journalclub - robust statistical fusion of image labels

8/2/2019 JournalClub - Robust Statistical Fusion of Image Labels

1/28

ROBUST STATISTICAL

FUSION OF IMAGE LABELSLandman et al

IEEE Transactions on Medical Imaging

VOL. 31, NO. 2, FEBRUARY 2012


2/28

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


3/28


4/28

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


5/28

Label inversion problem


6/28

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


7/28

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


8/28

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.

,,,,,,,,,,, , , ,

{, }


9/28

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)


10/28

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


11/28

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.


12/28

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


13/28

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


14/28


15/28

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


16/28

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)


17/28

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

+(|)

():

()


18/28

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


19/28

STAPLER algorithm

New E-step

,

(=)

(|):

(=) ):

where + is a global prior

+

New M-step

+

=

()

()


20/28

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


21/28


22/28

Result

Voxel-wise random simulated labels


23/28

Result

Boundary-random simulated labels


24/28

Result

Empirical simulation results

38 undergraduate raters

Raters labeled between 10 and 100 slices for the axial set


25/28

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.


26/28

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 , (|)


27/28

EM algorithm

Rewrite the equation

ln , ln

ln , ,

, , ln

,

ln ,

, (|

)

Thus,

+ arg max arg max , ln ,

arg max , ln ,


28/28

journalclub - robust statistical fusion of image labels

Documents