arXiv subjects

Ayush Bhandari

Publications and source records attributed to Ayush Bhandari.

28 records · Page 2Linked to original sources

arXiv2021

Unlimited Sampling from Theory to Practice: Fourier-Prony Recovery and Prototype ADC

Following the Unlimited Sampling strategy to alleviate the omnipresent dynamic range barrier, we study the problem of recovering a bandlimited signal from point-wise modulo samples, aiming to connect theoretical guarantees with hardware implementation considerations. Our starting point is a class of non-idealities that we observe in prototyping an unlimited sampling based analog-to-digital converter. To address these non-idealities, we provide a new Fourier domain recovery algorithm. Our approach is validated both in theory and via extensive experiments on our prototype analog-to-digital converter, providing the first demonstration of unlimited sampling for data arising from real hardware, both for the current and previous approaches. Advantages of our algorithm include that it is agnostic to the modulo threshold and it can handle arbitrary folding times. We expect that the end-to-end realization studied in this paper will pave the path for exploring the unlimited sampling methodology in a number of real world applications.

cs.IT↗

arXiv2021

The Modulo Radon Transform: Theory, Algorithms and Applications

Recently, experiments have been reported where researchers were able to perform high dynamic range (HDR) tomography in a heuristic fashion, by fusing multiple tomographic projections. This approach to HDR tomography has been inspired by HDR photography and inherits the same disadvantages. Taking a computational imaging approach to the HDR tomography problem, we here suggest a new model based on the Modulo Radon Transform (MRT), which we rigorously introduce and analyze. By harnessing a joint design between hardware and algorithms, we present a single-shot HDR tomography approach, which to our knowledge, is the only approach that is backed by mathematical guarantees. On the hardware front, instead of recording the Radon Transform projections that my potentially saturate, we propose to measure modulo values of the same. This ensures that the HDR measurements are folded into a lower dynamic range. On the algorithmic front, our recovery algorithms reconstruct the HDR images from folded measurements. Beyond mathematical aspects such as injectivity and inversion of the MRT for different scenarios including band-limited and approximately compactly supported images, we also provide a first proof-of-concept demonstration. To do so, we implement MRT by experimentally folding tomographic measurements available as an open source data set using our custom designed modulo hardware. Our reconstruction clearly shows the advantages of our approach for experimental data. In this way, our MRT based solution paves a path for HDR acquisition in a number of related imaging problems.

cs.IT↗

arXiv2020

On Unlimited Sampling and Reconstruction

Shannon's sampling theorem is one of the cornerstone topics that is well understood and explored, both mathematically and algorithmically. That said, practical realization of this theorem still suffers from a severe bottleneck due to the fundamental assumption that the samples can span an arbitrary range of amplitudes. In practice, the theorem is realized using so-called analog-to-digital converters (ADCs) which clip or saturate whenever the signal amplitude exceeds the maximum recordable ADC voltage thus leading to a significant information loss. In this paper, we develop an alternative paradigm for sensing and recovery, called the Unlimited Sampling Framework. It is based on the observation that when a signal is mapped to an appropriate bounded interval via a modulo operation before entering the ADC, the saturation problem no longer exists, but one rather encounters a different type of information loss due to the modulo operation. Such an alternative setup can be implemented, for example, via so-called folding or self-reset ADCs, as they have been proposed in various contexts in the circuit design literature. The key task that we need to accomplish in order to cope with this new type of information loss is to recover a bandlimited signal from its modulo samples. In this paper we derive conditions when perfect recovery is possible and complement them with a stable recovery algorithm. The sampling density required to guarantee recovery is independent of the maximum recordable ADC voltage and depends on the signal bandwidth only. Our recovery guarantees extend to measurements affected by bounded noise, which includes the case of round-off quantization. Numerical experiments validate our approach. Applications of the unlimited sampling paradigm can be found in a number of fields such as signal processing, communication and imaging.

