Stochastic multi-scale selection of the stopping criterion for ......Model selection A multi-scale...

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



in PET

Nicolai Bissantz


Overview



Model selection


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



in PET

Nicolai Bissantz


Overview



Model selection


Simulations

Conclusions

References

Positron Emission Tomography

Image source: Wikipedia



in PET

Nicolai Bissantz


Overview



Model selection


Simulations

Conclusions

References

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Hoffman phantom Sinogram

Reconstructed transaxial slice of brain.

Image source: Wikipedia

PET details



in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


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 ...



in PET

Nicolai Bissantz


Overview



Model selection


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, ...)



in PET

Nicolai Bissantz


Overview



Model selection


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



in PET

Nicolai Bissantz


Overview



Model selection


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



in PET

Nicolai Bissantz


Overview



Model selection


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)



in PET

Nicolai Bissantz


Overview



Model selection


Simulations

Conclusions

References

A multi-scale stopping rule.



in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


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 ...



in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


Simulations

Conclusions

References

20 40 60 80 100 1200

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600Iterate 14Iterate 47Phantom

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600Iterate 14Iterate 47Phantom

Profiles of the SMA iterate (14), max. SNR iterate (47), andphantom. Left: exact case, right: inexact case.



in PET

Nicolai Bissantz


Overview



Model selection


Simulations

Conclusions

References

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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.



in PET

Nicolai Bissantz


Overview



Model selection


Simulations

Conclusions

References

Results: Thorax phantom

Thorax phantom (left); SMA reconstructions: exact case (middle) andinexact case (right).



in PET

Nicolai Bissantz


Overview



Model selection


Simulations

Conclusions

References

Results: Thorax phantom

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in PET

Nicolai Bissantz


Overview



Model selection


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.



in PET

Nicolai Bissantz


Overview



Model selection


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

Stochastic multi-scale selection of the stopping criterion for ......Model selection A multi-scale...

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