This paper presents a signal processing algorithm that computes trajectories of drilled underground holes together with their uncertainties. The algorithm was tested on data obtained by an instrumented drilling rig, with which several holes were drilled. The collected data was preprocessed using hierarchical indexing to ensure easy processing later on. The algorithm uses Bezier curves and differential geometry to reconstruct the three-dimensional curve from its gradients, and describes a matrix-algebra framework for computing uncertainties and covariance matrices for the case of higher-dimensional curves.The novelty of the paper is the developed algebraic framework for the computation of uncertainties, as well as the comparison of the obtained results to two independent reference measurements to ensure accuracy. Furthermore, the developed algebraic framework of the covariance propagation allows for the computation of envelope curves, which shows the low uncertainty of the developed method, due to the high number of data samples used.

An undersampled autocorrelation-based spectrum sensing algorithm is proposed for orthogonal frequency division multiplexing (OFDM) signal detection in channels sampled below the Nyquist rate. The proposed algorithm reduces the complexity of autocorrelation-based detection by minimizing the number of unique multiplications required to compute the autocorrelation function, exploiting the inherent structure of the OFDM signal. This enables low-bandwidth cognitive radios to effectively detect wideband signals. The novelty of this work lies in a threshold design approach, which compensates for the performance loss due to undersampling and maintains reliable detection accuracy. Analytical and simulation results demonstrate that the proposed algorithm attains reasonable detection performance in the mid to high signal-to-noise ratio (SNR) scenarios, while also significantly reducing the false alarm probability under low SNR conditions.

This paper presents a new method for modelling periodic signals having an aperiodic trend, using the method of variable projection. It extends the commonly used four parameter sine wave model by permitting the background to be time varying; additionally, any number of harmonics of the periodic portion can be modelled. This paper focuses on using B-Splines to implement a piecewise polynomial model for the aperiodic portion of the signal. A thorough algebraic derivation of the method is presented, as well as a comparison to using global polynomial approximation. It is proven that B-Splines work better for modelling a more complicated aperiodic portion when compared to higher order polynomials. Furthermore, the piecewise polynomial model is capable of modelling the local signal variations produced by the interaction of a control system with a process in industrial applications. An added benefit of using the method of variable projection is the possibility to calculate the covariances of the linear coefficients of the model, enabling the calculation of confidence and prediction intervals. The method is tested on both real measurement data acquired in industrial processes, as well as synthetic data. The method shows promising results for the precise characterization of periodic signals embedded in highly complex aperiodic backgrounds. Finally, snippets of the m-code are provided, together with a toolbox for B-Splines, which permit the implementation of the complete computation.

This paper presents a real-time parameter identification algorithm for periodic signals, based on the recursive variable projection (RVP) algorithm. The recursive implementation enables the tracking of time-varying parameters. The signal model is linear with respect to the amplitude parameters while being nonlinear with respect to the phase and frequency. This feature motivates the use of a variable projection based approach. Its performance is tested using Monte Carlo simulations and the results are compared with those obtained by a multiobjective Gauss-Newton (MGN) algorithm. Furthermore, the RVP algorithm is applied to measurement data acquired by a MEMS accelerometer and it is demonstrated that it can successfully track time-varying linear and nonlinear parameters.

This paper presents a new approach to the detection of discontinuities in the n-th derivative of observational data. This is achieved by performing two polynomial approximations at each interstitial point. The polynomials are coupled by constraining their coefficients to ensure continuity of the model up to the (n − 1)-th derivative; while yielding an estimate for the discontinuity of the n-th derivative. The coefficients of the polynomials correspond directly to the derivatives of the approximations at the interstitial points through the prudent selection of a common coordinate system. The approximation residual and extrapolation errors are investigated as measures for detecting discontinuity. This is necessary since discrete observations of continuous systems are discontinuous at every point. It is proven, using matrix algebra, that positive extrema in the combined approximation-extrapolation error correspond exactly to extrema in the difference of the Taylor coefficients. This provides a relative measure for the severity of the discontinuity in the observational data. The matrix algebraic derivations are provided for all aspects of the methods presented here; this includes a solution for the covariance propagation through the computation. The performance of the method is verified with a Monte Carlo simulation using synthetic piecewise polynomial data with known discontinuities. It is also demonstrated that the discontinuities are suitable as knots for B-spline modelling of data. For completeness, the results of applying the method to sensor data acquired during the monitoring of heavy machinery are presented.

By focusing on the diverse range of applications that THz communications/technology can offer, specifically in the domain of advanced agricultural systems, we define a theoretical model for an approximation of the transmitted/reflected electric fields for THz radiation through plant leaves. The provided model can be modified to accommodate scenarios with single or multiple layers of leaves, given the permittivity of specific leaf types. In addition, we provide measured results of the permittivity of coffee leaves in the range of 0.3-2 THz. Based on the measured data and the provided model, we analyse the effect of various parameters, e.g., the leaf thickness, the separation distance between leaves and the transmission frequency, on the total transmitted/reflected fields. The results indicate the significant impact of the transmission frequency and the leaf attributes on degrading the THz signal, which can challenge the feasibility of the mentioned applications.

