Snapshot compressive imaging (SCI) is a compressed sensing (CS)-based high-speed imaging modality. Recent efforts have explored the underlying 3D representation from only an SCI image using neural radiance fields (NeRF), yet the training time, rendering computation cost, and reconstruction quality limitations are general issues that have limited wider adoption. This paper introduces SCI-Gaussian, the first 3D-aware SCI reconstruction based on 3D Gaussian splatting (3D-GS). This method utilizes an explicit 3D representation to achieve efficient and high-quality scene reconstruction. The motivation stems from the highly efficient representation and surprising quality of 3D-GS, despite when applied to SCI system, it encounters difficulties in generating point initialization for explicit Gaussians and accurate pose recovery from a single SCI measured image. Specifically, we effectively initialize these Gaussians through sampling a coarsely trained NeRF at various hash structures, then model the physical formation of the SCI measurement and jointly optimize Gaussians and camera trajectories with a bundle adjustment formulation during exposure time. Extensive experiments on synthetic and real-world datasets demonstrate that SCI-Gaussian outperforms the state-of-the-art (SOTA) methods, achieving comparable or better results with significantly 10× faster training and 1000× faster rendering speed than the most recent NeRF-based method.

Uncertainties for conventional nonparametric magnetotelluric (MT) impedance tensor estimates are typically quantified by analytic and asymptotic confidence bounds. Here, we consider MT impedance tensor confidence bounds for parametric estimates to both facilitate their interpretation and comparison to conventional nonparametric estimates. In addition to the standard asymptotic confidence bounds from system identification, we propose several resampling methods based on three strategies: subsampling (SB), moving block bootstrap (BS), and model-based resampling (MR). We applied these data-driven approaches to an MT dataset exhibiting strong interference to demonstrate resampling-based methods for accurate and reliable confidence bound determination, especially when the data is limited or exhibits poor data quality. This study also investigates various practical considerations, providing a viable and comprehensive approach for both parametric MT impedance tensor estimation and associated uncertainty determination.

Given the prevalence and intensity of cultural noise and its non-stationary characteristics, the uncertainties associated with estimated magnetotelluric (MT) impedances are crucial for MT interpretation and utilization. Here, we focus on MT impedance confidence interval estimates based on the parametric estimation method. In addition to the standard asymptotic estimates from system identification, we also consider two resampling methods for parametric estimation based on subsampling and moving block bootstrap strategies. We applied these approaches to a representative USArray site and the results show that the resampling based estimates have the potential to obtain more accurate and reliable estimated MT transfer impedances, especially when the data quality is poor.

Electromagnetic transfer functions (EMTFs) play a central role in interpreting the relationship between the magnetic and the electric fields, providing empirical information necessary for magnetotelluric (MT) technology. The proposed parametric method aims at estimating an EMTF based on system identification, making it possible to estimate EMTFs in the time domain with fewer parameters and data. Our prior studies have validated the effectiveness of the output-error (OE) model based parametric method with real MT data measured at USArray sites at a sampling frequency of 1 Hz. To further validate and explore the potential of this method for MT, in this study MT data at different sampling frequencies are considered, which can be used to consider the EMTF over a wider frequency band.

The computation required for a switching Kalman Filter (SKF) increases exponentially with the number of system operation modes. In this paper, a computationally tractable graph representation is proposed for a switching linear dynamic system (SLDS) along with the solution of a minimum-sum optimization problem for clustering to reduce the switching mode cardinality offline, before collecting measurements. It is shown that upon perfect mode detection, the induced error caused by mode clustering can be quantified exactly in terms of the dissimilarity measures in the proposed graph structure. Numerical results verify that clustering based on the proposed framework effectively reduces model complexity given uncertain mode detection and that the induced error can be well approximated if the underlying assumptions are satisfied.

Conventional magnetotelluric (MT) impedance tensor estimation methods are generally based on robust M-estimator spectral analysis along with remote reference (RR) processing to reduce the estimation biases caused by cultural noise. Here, we propose a parametric spectral estimation and system identification-based impedance tensor estimation method and validate it with three representative data segments measured at several USArray sites. The proposed parametric method has the potential to give an accurate and robust estimate of the electromagnetic transfer impedance without an RR while maintaining a smooth and continuous response in accordance with prior physical knowledge. The approach considers the geophysical problem in addition to signal processing and system identification, providing a statistically efficient strategy for electromagnetic transfer impedance estimation of periods from 10 to 104s.

