Image reconstruction, in common with many other inverse problems, is often mathematically ill-posed in the sense that solutions are neither stable nor unique. Ultrasonic image reconstruction is particularly notorious in this regard, with narrow transducer bandwidths and limited - sometimes sparsely sampled apertures posing formidable difficulties for conventional signal processing. To overcome these difficulties, some form of regularisation is mandatory, whereby the ill-posed problem is restated as a closely related, well-posed problem, and then solved uniquely. This thesis explores the application of maximum entropy (MaxEnt) regularisation to the problem of reconstructing complex-valued imagery from sparsely sampled coherent ultrasonic field data, with particular emphasis on three-dimensional problems in the non-destructive evaluation (NDE) of materials. MaxEnt has not previously been applied to this class of problem, and yet in comparison with many other approaches to image reconstruction, it emerges as the clear leader in terms of resolution and overall image quality. To account for this performance, it is argued that the default image model used with MaxEnt is particularly meaningful in cases of ultrasonic scattering by objects embedded in homogeneous media. To establish physical and mathematical insights into the forward problem, linear equations describing scattering from both penetrable and impenetrable objects are first derived using the Born and physical optics approximations respectively. These equations are then expressed as a shift-invariant computational model that explicitly incorporates sparse sampling. To validate this model, time-domain scattering responses are computed and compared with analytical solutions for a simple canonical test case drawn from the field of NDE. The responses computed via the numerical model are shown to accurately reproduce the analytical responses. To solve inverse scattering problems via MaxEnt, the robust Cambridge algorithm is generalised to the complex domain and extended to handle broadband (multiple-frequency) data. Two versions of the augmented algorithm are then compared with a range of other algorithms, including several linearly regularised algorithms and lastly, due to its acknowledged status as a competitor with MaxEnt in radio-astronomy, the non-linear CLEAN algorithm. These comparisons are made through simulated 3-D imaging experiments under conditions of both complete and sparse aperture sampling with low and high levels of additive Gaussian noise. As required in any investigation of inverse problems, the experimental confirmation of algorithmic performance is emphasised, and two common imaging geometries relevant to NDE are selected for this purpose. In monostatic synthetic aperture imaging experiments involving side-drilled holes in an aluminium plate and test objects immersed in H2O, MaxEnt image reconstruction is demonstrated to be robust against grating-lobe and side-lobe formation, in addition to temporal bandwidth restriction. This enables efficient reconstruction of 2-D and 3-D images from small numbers of discrete samples in the spatial and frequency domains. The thesis concludes with a description of the design and testing of a novel polyvinylidene fluoride (PVDF) bistatic array transducer that offers advantages over conventional point-sampled arrays in terms of construction simplicity and signal-to-noise ratio. This ultra-sparse orthogonal array is the only one of its kind yet demonstrated, and was made possible by MaxEnt signal processing.
| Identifer | oai:union.ndltd.org:ADTP/220811 |
| Date | January 1999 |
| Creators | Battle, David John |
| Publisher | University of Sydney. Electrical Engineering |
| Source Sets | Australiasian Digital Theses Program |
| Language | English, en_AU |
| Detected Language | English |
| Rights | Copyright Battle, David John;http://www.library.usyd.edu.au/copyright.html |
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