cs.IT↗

arXiv2018

Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain

Recovering a sparse signal from its low-pass projections in the Fourier domain is a problem of broad interest in science and engineering and is commonly referred to as super-resolution. In many cases, however, Fourier domain may not be the natural choice. For example, in holography, low-pass projections of sparse signals are obtained in the Fresnel domain. Similarly, time-varying system identification relies on low-pass projections on the space of linear frequency modulated signals. In this paper, we study the recovery of sparse signals from low-pass projections in the Special Affine Fourier Transform domain (SAFT). The SAFT parametrically generalizes a number of well known unitary transformations that are used in signal processing and optics. In analogy to the Shannon's sampling framework, we specify sampling theorems for recovery of sparse signals considering three specific cases: (1) sampling with arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels and, (3) recovery from Gabor transform measurements linked with the SAFT domain. Our work offers a unifying perspective on the sparse sampling problem which is compatible with the Fourier, Fresnel and Fractional Fourier domain based results. In deriving our results, we introduce the SAFT series (analogous to the Fourier series) and the short time SAFT, and study convolution theorems that establish a convolution--multiplication property in the SAFT domain.

cs.IT↗

arXiv2017

On Unlimited Sampling

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 are 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.

cs.IT↗

arXiv2017

Sampling Without Time: Recovering Echoes of Light via Temporal Phase Retrieval

This paper considers the problem of sampling and reconstruction of a continuous-time sparse signal without assuming the knowledge of the sampling instants or the sampling rate. This topic has its roots in the problem of recovering multiple echoes of light from its low-pass filtered and auto-correlated, time-domain measurements. Our work is closely related to the topic of sparse phase retrieval and in this context, we discuss the advantage of phase-free measurements. While this problem is ill-posed, cues based on physical constraints allow for its appropriate regularization. We validate our theory with experiments based on customized, optical time-of-flight imaging sensors. What singles out our approach is that our sensing method allows for temporal phase retrieval as opposed to the usual case of spatial phase retrieval. Preliminary experiments and results demonstrate a compelling capability of our phase-retrieval based imaging device.

cs.IT↗

arXiv2016

Shift-Invariant and Sampling Spaces Associated with the Special Affine Fourier Transform

The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. Shannon's sampling theorem, which is at the heart of modern digital communications, is a special case of sampling in shift-invariant spaces. Furthermore, it is well known that the Poisson summation formula is equivalent to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson summation formula. These results have been known to hold in the Fourier transform domain for decades and were recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed. The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short, self-contained proof of Shannon's theorem for functions bandlimited in the SAFT domain and then show that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFT domain. This interpretation of sampling leads to least-squares optimal sampling theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi-discrete convolution operators that are associated with the SAFT domain. We conclude the article with an application of fractional delay filtering of SAFT bandlimited functions.

cs.IT↗

arXiv2015

Convolution and Product Theorem for the Special Affine Fourier Transform

The Special Affine Fourier Transform or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Unlike the Fourier transform, the SAFT does not work well with the standard convolution operation. Recently, Q. Xiang and K. Y. Qin introduced a new convolution operation that is more suitable for the SAFT and by which the SAFT of the convolution of two functions is the product of their SAFTs and a phase factor. However, their convolution structure does not work well with the inverse transform in sofar as the inverse transform of the product of two functions is not equal to the convolution of the transforms. In this article we introduce a new convolution operation that works well with both the SAFT and its inverse leading to an analogue of the convolution and product formulas for the Fourier transform. Furthermore, we introduce a second convolution operation that leads to the elimination of the phase factor in the convolution formula obtained by Q. Xiang and K. Y. Qin.

cs.IT↗

arXiv2015

Super-Resolution in Phase Space

This work considers the problem of super-resolution. The goal is to resolve a Dirac distribution from knowledge of its discrete, low-pass, Fourier measurements. Classically, such problems have been dealt with parameter estimation methods. Recently, it has been shown that convex-optimization based formulations facilitate a continuous time solution to the super-resolution problem. Here we treat super-resolution from low-pass measurements in Phase Space. The Phase Space transformation parametrically generalizes a number of well known unitary mappings such as the Fractional Fourier, Fresnel, Laplace and Fourier transforms. Consequently, our work provides a general super- resolution strategy which is backward compatible with the usual Fourier domain result. We consider low-pass measurements of Dirac distributions in Phase Space and show that the super-resolution problem can be cast as Total Variation minimization. Remarkably, even though are setting is quite general, the bounds on the minimum separation distance of Dirac distributions is comparable to existing methods.

cs.IT↗

arXiv2014

Resolving Multi-path Interference in Time-of-Flight Imaging via Modulation Frequency Diversity and Sparse Regularization

Time-of-flight (ToF) cameras calculate depth maps by reconstructing phase shifts of amplitude-modulated 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.

cs.CV↗