Post on 19-Mar-2020
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Machine learning for Signature Verification
Sargur Srihari (with Harish Srinivasan and Matthew Beal)
Center of Excellence for Document Analysis and Recognition (CEDAR)
Department of Computer Science and EngineeringUniversity at Buffalo
State University of New York, USA
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Decision Task in SignatureVerification1:n Verification
Known Signatures Questioned (unknown) Signature
1.
2.… …… Questioned is
Genuine/Forgery/Unknown
Verification process
n.
n is small unlike in other machine learning domains
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Overview of Presentation1. Machine Learning Terminology2. Decision Task for Signatures3. Inference Strategies
1. Person-dependent 2. Person-independent
4. Methods of Comparing Distributions5. Similarity measure for Signatures6. Experimental Results7. Summary and Future Work
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Machine Learning Terminology
• Two stages in a machine learning system– Inference Stage
• Learning from samples• Training
– Decision Stage• Perform the task on given data• Execution
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General Learning and Special Learning
• Standard machine learning assumes large n which is not always the case in QD
• For signature analysis we introduce two types of inference: 1. Special (person dependent) 2. General (person independent)
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Person-dependent InferenceGenuine (Known) Signatures
},..,,{ 21 ngggG =
Questioned Signature Q
Decision is a one-class problem: whether Q belongs to G
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Person dependent Inference MethodGenuine Pairs (G x G)
InferenceSimilarity or distancefunction d(gi,gj)applied toknown pairsyields Genuine ( G x G)distribution
G x G = {.4, .1, .2, .3, .2, .1}n2 = 6
d
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Person dependent Decision MethodStep 1: Questioned vs Knowns yields Q x G Distribution
Questioned (Q)a
b
c
d
d(q,gi)yieldsn values
Q x G = {0, .4, .1, .2}
Step 2: Compare G x G and Q x G distributions
G x G = .4,.1,.2,.3,.2,.1
Statistical test tocompare distributions yields confidence of matchQ x G = 0,.4,.1,.2
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Person independent inferenceGenuinesignatures },..,{},,..,,{ 121 iimiim ggGGGGG ==
},..,{},,..,,{ 121 iimiim ffFFFFF ==Forgery(Impostor)signatures
Construct Genuine and Impostor distributions
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Genuine and Impostor Distributions
Impostor
Genuine
Person IndependentLearningGenuine: Gi x GiImpostor: Gi x Fi
Distance between pairs modeled as gamma distributions(since distances are positive)
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Person Independent DecisionCompare questioned againsteach of n genuines
Decision madeon each element ofQ x G
Impostor
Genuine
)/()/(log
dimpostorPdgenuinePLLR =
if LLR > α => Genuine, else Forgery (α learnt from ROC)For n knowns n LLR values are added (assuming independent knowns)
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Methods for Comparing Distributions
• Kolmogorov Smirnov (KS)• Kullback Leibler (KL) Divergence• Reverse KL• Symmetric KL• Jensen-Shannon KL
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Statistical TestsKolmogorov Smirnov (KS) Test: tell if two data sets drawn from same distribution.
Computes statistic D: max value of absolute difference between two cdfs.
Mapping to Probability (with which Q belongs to K)
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Kolmogorov Smirnov TestBased on Maximum difference between Two C.D.F.
G x G Distribution Q x G Distribution.4,.1,.2,.3,.2,.1 0,.4,.1,.2
sort sort
.1,.1,.2,.2,.3,.4 0,.1,.2,.4C. D. F.
0. 000. 200. 400. 60
0. 801. 001. 20
0. 00 1. 00 2. 00 3. 00 4. 00 5. 00Di st r i but i on
P(X
<= x
)
C. D. F.
0. 000. 200. 400. 60
0. 801. 001. 20
0. 00 1. 00 2. 00 3. 00 4. 00 5. 00Di st r i but i on
P (X
<=
x)
MaximumDifferenceBetweenTwoC.D.F. DKS = 0.333
DKS = 0.333 is mapped to Probability using formula on previous slide 89.98%Which is the confidence with which Q belongs to K
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Kullback Leibler Divergence
To measure similarities between distributions.B = total number of bins in the distribution. Pb and Qb are probabilities of the distributions in the bth bin. PKL = probability of similarity of the two distributions.
Combined KL and KS probabilistic measure = Average of the resulting probabilities of KL and KS.
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KL CalculationG x G Distribution Q x G Distribution
.4,.1,.2,.3,.2,.1 0,.4,.1,.2
Binning BinningSay B=2
Bins 0 – 0. 2 0.2 – 0.4
P.D.F. 4 / 6 = 0. 66
2 / 6 = 0 . 33
Bins 0 – 0. 2 0.2 – 0.4
P.D.F. 3 / 4 = 0. 75
1 / 4 = 0 . 25
DKL = (0.66* log(0.66/0.75) ) + (0.33 * log(0.33/0.25)) = 0.0072
PKL = exp(-0.0072) = 0.9928 = 99.28 %
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Other information theoretic measures
PKLKS : Combined KL and KS probabilistic measure = Average of the resulting probabilities of KL and KS.
Empirically proves to be thebest measure
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Data SetGenuine signatures (1320): 55 individuals , 24 signatures each
Forgery signatures (1320): 55 individuals , 24 signatures each
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Feature Vector
Gradient (12 bits): 111101111111Structural (12 bits): 000011001100Concavity (8 bits): 10100000
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( )
s"1" have rsboth vecto wherebits ofnumber theis s"0" have rsboth vecto wherebits ofnumber theis
10245.0 score Similarity
bits 10248481212 bits Total
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00
1100
CC
CC +×=
=××++=
*Described in paper at IWFHR, Tokyo, Nov. 2004
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Similarity/Distance measure
Similarity measure between two feature vectors
Hence transformto distance spacefrom features space
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Person independent verification accuracy
LLR distribution ROC curves
Each sub plot corresponds to training on different n
For n = 20 Error rate = 21.3 %
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Person dependent verification accuracyError vs Reject rateTable of error rates
For n = 20: Error rate = 16.40 %Cannot be used for n <4
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Summary• Two learning strategies introduced for small
sample learning with signatures
• Method 1:General Learning (Person Independent) • Inference
Does not require multiple( >1) GenuinesRequires forgery samples (of other writers)
• DecisionTwo-class problemBased on LLR from Gamma distributions
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Summary and Future Work• Method 2: Special Learning (Person dependent) • Inference
Requires multiple Genuine samples No forgery samples required DecisionOne-class problemBased on KSKL comparison of distributions
• Method 2 has 5% better accuracy with n=20
• Combination of methods needs investigation