Linear Back Projection Research Articles

Optical sensor configurations for process tomography

This paper describes an investigation into the optimum design of optical fibre sensing arrays to be incorporated in an optical tomographic measurement system for on-line monitoring of particles and droplets. Two approaches are considered to cover opaque and transparent materials; optical path length and optical attenuation. Four flow models are investigated: single-pixel flow representing a single particle or droplet, two-pixel flow as a simple check on aliasing in the reconstructed image, half flow representing half the sensing cross section filled with material and full flow, where the whole sensing cross section is full of material. Six projection geometries of the fibre sensors are considered. For tomographic imaging, the forward problem, which assumes particles are placed in specific places in the measurement cross section and calculates voltage outputs for the individual sensors, is modelled. The solutions from the forward problem are used to solve the inverse problem, which uses actual sensor voltage readings to estimate the spatial distribution of the material in the measurement cross section. The solution of the inverse problem is used to derive the linear back projection (LBP) and filtered LBP algorithms. In order to improve image quality, a hybrid reconstruction algorithm is implemented. This algorithm first checks if any sensors read zero and sets (locks for this estimation) all pixels associated with them to zero (no material). The algorithm then proceeds as for the LBP.

An image-reconstruction algorithm based on Landweber's iteration method for electrical-capacitance tomography

Electrical capacitance tomography (ECT) is a so-called `soft-field' tomography technique. The linear back-projection (LBP) method is used widely for image reconstruction in ECT systems. It is numerically simple and computationally fast because it involves only a single matrix-vector multiplication. However, the images produced by the LBP algorithm are generally qualitative rather than quantitative. This paper presents an image-reconstruction algorithm based on a modified Landweber iteration method that can greatly enhance the quality of the image when two distinct phases are present. In this algorithm a simple constraint is used as a regularization for computing a stabilized solution, with a better immunity to noise and faster convergence. Experimental results are presented.

Electrical capacitance tomography with square sensor

Electrical capacitance tomography (ECT) with circular sensors has previously been investigated. For some industrial applications such as circulating fluidised beds, square sensors are required. Research into this specific area has been carried out for the first time. To generate sensitivity maps, the Laplace equation is solved using a finite difference method. Both the linear back-projection algorithm and an iterative algorithm have been implemented for image reconstruction. Experimental results are promising.

PNEUMATIC CONVEYING SYSTEMS IN DENSE PHASE

A gamma ray tomograph has been designed for the study of the modes of flow in dense phase pneumatic pipelines. The instrument has six americium-241 gamma ray sources and thirty miniature sodium iodide detectors arranged symmetrically around a pipe section. The instrumentation system comprises a multi-channel preamplifier, a CAMAC crate containing multi-channel discriminators, scalers and a GPIB crate controller, and an IBM-compatible personal computer. The detector gains are individually controlled by analogue voltage outputs provided by multi-channel DAC interface cards. The detector count-rates are used with a filtered linear back-projection algorithm to estimate the distribution of solid material over the pipe cross-section at a rate of ten images per second in real time. Preliminary laboratory tests have been carried out on a fluidised bed formed inside a permeameter tube and a full-scale test pipeline and instrument room have been developed for conveying tests.

Measurement of sensitivity distributions of capacitance tomography sensors

Currently most electrical capacitance tomography systems employ the linear back-projection algorithm to reconstruct cross sectional images, the processing of which requires sensitivity maps for all electrode pairs. Usually these sensitivity maps are obtained by finite element analysis with accuracy limited by available data on sensor geometry. Alternatively they can be more accurately obtained by physical measurements so that the imaging accuracy can be improved. A test rig has been built at UMIST for this purpose. It can be controlled manually or by a computer. The experimental result shows the feasibility.

The use of entropic thresholding methods in reconstruction of capacitance tomography data

High-quality reconstruction of capacitance tomography data is of vital importance to ensure quantitative information from such systems. Several methods have been tested, such as linear back projection (LBP), representing a fast and crude method, and model-based reconstruction (MOR) which significantly improves the results, although at higher computational cost. The main problem of using LBP is the smoothing effect on sharp transitions in the dielectric constant distribution. The reconstruction quality could be improved by including a thresholding procedure in the reconstruction process. In this work, entropic thresholding methods are used. These methods are compared to previously published thresholding techniques such as Xie's method and an implicit model-based reconstruction method. A computer simulated eight-electrode capacitance tomography system has been used for the evaluation.

Non-recursive linear algorithms for optical imaging in diffusive media.

Optical imaging has been used to image phantoms, animals, and humans. It offers the potential for the production of functional images of tissues, such as oxygenation of brain during stroke. Fast algorithms are needed to allow diagnostically useful images to be generated under realistic conditions, including the likelihood that transmission geometries will not be possible. We proposed a linear algorithm, while less than ideal, may allow rapid reconstruction of images and avoid the pitfalls of recursive, nonlinear solutions. Such techniques may also facilitate the use of varied but physiologic imaging geometries. We found that linear backprojection tomography is feasible for clinical use. Conversion of the present mechanically scanning device to a clinical scanner should be possible with retention of the current processing algorithms. Such a clinical scanner should ultimately be able to generate images in less than one minute with centimeter resolution at the center of living human brain.

A new reconstruction algorithm for process tomography

In this work a new reconstruction algorithm for use with oil/gas pipe flow imaging has been developed. Accurate images of representative oil/gas distributions (that is, flow regimes) occurring in flow pipes has been obtained. A capacitance sensor system has been used. In the algorithm the oil/gas distribution has been represented by a set of parameters describing the oil/gas interface. A finite-element-based mathematical simulator of the multi-electrode capacitance sensor system has been developed. The simulator is capable of calculating the capacitances for a set of input parameters (namely for a given oil/gas distribution). The reconstruction algorithm calculates the image by finding the parameters that give fewest discrepancies between calculated and measured capacitances, using an optimization routine. All regimes tested in this work have been successfully reconstructed with the new algorithm, and the reconstructions are compared with images produced by the well-known linear back projection algorithm. The new algorithm has been tested using data from a capacitance imaging system; however, it can in principle be used with other imaging techniques.

A fast algorithm for backprojection with linear interpolation

In the filtered backprojection procedure for image reconstruction from projections, backprojection dominates the computation time. A simple algorithm that reduces the number of multiplications in linear interpolation and backprojection stage by 50%, with a small increase in the number of additions, is proposed. The algorithm performs the interpolation and backprojection of four views together. Examples of implementation are given and extension to interpolation of more than four views is discussed.

8-electrode capacitance system for two-component flow identification. Part 1: Tomographic flow imaging

An 8-electrode capacitance system for tomographic imaging of two-component flows has been designed based upon a 2-dimensional finite element model. Emphasis is made on the development of a linear back-projection image reconstruction algorithm. Some reconstructed images obtained using this algorithm are given. Limitations and possible future improvements of the system are discussed.

Tomographic imaging of two-component flow using capacitance sensors

Describes a tomographic flow imaging system based on capacitance sensors. It uses eight capacitance electrodes to perform a 'body scan' of the fluid-conveying pipe, and a linear back projection algorithm is developed to reconstruct the cross-sectional image of the two-component flow from the measured capacitance values. A prototype system has been successfully assembled and reconstructed images of static distribution models are presented. Possible industrial applications and practical limitations of the system are discussed.