary large amplitudes as long as some estimate of the signal norm is available when recovering. Numerical experiments that corroborate our theory indeed show that it is possible to perfectly recover function that takes values that are orders of magnitude higher than the ADC's threshold.

Time-of-flight (ToF) cameras calculate depth maps by reconstructing phase shifts of amplitudemodulated signals. For broad illumination or transparent objects, reflections from multiple scene points can illuminate a given pixel, giving rise to an erroneous depth map. We report here a sparsity regularized solution that separates K interfering components using multiple modulation frequency measurements. The method maps ToF imaging to the general framework of spectral estimation theory and has applications in improving depth profiles and exploiting multiple scattering.

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Multipath interference (MPI) is one of the major sources of both depth and amplitude measurement errors in Time-of-Flight (ToF) cameras. This problem has seen a lot of attention recently. In this work, we discuss the MPI problem within the framework spectral estimation theory and multifrequency measurements. As compared to previous approaches that consider up to two interfering paths, our model considers the general case of K-interfering paths. In the theoretical setting, we show that for the case of K-interfering paths of light, 2K + 1 frequency measurements suffice to recover the depth and amplitude values corresponding to each of the K optical paths. What singles out our method is the that our algorithm is non-iterative in implementation. This leads to a closed-form solution which is computationally attractive. Also, for the first time, we demonstrate the effectiveness of our model on an offthe-shelf Microsoft Kinect for the X-Box one.

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