stochastic multi-scale selection of the stopping criterion for ......model selection a multi-scale...
TRANSCRIPT
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Stochastic multi-scale selection of the stoppingcriterion for MLEM reconstructions in PET
Nicolai Bissantz
proudly presented by B. Mair and A. Munk
Ruhr-Universität Bochum, University of Florida,Georg August Universität Göttingen
Linz, October 29th, 2008
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Overview
- Positron Emission Tomogrophy
- Multi-scale analysis
- The multi-scale test statistic for PET images
- Simulation results: Application to Hoffman and thorax phantomdata
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Positron Emission Tomography
Image source: Wikipedia
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
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Hoffman phantom Sinogram
Reconstructed transaxial slice of brain.
Image source: Wikipedia
PET details
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
A discrete model for PET
The data are observations of Poisson random variables
Yi ∼ Poiss ([Af]i ) , i = 1, . . . ,m,
withA: projection matrix representing the scanner system response functionf: n-dimensional vector of emission intensities[Af]i : i th entry of the vector Af, i.e. the mean number of detectionsin the i th detector tube.We use the following image geometry:Image space: 128× 128 pixelDetector space (sinogram format): Siemens ECAT scanner: ν = 192angles (”views”) with N = 160 detectors per angle, so m = Nν.
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
The EM algorithm
I The observed data (number of detections in each detector tube),are “incomplete”. A set of complete data can be determined bythe number of detections in tube i which come from pixel Bj foreach i , j .
I The EM algorithm iteratively estimates the emission density inthe patient’s body:
Initial estimate: Uniform intensity.E-step: Estimate the complete data by its conditional
expectation given the incomplete data and thecurrent estimate.
M-step: Maximize the resulting complete datalog-likelihood from the E-step.
The SNR of the EM image estimates initially increase up to a certainiteration number, then gradually decrease as the iterations increase.Specifically, image noise increases with iteration, so that theEM-iterations become less smooth as the iterations increase.Other algorithms: Filtered Backprojection, Ordered Subsets EM,Penalized-Maximum Likelihood.
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Parameter selection
I Reconstruction methods for inverse problems depend on aregularization parameter, which is difficult to select.
I Typically, for PET data iterative EM-type algorithms are used ⇒We need to select a stopping index (for the iterations) asregularization parameter.
More generally: we need to define a method to choose the best modelamong a sequence of models, e.g. parametrized by the stopping indexin an iterative image reconstruction method.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Parameter selection (Veklerov & Llacer, 1987)
Basic idea: Transformation of the observations Yi (i = 1, . . . ,m)such that, if the null hypothesis holds, the transformedr.v. Zi are uniformly distributed on [0, 1]. Thistransformation depends on reconstructed image!
Model selection: Apply Pearson’s test for uniformity of the Zi , and, inconsequence, to test the reconstructed model.
Results: ”Exact case” (A is an exact model of the true systemresponse function): the method of V & L performs well,and yields a well-defined finite set of ”feasible iterates”.”Inexact case” (A estimated): method fails, producingin general either infinitely many or zero feasible iterates.
More ...
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Statistical Multi-Scale Analysis of the Residuals
I Assume the intensity µ = Af is ’large’, then
Yi − µi√µi≈ N(0, 1)
I Suppose we have achieved to reconstruct the ”true” f , i.e. f̂ = f .Then the standardized estimated residuals
R̂i =Yi − [Af̂]i√
[Af̂]i
, i = 1, . . . ,m
should ’behave like white noise’.
I What does this mean?
I Statistical Multiscale Analysis (SMA): Control the fluctuationbehaviour of residuals simultaneously on all ’scales’, i.e. partialsums (Siegmund/Yakir’00, Dümbgen/Spokoiny’01,Davies/Kovac’01, Boysen et al.’08, ...)
