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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

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Stochastic multi-scale 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

Stochastic multi-scale selection of the stopping criterion for MLEM reconstructions in PET

Nicolai Bissantz proudly presented by B. Mair and A. Munk

Ruhr-Universit¨ at Bochum, University of Florida, Georg August Universit¨ at G¨

ttingen

Linz, October 29th, 2008

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Stochastic multi-scale 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

Overview

Positron Emission Tomogrophy
Multi-scale analysis
The multi-scale test statistic for PET images
Simulation results: Application to Hoffman and thorax phantom

data

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Stochastic multi-scale 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

Positron Emission Tomography

Image source: Wikipedia

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Stochastic multi-scale 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

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Detector bin # Angle # 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 160 180

Hoffman phantom Sinogram Reconstructed transaxial slice of brain.

Image source: Wikipedia

PET details

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Stochastic multi-scale 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

A discrete model for PET

The data are observations of Poisson random variables Yi ∼ Poiss ([Af]i) , i = 1, . . . , m, with A: projection matrix representing the scanner system response function f: n-dimensional vector of emission intensities [Af]i : ith entry of the vector Af, i.e. the mean number of detections in the ith detector tube. We use the following image geometry: Image space: 128 × 128 pixel Detector space (sinogram format): Siemens ECAT scanner: ν = 192 angles (”views”) with N = 160 detectors per angle, so m = Nν.

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Stochastic multi-scale 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

The EM algorithm

◮ The observed data (number of detections in each detector tube),

are “incomplete”. A set of complete data can be determined by the number of detections in tube i which come from pixel Bj for each i, j.

◮ The EM algorithm iteratively estimates the emission density in

the patient’s body: Initial estimate: Uniform intensity. E-step: Estimate the complete data by its conditional expectation given the incomplete data and the current estimate. M-step: Maximize the resulting complete data log-likelihood from the E-step. 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.

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Stochastic multi-scale 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

Parameter selection

◮ Reconstruction methods for inverse problems depend on a

regularization parameter, which is difficult to select.

◮ Typically, for PET data iterative EM-type algorithms are used ⇒

We need to select a stopping index (for the iterations) as regularization parameter. More generally: we need to define a method to choose the best model among a sequence of models, e.g. parametrized by the stopping index in an iterative image reconstruction method.

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Stochastic multi-scale 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

Parameter selection (Veklerov & Llacer, 1987)

Basic idea: Transformation of the observations Yi (i = 1, . . . , m) such that, if the null hypothesis holds, the transformed r.v. Zi are uniformly distributed on [0, 1]. This transformation depends on reconstructed image! Model selection: Apply Pearson’s test for uniformity of the Zi, and, in consequence, to test the reconstructed model. Results: ”Exact case” (A is an exact model of the true system response function): the method of V & L performs well, and yields a well-defined finite set of ”feasible iterates”. ”Inexact case” (A estimated): method fails, producing in general either infinitely many or zero feasible iterates.

More ...

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Stochastic multi-scale 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

Statistical Multi-Scale Analysis of the Residuals

◮ Assume the intensity µ = Af is ’large’, then

Yi − µi √µi ≈ N(0, 1)

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Stochastic multi-scale 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

Statistical Multi-Scale Analysis of the Residuals

◮ Assume the intensity µ = Af is ’large’, then

Yi − µi √µi ≈ N(0, 1)

◮ Suppose we have achieved to reconstruct the ”true” f , i.e. ˆ

f = f . Then the standardized estimated residuals ˆ Ri = Yi − [Aˆ f]i

[Aˆ

f]i , i = 1, . . . , m should ’behave like white noise’.

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Stochastic multi-scale 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

Statistical Multi-Scale Analysis of the Residuals

◮ Assume the intensity µ = Af is ’large’, then

Yi − µi √µi ≈ N(0, 1)

◮ Suppose we have achieved to reconstruct the ”true” f , i.e. ˆ

f = f . Then the standardized estimated residuals ˆ Ri = Yi − [Aˆ f]i

[Aˆ

f]i , i = 1, . . . , m should ’behave like white noise’.

◮ What does this mean?

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Stochastic multi-scale 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

Statistical Multi-Scale Analysis of the Residuals

◮ Assume the intensity µ = Af is ’large’, then

Yi − µi √µi ≈ N(0, 1)

◮ Suppose we have achieved to reconstruct the ”true” f , i.e. ˆ

f = f . Then the standardized estimated residuals ˆ Ri = Yi − [Aˆ f]i

[Aˆ

f]i , i = 1, . . . , m should ’behave like white noise’.

◮ 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, ...)

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Stochastic multi-scale 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

Theory: Asymptotics for Gaussian r.v.’s

Theorem (Shao’95)

Let {Ym, m ≥ 1} be a sequence of i.i.d. standard normal random variables, S0 = 0 and Sn =

1≤j≤n Yj. Then we have

lim

m→∞ max 0≤j<m

max

1≤k≤m−j

Sj+k − Sj

2k log(m)

= 1 a.s..

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Stochastic multi-scale 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

The multi-scale test statistic for PET

◮ Here d = 1 for simplicity. ◮ In fact things are more involved. d = 2: Relplace intervals by

dyadic squares, wedgelets,...

◮ Poisson noise ◮ binning, ...

