Evaluation of segmentation

Example

Reference standard & segmentation

Segmentation performance

• Qualitative/subjective evaluation the easy way out, sometimes the only option

• Quantitative evaluation preferable in general• A wild variety of performance measures exists• Many measures are applicable outside the

segmentation domain as well• Focus here is on two class problems

Some terms

• Ground truth = the real thing• Gold standard = the best we can get• Bronze standard = gold standard with limitations• Reference standard = preferred term for gold

standard in the medical community

What to evaluate?

• Without reference standard, subjective or qualitative evaluation is hard to avoid

• Region/pixel based comparisons• Border/surface comparisons• (a selection of) Points• Global performance measures versus local

measures

Example

Reference standard & segmentation

What region to evaluate over?

Combination of reference and result

masked

true positive

true negative

false negative

false positive

False positives

False negatives

Confusion matrix (Contingency table)

Segmentation

Reference

negative positive

negative 191152 TN

3813 FP

positive 9764 FN

19648 TP

Do not get confused!

• False positives are actually negative• False negatives are actually positives

Confusion matrix (Contingency table)

Segmentation

Reference

negative positive

negative .852 TN

.017 FP

positive .044 FN

.088 TP

Accuracy, sensitivity, specificity

sensitivity = true positive fraction = 1 – false negative fraction = TP / (TP + FN)

specificity = true negative fraction = 1 – false positive fraction = TN / (TN + FP)

accuracy = (TP + TN) / (TP + TN + FP + FN)

Accuracy

• Range: from 0 to 1• Useful measure, but:• Depends on prior probability (prevalence); in

other words: on amount of background• Even ‘stupid’ methods can achieve high

accuracy (e.g. ‘all background’, or ‘most likely class’ systems)

Sensitivity & specificity

• Are intertwined• ‘stupid’ methods can achieve arbitrarily large

sensitivity/specificity at the expense of low specificity/sensitivity

• Do not depend on prior probability• Are useful when false positives and false

negatives have different consequences

N PN

N N

NN

P P

P

PP

PPNN

true positives (TP)false positives (FP)false negatives (FN)true negatives (TN)

sensitivity = true positive fraction = 1 – false negative fraction = TP / (TP + FN)

specificity = true negative fraction = 1 – false positive fraction = TN / (TN + FP)

accuracy = (TP+TN) / (TP+TN+FP+FN)

N PN

N N

NN

P P

P

PP

PPNN

true positives (TP) = 3false positives (FP) = 3false negatives (FN) = 2true negatives (TN) = 4

sensitivity = TP / (TP + FN) = 3 / 5 = 0.6

specificity = TN / (TN + FP) = 4 / 7 = 0.57

accuracy = (TP+TN) / (TP+TN+FP+FN) = 7 / 12 = 0.58

N PN

N N

NN

P P

P

PP

PP

NN

= 3= 3

= 2= 4

sensitivity = 3 / 5 = 0.6specificity = 4 / 7 = 0.57accuracy = 7 / 12 = 0.58

algorithm 1

N PN

P P

NP

P P

P

PP

PP

NN

= 4= 5

= 1= 2

sensitivity = 4 / 5 = 0.8specificity = 2 / 7 = 0.29accuracy = 6 / 12 = 0.5

algorithm 2

Which system is better?

Back to the retinal image…

resultreference

negative positive

negative .852 TN .017 FP

positive .044 FN .088 TP

Accuracy: 0.93949Sensitivity: 0.668027Specifity: 0.980443

Overlap = intersection / union = TP/(TP+FP+FN)

TPFN FP

TN

Reference Segmentation

Overlap

• Overlap ranges from 0 (no overlap) to 1 (complete overlap)

• The background (TN) is disregarded in the overlap measure

• Small objects with irregular borders have lower overlap values than big compact objects

Kappa

• Accuracy would not be zero if we used a system that is ‘guessing’

• A ‘guessing’ system should get a ‘zero’ mark (remember multiple choice exams…)

• Kappa is an attempt to measure ‘accuracy in excess of accuracy expected by chance’

Kappa Result

Reference

negative positive

negative 191152 3813 194965

positive 9764 19648 29412

200916 23461 224377

System positive rate:23461/224377 = .105

Total number of positives

True positives of a guessing system: .105 * 29412 = 3075… etcAccuracy guessingsystem: .792

System accuracy:(191152 + 19648)/ 224377 = .939

Kappa

• accguess = the accuracy of a randomly guessing system with a given positive (or negative) rate

• kappa = (acc – accguess) / (1 – accguess)• In our case: kappa = (.939 - .792)/(1 - .792)

= .707

Kappa

• Maximum value is 1, can be negative• A ‘guessing’ system has kappa = 0• ‘Stupid systems’ (‘all background’ or ‘most likely

class’) have kappa = 0• Systems with negative kappa have ‘worse than

chance’ performance

Positive/negative predictive value

• PPV and NPV depend on prevalence, contrary to sensitivity and specificity

ROC analysis

Evaluating algorithms

• Most algorithms can produce a continuous instead of a discrete output, monotonically related to the probability that a case is positive.

• Using a variable threshold on such a continuous output, a user can choose the (sensitivity, specificity) of the system. This is formalized in an ROC (receiver operator characteristic) analysis.

Reference standard & segmentation

Reference standard & soft segmentation

ROC analysis

Pn(x) Pp(x)

x

true positive fraction

true negative fraction false positive fraction

ROC curve

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

false pos itive probability

true

pos

itive

pro

babi

lity

true

posit

ive

fract

ion

sens

itivi

tyde

tect

ion

rate

false positive fraction1 - specificity

chance of false alarm

ROC curves

• Receiver Operating Characteristic curve• Originally proposed in radar detection theory• Formalizes the trade-off between sensitivity and specificity• Makes the discriminability and decision bias explicit• Each hard classification is one operating point on the ROC

curve

ROC curves

• A single measure for the performance of a system is the area under the ROC curve Az

• A system that randomly generates a label with probability p has an ROC curve that is a straight line from (0,0) to (1,1), Az = 0.5

• A perfect system has Az = 1• Az does not depend on prior probabilities

(prevalence)

ROC curves

• If one assumes Pn(x) and Pp(x) are Gaussian, two parameters determine the curve: the difference between the means and the ratio of the standards deviations. They can be estimated with a maximum-likelihood procedure.

• There are procedures to obtain confidence intervals for ROC curves and to test if the Az value of two curves are significantly different.

Intuitive meaning for Az

• Is there an intuitive meaning for Az?• Consider the two-alternative forced-choice

experiment: an observer is confronted with one positive and one negative case, both randomly chosen. The observer must select the positive case. What is the chance that the observer does this correctly?

Pn(x) Pp(x)

x

true positive fraction

x

pnz xPdxxPdxA )'( ' )(

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

false pos itive probability

true

pos

itive

pro

babi

lity

width false positive fraction column

x

pn xPdxxPdx )'( ' )( exp. AFC-2decision correct chance

Az as a segmentation performance measure

• Ranges from 0.5 to 1• Soft labeling is required (not easy for humans in

segmentation)• Independent of system threshold (operating

point) and prevalence (priors)• Depends on ‘amount of background’ though!

Summary

• Various pixel-based measures were considered for two class, hard (binary) classification results:– Accuracy– Sensitivity, specificity– Overlap– Kappa

• ROC