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Statistical Multi-Scale Analysis of the Residuals
I Assume the intensity µ = Af is ’large’, then
Yi − µi√µi≈ N(0, 1)
I Suppose we have achieved to reconstruct the ”true” f , i.e. f̂ = f .Then the standardized estimated residuals
R̂i =Yi − [Af̂]i√
[Af̂]i
, i = 1, . . . ,m
should ’behave like white noise’.
I What does this mean?
I Statistical Multiscale Analysis (SMA): Control the fluctuationbehaviour of residuals simultaneously on all ’scales’, i.e. partialsums (Siegmund/Yakir’00, Dümbgen/Spokoiny’01,Davies/Kovac’01, Boysen et al.’08, ...)
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Statistical Multi-Scale Analysis of the Residuals
I Assume the intensity µ = Af is ’large’, then
Yi − µi√µi≈ N(0, 1)
I Suppose we have achieved to reconstruct the ”true” f , i.e. f̂ = f .Then the standardized estimated residuals
R̂i =Yi − [Af̂]i√
[Af̂]i
, i = 1, . . . ,m
should ’behave like white noise’.
I What does this mean?
I Statistical Multiscale Analysis (SMA): Control the fluctuationbehaviour of residuals simultaneously on all ’scales’, i.e. partialsums (Siegmund/Yakir’00, Dümbgen/Spokoiny’01,Davies/Kovac’01, Boysen et al.’08, ...)
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Statistical Multi-Scale Analysis of the Residuals
I Assume the intensity µ = Af is ’large’, then
Yi − µi√µi≈ N(0, 1)
I Suppose we have achieved to reconstruct the ”true” f , i.e. f̂ = f .Then the standardized estimated residuals
R̂i =Yi − [Af̂]i√
[Af̂]i
, i = 1, . . . ,m
should ’behave like white noise’.
I What does this mean?
I Statistical Multiscale Analysis (SMA): Control the fluctuationbehaviour of residuals simultaneously on all ’scales’, i.e. partialsums (Siegmund/Yakir’00, Dümbgen/Spokoiny’01,Davies/Kovac’01, Boysen et al.’08, ...)
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Theory: Asymptotics for Gaussian r.v.’s
Theorem (Shao’95)Let {Ym,m ≥ 1} be a sequence of i.i.d. standard normal randomvariables, S0 = 0 and Sn =
∑1≤j≤n Yj . Then we have
limm→∞
max0≤j
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
The multi-scale test statistic for PET
I Here d = 1 for simplicity.
I In fact things are more involved. d = 2: Relplace intervals bydyadic squares, wedgelets,...
I Poisson noise
I binning, ...
Multi-scale test statistic: (Consider all averages of k-adjacentobservations: single, double,triple,...):
Dn = max0≤j
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
The general case (including Poisson noise)
Theorem (Steinebach’ 97)Let Y ; Y1, . . . ,Ym be a sequence of centered i.i.d. random variables withunit variance. Assume
inf {t : φ(t)
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
A multi-scale stopping rule.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
The multi-scale analysis (SMA) rule for thestopping index
Stop the iteration the first time the test statistic Dm falls below themedian of the simulated distribution of D̃m.
Otherwise,
I if the test statistic Dm is too large, we do not reproduce the datawell by our model,
I and if it is too small, we overfit the data.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Simulation Set-up
I 200 simulations of noisy data generated by 128× 128 Hoffmanphantom with 200K , 400K , 600K , 800K and 1M total counts,and Thorax phantom with 88K total counts for the exact andinexact cases.
I The system matrix in the inexact case was obtained by adding a±10% uniformly random gain to the true matrix.
I Dependence of the SMA-critical values on total counts wereinvestigated.
I Reconstructions of the Hoffman phantom obtained from stoppingthe EM algorithm at the iteration determined by the SMAmethod, and the iteration with the largest SNR were compared.
