final phd seminar
TRANSCRIPT
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Introduction Scalable Computation Informative Priors Conclusion
Bayesian Computational Methodsfor Spatial Analysis of Images
Matthew MooresMathematical Sciences School
Science & Engineering Faculty, QUT
PhD final seminarAugust 1, 2014
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Introduction Scalable Computation Informative Priors Conclusion
Acknowledgements
Principal supervisor: Kerrie Mengersen
Associate supervisor: Fiona Harden
Members of the Volume Analysis Tool project team at theRadiation Oncology Mater Centre (ROMC), Queensland Health:
Cathy Hargrave
Mike Poulsen
Tim Deegan
QHealth ethics HREC/12/QPAH/475 and QUT ethics 1200000724
Other co-authors:
Chris Drovandi
Clair Alston
Christian Robert
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Introduction Scalable Computation Informative Priors Conclusion
Outline
1 IntroductionImage-Guided RadiotherapyCone-Beam Computed TomographyAims & Objectives of the Thesis
2 Scalable ComputationDoubly-Intractable LikelihoodsPre-computation for ABC-SMCR package bayesImageS
3 Informative PriorsInformative Prior for µj and σ2
j
External FieldExperimental Results
4 Conclusion
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Introduction Scalable Computation Informative Priors Conclusion
Objectives
The overall objectives of the research are:
to develop a generative model of a digital image thatincorporates prior information,
to produce a computationally efficient implementation of thismodel, and
to apply the model to real world data in image-guidedradiotherapy and satellite remote sensing.
This reflects the parallel perspectives of statistical methods,computational algorithms, and applied bio- and geo-statistics.
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Introduction Scalable Computation Informative Priors Conclusion
Image-Guided Radiotherapy
Image courtesy of Varian Medical Systems, Inc. All rights reserved.
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Introduction Scalable Computation Informative Priors Conclusion
Radiotherapy Process
Before Treatment
fan-beamCT
MRI
contourstreatmentplan
QA
Daily Fractions (∼8 weeks)
positionpatient
cone-beamCT
deliverdose
off-lineanalysis
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Introduction Scalable Computation Informative Priors Conclusion
Radiotherapy Process
Before Treatment
fan-beamCT
MRI
contourstreatmentplan
QA
Daily Fractions (∼8 weeks)
positionpatient
cone-beamCT
deliverdose
off-lineanalysis
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Introduction Scalable Computation Informative Priors Conclusion
Segmentation of Anatomical Structures
Radiography courtesy of Cathy Hargrave, Radiation Oncology Mater Centre
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Introduction Scalable Computation Informative Priors Conclusion
Physiological Variability
Distribution of observed translations of the organs of interest:
Organ Ant-Post Sup-Inf Left-Right
prostate 0.1± 4.1mm −0.5± 2.9mm 0.2± 0.9mmseminal vesicles 1.2± 7.3mm −0.7± 4.5mm −0.9± 1.9mm
Volume variations in the organs of interest:
Organ Volume Gas
rectum 35− 140cm3 4− 26%bladder 120− 381cm3
Frank, et al. (2008) Quantification of Prostate and Seminal VesicleInterfraction Variation During IMRT. IJROBP 71(3): 813–820.
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Introduction Scalable Computation Informative Priors Conclusion
Cone-Beam Computed Tomography
(a) Fan-beam CT (b) Cone-beam CT
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Introduction Scalable Computation Informative Priors Conclusion
Distribution of Pixel Intensity
Hounsfield unit
Fre
quency
−1000 −800 −600 −400 −200 0 200
05000
10000
15000
(a) Fan-Beam CT
pixel intensity
Fre
quency
−1000 −800 −600 −400 −200 0 2000
5000
10000
15000
(b) Cone-Beam CT
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Introduction Scalable Computation Informative Priors Conclusion
Specific Aims I
The statistical aims of the research are:
M1 derivation and representation of informative priors forthe pixel labels.
M2 derivation of informative priors for additive Gaussiannoise from a previous image of the same subject.
M3 sequential Bayesian updating of this prior informationas more images are acquired.
The computational aims are:
C1 measuring the scalability of existing methods forBayesian inference with intractable likelihoods.
