Polarisation diversity of conformal arrays based on geometric algebra via convex optimisation

A simple and general design procedure is presented for the polarisation diversity of arbitrary conformal arrays; this procedure is based on the mathematical framework of geometric algebra and can be solved optimally using convex optimisation. Aside from being simpler and more direct than other derivations in the literature, this derivation is also entirely general in that it expresses the transformations in terms of rotors in geometric algebra which can easily be formulated for any arbitrary conformal array geometry. Convex optimisation has a number of advantages; solvers are widespread and freely available, the process generally requires a small number of iterations and a wide variety of constraints can be readily incorporated. The study outlines a two-step approach for addressing polarisation diversity in arbitrary conformal arrays: first, the authors obtain the array polarisation patterns using geometric algebra and secondly use a convex optimisation approach to find the optimal weights for the polarisation diversity problem. The versatility of this approach is illustrated via simulations of a 7×10 cylindrical conformal array. © 2012 The Institution of Engineering and Technology.

Direction and polarisation estimation using polarised cylindrical conformal arrays

In this study, the authors describe two-dimensional direction finding and signal polarisation estimation from a cylindrical conformal array consisting of directional and polarised antenna elements. Firstly, a simple and general transformation procedure, based on the mathematical framework of geometric algebra, is presented for arbitrary conformal arrays with polarised and directional antennas. Subsequently, the authors utilise the symmetry of cylindrical arrays to estimate signal parameters via rotational invariance techniques. The authors show how to iteratively estimate the azimuth and elevation angles of the incident signal, as well as its polarisation. To illustrate the versatility of this method, the results of simulations on a 3×4 cylindrical conformal array are shown and discussed. © 2012 The Institution of Engineering and Technology.

Heavy-ion conformal irradiation in the shallow-seated tumor therapy terminal at HIRFL

Basic research related to heavy-ion cancer therapy has been done at the Institute of Modern Physics (IMP), Chinese Academy of Sciences since 1995. Now a plan of clinical trial with heavy ions has been launched at IMP. First, superficially placed tumor treatment with heavy ions is expected in the therapy terminal at the Heavy Ion Research Facility in Lanzhou (HIRFL), where carbon ion beams with energy up to 100 MeV/u can be supplied. The shallow-seated tumor therapy terminal at HIRFL is equipped with a passive beam delivery system including two orthogonal dipole magnets, which continuously scan pencil beams laterally and generate a broad and uniform irradiation field, a motor-driven energy degrader and a multi-leaf collimator. Two different types of range modulator, ripple filter and ridge filter with which Guassian-shaped physical dose and uniform biological effective dose Bragg peaks can be shaped for therapeutic ion beams respectively, have been designed and manufactured. Therefore, two-dimensional and three-dimensional conformal irradiations to tumors can be performed with the passive beam delivery system at the earlier therapy terminal. Both the conformal irradiation methods have been verified experimentally and carbon-ion conformal irradiations to patients with superficially placed tumors have been carried out at HIRFL since November 2006.

Conformal irradiation to moving targets in heavy ion radiotherapy with raster-scanning beam delivery system: (II) feasibility experiments

Within the framework of the pilot heavy-ion therapy facility at GSI equipped with an active beam delivery system of advanced raster scanning technique, a feasibility study on actively conformal heavy-ion irradiation to moving tumors has been experimentally conducted. Laterally, real-time corrections to the beam scanning parameters by the raster scanner, leading to an active beam tracing, compensate for the lateral motion of a target volume. Longitudinally, a mechanically driven wedge energy degrader (called depth scanner) is applied to adjust the beam energy so as to locate the high-dose Bragg peak of heavy ion beam to the slice under treatment for the moving target volume. It has been experimentally shown that compensations for lateral target motion by the raster scanner and longitudinal target shift by the depth scanner are feasible.

Non-relativistic conformal symmetries in fluid mechanics

The symmetries of a free incompressible fluid span the Galilei group, augmented with independent dilations of space and time. When the fluid is compressible, the symmetry is enlarged to the expanded Schrodinger group, which also involves, in addition, Schrodinger expansions. While incompressible fluid dynamics can be derived as an appropriate non-relativistic limit of a conformally invariant relativistic theory, the recently discussed conformal Galilei group, obtained by contraction from the relativistic conformal group, is not a symmetry. This is explained by the subtleties of the non-relativistic limit.

Estimation of individual phases from the four-phase structure invariants in the single isomorphous replacement case

The probability distribution of the four-phase invariants in the case of single isomorphous replacement has been developed to estimate some individual phases. An example of its application to obtain the phases having special values of 0, pi or +/-pi /2 is given for a known protein structure in space group P2(1)2(1)2(1). The phasing procedure includes the determination of starting phases and an iterative calculation. The initial values of starting phases, which are required by the formula, can be obtained from the estimate of one-phase seminvariants and by specifying the origin and enantiomorph. In addition, the calculations lead to two sets of possible phases for each type of reflection by assigning arbitrarily an initial phase value. The present method provides a possibility for the multisolution technique to increase greatly the number of known phases while keeping the number of the trials quite small.

