Statistical approaches in quantitative positron emissiontomography
Statistics and Computing
Stochastic Approximation and Rate-Distortion Analysis for Robust Structure and Motion Estimation
International Journal of Computer Vision
High-Resolution Adaptive PET Imaging
IPMI '09 Proceedings of the 21st International Conference on Information Processing in Medical Imaging
Spatial resolution and noise properties of regularized motion-compensated image reconstruction
ISBI'09 Proceedings of the Sixth IEEE international conference on Symposium on Biomedical Imaging: From Nano to Macro
Bounds on the minimizers of (nonconvex) regularized least-squares
SSVM'07 Proceedings of the 1st international conference on Scale space and variational methods in computer vision
Covariance estimation for SAD block matching
SCIA'07 Proceedings of the 15th Scandinavian conference on Image analysis
Indirect density estimation using the iterative Bayes algorithm
Computational Statistics & Data Analysis
Propagation of blood function errors to the estimates of kinetic parameters with dynamic PET
Journal of Biomedical Imaging - Special issue on modern mathematics in biomedical imaging
Computational Statistics & Data Analysis
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Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore, investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators. This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters. We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics. We also describe a “plug-in” approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement. The approximations should be useful in a wide range of estimation problems