C2 development and implementation of improvedalgorithms for fast, approximate inference in imageanalysis.
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Introduction Scalable Computation Informative Priors Conclusion
Specific Aims II
The applied aims are:
A1 To classify pixels in cone-beam CT scans ofradiotherapy patients according to tissue type.
A2 To demonstrate the broad applicability of thesemethods by classifying pixels in satellite imageryaccording to land use or abundance of phytoplankton.
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Introduction Scalable Computation Informative Priors Conclusion
Research Progress
1 Moores, Hargrave, Harden & Mengersen (2014). Segmentation ofcone-beam CT using a hidden Markov random field with informativepriors. Journal of Physics: Conference Series 489:012076.
2 Moores & Mengersen (2014). Bayesian approaches to spatial inference:modelling and computational challenges and solutions. To appear in AIPConference Proceedings.
3 Moores, Drovandi, Mengersen & Robert. Pre-processing for approximateBayesian computation in image analysis. Statistics & Computing(Submitted: March 2014, Revised: June 2014).
4 Moores, Hargrave, Harden & Mengersen. An external field prior for thehidden Potts model with application to cone-beam computed tomography.Computational Statistics & Data Analysis (currently in revision).
5 Moores, Alston & Mengersen. Scalable Bayesian inference for the inversetemperature of a hidden Potts model. (In Prep).
6 Moores, Hargrave, Deegan, Poulsen, Harden & Mengersen. Multi-objectsegmentation of cone-beam CT using a hidden MRF with external fieldprior. (In Prep).
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Introduction Scalable Computation Informative Priors Conclusion
hidden Markov random field
Joint distribution of observed pixel intensities yi ∈ yand latent labels zi ∈ z:
Pr(y, z|µ,σ2, β) ∝ L(y|µ,σ2, z)π(z|β) (1)
Additive Gaussian noise:
yi|zi=jiid∼ N
(µj , σ
2j
)(2)
Potts model:
π(zi|zi∼`, β) =exp {β
∑i∼` δ(zi, z`)}∑k
j=1 exp {β∑
i∼` δ(j, z`)}(3)
Potts (1952) Proceedings of the Cambridge Philosophical Society 48(1)
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Introduction Scalable Computation Informative Priors Conclusion
Inverse Temperature
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Introduction Scalable Computation Informative Priors Conclusion
Doubly-intractable likelihood
p(β|z) = C(β)−1π(β) exp {β S(z)} (4)
The normalising constant of the Potts model has computationalcomplexity of O(n2kn), since it involves a sum over all possiblecombinations of the labels z ∈ Z:
C(β) =∑z∈Z
exp {β S(z)} (5)
S(z) is the sufficient statistic of the Potts model:
S(z) =∑i∼`∈L
δ(zi, z`) (6)
where L is the set of all unique neighbour pairs.
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Introduction Scalable Computation Informative Priors Conclusion
Expectation of S(z)
exact expectation of S(z) for n=12 and k=
β
E(S
(z))
5
10
15
1 2 3 4
2
3
4
(a) n = 12 & k ∈ 2, 3, 4
exact expectation of S(z) for k=3 and n=
β
E(S
(z))
5
10
15
1 2 3 4
4
6
9
12
(b) k = 3 & n ∈ 4, 6, 9, 12
Figure: Distribution of Ez|β [S(z)]
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Introduction Scalable Computation Informative Priors Conclusion
Standard deviation of S(z)
exact standard deviation of S(z) for n=12 and k=
β
σ(S
(z))
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1 2 3 4
2
3
4
(a) n = 12 & k ∈ 2, 3, 4
exact standard deviation of S(z) for k=3 and n=
β
σ(S
(z))
0.0
0.5
1.0
1.5
2.0
2.5
1 2 3 4
4
6
9
12
(b) k = 3 & n ∈ 4, 6, 9, 12
Figure: Distribution of σz|β [S(z)]
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Introduction Scalable Computation Informative Priors Conclusion
Approximate Bayesian Computation
Algorithm 1 ABC rejection sampler
1: for all iterations t ∈ 1 . . . T do2: Draw independent proposal β′ ∼ π(β)3: Generate w ∼ f(·|β′)4: if |S(w)− S(z)| < ε then5: set βt ← β′
6: else7: set βt ← βt−1
8: end if9: end for
Grelaud, Robert, Marin, Rodolphe & Taly (2009) Bayesian Analysis 4(2)Marin & Robert (2014) Bayesian Essentials with R §8.3
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Introduction Scalable Computation Informative Priors Conclusion
Pre-computation Step
The distribution of the summary statistics f(S(w)|β) isindependent of the observed data y
By simulating pseudo-data for values of β, we can create abinding function φ(β) for an auxiliary model fA(S(w)|φ(β))
This binding function can be reused across multiple datasets,amortising its computational cost
By replacing S(w) with approximate values drawn from ourauxiliary model, we avoid the need to simulate pseudo-data duringmodel fitting.
