Linear inverse problems are ubiquitous in signal processing and computational imaging. The prototypical problem is to recover a signal from noisy linear measurements. A typical optimization-based approach is to minimize the sum of a data-fidelity loss and a regularization function. The data-fidelity function ensures consistency with the measurements and the regularization function imparts desired properties to the solution. Convex regularization functions are typically preferred as one can provide theoretical guarantees. However, the convex-nonconvex (CNC) framework, which employs a convex data-fidelity loss and a nonconvex regularization function has been shown to be superior in terms of the quality of signal recovery. In this paper, we consider model-based and data-driven nonconvex regularization objectives to solve linear inverse problems. We consider the denoising problem and propose a constructive approach to design weakly convex regularization functions by minimizing a measure of maximum concavity. Our design approach captures known model-based regularization functions, including those that promote sparsity, and also includes learnable convolutional neural networks. Minimization of the objective follows first-order gradient-based methods. Our approach ensures that the overall reconstruction technique is provably convergent. We show that it outperforms state-of-the-art model-based techniques and is comparable to the benchmark learningbased methods. Crucially, our technique results in reconstructions with fewer artifacts compared to the state-of-the-art learning-based methods. Our reconstruction approach reduces the number of network parameters to be learnt for similar neural network architectures making it easier/faster to train.

The unlimited sampling framework (USF) is a computational sensing paradigm that addresses the practical bottleneck pertaining to finite dynamic range and quantization resolution of standard analog-to-digital converters (ADCs). The essence of unlimited sampling is to capture high-dynamic range (HDR) signals using practical sensors by folding the signal within the dynamic range of the sensor, followed by leveraging computational techniques for reconstruction. In this paper, we consider unlimited sampling using neuromorphic/event-driven encoders that enable acquisition of HDR signals as they simultaneously fold the signal and keep track of the folding instants using events. At ICASSP 2024, we proposed neuromorphic unlimited sampling for bandlimited signals. Herein, we extend the theory to include signals in principal shift-invariant spaces, and show that perfect reconstruction is possible without an oversampling requirement on the ADC. Samples of the folded signal along with the events are sufficient for perfect reconstruction with sampling rates that are independent of the dynamic-range of the ADC. Within the proposed technique, a large class of smooth signals that lie outside any shift-invariant space can be accommodated with the reconstruction error decreasing as the sampling interval is reduced. On the experimental front, we consider acquisition of HDR video signals using an array of neuromorphic unlimited samplers and demonstrate accurate reconstruction. The proposed neuromorphic design for HDR video acquisition does not require oversampling of the ADC because the HDR information is indirectly captured in a compressed form using binary events.

Inverse problems form the backbone of modern signal/image processing and computational imaging, where signal reconstruction from corrupted measurements follows an optimization problem. The objective function is the sum of a data-fidelity term and a regularization functional that enforces desired properties in the reconstruction. The adversarial regularization (AR) framework is an unsupervised, data-driven approach for solving inverse problems, where the regularization function is learnt adversarially as a critique between the ground-truth distribution and the distribution of unregularized reconstructions. Thereafter, the solution to the regularized inverse problem follows an iterative technique. In this paper, we analyze the AR framework from a variational perspective, and, using Euler-Lagrange conditions, obtain the optimal regularization function in closed-form. The overall objective function is smooth and readily amenable to gradient descent minimization. We introduce momentum into the iterates as a natural extension to accelerate convergence. Since the optimal solutions are obtained in closed-form, our approach to solving inverse problems does not require prior training whilst being data-driven. We demonstrate the proposed technique on image deconvolution and show that the reconstruction performance of the proposed techniques measured in terms of peak signal-to-noise ratio (PSNR) and structural similarity index metric (SSIM) are identical to the learnt counterparts.

Unlimited sampling is a computational sensing paradigm for high-dynamic range (HDR) acquisition of continuous-time signals. In standard analog-to-digital converters (ADC), a fixed input dynamic range is accommodated by clipping or saturation of the signal outside the dynamic range. In unlimited sampling, signal that lies outside the fixed dynamic range is folded back, thereby preserving the signal dynamic range. The folding is achieved using a self-reset ADC (SR-ADC), which performs a continuous-time modulo operation. In this paper, we use a neuromorphic encoder, which is an opportunistic and event-driven sampling device, and propose a new technique for unlimited sampling. We show that the neuromorphic encoder folds the signal that lies outside the dynamic range and simultaneously records a compressed representation of the error signal. Unlimited sampling is achieved by measuring the folded signal. We analyze sampling and reconstruction of finite-energy, bandlimited signals and show that perfect reconstruction is possible using uniform samples acquired at the Nyquist rate of the signal. The reconstruction technique operates in real-time, and can be readily extended to larger classes of continuous-time signals. We demonstrate the performance of our technique and report comparisons with state-of-the-art techniques using simulations.

Image restoration involves solving an optimization problem where the objective function is the sum of a data-fidelity term and a regularization functional that incorporates a desired image prior. Solving the optimization problem using proximal methods results in iterative algorithms that require computing a gradient step corresponding to the data-fidelity loss and a proximal update corresponding to enforcing the image prior. In this paper, we develop a novel formulation for image restoration considering a generalized data-fidelity loss and a convex regularization function that enforces a desired image prior, and we solve the problem using proximal gradient method. The choice of the data-fidelity loss is such that the adjoint operator is reminiscent of Wiener filtering when the forward operator is a convolutional operator (for instance, a shift-invariant blur kernel). The proposed gradient update ensures that the iterates remain in the solution-space of the linear measurement constraints. We further propose the plug-and-play counterpart of the restoration technique, which allows one to leverage off-the-shelf data-driven denoisers in place of the proximal operator. Experimental validations carried out on BSD500, Brodatz, Urban100, and DIV2K datasets show that the proposed technique gives rise to superior image reconstruction quality compared with the state-of-the-art techniques, with the performance measured in terms of peak signal-to-noise ratio (PSNR) and structural similarity index metric (SSIM), with comparable computational complexity.