The conventional approach to estimate the magnetotelluric (MT) impedance tensor from surface measured magnetic and geomagnetic electrical field time series relies on a frequency domain formulation and measurements from a remote reference site to mitigate correlated, systematic noise. Guided by system identification principles, we propose a parametric, time-domain approach that potentially offers greater statistical efficiency and noise resilience. Based on a study from measured data, we demonstrate that the parametric approach can yield estimates of the MT impedance with as little as one day of data and without the need of a remote reference site.

Switching Kalman Filters (SKF) are well known for solving switching linear dynamic system (SLDS), i.e., piece-wise linear estimation problems. Practical SKFs are heuristic, approximate filters and require more computational resources than a single-mode Kalman filter (KF). On the other hand, applying a single-mode mismatched KF to an SLDS results in erroneous estimation. This paper quantifies the average error an SKF can eliminate compared to a mismatched, single-mode KF before collecting measurements. Derivations of the first and second moments of the estimators errors are provided and compared. One can use these derivations to quantify the average performance of filters beforehand and decide which filter to run in operation to have the best performance in terms of estimation error and computation complexity. We further provide simulation results that verify our mathematical derivations.

Remote sensing and astronomical image formation is often complicated by deficiencies in measurement quality, density, or diversity. Penalized likelihood methods can incorporate additional first-principles physical prior knowledge and improve the image reconstructions, but a systematic bias is unavoidable as a consequence. This work derives theory to understand the bias and develops a computational tool to probe its effect on the reconstructed image and bound resolution limits. Though the focus is on image formation, the contributions of this paper apply to any inference problem that can be expressed under the linear state-space signal model.

Dynamic tomography is concerned with the image formation of a temporally changing object from its line integral projections. The problem remains challenging because of its high dimensionality. In this paper, we identify a sufficient class of dynamic tomography problems that can be solved by a state estimator that requires only linear shift-invariant filtering operations. This class includes rigid-body motion, common in biomedical imaging scenarios. The new state estimator is far less computationally demanding than classic methods such as the Kalman filter. Whereas the Kalman filter requires O(N2) memory storage and O(N3) processing for anN-dimensional problem, the state estimator derived in this work requires only O(N) storage and O(N logN) processing.

We address the image formation of a dynamic object from projections by formulating it as a state estimation problem. The problem is solved with the ensemble Kalman filter (EnKF), a Monte Carlo algorithm that is computationally tractable when the state dimension is large. In this paper, we first rigorously address the convergence of the EnKF. Then, the effectiveness of the EnKF is demonstrated in a numerical experiment where a highly variable object is reconstructed from its projections, an imaging modality not yet explored with the EnKF. The results show that the EnKF can yield estimates of almost equal quality as the optimal Kalman filter but at a fraction of the computational effort. Further experiments explore the rate of convergence of the EnKF, its performance relative to an idealized particle filter, and implications of modeling the system dynamics as a random walk.

This paper formally addresses the asymptotic convergence of the ensemble Kalman filter (EnKF), a state estimation procedure that, when combined with a technique called localization, provides computationally tractable solutions to large-dimensional state estimation problems. The proof presented in this paper shows that the estimates given by the EnKF converge to the optimal estimates given by the Kalman filter (KF) and provides a formal justification for the use of the EnKF in dynamic remote sensing image formation. The implications of the proof are twofold: it shows that the EnKF converges to a well-defined limit and provides a formal argument that the EnKF is in fact a Monte Carlo algorithm that converges to the KF.

We address the reconstruction of a physically evolving unknown from tomographic measurements by formulating it as a state estimation problem. The approach presented in this paper is the localized ensemble Kalman filter (LEnKF); a Monte Carlo state estimation procedure that is computationally tractable when the state dimension is large. We establish the conditions under which the LEnKF is equivalent to the Gaussian particle filter. The performance of the LEnKF is evaluated in a numerical example and is shown to give state estimates of almost equal quality as the optimal Kalman filter but at a 95% reduction in computation.

Due to inherent factors such as a small and fragmented market and rapid hardware obsolescence, the conventional textbook is inadequate for DSP laboratory education. Freely available open-content materials that enable and promote both local customization and further development by a community of educators offers a fresh approach to lab text development that can surmount these barriers. We overview a joint effort under the aegis of the Connexions Project to develop a large pool of DSP lab modules sufficient to serve as the complete, stand-alone text for several types of DSP lab courses.