Canonical solution groups of a homomorphism : manipulator kinematics, nonparametric regression and distributed object systems

The kinematic mapping of a rigid open-link manipulator is a homomorphism between Lie groups. The homomorphisrn has solution groups that act on an inverse kinematic solution element. A canonical representation of solution group operators that act on a solution element of three and seven degree-of-freedom (do!) dextrous manipulators is determined by geometric analysis. Seven canonical solution groups are determined for the seven do! Robotics Research K-1207 and Hollerbach arms. The solution element of a dextrous manipulator is a collection of trivial fibre bundles with solution fibres homotopic to the Torus. If fibre solutions are parameterised by a scalar, a direct inverse funct.ion that maps the scalar and Cartesian base space coordinates to solution element fibre coordinates may be defined. A direct inverse pararneterisation of a solution element may be approximated by a local linear map generated by an inverse augmented Jacobian correction of a linear interpolation. The action of canonical solution group operators on a local linear approximation of the solution element of inverse kinematics of dextrous manipulators generates cyclical solutions. The solution representation is proposed as a model of inverse kinematic transformations in primate nervous systems. Simultaneous calibration of a composition of stereo-camera and manipulator kinematic models is under-determined by equi-output parameter groups in the composition of stereo-camera and Denavit Hartenberg (DH) rnodels. An error measure for simultaneous calibration of a composition of models is derived and parameter subsets with no equi-output groups are determined by numerical experiments to simultaneously calibrate the composition of homogeneous or pan-tilt stereo-camera with DH models. For acceleration of exact Newton second-order re-calibration of DH parameters after a sequential calibration of stereo-camera and DH parameters, an optimal numerical evaluation of DH matrix first order and second order error derivatives with respect to a re-calibration error function is derived, implemented and tested. A distributed object environment for point and click image-based tele-command of manipulators and stereo-cameras is specified and implemented that supports rapid prototyping of numerical experiments in distributed system control. The environment is validated by a hierarchical k-fold cross validated calibration to Cartesian space of a radial basis function regression correction of an affine stereo model. Basic design and performance requirements are defined for scalable virtual micro-kernels that broker inter-Java-virtual-machine remote method invocations between components of secure manageable fault-tolerant open distributed agile Total Quality Managed ISO 9000+ conformant Just in Time manufacturing systems.

The development of constrained source localisation algorithms for human brain imaging

This work sets out to evaluate the potential benefits and pit-falls in using a priori information to help solve the Magnetoencephalographic (MEG) inverse problem. In chapter one the forward problem in MEG is introduced, together with a scheme that demonstrates how a priori information can be incorporated into the inverse problem. Chapter two contains a literature review of techniques currently used to solve the inverse problem. Emphasis is put on the kind of a priori information that is used by each of these techniques and the ease with which additional constraints can be applied. The formalism of the FOCUSS algorithm is shown to allow for the incorporation of a priori information in an insightful and straightforward manner. In chapter three it is described how anatomical constraints, in the form of a realistically shaped source space, can be extracted from a subject’s Magnetic Resonance Image (MRI). The use of such constraints relies on accurate co-registration of the MEG and MRI co-ordinate systems. Variations of the two main co-registration approaches, based on fiducial markers or on surface matching, are described and the accuracy and robustness of a surface matching algorithm is evaluated. Figures of merit introduced in chapter four are shown to given insight into the limitations of a typical measurement set-up and potential value of a priori information. It is shown in chapter five that constrained dipole fitting and FOCUSS outperform unconstrained dipole fitting when data with low SNR is used. However, the effect of errors in the constraints can reduce this advantage. Finally, it is demonstrated in chapter six that the results of different localisation techniques give corroborative evidence about the location and activation sequence of the human visual cortical areas underlying the first 125ms of the visual magnetic evoked response recorded with a whole head neuromagnetometer.

