Sensing Beyond Barriers via Non-Linearities: Theory, Algorithms and Applications

Digital data capture is the backbone of all modern-day systems and the "Digital Revolution" has been aptly termed as the Third Industrial Revolution. Underpinning the digital representation is the Shannon-Nyquist sampling theorem and more recent developments such as compressive sensing approaches. The fact that there is a physical limit to which sensors can measure amplitudes poses a fundamental bottleneck when it comes to leveraging the performance guaranteed by recovery algorithms. In practice, whenever a physical signal exceeds the maximum recordable range, the sensor saturates, resulting in permanent information loss. Examples include (a) dosimeter saturation during the Chernobyl reactor accident, reporting radiation levels far lower than the true value and (b) loss of visual cues in self-driving cars coming out of a tunnel (due to sudden exposure to light).

To reconcile this gap between theory and practice, we have introduced the Unlimited Sensing framework or the USF that is based on a co-design of hardware and algorithms. On the hardware front, our work is based on a radically different analog-to-digital converter (ADC) design, which allows for the ADCs to produce modulo or folded samples. On the algorithms front, we develop new, mathematically guaranteed recovery strategies.

In the context of the USF, our goal is to expand the frontiers of sensing and imaging beyond the restrictions imposed by conventional sampling architectures. For this purpose we resort to non-linear acquisition strategies in the sensing pipeline. Three main frontiers are considered:

(1) Dynamic Range Barrier.
Given modulo samples, here, we study the mathematical aspects of recovery of signals that belong to shift-invariant spaces (SIS). Within the SIS model, we will study (a) wavelet and spline families which are the key to modeling images and (b) multi-band signals that naturally arise in applications such as radar and radio communication. We also develop robust reconstruction algorithms for recovery from modulo samples that are validated on customized hardware. There on, we extend the utility of such algorithms for one-bit modulo sampling.

(2) Resolution Barrier.
Recovering spikes from low-pass filtered measurements is a classical problem and is known as super-resolution. However, in many practical cases of interest, the pulse or filter may be unknown due to a lack of calibration or physical properties of propagation and transmission. In the USF context, we pose and study the blind sparse super-resolution problem and extend this case when the acquisition pipeline consists of one-bit modulo architecture. This line of work finds applications in time-of-flight imaging, terahertz spectroscopy and photo-acoustic tomography.

(3) Imaging-related Barrier.
We develop efficient reconstruction algorithms for multi-dimensional signals that live on a manifold. This generalizes the HDR image recovery problem. We also develop efficient reconstruction algorithms for Modulo Radon Transform enabling HDR tomography.

Our algorithms are validated on experimentally acquired data with the help of inter-disciplinary and multi-university collaborations.