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Parametric measures to estimate and predict performance of identification techniques
Amos Y. Johnson & Aaron Bobick
STATISTICAL METHODS FOR COMPUTATIONAL EXPERIMENTS
IN VISUAL PROCESSING & COMPUTER VISIONNIPS 2002
Setup – for example
Given a particular human identification technique
Setup – for example
Given a particular human identification technique This technique measures 1 feature (q) from n individuals
n321 q ... q q qx
- 1D Feature Space -
Setup – for example
Given a particular human identification technique This technique measures 1 feature (q) from n individuals Measure the feature again
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Setup – for example
Given a particular human identification technique This technique measures 1 feature (q) from n individuals Measure the feature again
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Gallery
Probe
Setup – for example
Given a particular human identification technique This technique measures 1 feature (q) from n individuals Measure the feature again
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Gallery
Probe
For template
Target
Setup – for example
Given a particular human identification technique This technique measures 1 feature (q) from n individuals Measure the feature again
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Gallery
Probe
For template
Target Imposters
Question
For a given human identification technique, how should identification performance be evaluated?
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Gallery
Probe
For template
Target Imposters
Possible ways to evaluate performance
For a given classification threshold, compute False accept rate (FAR) of impostors Correct accept rate (HIT) of genuine targets
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Gallery
Probe
For template
Target Imposters
Possible ways to evaluate performance
For various classification thresholds, plot Multiple FAR and HIT rates (ROC curve)
Possible ways to evaluate performance
For various classification thresholds, plot Multiple FAR and HIT rates (ROC curve) Compute area under a ROC curve (AUROC)
Probability of correctclassification
Possible ways to evaluate performance
For various classification thresholds, plot Multiple FAR and HIT rates (ROC curve) Compute 1 - area under a ROC curve (1 -AUROC)
Probability of incorrectclassification
Problem Database size
If the database is not of sufficient size, then results may not estimate or predict performance on a larger population of people.
1 - AUROC
Our Goal
To estimate and predict identification performance with a small number subjects
1 - AUROC
Our Solution
Derive two parametric measures Expected Confusion (EC) Transformed Expected-Confusion (EC*)
Our Solution
Derive two parametric measures Expected Confusion (EC) Transformed Expected-Confusion (EC*)
Probability that an imposter’s feature vector is withinthe measurement variation of a target’s template
Our Solution
Derive two parametric measures Expected Confusion (EC) Transformed Expected-Confusion (EC*)
Probability that an imposter’s feature vector is closer to a target’s template, than the target’s feature vector
Our Solution
Derive two parametric measures Expected Confusion (EC) Transformed Expected-Confusion (EC*)
EC* = 1 - AUROC
Expected Confusion
Probability that an imposter’s feature vector is within the measurement variation of a target’s template
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Gallery
Probe
For template
Target Imposters
Expected Confusion - Uniform
The templates of the n individuals, are from an uniform density Pp(x) = 1/n
- 1D Feature Space -
n321 q ... q q qx
P(x)
1/nPp(x)
Expected Confusion - Uniform
The measurement variation of a template is also uniform Pi(x) = 1/m
- 1D Feature Space -
n321 q ... q q qx
P(x)
1/nPp(x)
1/mPi(x)
Expected Confusion - Uniform
The probability that an imposter’s feature vector is within the measurement variation of template q3 is the area of overlap
True if m << n
- 1D Feature Space -
n321 q ... q q qx
P(x)
1/nPp(x)
1/mPi(x)
n
mA )q( 3
Expected Confusion - Uniform
The probability that an imposter’s feature vector is within the measurement variation of any template q
True if m << n
n321 q ... q q qx
P(x)
1/nPp(x)
1/mPi(x)
n
mA )q( 3
n
mdqPAEC p
)q()q(
Following the same analysis, for the multidimensional Gaussian case
Expected Confusion - Gaussian