Wood (2010) Nature 466Cabras, Castellanos & Ruli (2014) Metron (to appear)
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Introduction Scalable Computation Informative Priors Conclusion
Simulation from f(·|β)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
10
15
20
25
30
β
E(S
(z))
(a) Ez|β (S(w))
0.0 0.5 1.0 1.5 2.0 2.5 3.00
12
34
β
σ(S
(z))
(b) σz|β (S(w))
Figure: Approximation of S(w)|β using 1000 iterations ofSwendsen-Wang (discarding 500 as burn-in)
Swendsen & Wang (1987) Physical Review Letters 58
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Introduction Scalable Computation Informative Priors Conclusion
Piecewise linear model
0.0 0.5 1.0 1.5 2.0 2.5 3.0
10
00
01
50
00
20
00
02
50
00
30
00
0
β
ES
(z)
(a) φµ(β)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05
01
00
15
02
00
25
03
00
35
0
β
σS
(z)
(b) φσ(β)
Figure: Binding functions for S(w) | β with n = 56, k = 3
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Introduction Scalable Computation Informative Priors Conclusion
Scalable ABC-SMC for the hidden Potts model
Algorithm 2 ABC-SMC using precomputed fA(S(w)|φ(β))
1: Draw N particles β′i ∼ π0(β)
2: Draw N ×M statistics S(wi,m) ∼ N(φµ(β′i), φσ(β′i)
2)
3: repeat4: Update S(zt)|y, πt(β)5: Adaptively select ABC tolerance εt6: Update importance weights ωi for each particle7: if effective sample size (ESS) < Nmin then8: Resample particles according to their weights9: end if
10: Update particles using random walk proposal(with adaptive RWMH bandwidth σ2
t )11: until naccept
N < 0.015 or εt < 10−9 or t ≥ 100
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Introduction Scalable Computation Informative Priors Conclusion
Accuracy of posterior estimates for β
0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
β
po
ste
rio
r d
istr
ibu
tio
n
(a) pseudo-data (M=50)
0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
β
po
ste
rio
r d
istr
ibu
tio
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(b) pre-computed (M=200)
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Introduction Scalable Computation Informative Priors Conclusion
Improvement in runtime
Pseudo−data Pre−computed
0.5
1.0
2.0
5.0
10
.02
0.0
50
.01
00
.0
algorithm
ela
pse
d t
ime
(h
ou
rs)
(a) elapsed (wall clock) time
Pseudo−data Pre−computed
51
02
05
01
00
20
05
00
algorithm
CP
U t
ime
(h
ou
rs)
(b) CPU time
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Introduction Scalable Computation Informative Priors Conclusion
bayesImageS
An R package for Bayesian image segmentationusing the hidden Potts model:
RcppArmadillo for fast computation in C++
OpenMP for parallelism�l i b r a r y ( bayes ImageS )p r i o r s ← l i s t ("k"=3,"mu"=rep ( 0 , 3 ) , "mu.sd"=sigma ,
"sigma"=sigma , "sigma.nu"=c ( 1 , 1 , 1 ) , "beta"=c ( 0 , 3 ) )mh ← l i s t ( a l g o r i t h m="pseudo" , bandwidth =0.2)r e s u l t ← mcmcPotts ( y , ne igh , b lock , NULL ,
55000 ,5000 , p r i o r s , mh)
Eddelbuettel & Sanderson (2014) RcppArmadillo: Accelerating R withhigh-performance C++ linear algebra. CSDA 71
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Introduction Scalable Computation Informative Priors Conclusion
Bayesian computational methods
bayesImageS supports methods for updating the latent labels z:
Chequerboard updating (Winkler 2003)
Swendsen-Wang (1987)
and also methods for updating the inverse temperature β:
Pseudolikelihood (Ryden & Titterington 1998)
Path Sampling (Gelman & Meng 1998)
Exchange Algorithm (Murray, Ghahramani & MacKay 2006)
Approximate Bayesian Computation (Grelaud et al. 2009)
