A framework of incorporating variance estimates into wavelet analysis in PET metabolism and receptor studies

Abstract

Recently, methods have been developed to estimate the variance of parameter estimates derived using wavelet methods 1. Previous methods of analysis in the wavelet domain 2, 3, 4 had only yielded estimates of the parameters after transformation back into the image domain. The development of methods which return the additional variance information present the opportunity for further analyses which incorporate the attractive properties of wavelet based methods. The underlying principle of the method of variance estimation is that whilst the variance parameters from wavelet space cannot be easily transformed back into image space, as they are a matrix (2nd order) as opposed to the intrinsic vector (first order) of the parameter estimates, there do exist first order estimates associated with the variance which can easily be transformed, namely the residuals. Thus by transforming back the residuals subject to the equivalent shrinkage of that of the parameter estimates, it is possible to determine the variance of the parameter estimates in the image domain. These methods have been derived and validated on linear models, and linearly transformed data. The figure 1 shows one such PET FDG study and its error variance after thresholding in the wavelet domain. The ability to use these methods with common non-linear PET compartmental models will be discussed further in the abstract by Su, HR et al. In neuropsychology, simple association has been considered the lowest level of evidence for functional segregation, the higher levels being respectively dissociation and double dissociation. The determination of dissociation requires the within-brain comparison of task related effects between two or more locations. Jernigan et al 5 suggested the use of t-statistics for point-wise comparisons between brain locations to assess dissociation. However, statistical maps are characterized by high noise levels and require regularization to allow a detailed assessment of regional variability. Wavelet methods are natural candidates for the regularization task. They provide a parsimonious multiresolution characterization of the signal and efficiently whiten the correlated noise processes of functional images. With the introduction of the error estimates for wavelet methods, dissociation investigation now becomes possible. The error variances allow surrogates for t-value maps to be derived. However, not only is it of importance to know the error variances but it is also of interest to characterise the underlying distribution with a view to integration of the methodology into the hypothesis testing framework. The error estimates are the first step in a characterisation of these distributions which can in turn be used to perform hypothesis testing.