Abstract. The choice of the sensing matrix is crucial in compressed sensing. Random Gaussian sensing matrices satisfy the restricted isometry property, which is crucial for solving the sparse recovery problem using convex optimization techniques. However, tight-frame sensing matrices result in minimum mean-squared-error recovery given oracle knowledge of the support of the sparse vector. If the sensing matrix is not tight, could one achieve the recovery performance assured by a tight frame by suitably designing the recovery strategy? This is the key question addressed in this paper. We consider the analysis-sparse [Formula: see text]-minimization problem with a generalized [Formula: see text]-norm-based data-fidelity and show that it effectively corresponds to using a tight-frame sensing matrix. The new formulation offers improved performance bounds when the number of nonzeros is large. One could develop a tight-frame variant of a known sparse recovery algorithm using the proposed formalism. We solve the analysis-sparse recovery problem in an unconstrained setting using proximal methods. Within the tight-frame sensing framework, we rescale the gradients of the data-fidelity loss in the iterative updates to further improve the accuracy of analysis-sparse recovery. Experimental results show that the proposed algorithms offer superior analysis-sparse recovery performance. Proceeding further, we also develop deep-unfolded variants, with a convolutional neural network as the sparsifying operator. On the application front, we consider compressed sensing image recovery. Experimental results on Set11, BSD68, Urban100, and DIV2K datasets show that the proposed techniques outperform the state-of-the-art techniques, with performance measured in terms of peak signal-to-noise ratio and structural similarity index metric.

Time-encoding of continuous-time signals is an alternative sampling paradigm to Shannon sampling. In time-encoding or event-driven sampling, the signal is encoded using a sequence of time instants corresponding to an event. In this paper, we propose multichannel time-encoding of signals with a finite-rate-of-innovation (FRI) in single-input-multi-output (SIMO) and multi-input-multi-output (MIMO) configurations using the integrate-and-fire model. We demonstrate perfect reconstruction of FRI signals with common support from MIMO time-encoded measurements using a joint estimation technique, and perfect reconstruction of FRI signals from SIMO time-encoded measurements with reduced sampling requirement as compared to the single channel case. We provide sufficient conditions for perfect reconstruction with sampling requirement of the order of the rate of innovation of the signal. We substantiate our claims using simulations on noise-free and noisy measurements.

Time-encoding or event-driven sampling of continuous-time signals is an alternative paradigm to uniform sampling. In this sampling scheme, the signal is encoded by a sequence of time-instants as opposed to a sequence of amplitudes in uniform sampling. Time-encoding is opportunistic by design – measurements are taken only when the signal exhibits significant variability. Consequently, the measurements are sparse, noise-robust, and require low power. However, standard processing and reconstruction methods do not apply. In this paper, we introduce a new time-encoding machine, namely, differentiate-and-fire time-encoding machine (DIF-TEM) inspired by the functioning of the human visual system. A DIF-TEM can be tuned to provide sampling sets with variable densities – sparse sets that mimic dynamic vision sensors (neuromorphic cameras) or dense sets that mimic classical time-encoding machines. We propose kernel-based time-encoding of finite-rate-of-innovation (FRI) signals using DIF-TEM via Fourier-domain analysis. We show that DIF-TEM measurements are sufficient for perfect signal reconstruction under certain conditions. We provide simulation results to substantiate our claims.

Time-based sampling of continuous-time signals is an alternative to Shannon's sampling paradigm in which the signal is encoded using a sequence of nonuniform time instants. The standard methods for reconstructing signals in bandlimited and shift-invariant spaces from their nonuniform measurements employ alternating projections algorithms. In this paper, we consider the problem of sampling and perfect reconstruction of periodic finite-rate-of-innovation (FRI) signals using crossing-time-encoding machine (C-TEM) and integrate-and-fire TEM (IF-TEM). We formulate the reconstruction problem in the frequency domain and develop techniques to compute the Fourier coefficients, which contain the unknown parameters of the signal in the form of a sum of weighted complex exponentials. The parameters are then estimated using high-resolution spectral estimation techniques. Unlike state-of-the-art methods, the proposed method is generalized to incorporate reconstruction of periodic FRI signals consisting of weighted and shifted versions of an arbitrary pulse with arbitrarily close delays, and is compatible with a large class of sampling kernels. We provide sufficient conditions for sampling and perfect reconstruction using C-TEM and IF-TEM. We present simulation results to support our claims. We also discuss an extension to the sampling of aperiodic FRI signals.

Fourier descriptors are used to parametrically represent closed contours. In practice, a finite set of Fourier descriptors can model a large class of smooth contours. In this paper, we propose a method for estimating the Fourier descriptors of a given contour from its partial samples. We take a sampling-theoretic approach to model the x and y coordinate functions of the shape and express them as a sum of weighted complex exponentials, which belong to the class of finite-rate-of-innovation (FRI) signals. The weights represent the Fourier descriptors of the shape. We use the FRI framework to estimate the shape parameters reliably from noisy and partial measurements. We model non-uniformities in sampling using the sampling jitter model and employ a prefiltering process to reduce the effect of measurement noise and jitter. The average sampling interval is estimated by a block annihilating filter, which is then followed by the estimation of Fourier descriptors using least-squares fitting. We demonstrate the robustness of the proposed algorithm to noise and sampling jitter. Monte Carlo performance analysis shows that the variances of the estimators are close to the Cramér-Rao lower bounds. We present results for outlining shapes in synthetic as well as real images.