),()q( ppp Np : Population density
),q()( ii Nxp : Measurement variation
Expected Confusion - Gaussian
Following the same analysis, for the multidimensional Gaussian case True if the measurement variation is significantly less then the population variation
2/1
2/1
||
||EC
p
i
Probability that an imposter’s feature vector is within the measurement variation of a target’s template
Expected Confusion - Gaussian
Relationship to other metrics Mutual Information
The negative natural log of the EC is the mutual information of two Gaussian densities
)|ln(|)|ln(|)||
||ln( 2/12/1
2/1
2/1
ipp
i
Transformed Expected-Confusion
Probability that an imposter’s feature vector is closer to a target’s template, than the target’s feature vector
- 1D Feature Space -
n321 q ... q q qx
''''
n321q ... q q q
Gallery
Probe
For template
Target Imposters
Transformed Expected-Confusion
First: We find the probability that a target’s feature vector is some distance k away from its template
n321 q ... q q qx
''''
n321q ... q q q
For template
Target Imposters
k
)( dkkpqt
n321 q ... q q qx
''''
n321q ... q q q
For template
Target Imposters
k
)( dkkpqt
Transformed Expected-Confusion
Second: We find the probability that an imposter’s feature vector is less than or equal to that distance k
k
qim dvvp
0
)(
n321 q ... q q qx
''''
n321q ... q q q
Target Imposters
k
Transformed Expected-Confusion
Therefore: The probability that an imposter’s feature is closer to the target’s template, than the target’s feature (for a distance k) is
dkdqdvvpkpqpk
qim
qtp
00
)()()(
n321 q ... q q qx
''''
n321q ... q q q
Target Imposters
k
Transformed Expected-Confusion
Therefore: The probability that an imposter’s feature is closer to the target’s template, than the target’s feature (for any distance k) is
dkdqdvvpkpqpk
qim
qtp
00
)()()(
x
''''
n321q ... q q q
Transformed Expected-Confusion
Therefore: The expected value of this probability over all target’s templates is
dkdqdvvpkpqpk
qim
qtp
00
)()()(
n321 q ... q q q
Transformed Expected-Confusion
Next: Replace the density of the distance between a target’s feature-vectors and its template q
dkdqdvvpkpqpECk
qim
qtp
00
)()()(*
)(kpt
Transformed Expected-Confusion
Answer: Probability that an imposter’s feature vector is closer to a target’s template, than the target’s feature vector
dkdqdvvpkpqpECk
qim
qtp
00
)()()(*
Transformed Expected-Confusion
This probability can be shown to be one minus the area under a ROC curve
Following the analysis of Green and Swets (1966)
dkdqdvvpkpqpECk
qim
qtp
00
)()()(*
Transformed Expected-Confusion
Integrate: With these assumptions
dkdqdvvpkpqpECk
qim
qtp
00
)()()(*
Transformed Expected-Confusion
Integrate: With these assumptions
dkdqdvvpkpqpECk
qim
qtp
00
)()()(*
),;( ppqN
Transformed Expected-Confusion
Integrate: With these assumptions
dkdqdvvpkpqpECk
qim
qtp
00
)()()(*
),;( ppqN
2
2
2)12/(
2/)2(
)2/(2i
k
ddi
d
ed
k
Transformed Expected-Confusion
Integrate: With these assumptions
),;( ppqN d
dp kVqp )(
2
2
2)12/(
2/)2(
)2/(2i
k
ddi
d
ed
k
dkdqdvvpkpqpECk
qim
qtp
00
)()()(*
Transformed Expected-Confusion
Integrate: Probability that an imposter’s feature vector is closer to a target’s template, than the target’s feature vector
dkdqdvvpkpqpECk
qim
qtp
00
)()()(* ECd
dVd
d
)2/()2(
)!1(2/
Transformed Expected-Confusion
Compare: EC* with 1 - AUROC
EC* = 1 - AUROC
Conclusion
Derive two parametric measures Expected Confusion
(EC) Transformed
Expected-Confusion (EC*)
Probability that an imposter’s feature vector is closer
to a target’s template, than the target’s feature vector
Conclusion
Derive two parametric measures Expected Confusion
(EC) Transformed
Expected-Confusion (EC*)
Probability that an imposter’s feature vector is within
the measurement variation of a target’s template
Probability that an imposter’s feature vector is closer
to a target’s template, than the target’s feature vector
Conclusion
Derive two parametric measures Expected Confusion
(EC) Transformed
Expected-Confusion (EC*)
Probability that an imposter’s feature vector is within
the measurement variation of a target’s template
Probability that an imposter’s feature vector is closer
to a target’s template, than the target’s feature vector
Future Work
Developing a mathematical model of the cumulative match characteristic (CMC) curve Benefit: To predict how the CMC curve
changes as more subjects are added
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