Wavelet-based multiscale stochastic models for efficient tomographic discrimination of fractal fields

Proposes a technique for discrimination of fractal fields with different fractal dimensions, directly from the noisy and sparse tomographic projection data. This application is motivated from the medical field, where a change in fractal dimension is used to differentiate between normal and abnormal conditions in many different contexts, including diagnosis of liver abnormalities. The conventional method for discrimination of fractal fields from tomographic data is based on the calculation of the slope of the power spectra of the corresponding projections. This method, derived from the Radon transform results, breaks down in case the projection data are sparse and/or noisy. In order to avoid any restrictions on the duality and quantity of the projection data, we formulate our discrimination problem in a discrete hypothesis testing framework, the solution to which is given by the maximum-log-likelihood discrimination rule. The problem of discriminating fractal fields through likelihood calculations is, however, complicated by the fact that inverses and determinants of large, full, and generally ill conditioned fractal-field data covariance matrices are required. We show that these complications in the likelihood calculations can be removed by a transformation to the multiscale framework. The multiscale data covariance matrices are sparse and in addition, are naturally partitioned into ill conditioned coarsest scale approximation blocks and relatively well conditioned multiscale detail blocks. We simplify our likelihood calculations by using the class of multiscale stochastic models defined on trees to realize accurate approximations of the detail block of the data covariance matrices