Stochastic multi-scale selection of the stopping Nicolai Bissantz - PowerPoint PPT Presentation

Stochastic multi-scale selection of the stopping criterion for MLEM reconstructions in PET Stochastic multi-scale selection of the stopping Nicolai Bissantz criterion for MLEM reconstructions in PET proudly presented by B. Mair and A. Munk Overview Positron Emission Nicolai Bissantz Tomography Image reconstruction methods for PET data proudly presented by B. Mair and A. Munk Model selection A multi-scale stopping rule Ruhr-Universit¨ at Bochum, University of Florida, Georg August Universit¨ at G¨ ottingen Simulations Conclusions References Linz, October 29 th , 2008

Stochastic multi-scale Overview selection of the stopping criterion for MLEM reconstructions in PET Nicolai Bissantz proudly presented by B. Mair and A. Munk Overview Positron Emission - Positron Emission Tomogrophy Tomography Image reconstruction - Multi-scale analysis methods for PET data Model selection - The multi-scale test statistic for PET images A multi-scale stopping - Simulation results: Application to Hoffman and thorax phantom rule Simulations data Conclusions References

Stochastic multi-scale Positron Emission Tomography selection of the stopping criterion for MLEM reconstructions in PET Nicolai Bissantz proudly presented by B. Mair and A. Munk Overview Positron Emission Tomography Image reconstruction methods for PET data Model selection A multi-scale stopping rule Simulations Conclusions References Image source: Wikipedia

1 Stochastic multi-scale selection of the 0.9 stopping criterion for MLEM reconstructions 0.8 in PET 20 0.7 Nicolai Bissantz 40 0.6 60 0.5 80 proudly presented by Angle # 0.4 B. Mair and A. Munk 100 120 0.3 Overview 140 0.2 160 Positron Emission 0.1 180 Tomography 20 40 60 80 100 120 140 160 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Detector bin # Image reconstruction Hoffman phantom Sinogram methods for PET data Model selection A multi-scale stopping rule Simulations Conclusions References Reconstructed transaxial slice of brain. Image source: Wikipedia PET details

Stochastic multi-scale A discrete model for PET selection of the stopping criterion for MLEM reconstructions in PET Nicolai Bissantz proudly presented by The data are observations of Poisson random variables B. Mair and A. Munk Overview Y i ∼ Poiss ([ A f ] i ) , i = 1 , . . . , m , Positron Emission Tomography with Image reconstruction methods for PET data A : projection matrix representing the scanner system response function Model selection f : n-dimensional vector of emission intensities A multi-scale stopping [ A f ] i : i th entry of the vector A f , i.e. the mean number of detections rule in the i th detector tube. Simulations Conclusions We use the following image geometry: References Image space: 128 × 128 pixel Detector space (sinogram format): Siemens ECAT scanner: ν = 192 angles (”views”) with N = 160 detectors per angle, so m = N ν .

Stochastic multi-scale The EM algorithm selection of the stopping criterion for MLEM reconstructions in PET ◮ The observed data (number of detections in each detector tube), Nicolai Bissantz are “incomplete”. A set of complete data can be determined by proudly presented by the number of detections in tube i which come from pixel B j for B. Mair and A. Munk each i , j . Overview ◮ The EM algorithm iteratively estimates the emission density in Positron Emission Tomography the patient’s body: Image reconstruction Initial estimate: Uniform intensity. methods for PET data E-step: Estimate the complete data by its conditional Model selection expectation given the incomplete data and the A multi-scale stopping rule current estimate. Simulations M-step: Maximize the resulting complete data Conclusions log-likelihood from the E-step. References The SNR of the EM image estimates initially increase up to a certain iteration number, then gradually decrease as the iterations increase. Specifically, image noise increases with iteration, so that the EM-iterations become less smooth as the iterations increase. Other algorithms: Filtered Backprojection, Ordered Subsets EM, Penalized-Maximum Likelihood.

Stochastic multi-scale Parameter selection selection of the stopping criterion for MLEM reconstructions in PET Nicolai Bissantz proudly presented by B. Mair and A. Munk Overview ◮ Reconstruction methods for inverse problems depend on a Positron Emission regularization parameter, which is difficult to select. Tomography Image reconstruction ◮ Typically, for PET data iterative EM-type algorithms are used ⇒ methods for PET data We need to select a stopping index (for the iterations) as Model selection regularization parameter. A multi-scale stopping rule More generally: we need to define a method to choose the best model Simulations among a sequence of models, e.g. parametrized by the stopping index Conclusions References in an iterative image reconstruction method.

