In particulate materials, such as emulsions and granular media, a "jammed" system results if particles are packed together so that all particles are touching their neighbours, provided the density is sufficiently high. This paper studies through experiment, theory and simulation, the forces that particles exert upon one another in such a jammed state. Confocal microscopy of a compressed polydisperse emulsion provides a direct 3D measurement of the dispersed phase morphology within the bulk of the sample. This allows the determination of the probability distribution of interdroplet forces, P(f) where f is the magnitude of the force, from local droplet deformations. In parallel, the simplest form of the Boltzmann equation for the probability of force distributions predicts P(f) to be of the form e(-f/p), where p is proportional to the mean force f for large forces. This result is in good agreement with experimental and simulated data.

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Dimensional inspection of engineering components comprising free-form surfaces demands accurate measurement of a large number of discrete points, such that the actual shape may be fully characterised. This paper presents a methodology for CAD-based measurement of such components using a coordinate measuring machine equipped with a touch-trigger probe. The main shortcomings of the conventional methodology have been identified to be in relation to registration and probe radius compensation. The proposed measurement process involves the following main steps: registration, definition of measurement points, probe path generation, path optimisation and verification, measurement and probe radius compensation. By employing the CAD model at every step, the implemented methodology maximises the measurement accuracy and this is verified through a detailed simulation study. In addition, the implemented tools for CMM programming achieve accurate control of the overall measurement process and provide a high level of confidence when dealing with complex component geometry.

Fast and accurate fitting of non-uniform rational B-spline (NURBS) curves and surfaces through large sets of measured data is an important problem in applications such as reverse engineering and geometric modelling. This paper presents a method for realising significant improvements in the computational efficiency of this task. The basic idea is that the sparsity structures of the relevant matrices that are specific to the problem of NURBS fitting can be precisely defined and that full exploitation of these structures leads to significant savings in both computational and storage requirements. These savings allow for a large number of control points to be used in order to define the surface and consequently to improve the accuracy of shape representation. The achieved computational complexity is linear in both the number of measured points and the number of control points while the storage requirements of the algorithm are linear with the number of control points only. The complexity analysis, as well as the analysis of actual running times is presented. The results demonstrate that, using this approach, highly complex shapes may be modelled accurately with a single NURBS surface.

ObjectA new method for 3D localization of N fiducial markers from 1D projections is presented and analysed. It applies to semi-active markers and active markers using a single receiver channel.Materials and methodsThe novel algorithm computes candidate points using peaks in three optimally selected projections and removes fictitious points by verifying detected peaks in additional projections. Computational complexity was significantly reduced by avoiding cluster analysis, while higher accuracy was achieved by using optimal projections and by applying Gaussian interpolation in peak detection. Computational time, accuracy and robustness were analysed through Monte Carlo simulations and experiments. The method was employed in a prototype MRI guided prostate biopsy system and used in preclinical experiments.ResultsThe computational time for 6 markers was better than 2 ms, an improvement of up to 100 times, compared to the method by Flask et al. (J Magn Reson Imaging 14(5):617–627, 2001). Experimental maximum localization error was lower than 0.3 mm; standard deviation was 0.06 mm. Targeting error was about 1 mm. Tracking update rate was about 10 Hz.ConclusionThe proposed method is particularly suitable in systems requiring any of the following: high frame rate, tracking of three or more markers, data filtering or interleaving.