Sequential Monte Carlo (ABC-SMC) with pre-computation(Del Moral, Doucet & Jasra 2012; Moores et al. 2014)
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Introduction Scalable Computation Informative Priors Conclusion
Electron Density phantom
(a) CIRS Model 062 ED phantom (b) Helical, fan-beam CT scanner
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Introduction Scalable Computation Informative Priors Conclusion
Regression Adjustment
0 1 2 3 4
−1
00
0−
80
0−
60
0−
40
0−
20
00
20
0
Electron Density
Ho
un
sfie
ld u
nit
(a) Fan-Beam CT
0 1 2 3 4
−1
00
0−
80
0−
60
0−
40
0−
20
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20
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Electron Density
pix
el in
ten
sity
(b) Cone-Beam CT
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Introduction Scalable Computation Informative Priors Conclusion
Distribution of Pixel Intensities
Hounsfield units
De
nsity
−1000 −500 0 500 1000
0.0
00
0.0
01
0.0
02
0.0
03
0.0
04
0.0
05
0.0
06
(a) Fan-beam CT
Pixel intensity
De
nsity
−1000 −500 0 500 1000
0.0
00
0.0
01
0.0
02
0.0
03
0.0
04
(b) Cone-beam CT
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Introduction Scalable Computation Informative Priors Conclusion
Priors for additive Gaussian noise
Tissue Type Density π(µj)
gas 0.63 -889.74adipose 3.17 -155.03RECT WALL 3.25 29.04BLADDER 3.39 76.75SEM VES 3.40 81.48PROSTATE 3.45 99.25muscle 3.48 110.99spongy bone 3.73 197.75dense bone 4.86 595.37
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Introduction Scalable Computation Informative Priors Conclusion
Treatment Plan
−50 0 50
15
02
00
25
0
right−left (mm)
po
ste
rio
r−a
nte
rio
r (m
m)
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Introduction Scalable Computation Informative Priors Conclusion
External Field
p(zi|zi∼`, β,µ,σ2, yi) =exp {αi,zi + π(αi,zi)}∑kj=1 exp {αi,j + π(αi,j)}
π(zi|zi∼`, β)
(7)Isotropic translation:
π(αi,j) = log
1
nj
∑h∈j
φ(∆(h, i)|µ∆ = 1.2, σ2
∆ = 7.32) (8)
where
nj is the number of voxels in object j
h ∈ j are the voxels in object j
∆(u, v) is the Euclidean distance between the coordinates ofpixel u and pixel v
µ∆, σ2∆ are parameters that describe the level of spatial
variability of the object j
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Introduction Scalable Computation Informative Priors Conclusion
External Field II
External field prior for the ED phantom (σ∆ = 7.3mm)
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Introduction Scalable Computation Informative Priors Conclusion
Anisotropy
αi(prostate) ∼ MVN
0.1−0.50.2
,4.12 0 0
0 2.92 00 0 0.92
(a) Bitmask (b) External Field
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Introduction Scalable Computation Informative Priors Conclusion
Seminal Vesicles
αi(SV) ∼ MVN
1.2−0.7−0.9
,7.32 0 0
0 4.52 00 0 1.92
(a) Bitmask (b) External Field
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Introduction Scalable Computation Informative Priors Conclusion
External Field
Organ- and patient-specific external field (slice 49, 16mm Inf)
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Introduction Scalable Computation Informative Priors Conclusion
Preliminary Results
−300 −250 −200 −150
15
02
00
25
03
00
right−left (mm)
po
ste
rio
r−a
nte
rio
r (m
m)
(a) Cone-Beam CT
−300 −250 −200 −150
15
02
00
25
03
00
right−left (mm)
po
ste
rio
r−a
nte
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r (m
m)
(b) Segmentation
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Introduction Scalable Computation Informative Priors Conclusion
ED phantom experiment
