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

数字数据采集是所有现代系统的支柱，“数字革命”被恰当地称为第三次工业革命。数字表示的基础是香农-奈奎斯特采样定理和更新的发展，如压缩传感方法。传感器可以测量幅度的物理限制这一事实，在利用恢复算法保证的性能时，构成了一个根本的瓶颈。实际上，每当物理信号超过最大可记录范围时，传感器就会饱和，导致永久性信息丢失。例子包括(A)切尔诺贝利反应堆事故期间剂量计饱和，报告的辐射水平远远低于真实值，以及(B)自动驾驶汽车从隧道出来(由于突然暴露在光中)失去视觉线索。为了调和理论和实践之间的差距，我们引入了无限感知框架或USF，它基于硬件和算法的联合设计。在硬件方面，我们的工作基于一种完全不同的模数转换器(ADC)设计，允许ADC产生模数或折叠样本。在算法方面，我们开发了新的、数学上有保证的恢复策略。在美国联邦的背景下，我们的目标是扩大传感和成像的前沿，超越传统采样架构施加的限制。为此，我们在传感流水线中采用了非线性捕获策略。主要考虑了三个方面：(1)动态范围障碍。在给定模样的情况下，我们研究了属于移位不变空间(SIS)的信号恢复的数学方面。在SIS模型中，我们将研究(A)小波族和样条族，它们是图像建模的关键，以及(B)在雷达和无线电通信等应用中自然产生的多频带信号。我们还开发了稳健的重建算法，用于从模样本中恢复，并在定制的硬件上进行了验证。在此基础上，扩展了这种算法在一比特模采样中的应用。(2)分辨率障碍：从低通滤波测量中恢复尖峰是一个经典的问题，被称为超分辨率。然而，在许多感兴趣的实际情况中，由于缺乏传播和传输的校准或物理属性，脉冲或滤波器可能是未知的。在USF的背景下，我们提出并研究了盲稀疏超分辨问题，并在捕获流水线由一比特模结构组成的情况下进行了扩展。这项工作在飞行时间成像、太赫兹光谱学和光声断层成像中得到了应用。(3)成像相关障碍：针对存在于流形上的多维信号，我们提出了高效的重建算法。这概括了HDR图像恢复问题。我们还开发了高效的模Radon变换重建算法，使HDR层析成像成为可能。在跨学科和多大学合作的帮助下，我们的算法在实验获得的数据上得到了验证。

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.