Multi-scale test statistic: (Consider all averages of k-adjacent

bservations: single, double,triple,...):

Dn = max

0≤j<m

max

1≤k≤m−j

|Sj+k − Sj| kα(k/ log(m)), where Sj+k − Sj = j+k

i=j+1 ˆ

Ri are the partial sums.

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Stochastic multi-scale 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

The general case (including Poisson noise)

Theorem (Steinebach’ 97)

Let Y ; Y1, . . . , Ym be a sequence of centered i.i.d. random variables with unit variance. Assume inf {t : φ(t) < ∞} < 0 < sup {t : φ(t) < ∞} , where φ(t) = E exp(tY ) and consider the Chernoff function of Y ρ(y) = inft≥0φ(t) exp(−ty). Moreover, define, for c > 0, the ”inverse Chernoff function” α(c) = sup {y ≥ 0 : ρ(y) ≥ exp(−1/c)} . Then lim

m→∞ max 0≤j<m

max

1≤k≤m−j

Sj+k − Sj kα(k/ log m) = 1 a.s..

There is also a multidimensional (cubes) (Kabluchko,M., 08a) and distributional version (Kabluchko,M’08b).

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Stochastic multi-scale 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

A multi-scale stopping rule.

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Stochastic multi-scale 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

The multi-scale analysis (SMA) rule for the stopping index

Stop the iteration the first time the test statistic Dm falls below the median of the simulated distribution of Dm. Otherwise,

◮ if the test statistic Dm is too large, we do not reproduce the data

well by our model,

◮ and if it is too small, we overfit the data.

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Stochastic multi-scale 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

Simulation Set-up

◮ 200 simulations of noisy data generated by 128 × 128 Hoffman

phantom with 200K, 400K, 600K, 800K and 1M total counts, and Thorax phantom with 88K total counts for the exact and inexact cases.

◮ The system matrix in the inexact case was obtained by adding a

±10% uniformly random gain to the true matrix.

◮ Dependence of the SMA-critical values on total counts were

investigated.

◮ Reconstructions of the Hoffman phantom obtained from stopping

the EM algorithm at the iteration determined by the SMA method, and the iteration with the largest SNR were compared.

◮ Reconstructions of thef Thorax phantom determined by the SMA

and LV methods were quantitatively compared by using the SNR and contrast coefficients for a hot region.

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Stochastic multi-scale 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 Quality Measures

SNR(ˆ f ) = 20 log f 2 ˆ f − f 2 CRC(A, ˆ f ) = ˆ f (A)/ˆ f (B) − 1 f (A)/f (B) − 1 where B is a subset of the background.

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Stochastic multi-scale 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

Dependence of the multi-scale test statistic on iteration number and noise level

10 20 30 40 0.8 1 1.2 1.4 1.6 1.8 2 1M total counts Iteration M 10 20 30 40 0.8 1 1.2 1.4 1.6 1.8 2 800K total counts Iteration M 10 20 30 40 0.8 1 1.2 1.4 1.6 1.8 2 600K total counts Iteration M 10 20 30 40 0.8 1 1.2 1.4 1.6 1.8 2 400K total counts Iteration M 10 20 30 40 0.8 1 1.2 1.4 1.6 1.8 2 200K total counts Iteration M

SMA test statistic Dm for each iteration number i of the EM algorithm applied to Hoffman phantom data with various total counts.

(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) EM iterate with max. SNR: inexact case.

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Stochastic multi-scale 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

20 40 60 80 100 120 100 200 300 400 500 600 Iterate 14 Iterate 47 Phantom 20 40 60 80 100 120 100 200 300 400 500 600 Iterate 14 Iterate 47 Phantom

Profiles of the SMA iterate (14), max. SNR iterate (47), and

phantom. Left: exact case, right: inexact case.

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Stochastic multi-scale 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

0.2 0.4 0.6 100 200 300 400 500 0.2 0.4 0.6 100 200 300 400 500 1M total counts 0.2 0.4 0.6 100 200 300 400 500 0.2 0.4 0.6 50 100 Profile at y−axis=0.27 0.2 0.4 0.6 50 100 Profile at y−axis=0.5 200000 total counts 0.2 0.4 0.6 50 100 Profile at y−axis=0.76

True profiles of Hoffman phantom (thin full curves, grey) compared with SMS stopped iterate is (full curves, red), and with maximal SNR stopped iterate (dashed curves, blue), for total counts 200K and 1M.

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Stochastic multi-scale 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

Results: Thorax phantom

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

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Stochastic multi-scale 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

Results: Thorax phantom

10 20 30 40 50 −30 −20 −10 10 20 30 40 Iteration All Iterations LV Iteration SMAP Iteration 10 20 30 40 50 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 Iteration All Iterations LV Iteration SMAP Iteration

Signal-to-Noise Ratio Contrast Recovery Coefficient

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Stochastic multi-scale 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

Conclusions

◮ We propose a fully data-driven method (SMA) for obtaining

regularized solutions of Poisson linear inverse problems by early termination of iterative algorithms.

◮ It depends only on the median of a simulated test statistic, which

depends only weakly on the total counts ⇒ few simulated values will be sufficient for a wide range of applications.

◮ The SMA method was tested on the EM algorithm for

reconstructing PET images from simulated noisy data with and without modeling errors in the system matrix.

◮ In these simulations, the SMA iterates had a SNR of

approximately 80% of the maximal SNR available from all EM iterates, and had larger SNR and contrast coefficients for a hot region than the LV method.

SLIDE 27

Stochastic multi-scale 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

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.