Magnetoencephalography:technical improvements in image co-registration and studies of visual cortical oscillations

The work presented in this thesis is divided into two distinct sections. In the first, the functional neuroimaging technique of Magnetoencephalography (MEG) is described and a new technique is introduced for accurate combination of MEG and MRI co-ordinate systems. In the second part of this thesis, MEG and the analysis technique of SAM are used to investigate responses of the visual system in the context of functional specialisation within the visual cortex. In chapter one, the sources of MEG signals are described, followed by a brief description of the necessary instrumentation for accurate MEG recordings. This chapter is concluded by introducing the forward and inverse problems of MEG, techniques to solve the inverse problem, and a comparison of MEG with other neuroimaging techniques. Chapter two provides an important contribution to the field of research with MEG. Firstly, it is described how MEG and MRI co-ordinate systems are combined for localisation and visualisation of activated brain regions. A previously used co-registration methods is then described, and a new technique is introduced. In a series of experiments, it is demonstrated that using fixed fiducial points provides a considerable improvement in the accuracy and reliability of co-registration. Chapter three introduces the visual system starting from the retina and ending with the higher visual rates. The functions of the magnocellular and the parvocellular pathways are described and it is shown how the parallel visual pathways remain segregated throughout the visual system. The structural and functional organisation of the visual cortex is then described. Chapter four presents strong evidence in favour of the link between conscious experience and synchronised brain activity. The spatiotemporal responses of the visual cortex are measured in response to specific gratings. It is shown that stimuli that induce visual discomfort and visual illusions share their physical properties with those that induce highly synchronised gamma frequency oscillations in the primary visual cortex. Finally chapter five is concerned with localization of colour in the visual cortex. In this first ever use of Synthetic Aperture Magnetometry to investigate colour processing in the visual cortex, it is shown that in response to isoluminant chromatic gratings, the highest magnitude of cortical activity arise from area V2.

Neuro-electromagnetic imaging of the human visual cortex

Methods of solving the neuro-electromagnetic inverse problem are examined and developed, with specific reference to the human visual cortex. The anatomy, physiology and function of the human visual system are first reviewed. Mechanisms by which the visual cortex gives rise to external electric and magnetic fields are then discussed, and the forward problem is described mathematically for the case of an isotropic, piecewise homogeneous volume conductor, and then for an anisotropic, concentric, spherical volume conductor. Methods of solving the inverse problem are reviewed, before a new technique is presented. This technique combines prior anatomical information gained from stereotaxic studies, with a probabilistic distributed-source algorithm to yield accurate, realistic inverse solutions. The solution accuracy is enhanced by using both visual evoked electric and magnetic responses simultaneously. The numerical algorithm is then modified to perform equivalent current dipole fitting and minimum norm estimation, and these three techniques are implemented on a transputer array for fast computation. Due to the linear nature of the techniques, they can be executed on up to 22 transputers with close to linear speedup. The latter part of the thesis describes the application of the inverse methods to the analysis of visual evoked electric and magnetic responses. The CIIm peak of the pattern onset evoked magnetic response is deduced to be a product of current flowing away from the surface areas 17, 18 and 19, while the pattern reversal P100m response originates in the same areas, but from oppositely directed current. Cortical retinotopy is examined using sectorial stimuli, the CI and CIm ;peaks of the pattern onset electric and magnetic responses are found to originate from areas V1 and V2 simultaneously, and they therefore do not conform to a simple cruciform model of primary visual cortex.

Effective electromagnetic noise cancellation with beamformers and synthetic gradiometry in shielded and partly shielded environments

The major challenge of MEG, the inverse problem, is to estimate the very weak primary neuronal currents from the measurements of extracranial magnetic fields. The non-uniqueness of this inverse solution is compounded by the fact that MEG signals contain large environmental and physiological noise that further complicates the problem. In this paper, we evaluate the effectiveness of magnetic noise cancellation by synthetic gradiometers and the beamformer analysis method of synthetic aperture magnetometry (SAM) for source localisation in the presence of large stimulus-generated noise. We demonstrate that activation of primary somatosensory cortex can be accurately identified using SAM despite the presence of significant stimulus-related magnetic interference. This interference was generated by a contact heat evoked potential stimulator (CHEPS), recently developed for thermal pain research, but which to date has not been used in a MEG environment. We also show that in a reduced shielding environment the use of higher order synthetic gradiometry is sufficient to obtain signal-to-noise ratios (SNRs) that allow for accurate localisation of cortical sensory function.

