Projected gradient techniques

Ramanathan and Ackerman (1999) have shown that solid-state 31P NMR imaging can be used to measure quantitatively the mass of hydroxyapatite in the presence of bone hence to follow non-invasively the resorption and remodeling of calcium phosphate implants in vivo. A three-dimensional projection reconstruction technique has been used to record NMR images in the presence of a fixed amplitude field gradient, the direction of which was varied uniformly over the unit sphere. Chemical selection was achieved using differences in T1 relaxation time of neighbouring protons as the synthetic hydroxyapatite has a shorter T1 (1.8 s at 4.7 T) compared to bone (approximately 15 s at 4.7 T in vivo, 42 s ex vivo). The experimental results demonstrated that a linear relationship exists between image intensity and HAp density. 

It should be noted that the force constant matrix can be calculated at any geometry, but the transformation to nonnal coordinates is only valid at a stationary point, i.e. where the first derivative is zero. At a non-stationary geometry, a set of 3A—7 generalized frequencies may be defined by removing the gradient direction from the force constant matrix (for example by projection techniques, eq. (13.17)) before transformation to normal coordinates. 

To create a two-dimensional image, two gradients are applied along the X- and y-directions, and a series of one-dimensional images recorded in different directions in the xy-plane. A technique known as back-projection... 

Accountability of the variances and covariances of the responses makes this optimization procedure particularly noteworthy. From the formulator s viewpoint, the distance criterion could lead to an unacceptable optimum if the formulation levels at the optimum produce response values in an undesirable property range. Khuri and Conlon mentioned the possible use of this procedure for multiresponse mixture optimization although no elaboration or examples were given. The reliance of this method on the gradient projection technique could present difficulties with component level compensation if applied to formulations. 

The economics of PRO systems using brines and fresh water sources and current membranes are more favorable, with estimated power outputs as high as 200 watt/m. However, surface brines exist in deserts where there is limited fresh water, and brines that might be produced from salt domes pose a difficult effluent disposal problem. If PRO systems can be produced at an installed cost of 100/m2 of membrane, the projected economics are competitive with other power-generating techniques. This appears to be the only salinity gradient resource worthy of further study. 

A novel gradient-based optimisation framework for large-scale steady-state input/output simulators is presented. The method uses only low-dimensional Jacobian and reduced Hessian matrices calculated through on-line model-reduction techniques. The typically low-dimensional dominant system subspaces are adaptively computed using efficient subspace iterations. The corresponding low-dimensional Jacobians are constructed through a few numerical perturbations. Reduced Hessian matrices are computed numerically from a 2-step projection, firstly onto the dominant system subspace and secondly onto the subspace of the (few) degrees of freedom. The tubular reactor which is known to exhibit a rich parametric behaviour is used as an illustrative example. 

Fig. 6.1.4 Gradient paths for 3D reconstruction from projections. Only half a hemisphere is covered by the gradient paths, because signal for negative gradient values can be acquired by time inversion in echo techniques, (a) 3D space can be covered by a set of 2D projections, so that the 2D algorithm can be applied in two steps, (b) Optimization of the point density in 3D k space requires an integral approach to 3D reconstruction from projections. Adapted from [Lail] with permission from Institute of Physics.

Different approaches can be taken to obtain radial images. Radial field gradients can be applied by the use of dedicated hardware [Hakl, Leel, Lee2]. Alternatively, a 2D image can be reconstructed from one projection by the backprojection technique, and a radial cross-section can be taken through it. The most direct way to access the radial image from a projection consists in computing the inverse Hankel transformation (cf. Section 4.4.2) of the FID measured in Cartesian k space (cf. Fig. 4.4.1) [Majl]. But in practice, the equivalent route via Fourier transformation of the FID and subsequent inverse Abel transformation (cf. Section 4.4.3) is preferred because established phase and baseline correction routines can be used in the calculation of the projection as an intermediate result. 

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