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Evaluation of segmentation

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Evaluation of segmentation

- Qualitative/subjective evaluation ïƒ 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

- 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

- 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

masked

true positive

true negative

false negative

false positive

Segmentation

Reference

- False positives are actually negative
- False negatives are actually positives

Segmentation

Reference

sensitivity = true positive fraction

= 1 â€“ false negative fraction

= TP / (TP + FN)

specificity = true negative fraction

= 1 â€“ false positive fraction

= TN / (TN + FP)

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

- Range: from 0 to 1
- Useful measure, but:
- Depends on prior probability (prevalence); in other words: on amount of background
- Even â€˜stupidâ€™ methods can achieve high accuracy (e.g. â€˜all backgroundâ€™, or â€˜most likely classâ€™ systems)

- Are intertwined
- â€˜stupidâ€™ 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

P

N

N

P

P

P

N

N

P

N

N

P

P

true positives (TP)

sensitivity = true positive fraction

= 1 â€“ false negative fraction

= TP / (TP + FN)

P

false positives (FP)

N

false negatives (FN)

N

true negatives (TN)

specificity = true negative fraction

= 1 â€“ false positive fraction

= TN / (TN + FP)

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

P

N

N

P

P

P

N

N

P

N

N

P

P

true positives (TP) = 3

P

false positives (FP) = 3

N

false negatives (FN) = 2

N

true 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

P

P

N

P

N

P

P

P

P

P

P

P

N

N

N

N

P

P

P

N

N

N

P

P

sensitivity = 4 / 5 = 0.8

P

N

= 4

= 1

algorithm 2

specificity = 2 / 7 = 0.29

P

N

= 5

= 2

accuracy = 6 / 12 = 0.5

sensitivity = 3 / 5 = 0.6

P

N

= 3

= 2

algorithm 1

specificity = 4 / 7 = 0.57

P

N

= 3

= 4

accuracy = 7 / 12 = 0.58

Which system is better?

Accuracy: 0.93949

Sensitivity: 0.668027

Specifity: 0.980443

Reference

Segmentation

TP

FN

FP

TN

- 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

- Accuracy would not be zero if we used a system that is â€˜guessingâ€™
- A â€˜guessingâ€™ system should get a â€˜zeroâ€™ mark (remember multiple choice examsâ€¦)
- Kappa is an attempt to measure â€˜accuracy in excess of accuracy expected by chanceâ€™

System accuracy:

(191152 + 19648)/

224377 = .939

Total number of positives

True positives of a

guessing system:

.105 * 29412 = 3075

â€¦ etc

Accuracy guessing

system: .792

System positive rate:

23461/224377 = .105

- accguess = the accuracy of a randomly guessing system with a given positive (or negative) rate
- kappa = (acc â€“ accguess) / (1 â€“ accguess)
- In our case: kappa = (.939 - .792)/(1 - .792) = .707

- Maximum value is 1, can be negative
- A â€˜guessingâ€™ system has kappa = 0
- â€˜Stupid systemsâ€™ (â€˜all backgroundâ€™ or â€˜most likely classâ€™) have kappa = 0
- Systems with negative kappa have â€˜worse than chanceâ€™ performance

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

- 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.

Pp(x)

Pn(x)

true positive fraction

x

false positive fraction

true negative fraction

true positive fraction

sensitivity

detection rate

false positive fraction

1 - specificity

chance of false alarm

- 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

- 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)

- 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.

- 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?

Pp(x)

Pn(x)

x

true positive fraction

width false positive fraction column

- 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 â€˜amount of backgroundâ€™ though!

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

- ROC