Determining planar multiple sound-soft obstacles from scattered acoustic fields

An inverse problem is considered where the structure of multiple sound-soft planar obstacles is to be determined given the direction of the incoming acoustic field and knowledge of the corresponding total field on a curve located outside the obstacles. A local uniqueness result is given for this inverse problem suggesting that the reconstruction can be achieved by a single incident wave. A numerical procedure based on the concept of the topological derivative of an associated cost functional is used to produce images of the obstacles. No a priori assumption about the number of obstacles present is needed. Numerical results are included showing that accurate reconstructions can be obtained and that the proposed method is capable of finding both the shapes and the number of obstacles with one or a few incident waves.

CROW bottles

A coupled resonator optical waveguide (CROW) bottle is a bottle-shaped nonuniform distribution of resonator and coupling parameters. This Letter solves the inverse problem for a CROW bottle, i.e., develops a simple analytical method that determines a CROW with the required group delay and dispersion characteristics. In particular, the parameters of CROWs exhibiting the group delay with zero dispersion (constant group delay) and constant dispersion (linear group delay) are found. © 2014 Optical Society of America.

Source localization methods

One of the most pressing demands on electrophysiology applied to the diagnosis of epilepsy is the non-invasive localization of the neuronal generators responsible for brain electrical and magnetic fields (the so-called inverse problem). These neuronal generators produce primary currents in the brain, which together with passive currents give rise to the EEG signal. Unfortunately, the signal we measure on the scalp surface doesn't directly indicate the location of the active neuronal assemblies. This is the expression of the ambiguity of the underlying static electromagnetic inverse problem, partly due to the relatively limited number of independent measures available. A given electric potential distribution recorded at the scalp can be explained by the activity of infinite different configurations of intracranial sources. In contrast, the forward problem, which consists of computing the potential field at the scalp from known source locations and strengths with known geometry and conductivity properties of the brain and its layers (CSF/meninges, skin and skull), i.e. the head model, has a unique solution. The head models vary from the computationally simpler spherical models (three or four concentric spheres) to the realistic models based on the segmentation of anatomical images obtained using magnetic resonance imaging (MRI). Realistic models – computationally intensive and difficult to implement – can separate different tissues of the head and account for the convoluted geometry of the brain and the significant inter-individual variability. In real-life applications, if the assumptions of the statistical, anatomical or functional properties of the signal and the volume in which it is generated are meaningful, a true three-dimensional tomographic representation of sources of brain electrical activity is possible in spite of the ‘ill-posed’ nature of the inverse problem (Michel et al., 2004). The techniques used to achieve this are now referred to as electrical source imaging (ESI) or magnetic source imaging (MSI). The first issue to influence reconstruction accuracy is spatial sampling, i.e. the number of EEG electrodes. It has been shown that this relationship is not linear, reaching a plateau at about 128 electrodes, provided spatial distribution is uniform. The second factor is related to the different properties of the source localization strategies used with respect to the hypothesized source configuration.

Identification of a time-dependent bio-heat blood perfusion coefficient

In the paper the identification of the time-dependent blood perfusion coefficient is formulated as an inverse problem. The bio-heat conduction problem is transformed into the classical heat conduction problem. Then the transformed inverse problem is solved using the method of fundamental solutions together with the Tikhonov regularization. Some numerical results are presented in order to demonstrate the accuracy and the stability of the proposed meshless numerical algorithm.

Source localization of reaction-diffusion models for brain tumors

We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures.

On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green’s functions, and properties of these equations are shown in an L2-setting. An effective way of discretizing these boundary integral equations based on the Nystr¨om method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.

Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy–Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367–4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in Johansson et al. (2011) and show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.

3D reconstruction of complex dielectric function of the glass during the femtosecond micro-fabrication

We measure complex amplitude of scattered wave in the far field, and justify theoretically and numerically solution of the inverse scattering problem. This allows single-shot reconstructing of dielectric function distribution during direct femtosecond laser micro-fabrication.

3D Reconstruction of the complex dielectric function of glass during femtosecond laser micro-fabrication

We report on a new technique to reconstruct the 3D dielectric function change in transparent dielectric materials and the application of the technique for on-line monitoring of refractive index modification in BK7 glass during direct femtosecond laser microfabrication. The complex optical field scattered from the modified region is measured using two-beam, single-shot interferogram and the distribution of the modified refractive index is reconstructed by numerically solving the inverse scattering problem in Born approximation. The optical configuration suggested is further development of digital holographic microscopy (DHM). It takes advantage of high spatial resolution and almost the same optical paths for both interfering beams, and allows ultrafast time resolution.