This dissertation investigates three topics related to imagereconstruction from list-mode Anger camera data. Our mainfocus is the processing of photomultiplier-tube (PMT)signals directly into images. First we look at the use of list-mode calibration data toreconstruct a non-parametric likelihood model relating theobject to the data list. The reconstructed model can thenbe combined with list-mode object data to produce amaximum-likelihood (ML) reconstruction, an approach we calldouble list-mode reconstruction. This trades off reducedprior assumptions about the properties of the imaging systemfor greatly increased processing time and increaseduncertainty in the reconstruction. Second we use the list-mode expectation-maximization (EM)algorithm to reconstruct planar projection images directlyfrom PMT data. Images reconstructed by EM are compared withimages produced using the faster and more common techniqueof first producing ML position estimates, then histogramingto form an image. A mathematical model of the human visualsystem, the channelized Hotelling observer, is used tocompare the reconstructions by performing the Rayleigh task,a traditional measure of resolution. EM is found to producehigher resolution images than the histogram approach,suggesting that information is lost during the positionestimation step. Finally we investigate which linear parameters of an objectare estimable, in other words may be estimated without biasfrom list-mode data. We extend the notion of a linearsystem operator, familiar from binned-mode systems, tolist-mode systems, and show the estimable parameters aredetermined by the range of the adjoint of the systemoperator. As in the binned-mode case, the list-modesensitivity functions define ``natural pixels'' with whichto reconstruct the object.

In magnetic resonance imaging (MRI), sampling methods that lead to incomplete data coverage of k-space are used to accelerate imaging and reduce overall scan time. Non-Cartesian sampling trajectories such as radial, spiral, and random trajectories are employed to facilitate advanced imaging techniques, such as compressed sensing, or to provide more efficient coverage of k-space for a shorter scan period. When k-space is undersampled or unevenly sampled, traditional methods of transforming Fourier data to obtain the desired image, such as the FFT, may no longer be applicable. The Fourier reconstruction of optical interferometer data (FROID) algorithm is a novel reconstruction method developed by A. R. Hajian that has been successful in the field of optical interferometry in reconstructing images from sparsely and unevenly sampled data. It is applicable to cases where the collected data is a Fourier representation of the desired image or spectrum. The framework presented allows for a priori information, such as the positions of the sampled points, to be incorporated into the reconstruction of images. Initially, FROID assumes a guess of the real-valued spectrum or image in the form of an interpolated function and calculates the corresponding integral Fourier transform. Amplitudes are then sampled in the Fourier space at locations corresponding to the acquired measurements to form a model dataset. The guess spectrum or image is then adjusted such that the model dataset in the Fourier space is least squares fitted to measured values. In this thesis, FROID has been adapted and implemented for use in MRI where k-space is the Fourier transform of the desired image. By forming a continuous mapping of the image and modelling data in the Fourier space, a comparison and optimization with respect to data acquired in k-space that is either undersampled or irregularly sampled can be performed as long as the sampling positions are known. To apply FROID to the reconstruction of magnetic resonance images, an appropriate objective function that expresses the desired least squares fit criteria was defined and the model for interpolating Fourier data was extended to include complex values of an image. When an image with two Gaussian functions was tested, FROID was able to reconstruct images from data randomly sampled in k-space and was not restricted to data sampled evenly on a Cartesian grid. An MR image of a bone with complex values was also reconstructed using FROID and the magnitude image was compared to that reconstructed by the FFT. It was found that FROID outperformed the FFT in certain cases even when data were rectilinearly sampled.

Breast cancer continues to be the most frequently diagnosed cancer in Canadian women. Currently, mammography is the clinically accepted best modality for breast cancer detection and the regular use of screening has been shown to contribute to the reduced mortality. However, mammography suffers from several drawbacks which limit its sensitivity and specificity. As a potential solution, digital breast tomosynthesis (DBT) uses a limited number (typically 10-20) of low-dose x-ray projections to produce a three-dimensional tomographic representation of the breast. The reconstruction of DBT images is challenged by such incomplete sampling. The purpose of this thesis is to evaluate the effect of image acquisition parameters on image quality of DBT for various reconstruction techniques and to optimize these, with three specific goals: A) Develop a better power spectrum estimator for detectability calculation as a task-based image quality index; B) Develop a paired-view algorithm for artifact removal in DBT reconstruction; and C) Increase dose efficiency in DBT by reducing random noise. A better power spectrum estimator was developed using a multitaper technique, which yields reduced bias and variance in estimation compared to the conventional moving average method. This gives us an improved detectability measurement with finer frequency steps. The paired-view scheme in DBT reconstruction provides better image quality than the commonly used sequential method. A simple ordering like the “side-to-side” method can achieve less artifact and higher image quality in reconstructed slices. The new denoising algorithm developed was applied to the projection views acquired in DBT before reconstruction. The random noise was markedly removed while the anatomic details were maintained. With the help of this artifact-removal technique used in reconstruction and the denoising method employed on the projection views, the image quality of DBT is enhanced and lesions should be more readily detectable.

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.

Thesis (Ph.D.)--Rochester Institute of Technology, 2008. / Typescript. Includes bibliographical references (leaves [331]-346).

Thesis (Ph.D.) -- University of Adelaide, School of Computer Science, 2008. / "December, 2007" Bibliography: p. 181-185. Also available in print form.

Thesis (M.S.)--Duquesne University, 2004. / Title from document title page. Abstract included in electronic submission form. Includes bibliographical references (p. 52-53) and abstract.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 86-88). Also available in electronic version. Access restricted to campus users.

Thesis (M.S. in Applied Physics)--Naval Postgraduate School, Dec. 2004. / Thesis Advisor(s): Christopher Olsen. Includes bibliographical references (p. 83-84). Also available online.

Thesis (M. S.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Smith, Mark J. T.; Committee Co-Chair: Mersereau, Russell M.; Committee Member: Anderson, David; Committee Member: Lanterman, Aaron; Committee Member: Rosen, Gail; Committee Member: Wardi, Yorai.