Generalized methods and solvers for noise removal from piecewise constant signals. I. Background theory

Removing noise from piecewise constant (PWC) signals is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need to be separated into stratigraphic zones, and in biophysics, jumps between molecular dwell states have to be extracted from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited. This paper (part I, the first of two) shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following and coordinate descent. In the second paper, part II, we introduce novel PWC denoising methods, and comparisons between these methods performed on synthetic and real signals, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.

Dynamics of on-line gradient descent learning for multilayer neural networks

We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is presented and used to investigate the role of the learning rate in controlling the evolution and convergence of the learning process.

Computationally efficient natural gradient descent

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Crystal structure and magnetic behaviour of a five-coordinate iron(III) complex of pyridoxal-4-methylthiosemicarbazone

The crystal structure and magnetic properties of a penta-coordinate iron(III) complex of pyridoxal-4-methylthiosemicarbazone, [Fe(Hmthpy)Cl](CHCHSO), are reported. The synthesised ligand and the metal complex were characterised by spectroscopic methods (H NMR, IR, and mass spectroscopy), elemental analysis, and single crystal X-ray diffraction. The complex crystallises as dark brown microcrystals. The crystal data determined at 100(1) K revealed a triclinic system, space group P over(1, ¯) (Z = 2). The ONSCl geometry around the iron(III) atom is intermediate between trigonal bipyramidal and square pyramidal (t = 0.40). The temperature dependence of the magnetic susceptibility (5-300 K) is consistent with a high spin Fe(III) ion (S = 5/2) exhibiting zero-field splitting. Interpretation of these data yielded: D = 0.34(1) cm and g = 2.078(3). © 2007 Elsevier B.V. All rights reserved.

An exploration into measurement consistency on coordinate measuring machines

In high precision industry, the measurement of geometry is often performed using coordinate measuring machines (CMMs). Measurements on CMMs can occur at many places within a long and global supply chain. In this context it is a challenge to control consistency, so that measurements are applied with appropriate levels of rigour and achieve comparable results, wherever and whenever they are performed. In this paper, a framework is outlined in which consistency is controlled through measurement strategy, such as the number and location of measurement points. The framework is put to action in a case study, demonstrating the usefulness of the approach and highlighting the dangers of imposing rigid measurement strategies across the supply chain, even if linked to standardised manufacturing processes. Potential mitigations, and the requirements for future research, are outlined.

Estimation of uncertainty in three-dimensional coordinate measurement by comparison with calibrated points

This paper details a method of estimating the uncertainty of dimensional measurement for a three-dimensional coordinate measurement machine. An experimental procedure was developed to compare three-dimensional coordinate measurements with calibrated reference points. The reference standard used to calibrate these reference points was a fringe counting interferometer with a multilateration-like technique employed to establish three-dimensional coordinates. This is an extension of the established technique of comparing measured lengths with calibrated lengths. Specifically a distributed coordinate measurement device was tested which consisted of a network of Rotary-Laser Automatic Theodolites (R-LATs), this system is known commercially as indoor GPS (iGPS). The method was found to be practical and was used to estimate that the uncertainty of measurement for the basic iGPS system is approximately 1 mm at a 95% confidence level throughout a measurement volume of approximately 10 m × 10 m × 1.5 m. © 2010 IOP Publishing Ltd.