Stochastic multi-scale Parameter selection (Veklerov & Llacer, 1987) selection of the stopping criterion for MLEM reconstructions in PET Nicolai Bissantz Basic idea: Transformation of the observations Y i ( i = 1 , . . . , m ) proudly presented by B. Mair and A. Munk such that, if the null hypothesis holds, the transformed Overview r.v. Z i are uniformly distributed on [0 , 1]. This Positron Emission transformation depends on reconstructed image! Tomography Image reconstruction Model selection: Apply Pearson’s test for uniformity of the Z i , and, in methods for PET data consequence, to test the reconstructed model. Model selection Results: ”Exact case” ( A is an exact model of the true system A multi-scale stopping rule response function): the method of V & L performs well, Simulations and yields a well-defined finite set of ”feasible iterates”. Conclusions ”Inexact case” ( A estimated): method fails, producing References in general either infinitely many or zero feasible iterates. More ...

Stochastic multi-scale Statistical Multi-Scale Analysis of the Residuals selection of the stopping criterion for MLEM reconstructions in PET ◮ Assume the intensity µ = A f is ’large’, then Nicolai Bissantz Y i − µ i proudly presented by ≈ N (0 , 1) √ µ i B. Mair and A. Munk Overview Positron Emission Tomography Image reconstruction methods for PET data Model selection A multi-scale stopping rule Simulations Conclusions References

Stochastic multi-scale Statistical Multi-Scale Analysis of the Residuals selection of the stopping criterion for MLEM reconstructions in PET ◮ Assume the intensity µ = A f is ’large’, then Nicolai Bissantz Y i − µ i proudly presented by ≈ N (0 , 1) √ µ i B. Mair and A. Munk Overview Positron Emission Tomography ◮ Suppose we have achieved to reconstruct the ”true” f , i.e. ˆ f = f . Image reconstruction Then the standardized estimated residuals methods for PET data Model selection R i = Y i − [ A ˆ f ] i A multi-scale stopping ˆ � , i = 1 , . . . , m rule [ A ˆ f ] i Simulations Conclusions should ’behave like white noise’. References

Stochastic multi-scale Statistical Multi-Scale Analysis of the Residuals selection of the stopping criterion for MLEM reconstructions in PET ◮ Assume the intensity µ = A f is ’large’, then Nicolai Bissantz Y i − µ i proudly presented by ≈ N (0 , 1) √ µ i B. Mair and A. Munk Overview Positron Emission Tomography ◮ Suppose we have achieved to reconstruct the ”true” f , i.e. ˆ f = f . Image reconstruction Then the standardized estimated residuals methods for PET data Model selection R i = Y i − [ A ˆ f ] i A multi-scale stopping ˆ � , i = 1 , . . . , m rule [ A ˆ f ] i Simulations Conclusions should ’behave like white noise’. References ◮ What does this mean?

Stochastic multi-scale Statistical Multi-Scale Analysis of the Residuals selection of the stopping criterion for MLEM reconstructions in PET ◮ Assume the intensity µ = A f is ’large’, then Nicolai Bissantz Y i − µ i proudly presented by ≈ N (0 , 1) √ µ i B. Mair and A. Munk Overview Positron Emission Tomography ◮ Suppose we have achieved to reconstruct the ”true” f , i.e. ˆ f = f . Image reconstruction Then the standardized estimated residuals methods for PET data Model selection R i = Y i − [ A ˆ f ] i A multi-scale stopping ˆ � , i = 1 , . . . , m rule [ A ˆ f ] i Simulations Conclusions should ’behave like white noise’. References ◮ What does this mean? ◮ Statistical Multiscale Analysis (SMA): Control the fluctuation behaviour of residuals simultaneously on all ’scales’, i.e. partial sums (Siegmund/Yakir’00, D¨ umbgen/Spokoiny’01, Davies/Kovac’01, Boysen et al.’08, ...)

Stochastic multi-scale Theory: Asymptotics for Gaussian r.v.’s selection of the stopping criterion for MLEM reconstructions in PET Nicolai Bissantz proudly presented by B. Mair and A. Munk Overview Theorem (Shao’95) Positron Emission Tomography Let { Y m , m ≥ 1 } be a sequence of i.i.d. standard normal random Image reconstruction variables, S 0 = 0 and S n = � methods for PET data 1 ≤ j ≤ n Y j . Then we have Model selection A multi-scale stopping S j + k − S j rule m →∞ max lim max � = 1 a.s. . Simulations 0 ≤ j < m 1 ≤ k ≤ m − j 2 k log( m ) Conclusions References