27 cone-beam CT scans of the ED phantom
Cropped to 376× 308 pixels and 23 slices(330× 270× 46 mm)
Inner ring of inserts rotated by between 0◦ and 16◦
2D displacement of between 0mm and 25mmIsotropic external field prior with σ∆ = 7.3mm
9 component Potts model
8 different tissue types, plus water-equivalent backgroundPriors for noise parameters estimated from 28 fan-beam CTand 26 cone-beam CT scans
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Introduction Scalable Computation Informative Priors Conclusion
Image Segmentation
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Introduction Scalable Computation Informative Priors Conclusion
Quantification of Segmentation Accuracy
Dice similarity coefficient:
DSCg =2× |g ∩ g||g|+ |g|
(9)
where
DSCg is the Dice similarity coefficient for label g
|g| is the count of pixels that were classified with thelabel g
|g| is the number of pixels that are known to trulybelong to component g
|g ∩ g| is the count of pixels in g that were labeled correctly
Dice (1945) Measures of the amount of ecologic association between species.Ecology 26(3): 297–302.
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Introduction Scalable Computation Informative Priors Conclusion
Results
Tissue Type Simple Potts External Field
Lung (inhale) 0.507± 0.053 0.868± 0.011Lung (exhale) 0.169± 0.006 0.839± 0.008Adipose 0.048± 0.006 0.713± 0.041Breast 0.057± 0.017 0.748± 0.007Water 0.123± 0.134 0.954± 0.004Muscle 0.071± 0.004 0.758± 0.016Liver 0.075± 0.011 0.662± 0.033Spongy Bone 0.094± 0.020 0.402± 0.175Dense Bone 0.013± 0.001 0.297± 0.201
Table: Segmentation Accuracy (Dice Similarity Coefficient ±σ)
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Introduction Scalable Computation Informative Priors Conclusion
Discussion
Contributions of this thesis:
M1 External field prior for representing spatialinformation in the hidden Potts model
M2 Regression model for adjusting priors for the noiseparameters µj and σ2
j
C2 Pre-computation for ABC-SMC leads to two ordersof magnitude faster computation
A1 Application to cone-beam CT scans of the EDphantom and radiotherapy patient data from theRadiation Oncology Mater Centre
Not discussed in this talk:
M3 Sequential Bayesian updating of the external fieldprior
C1 Scalability experiments with other algorithms fordoubly-intractable likelihoods
A2 Application to satellite remote sensing
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Introduction Scalable Computation Informative Priors Conclusion
Ongoing & Future Work
Complete the analysis of the patient data and submit journalarticle to ANZ J. Stat.
Model object boundaries (eg. for bony anatomy) and spatialcorrelation between objects
Model spatially-correlated noise and artefacts in cone-beamCT scans
Collaboration with Antonietta Mira & Alberto Caimo (USI,Switzerland) on pre-computation for ERGM
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ED phantom inserts
Tissue Type Electron Density Diameter(×1023/cc) (cm)
Lung (inhale) 0.634 3.05Lung (exhale) 1.632 3.05Adipose 3.170 3.05Breast 3.261 3.05Water 3.340 *Muscle 3.483 3.05Liver 3.516 3.05Spongy Bone 3.730 3.05Dense Bone 4.862 1.00
Table: Properties of the CIRS Model 062 ED phantom
* overall dimensions are 33cm× 27cm× 5cm
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Cone-beam CT reconstructed images
Half-fan acquisition mode: FOV 450mm × 450mm × 137mm(Kan, Leung, Wong & Lam 2008)
reconstructed from 650-700 projections (Varian .HND files)
512 × 512 pixels with 2mm slice width (70-80 slices)
∼ 20 million voxels
70-80MB DICOM image stack