The possibility of improving the generalization capability of a neural network by introducing additive noise to the training samples is discussed. The network considered is a feedforward layered neural network trained with the back-propagation algorithm. Back-propagation training is viewed as nonlinear least-squares regression and the additive noise is interpreted as generating a kernel estimate of the probability density that describes the training vector distribution. Two specific application types are considered: pattern classifier networks and estimation of a nonstochastic mapping from data corrupted by measurement errors. It is not proved that the introduction of additive noise to the training vectors always improves network generalization. However, the analysis suggests mathematically justified rules for choosing the characteristics of noise if additive noise is used in training. Results of mathematical statistics are used to establish various asymptotic consistency results for the proposed method. Numerical simulations support the applicability of the training method.

In x-ray tomography, the structure of a three-dimensional body is reconstructed from a collection of projection images of the body. Medical CT imaging does this using an extensive set of projections from all around the body. However, in many practical imaging situations only a small number of truncated projections are available from a limited angle of view. Three-dimensional imaging using such data is complicated for two reasons: (i) typically, sparse projection data do not contain sufficient information to completely describe the 3D body, and (ii) traditional CT reconstruction algorithms, such as filtered backprojection, do not work well when applied to few irregularly spaced projections. Concerning (i), existing results about the information content of sparse projection data are reviewed and discussed. Concerning (ii), it is shown how Bayesian inversion methods can be used to incorporate a priori information into the reconstruction method, leading to improved image quality over traditional methods. Based on the discussion, a low-dose three-dimensional x-ray imaging modality is described.

Diagnostic and operational tasks in dental radiology often require three-dimensional information that is difficult or impossible to see in a projection image. A CT-scan provides the dentist with comprehensive three-dimensional data. However, often CT-scan is impractical and, instead, only a few projection radiographs with sparsely distributed projection directions are available. Statistical (Bayesian) inversion is well-suited approach for reconstruction from such incomplete data. In statistical inversion, a priori information is used to compensate for the incomplete information of the data. The inverse problem is recast in the form of statistical inference from the posterior probability distribution that is based on statistical models of the projection data and the a priori information of the tissue. In this paper, a statistical model for three-dimensional imaging of dentomaxillofacial structures is proposed. Optimization and MCMC algorithms are implemented for the computation of posterior statistics. Results are given with in vitro projection data that were taken with a commercial intraoral x-ray sensor. Examples include limited-angle tomography and full-angle tomography with sparse projection data. Reconstructions with traditional tomographic reconstruction methods are given as reference for the assessment of the estimates that are based on the statistical model.

Pattern classification using neural networks and statistical methods is discussed. We give a tutorial overview in which popular classifiers are grouped into distinct categories according to their underlying mathematical principles; also, we assess what makes a classifier neural. The overview is complemented by two case studies using handwritten digit and phoneme data that test the performance of a number of most typical neural-network and statistical classifiers. Four methods of our own are included: reduced kernel discriminant analysis, the learning k-nearest neighbors classifier, the averaged learning subspace method, and a version of kernel discriminant analysis.

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