Sampling is classically performed by recording the amplitude of the input at given time instants; however, sampling and reconstructing a signal using multiple devices in parallel becomes a more difficult problem to solve when the devices have an unknown shift in their clocks.Alternatively, one can record the times at which a signal (or its integral) crosses given thresholds. This can model integrateand-fire neurons, for example, and has been studied by Lazar and Tóth under the name of "Time Encoding Machines". This sampling method is closer to what is found in nature.In this paper, we show that, when using time encoding machines, reconstruction from multiple channels has a more intuitive solution, and does not require the knowledge of the shifts between machines. We show that, if single-channel time encoding can sample and perfectly reconstruct a 2Ω-bandlimited signal, then M -channel time encoding can sample and perfectly reconstruct a signal with M times the bandwidth. Furthermore, we present an algorithm to perform this reconstruction and prove that it converges to the correct unique solution, in the noiseless case, without knowledge of the relative shifts between the machines. This is quite unlike classical multi-channel sampling, where unknown shifts between sampling devices pose a problem for perfect reconstruction.

In event-based sensing, many sensors independently and asynchronously emit events when there is a change in their input. Event-based sensing can present significant improvements in power efficiency when compared to traditional sampling, because (1) the output is a stream of events where the important information lies in the timing of the events, and (2) the sensor can easily be controlled to output information only when interesting activity occurs at the input.Moreover, event-based sampling can often provide better resolution than standard uniform sampling. Not only does this occur because individual event-based sensors have higher temporal resolution [1], it also occurs because the asynchrony of events within a sensor and therefore across sensors allows for less redundant and more informative encoding. We would like to explain how such curious results come about.To do so, we use ideal time encoding machines as a proxy for event-based sensors. We explore time encoding of signals with low rank structure, and apply the resulting theory to video. We then see how the asynchronous firing across time encoding machines can couple spatial sampling density with temporal resolution, leading to better reconstruction, whereas, in frame-based video, temporal resolution depends solely on the frame-rate and spatial resolution solely on the pixel grid used.

Traditional sampling involves encoding a signal through (time, value)-pairs. In contrast, time encoding machines (TEMs) characterize a signal by recording time points which depend on the integral of the signal over time. We study multi-channel TEMs where channels have shifted values for their integrators. We show that M channels can enable recovery of bandlimited signals with M times the bandwidth of that allowed in the single channel case. Moreover, our recovery algorithm is linear, even when the shift between the integrators of the TEMs is unknown. This is in stark contrast to traditional multi-channel sampling, where complicated non-linear methods are required to recover the unknown time shift between channels.

It often happens that we are interested in reconstructing an unknown signal from partial measurements. Also, it is typically assumed that the location (temporal or spatial) of each sample is known and that the only distortion present in the observations is due to additive measurement noise. However, there are some applications where such location information is lost. In this paper, we consider the situation in which the order of noisy samples, taken from a linear measurement system, is missing. Previous work on this topic has only considered the noiseless case and exhaustive search combinatorial algorithms. We propose a much more efficient algorithm based on a geometrical viewpoint of the problem. We also study the uniqueness of the solution under different choices of the sampling matrix and its robustness to noise for the case of two-dimensional signals. Finally we provide simulation results to confirm the theoretical findings of the paper.

Conventional sampling focuses on encoding and decoding bandlimited signals by recording signal amplitudes at known time points. Alternately, sampling can be approached using biologically-inspired schemes. Among these are integrateand-fire time encoding machines (IF-TEMs). They behave like simplified versions of spiking neurons and encode their input using spike times rather than amplitudes.Moreover, when multiple of these neurons jointly process a set of mixed signals, they form one layer in a feedforward spiking neural network. In this paper, we investigate the encoding and decoding potential of such a layer.We propose a setup to sample a set of bandlimited signals, by mixing them and sampling the result using different IF-TEMs. We provide conditions for perfect recovery of the set of signals from the samples in the noiseless case, and suggest an algorithm to perform the reconstruction.