This paper presents three new methods for regularizing the least squares solution of the reconstruction of a surface from its gradient field: firstly, the spectral regularization based on discrete generalized Fourier series (e.g., Gram Polynomials, Haar Functions, etc.); secondly, the Tikhonov regularization applied directly to the 2D domain problem; and thirdly, the regularization via constraints such as arbitrary Dirichlet boundary conditions. It is shown that the solutions to the aforementioned problems all satisfy Sylvester Equations, which leads to substantial computational gains; specifically, the solution of the Sylvester Equation is direct (non-iterative) and for an m × n surface is of the same complexity as computing an SVD of the same size, i.e., an O (n3) algorithm. In contrast, state-of-the-art algorithms are based on large-scale-linear-solvers, and use iterative techniques based on an O [n6) linear sub-step. To emphasize this improvement, it is demonstrated that the new algorithms are upwards of 104(ten-thousand) times faster than the state-of-the-art techniques incorporating regularization. In fact, the new algorithms allow for the realtime regularized reconstruction of surfaces on the order of megapixels, which is unprecedented for this computer vision problem.

This paper presents a new method for the reconstruction of a surface from its x and y gradient field, measured, for example, via Photometric Stereo. The new algorithm produces the unique discrete surface whose gradients are equal to the measured gradients in the global vertical-distance least-squares sense. We show that it has been erroneously believed that this problem has been solved before via the solution of a Poisson equation. The numerical behaviour of the algorithm allows for reliable surface reconstruction on exceedingly large scales, e.g., full digital images; moreover, the algorithm is direct, i.e., non-iterative. We demonstrate the algorithm with synthetic data as well as real data obtained via photometric stereo. The algorithm does not exhibit a low-frequency bias and is not unrealistically constrained to arbitrary boundary conditions as in previous solutions. In fact, it is the first algorithm which can reconstruct a surface of polynomial degree two or higher exactly. It is hence the first viable algorithm for online industrial inspection where real defects (as opposed to phantom defects) must be identified in a robust manner.

There is a prevalent myth in Computer Vision that the gradient weighted algebraic distance, the so-called “Sampson Error,” is a first order approximation of the distance from a point to a curve or surface. In truth, however, it is the exact geometric distance to the first order approximation of the curve. The linguistic difference is subtle, but mathematically, the two statements are at odds. In this paper, we derive the actual first order approximation of the Mahalanobis distance to a curve, a special case of which is the geometric distance. Furthermore, we show that it too, like the Sampson error, is a weighted algebraic distance. The first order distance introduces an increase in computational effort (total “flops”), which is the inevitable cost of a better approximation; however, since it too is an explicit expression, it has the same computational complexity as the Sampson error. Numerical testing shows that the first order distance performs an order of magnitude better than the Sampson error in terms of relative error with respect to the geometric and Mahalanobis distances. Our results suggest that the first order distance is an exceptional candidate cost function for approximate maximum likelihood fitting.

A new method for the non-iterative computation of a homography matrix is described. Rearrangement of the equations leads to a block partitioned sparse matrix, facilitating a residualization based on orthogonal matrix projections. This improves the handling of the error structure of the linear system of equations. The vanishing line is treated as the principal component in the estimation process. This estimate is more robust, since the position of the vanishing line depends only on the relative position and orientation of the camera to the observed plane, and is invariant to the structure of the points observed on the plane. A flop-count indicates that the new method is 11 times faster for four point correspondences, and converges to a factor of 5 for a large number of points. Furthermore, a new non-iterative method of treating error in both images is derived. Combining the forward H and reverse G projections in a suitable manner eliminates the systematic bias of the estimation, and the first order error: a strict bound on the error reduction is derived. This can be achieved faster than a classical DLT due to the improved numerical efficiency. Results of Monte-Carlo simulations are presented to verify the performance. 1

A new method to fit specific types of conics to scattered data points is introduced. Direct, specific fitting of ellipses and hyperbolæ is achieved by imposing a quadratic constraint on the conic coefficients, whereby an improved partitioning of the design matrix is devised so as to improve computational efficiency and numerical stability by eliminating redundant aspects of the fitting procedure. Fitting of parabolas is achieved by determining an orthogonal basis vector set in the Grassmannian space of quadratic conic forms. The linear combination of the basis vectors which fulfills the parabolic condition and has a minimum residual is determined using Lagrange multipliers. 1

The Cheshire II online catalog system was designed to provide a bridge between the realms of purely bibliographical information and the rapidly expanding full-text and multi-media collections available online. It is based on a number of national and international standards for data description, communication, and interface technology. The system uses a client-server architecture with X window client communication with an SGML-based probabilistic search engine using the Z39.50 information retrieval protocol. © 1996 John Wiley & Sons, Inc.

The authors present a modified direct-digital synthesizer which uses noise shaping to reduce the effects of phase-accumulator truncation on the output spectrum. The discrete spectral disturbances associated with this truncation error are strongly reduced with the proposed method, making the synthesizer suitable for high-performance signal processing. The proposed architecture uses first-order noise shaping. Higher order noise shaping can also be used if required. The modified circuit offers considerable improvement in the synthesizer performance when the generated frequency is low with respect to the clock frequency.>

A circuit is described which uses one oversampling frequency synthesizer in an interleaved mode to generate dual-tone multifrequency (DTMF) telephone dialing tones. The synthesizer is based on direct digital synthesis using phase accumulation. The digital-to-analog conversion is carried out using a first-order sigma-delta converter. This sigma-delta converter requires only a simple second-order RC filter, eliminating the need for a switched capacitor filter. Measurements on the circuit confirm that all specifications of the DTMF are fulfilled.>