Quantifying Fingerprint Evidence using Bayesian Alignment

Transcription

1 Quantifying Fingerprint Evidence using Bayesian Alignment Peter Forbes Joint work with Steffen Lauritzen and Jesper Møller Department of Statistics University of Oxford UCL CSML Lunch Talk 14 February 2014

2 History of fingerprints Fingerprints have been used to authenticate legal documents in China since 300 BC Scottish missionary Henry Faulds first used fingerprints for forensic identification in 1880 Sir Francis Galton established in 1892 that fingerprints are invariant over time and crudely estimated that Probability two fingerprints are identical = 1/68 billion Yet crime scene prints are often partial and blurry

3 Strength of evidence For its entire history, fingerprint evidence has been presented categorically Of all the methods of identification, fingerprinting alone has proved to be both infallible and feasible (FBI training manual, 1963) From a statistical viewpoint, the scientific foundation for fingerprint individuality is incredibly weak... there has been much speculation and little data. None of the models has been subjected to testing, which is of course the basic element of the scientific approach (Stoney, 2001)

4 Motivation Fingerprint evidence is forensic acme: UK has 330,000 crime scene prints collected each year and 34,000 identifications Recent push to follow DNA evidence and quantify the uncertainty (Neumann et al., 2012) My model attempts this by computing the likelihood ratio between two competing models: H p : A and B originate from the same finger H d : A and B originate from independent fingers. Test using a public dataset from the NIST-FBI (Garris and McCabe, 2000)

5 Fingerprints as minutia sets Figure: An example fingerprint with minutiae labelled Minutiae are points where epidermal ridges end or bifurcate Represent with m = (r C, s S 1, t {0, 1}), where S 1 is the unit circle embedded in C Most fingers have around 135 minutiae

6 Preliminaries Define the complex Normal distribution on C n as Z CN n (µ, Σ) Re(Z µ), Im(Z µ) iid N n(0, Σ/2) with density ϕ n (z; µ, Σ) Define the root von Mises distribution on S 1 as X, Y iid rvm(κ) Pr(XY ) = exp{κre(x Ȳ )}/{2πI 0 (κ)} A set of points A V form a marked Poisson point process with rate ρ and mark distribution g (denoted MPPP(ρ, g)) iff Number of points in any two disjoint regions are independent Number of points in v V Poisson( v ρ(r)dr) Each point has a mark, and marks are iid with density g Conditional on A, Pr(A) r A ρ(r)g(mark(r))

7 Model for latent finger Like Green and Mardia (2006), we view the observed point sets as partial, distorted copies of a latent true point set Latent minutiae are distributed as MPPP(ρ, g) on C with ρ(r) = ρ 0 ϕ 1 (r; 0, σ 2 ) and g(s, t) = p t (1 p) 1 t /(2π) for ρ 0 > 0, σ > 0 and p (0, 1)

8 Model for observed minutia sets m l observed w.p. q A {m a = (r a, s a, t a )} r a r a + CN(0, ω 2 ) s a s a + rvm(κ) Latent finger {m l = (r l, s l, t l )} Independent Binomial Thinning q A, q B (0, 1) Independent Observation Errors ω, κ > 0 m l observed w.p. q B {m b = (r b, s b, t b )} r b r b + CN(0, ω 2 ) s b s b + rvm(κ) r a ψ A (r a + τ A ) s b ψ A s b Rigid motion τ A, τ B C 2 ψ A, ψ B S 1 r a ψ B (r a + τ B ) s b ψ B s b Fingerprint Fingermark A = {(r a, s a, t a )} Observed minutia sets B = {(r b, s b, t b )}

9 Introduction Model Example fingerprint Algorithm Results Future work References

10 Example high-quality fingermark

11 Introduction Model Algorithm Example overlaid with a rigid motion Results Future work References

12 Finding the densities Desire the likelihood ratio LR = Pr(A, B H p) Pr(A, B H d ) First we need to find the densities under H d and H p These will depend on the constants ρ 0, p, ω, κ Will also depend on the variables θ = (q A, q B, τ A, τ B, ψ, σ)

