Jonathan M. Nichols

Compressed sensing for practical optical imaging systems: a tutorial

The emerging field of compressed sensing (CS, also referred to as compressive sampling)1, 2 has potentially powerful implications for the design of optical imaging devices. In particular, compressed sensing theory suggests that one can recover a scene at a higher resolution than is dictated by the pitch of the focal plane array. This rather remarkable result comes with some important caveats however, especially when practical issues associated with physical implementation are taken into account. This tutorial discusses compressed sensing in the context of optical imaging devices, emphasizing the practical hurdles related to building such devices and offering suggestions for overcoming these hurdles. Examples and analysis specifically related to infrared imaging highlight the challenges associated with large format focal plane arrays and how these challenges can be mitigated using compressed sensing ideas.

A Unified Approach to Attractor Reconstruction

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e., attractor reconstruction. The process has focused primarily on intuitive, heuristic, and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several longstanding, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.

Beating Nyquist with light: a compressively sampled photonic link

We report the successful demonstration of a compressively sampled photonic link. The system takes advantage of recent theoretical developments in compressive sampling to enable signal recovery beyond the Nyquist limit of the digitizer. This rather remarkable result requires that (1) the signal being recovered has a sparse (low-dimensional) representation and (2) the digitized samples be incoherent with this representation. We describe an all-photonic system architecture that meets these requirements and then show that 1 GHz harmonic signals can be faithfully reconstructed even when digitizing at 500 MS/s, well below the Nyquist rate.

Bragg grating-based fibre optic sensors in structural health monitoring

This work first considers a review of the dominant current methods for fibre Bragg grating wavelength interrogation. These methods include WDM interferometry, tunable filter (both Fabry–Perot and acousto-optic) demultiplexing, CCD/prism technique and a newer hybrid method utilizing Fabry–Perot and interferometric techniques. Two applications using these techniques are described: hull loads monitoring on an all-composite fast patrol boat and bolt pre-load loss monitoring in a composite beam in conjunction with a state-space modelling data analysis technique.

Structural health monitoring through chaotic interrogation

The field of vibration based structural health monitoring involves extracting a ‘feature’ which robustly quantifies damage induced changes to the structure in the presence of ambient variation, that is, changes in ambient temperature, varying moisture levels, etc. In this paper, we present an attractor-based feature derived from the field of nonlinear time-series analysis. Emphasis is placed on the use of chaos for the purposes of system interrogation. The structure is excited with the output of a chaotic oscillator providing a deterministic (low-dimensional) input. Use is made of the Kaplan–Yorke conjecture in order to ‘tune’ the Lyapunov exponents of the driving signal so that varying degrees of damage in the structure will alter the state space properties of the response attractor. The average local attractor variance ratio (ALAVR) is suggested as one possible means of quantifying the state space changes. Finite element results are presented for a thin aluminum cantilever beam subject to increasing damage, as specified by weld line separation, at the clamped end. Comparisons of the ALAVR to two modal features are evaluated through the use of a performance metric.

Using state space predictive modeling with chaotic interrogation in detecting joint preload loss in a frame structure experiment

This work explores the role of steady-state dynamic analysis in the vibration-based structural health monitoring field. While more traditional approaches focus on transient or stochastic vibration analysis, the method described here utilizes a geometric portrait of system dynamics to extract information about the steady-state response of the structure to sustained excitation. The approach utilizes the fundamental properties of chaotic signals to produce low-dimensional response data which are then analyzed for features which indicate the degree to which the dynamics have been altered by damage. A discussion of the fundamental issues involved in the approach is presented along with experimental evidence of the approachs ability to discriminate among several damage scenarios.

Use of data-driven phase space models in assessing the strength of a bolted connection in a composite beam

This work explores the role of empirical dynamical models in deducing the level of preload loss in a bolted connection. Specifically, we examine the functional relationship between data gleaned from locations on either side of the connection using nonlinear predictive models. This relationship, as quantified by a measure of prediction error, changes as a function of bolt loosening, thus allowing both the presence and magnitude of the axial load to be identified. The models are based on a phase space portrayal of the system dynamics and require only that the structures response be low dimensional. The technique is demonstrated experimentally on a composite beam fastened to steel plates with four instrumented bolts. Results are compared to a similar approach using an auto-regressive (AR) modeling technique.

The Bispectrum and Bicoherence for Quadratically Nonlinear Systems Subject to Non-Gaussian Inputs

In the analysis of data from nonlinear systems both the bispectrum and the bicoherence have emerged as useful tools. Both are frequently used to detect the influence of a nonlinear system on the joint probability distribution of the system input. Previous work has provided an analytical expression for the bispectrum of a quadratically nonlinear system output if the input is stationary, jointly Gaussian distributed. This work significantly generalizes the previous analysis by providing an analytical expression for the bispectrum of the response of quadratically nonlinear systems subject to stationary, jointly non-Gaussian inputs possessing arbitrary auto-correlation function. The expression is then used to determine the optimal input probability density function for detecting a quadratic nonlinearity in a second-order system. It is also shown how the expression can be used to design an optimal nonlinear filter for detecting deviations from normality in the probability density of a signal.

Calculation of Differential Entropy for a Mixed Gaussian Distribution

In this work, an analytical expression is developed for the differential entropy of a mixed Gaussian distribution. One of the terms is given by a tabulated function of the ratio of the distribution parameters.

Handbook of Differential Entropy

One of the main issues in communications theory is measuring the ultimate data compression possible using the concept of entropy. While differential entropy may seem to be a simple extension of the discrete case, it is a more complex measure that often requires a more careful treatment. Handbook of Differential Entropy provides a comprehensive introduction to the subject for researchers and students in information theory. Unlike related books, this one brings together background material, derivations, and applications of differential entropy. The handbook first reviews probability theory as it enables an understanding of the core building block of entropy. The authors then carefully explain the concept of entropy, introducing both discrete and differential entropy. They present detailed derivations of differential entropy for numerous probability models and discuss challenges with interpreting and deriving differential entropy. They also show how differential entropy varies as a function of the model variance. Focusing on the application of differential entropy in several areas, the book describes common estimators of parametric and nonparametric differential entropy as well as properties of the estimators. It then uses the estimated differential entropy to estimate radar pulse delays when the corrupting noise source is non-Gaussian and to develop measures of coupling between dynamical system components.