Probability distribution of the four-phase structure invariants of isomorphously related structure factors and its applications

The probability distribution of the four-phase structure invariants (4PSIs) involving four pairs of structure factors is derived by integrating the direct methods with isomorphous replacement (IR). A simple expression of the reliability parameter for 16 types of invariant is given in the case of a native protein and a heavy-atom derivative. Test calculations on a protein and its heavy-atom derivative using experimental diffraction data show that the reliability for 4PSI estimates is comparable with that for the three-phase structure invariants (3PSIs), and that a large-modulus invariants method can be used to improve the accuracy.

General expression for probabilistic estimation of multiphase structure invariants in the case of a native protein and multiple derivatives. Application to estimates of the three-phase structure invariants

Concise probabilistic formulae with definite crystallographic implications are obtained from the distribution for eight three-phase structure invariants (3PSIs) in the case of a native protein and a heavy-atom derivative [Hauptman (1982). Acta Cryst. A38, 289-294] and from the distribution for 27 3PSIs in the case of a native and two derivatives [Fortier, Weeks & Hauptman (1984). Acta Cryst. A40, 646-651]. The main results of the probabilistic formulae for the four-phase structure invariants are presented and compared with those for the 3PSIs. The analysis directly leads to a general formula of probabilistic estimation for the n-phase structure invariants in the case of a native and m derivatives. The factors affecting the estimated accuracy of the 3PSIs are examined using the diffraction data from a moderate-sized protein. A method to estimate a set of the large-modulus invariants, each corresponding to one of the eight 3PSIs, that has the largest \Delta\ values and relatively large structure-factor moduli between the native and derivative is suggested, which remarkably improves the accuracy, and thus a phasing procedure making full use of all eight 3PSIs is proposed.

Direct Computation of 3D Shape Invariants and the Focus of Expansion

Structure from motion often refers to the computation of 3D structure from a matched sequence of images. However, a depth map of a surface is difficult to compute and may not be a good representation for storage and recognition. Given matched images, I will first show that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple difference of slopes of line-segments in one image. Using this result, local surface patches can be classified as convex, concave, parabolic (cylindrical), hyperbolic (saddle point) or planar. At the same time the translational component of the optical flow is obtained, from which the focus of expansion can be computed.

Recognition of Topological Invariants by Iterative Arrays

A study is made of the recognition and transformation of figures by iterative arrays of finite state automata. A figure is a finite rectangular two-dimensional array of symbols. The iterative arrays considered are also finite, rectangular, and two-dimensional. The automata comprising any given array are called cells and are assumed to be isomorphic and to operate synchronously with the state of a cell at time t+1 being a function of the states of it and its four nearest neighbors at time t. At time t=0 each cell is placed in one of a fixed number of initial states. The pattern of initial states thus introduced represents the figure to be processed. The resulting sequence of array states represents a computation based on the input figure. If one waits for a specially designated cell to indicate acceptance or rejection of the figure, the array is said to be working on a recognition problem. If one waits for the array to come to a stable configuration representing an output figure, the array is said to be working on a transformation problem.

A comparison of conformal and intensity-modulated techniques for oesophageal radiotherapy.

Background and purpose: To investigate the potential of intensity-modulated radiotherapy (IMRT) to reduce lung irradiation in the treatment of oesophageal carcinoma with radical radiotherapy.Materials and methods: A treatment planning study was performed to compare two-phase conformal radiotherapy (CFRT) with IMRT in five patients. The CFRT plans consisted of anterior, posterior and bilateral posterior oblique fields, while the IMRT plans consisted of either nine equispaced fields (9F), or four fields (4F) with orientations equal to the CFRT plans. IMRT plans with seven, five or three equispaced fields were also investigated in one patient. Treatment plans were compared using dose-volume histograms and normal tissue complication probabilities.Results: The 9F IMRT plan was unable to improve on the homogeneity of dose to the planning target volume (PTV), compared with the CFRT plan (dose range, 16.9+/-4.5 (1 SD) vs. 12.4+/-3.9%; P=0.06). Similarly, the 9F IMRT plan was unable to reduce the mean lung dose (11.7+/-3.2 vs. 11.0+/-2.9 Gy; P=0.2). Similar results were obtained for seven, five and three equispaced fields in the single patient studied. The 4F IMRT plan provided comparable PTV dose homogeneity with the CFRT plan (11.8+/-3.3 vs. 12.4+/-3.9%; P=0.6), with reduced mean lung dose (9.5+/-2.3 vs 11.0+/-2.9 Gy; P=0.001).Conclusions: IMRT using nine equispaced fields provided no improvement over CFRT. This was because the larger number of fields in the IMRT plan distributed a low dose over the entire lung. In contrast, IMRT using four fields equal to the CFRT fields offered an improvement in lung sparing. Thus, IMRT with a few carefully chosen field directions may lead to a modest reduction in pneumonitis, or allow tumour dose escalation within the currently accepted lung toxicity.