JournalClub - Robust Statistical Fusion of Image Labels
Transcript of JournalClub - Robust Statistical Fusion of Image Labels
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ROBUST STATISTICAL
FUSION OF IMAGE LABELSLandman et al
IEEE Transactions on Medical Imaging
VOL. 31, NO. 2, FEBRUARY 2012
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Labeling in MR image analysis
Labeling Problem
Identifying class membership of voxels
Currently no true answer
Manual Voxel-by-voxel Labeling
Considered as a gold standard
Exceptionally time consuming and resource intensive
Difference in interpretation between raters
Validating automatic or semi-automatic methods
The study of structures for which no automated method exists
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Limitation of STAPLE algorithm
STAPLE requires that all raters delineate all voxelswithin a given region
Raters are often requested to label datasets more thanoncein order to establish a measure of intra-raterreliability.
Raters are often divided into a class of experts and
novices
Label inversion problem due to highly inaccurate raters
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Label inversion problem
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STAPLER algorithm
Algorithm
Estimate a maximum a posterioriof both rater reliability and truelabels in the Expectation Maximization framework
Evaluation
Random rater simulation
Boundary random rater simulation
Simulations to characterize the occurrence of label inversionproblem
Note Minimally trained raters and large number of participants
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STAPLER algorithm
How to estimate true labels?
Majority voting
:
How to estimate rater performance?
Rater performance is not considered to be perfect but to be fuzzy.
| : [ , , ,,()]
Confusion matrix (hidden variable in EM framework)
1 2
1 0.9 0.2
2 0.1 0.8
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EM algorithm
Expectation Maximization
Parameter estimation in probabilistic models with incomplete data
Computes iteratively the Maximum Likelihood estimationwith theassumption of hidden variable
Toy Example: Coin flipping with two different coins, A and B
{, } : 10 trials of flipping.
,,,,,,,,,,, , , ,
{, }
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EM algorithm
Assume that we know which coin is flipped, for example:
How to estimate , maximizing the likelihood of(| , )?
A: 24 Head 6 Tail
B: 9 Head 11 Tail
Coin Results
B H T T T H H T H T H (5H 5T)
AH H H H T H H H H H (9H 1T)
A H H H H T H T H H H (8H 2T)
B H T T T T H H T H T (4H 6T)
A H T H H T T H H H H (7H 3T)
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EM algorithm
+ 0.8
+ 0.45
Maximum Likelihood Estimation
arg max
(| , )
30
24
1
24
1 6
1
1
24 24 6 2 43 0 0
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EM algorithm
What if we dont know which coin is used?
{, , , } is called hidden variable or latent factor
Maximizing log with respect to
Efficient iterative process and guarantees to converge
Repeat E-step and M-step until the algorithm converges
E-step: , [log (, |)]
M-step: argmax , [log (, |)]
EM algorithm becomes very slow as the number of variables increases.
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EM algorithm
Begin with initial parameters
, 0.6, 0.5
Compute the probability of the event for A and B
Event (, , ) (, , ) ( |, )
#1 (5H 5T) 0.2007
(105 0.60.4)
0.2495
(105 0.50.5)0.2/(0.2+0.25)=0.45
#2 (9H 1T) 0.04 0.0098#3 (8H 2T) 0.1209 0.0439
#4 (4H 6T) 0.1115 0.2051
#5 (7H 3T) 0.2150 0.1172
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EM algorithm
Compute expectation (E-step)
Estimate new parameter (M-step)
.
.+. 0.71,
.
.+. 0.58
Event ( |, ) ( |, ) ( , ) ( , )
#1 (5H 5T) 0.45 0.55 2.2H 2.2T 2.8H 2.8T
#2 (9H 1T) 0.80 0.2 7.2H 0.8T 1.8H 0.2T
#3 (8H 2T) 0.73 0.27 5.9H 1.5T 2.1H 0.5T
#4 (4H 6T) 0.35 0.65 1.4H 2.1T 2.6H 3.9T
#5 (7H 3T) 0.65 0.35 4.5H 1.9T 2.5H 1.1T
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Label Fusion
An image of N voxels
: the number of training voxels
: the number of undetermined voxels
, , arethe set of all voxels
{0,2,3, , L 1}is all possible labels
{, , }is a collection of raters
represents the r-th observation of voxel iby raterj
represents the hidden true segmentation
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STAPLE algorithm
Estimate true segmentation in a probabilistic framework
: [ , , ,,()]
:
Probability distribution function of the true label
() ( matrix) represents the probability that voxel ihas true
label son the k-th iteration
Hidden variable that control the p.d.f
(|) ( matrix) represents the probability that raterj
observes label swhen the true label is son the k-th iteration (raterperformance)
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STAPLE algorithm
EM algorithmto estimate the hidden true segmentation
E-step: the calculation of the conditional probability ofthe true segmentation
() , (
=)
(|)
(=) (|)
M-step: the calculation of the rater performanceparameters
+(|)
():
()
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STAPLE algorithm
Example
1 ...
...+... 0.9545
2 ...
...+... 0.6667
Rater A B C
Label 1 1 1 2
Label 2 2 1 1
A 1 2
1 0.9 0.2
2 0.1 0.8
B 1 2
1 0.7 0.4
2 0.3 0.6
C 1 2
1 0.5 0.5
2 0.5 0.5
1 1 .9545
0.9545 0.3333
2 1 .3333
0.9545 0.3333
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STAPLER algorithm
New E-step
,
(=)
(|):
(=) ):
where + is a global prior
+
New M-step
+
=
()
()
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Results
Data
A high-resolution MPRAGE
149 x 81 x 39 voxels
0.82 x 0.82 x 1.5 mm resolution
Expert labeled the cerebellum from each dataset with 12 divisionsof the cerebellar hemispheres
Simulation
Voxel-wise random raters
Boundary random raters Evaluation
Jaccard Index
Dice coefficient
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Result
Voxel-wise random simulated labels
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Result
Boundary-random simulated labels
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Result
Empirical simulation results
38 undergraduate raters
Raters labeled between 10 and 100 slices for the axial set
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Discussion
STAPLER extends the applicability of the STAPLEtechnique to common research situations with missing,partial, and repeated data.
STAPLER facilitates use of training data and reliability
priors to improve accuracy.
STAPLER enables parallel manual labeling and reducesdetrimental impacts
STAPLER can readily be augmented by introducingspatially adaptive, unconditional label probabilities.
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EM algorithm
Log likelihood function defined as
log (|)
log is strictly concave and also strictly increasing because
< 0
Definition of convex function
(1 ) (1 )()
Compute an updated estimate such that,
ln ln (|)
Introduce a hidden variable , (|)
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EM algorithm
Rewrite the equation
ln , ln
ln , ,
, , ln
,
ln ,
, (|
)
Thus,
+ arg max arg max , ln ,
arg max , ln ,
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