In additive component model estimation problems, the number of additive components (model order) and values of the model parameters in each of the additive components are estimated. Traditional methods typically estimate parameters for a set of models with fixed order; parameter estimation is performed over a continuous space when parameters are not discrete. The model order is estimated as the minimizer, over the set of fixed model orders, of a cost function compromising between signal fit to measurements and model complexity.
This dissertation explores dictionary-based estimation methods for joint model order and parameter estimation. In dictionary estimation, the continuous parameter space is discretized, forming a dictionary. Each column of the dictionary is a model component at a sampled parameter value, and a linear combination of a subset of columns is used to represent the model. It is assumed that the model consists of a small number of components, and a sparse reconstruction algorithm is used to select a sparse superposition of columns to represent the signal. The number of columns selected is the estimated model order, and the parameters of each column are the parameter estimates.
We examine both static and dynamic dictionary-based estimation methods. In static estimation, the dictionary is fixed, while in dynamic estimation, dictionary parameters adapt to the data. We propose two new dynamic dictionary-based estimation algorithms and examine the performance of both static and dynamic algorithms in terms of model order probability and parameter estimation error when dictionaries are highly correlated. Highly correlated dictionaries arise from using closely spaced parameter samples in dictionary formation; we propose a method for selecting algorithm settings based on an information criterion. We show the following results: 1) dictionary-based estimation methods are capable of performance comparable to the Cramer Rao lower bound and to traditional benchmark estimation algorithms over a wide range of signal-to-noise ratios; 2) in the complex exponential model, dictionary-based estimation can superresolve closely spaced frequencies, and 3) dynamic dictionary methods overcome parameter estimation bias caused by quantization error in static dictionary-based estimation.
We apply dictionary-based estimation to the problem of 3D synthetic aperture radar (SAR) imaging. Traditional 3D SAR image formation requires collection of data over a large contiguous sector of azimuth-elevation aspect angles; this collection is difficult or impossible to obtain in practice. We show that dictionary-based estimation can be used to produce well-resolved, wide-angle 3D SAR images from sparse, irregular flight paths.