I Reconstructions of thef Thorax phantom determined by the SMAand LV methods were quantitatively compared by using the SNRand contrast coefficients for a hot region.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Image Quality Measures
SNR(f̂ ) = 20 log‖f ‖2‖f̂ − f ‖2
CRC (A, f̂ ) =f̂ (A)/f̂ (B)− 1f (A)/f (B)− 1
where B is a subset of the background.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Dependence of the multi-scale test statistic oniteration number and noise level
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SMA test statistic Dm for each iteration number i of the EMalgorithm applied to Hoffman phantom data with various total counts.
More on stopping rules ...
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
(a) (b) (c)
(d) (e)(a) Hoffman phantom; (b) SMA iterate: exact case; (c) SMA iterate:inexact case; (d) EM iterate with max. SNR: exact case; (e) EMiterate with max. SNR: inexact case.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
20 40 60 80 100 1200
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Profiles of the SMA iterate (14), max. SNR iterate (47), andphantom. Left: exact case, right: inexact case.
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
0.2 0.4 0.60
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True profiles of Hoffman phantom (thin full curves, grey) comparedwith SMS stopped iterate is (full curves, red), and with maximal SNRstopped iterate (dashed curves, blue), for total counts 200K and 1M.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Results: Thorax phantom
Thorax phantom (left); SMA reconstructions: exact case (middle) andinexact case (right).
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Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Results: Thorax phantom
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Signal-to-Noise Ratio Contrast Recovery Coefficient
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
Conclusions
I We propose a fully data-driven method (SMA) for obtainingregularized solutions of Poisson linear inverse problems by earlytermination of iterative algorithms.
I It depends only on the median of a simulated test statistic, whichdepends only weakly on the total counts ⇒ few simulated valueswill be sufficient for a wide range of applications.
I The SMA method was tested on the EM algorithm forreconstructing PET images from simulated noisy data with andwithout modeling errors in the system matrix.
I In these simulations, the SMA iterates had a SNR ofapproximately 80% of the maximal SNR available from all EMiterates, and had larger SNR and contrast coefficients for a hotregion than the LV method.
-
Stochastic multi-scaleselection of the
stopping criterion forMLEM reconstructions
in PET
Nicolai Bissantz
proudly presented byB. Mair and A. Munk
Overview
Positron EmissionTomography
Image reconstructionmethods for PET data
Model selection
A multi-scale stoppingrule
Simulations
Conclusions
References
References
N. Bissantz, B. Mair, and A. Munk, ”A multi-scale stopping criterion for MLEM
reconstructions in PET,” IEEE Nucl. Sci. Symp. Conf. Rec.,vol. 6, 3376-3379, 2006.
E. Veklerov, and J. Llacer, ”Stopping rule for the MLE algorithm based on statistical
hypothesis testing,” IEEE Trans. Med. Imag., vol. 6, no. 4, pp. 313–319, 1987.
J. Llacer, and E. Veklerov, ”Feasible images and practical stopping rules for iterative
algorithms in emission tomography,” IEEE Trans. Med. Imag., vol. 8, pp. 186–193, 1989.
P. L. Davies, and A. Kovac, ”Local extremes, runs, strings and multiresolution,” Ann.
Stat., vol. 29, no. 1, 1–65, 2001.
Kabluchko, Z., Munk, A., ”Exact convergence rate for the maximum of standardized
Gaussian increments,” Elect. Comm. in Probab., vol. 13, 302-310, 2008.
Kabluchko, Z. and Munk, A., ”Shao’s theorem on the maximum of standardized random
walk increments for multidimensional arrays,” ESAIM Prob. Stat., 2008, to appear.
J. Steinebach, ”On a conjecture of Revesz and its analogue for renewal processes,” in
Asymptotic Methods in Probability and Statistics (ICAMPS 1997), Carleton Univ.,Ottawa, Ontario, Canada, July 1997, pp. 311–322.
OverviewPositron Emission TomographyImage reconstruction methods for PET dataModel selectionA multi-scale stopping ruleSimulationsConclusionsReferences