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This thesis is an investigation of methods for processing multidimensional signals acquired using modern tomography systems that have an anisotropic or spatially variant response function. The main result of this research is the discovery of a new method to obtain better estimators of an unknown spatial intensity distribution by incorporating detailed knowledge about the tomograph system response function and statistical properties of the acquired signal into a mathematical model. In this model, data acquisition is represented by a linear transformation, $F\sb{\theta k}$, of functions, b, on a continuous domain object space to functions, $p\sb{\theta k}$, on a discrete domain observation space as described by $p\sb{\theta k}$ = $F\sb{\theta k}\cdot b$ = $\int\sb{\IR}dy\int\sb{\IR}dx\ f\sb{\theta k}(x,y)b(x,y)$. This data acquisition model and the singular value decomposition, $F\sb{\theta k} = U\sb{\theta k}S\cdot V\sp{T}$, of this model into the left singular vectors, $U\sb{\theta k}$, right singular functions, V, and the singular value operator, S, form a common, unifying structure for the three areas of this work: identification, estimation, and optimization. For system identification, the model is easily adapted to represent tomography systems that are spatially invariant, spatially variant with symmetry, or spatially variant without symmetry. Least squares, normal maximum likelihood, and Poisson maximum likelihood estimators and the corresponding covariance have been formulated to compensate for all three types of spatial responses. While bias is reduced by these estimators, undesirable statistical and systematic fluctuations can result due to pixelization effects. To reduce these fluctuations new estimators such as the filtered singular value least squares estimator $\ b = V\cdot S\sp{T}D(S\cdot S\sp{T})\sp+U\sp{T}\underline p$ with diagonal filter matrix, D, were developed. These new estimators use an orthonormal pixel basis decomposition of the unknown spatial distribution to eliminate systematic error and minimize statistical errors that occur using square or polar pixel bases. Characterizing the effects which sampling density has on reconstructed image resolution and noise using singular value analysis has led to a method to optimize sampling strategies to obtain better estimates of the unknown spatial distribution given bounds for noise and resolution. Symmetries in the data acquisition process lead to very efficient implementations of the new inverse problem algorithms.

Spatially variant tomographic imaging: Estimation, identification, and optimization