Signal detection in medical imaging

Abstract

The goal of this research is to develop computational methods for predicting how a given medical imaging system and reconstruction algorithm will perform when mathematical observers for tumor detection use the resulting images. Here the mathematical observer is the ideal observer, which sets an upper limit to the performance as measured by the Bayesian risk or receiver operating characteristic analysis. This dissertation concentrates on constructing the ideal observer in complex detection problems and estimating its performance. Thus the methods reported in this dissertation can be used to approximate the ideal observer in real medical images. We define our detection problem as a two-hypothesis detection task where a known signal is superimposed on a random background with complicated distributions and embedded in independent Poisson noise. The first challenge of this detection problem is that the distribution of the random background is usually unknown and difficult to estimate. The second challenge is that the calculation of the ideal observer is computationally intensive for non stylized problems. In order to solve these two problems, our work relies on multiresolution analysis of images. The multiresolution analysis is achieved by decomposing an image into a set of spatial frequency bandpass images so each bandpass image represents information about a particular fitness of detail or scale. Connected with this method, we will use three types of image representation by invertible linear transforms. They are the orthogonal wavelet transform, pyramid transform and independent component analysis. Based on the findings from human and mammalian vision, we can model textures by using marginal densities of a set of spatial frequency bandpass images. In order to estimate the distribution of an ensemble of images given the empirical marginal distributions of filter responses, we can use the maximum entropy principle and get a unique solution. We find that the ideal observer calculates a posterior mean of the ratio of conditional density functions, or the posterior mean of the ratio of two prior density functions, both of which are high dimensional integrals and have no analytic solution usually. But there are two ways to approximate the ideal observer. The first one is a classic decision process; that is, we construct a classifier following feature extraction steps. We use the integrand of the posterior mean as features, which are calculated at the estimated background close to the posterior mode. The classifier combines these features to approximate the integral (or the ideal observer). Finally, if we know both the conditional density function and the prior density function then we can also approximate the high dimensional integral by Monte Carlo integration methods. Since the calculation of the posterior mean is usually a very high dimensional integration problem, we must construct a Markov chain, which can explore the posterior distribution efficiently. We will give two proposal functions. The first proposal function is the likelihood function of random backgrounds. The second method makes use of the multiresolution representation of the image by decomposing the image into a set of spatial frequency bands. Sampling one pixel in each band equivalently updates a cluster of pixels in the neighborhood of the pixel location in the original image.