13 Finding the densities Under H d, A and B are from independent latent fingers. By integrating over the latent minutia, we have Pr(A, B θ, H d ) ρ A (r a )g(s a, t a ) (r a,s a,t a) A ρ B (r b )g(s b, t b ) where ρ A and ρ B are given by (r b,s b,t b ) B ρ A (r a ) = ρ 0 q A ϕ 1 (r a ; τ A, σ 2 + ω 2 ) ρ B (r b ) = ρ 0 q B ϕ 1 (r b ; τ B, σ 2 + ω 2 ) This can be integrated analytically over θ

14 Finding the densities Under H p we observe A = M 10 {m a : (m a, m b ) M 11 } and B = M 01 {m b : (m a, m b ) M 11 } where M 10 MPPP((1 q B )ρ A ( ), g) are the points observed in A not B M 01 MPPP((1 q A )ρ B ( ), g) are the points observed in B not A M 11 MPPP(ρ 11, g 11 ) are the points observed in both A and B Note M 11 is a MPPP on C 2. Integrating over the latent true minutia (r, s, t), we have (( ) ra ρ 11 (r a, r b ) = ρ 0 q A q B ϕ 2 ; r b ( τa τ B ) ( σ, 2 + ω 2 σ 2 )) ψ σ 2 ψ σ 2 + ω 2 g 11 (s a, t a, s b, t b ) = I(t a = t b ) pta (1 p) 1 ta 4π 2 exp{κre(s a s b ψ)} I 0 (κ) where ψ = ψ A ψb is the rotation between A and B

15 Finding the densities The density under H p is thus Pr(A, B θ, H p ) (1 q B )ρ A (r a )g(s a, t a ) (r a,s a,t a) M 01 (1 q A )ρ B (r b )g(s b, t b ) (r b,s b,t b ) M 10 (r a,s a,t a) (r b,s b,t b ) M 11 ρ 11 (r a, r b )g 11 (s a, t a, s b, t b ) but we only observe A = M 10 {m a : (m a, m b ) M 11 } and B = M 01 {m b : (m a, m b ) M 11 }! Need to treat the matching ξ between A and B as an unknown variable and sum over its possible values

16 Computing the likelihood ratio τ A, τ B, ψ, σ are assigned (improper) flat priors π, which ensures LR is invariant under similarity transformations q A has a conjugate Beta prior with hyperparameters α, β q B has a flat prior ρ 0, p, ω, κ, α, β are constants to be estimated by MLE Find LR by marginalizing over θ and ξ ξ Pr(A, B, ξ θ, Hp )π(θ)dθ LR = Pr(A, B θ, Hd )π(θ)dθ Sum in numerator contains many terms min( A, B ) n=0 Use MCMC to approximate the LR A! B! n!( A n)!( B n)! 10100

17 An estimate for LR Define the joint distribution of (θ, ξ, H) by { p 0 Pr(A, B, ξ θ, H p )π(θ) if H = H p, Pr(θ, ξ, H, A, B) = (1 p 0 )Pr(A, B H d )q(ξ θ)q(θ) if H = H d. where p 0 (0, 1) and the densities q are chosen to promote good mixing of our MCMC over the model space By sampling from this, we can approximate LR by replacing the below expectation with its sample average: LR = Pr(A, B H p) Pr(A, B H d ) = p 0 1 p 0 E θ,ξ,h A,B [I (H = H p )] E θ,ξ,h A,B [I (H = H d )].

18 Tuning the sampler To accurately estimate LR we must switch models often, i.e. p 0 Pr(A, B, ξ θ, H p )π(θ) (1 p 0 )Pr(A, B H d )q(ξ θ)q(θ) We have severe problems with local modes under H p due to the high dimensionality of ξ Attempting to tune q(θ) often resulted in tuning to the current local mode, so we use a fixed diffuse distribution We choose q(ξ θ) to approximate Pr(θ, ξ A, B, H p ) Put some arbitrary ordering on A and let ξ α = {(a, b) ξ : a < α}, B α = {b B : (a, b) ξ α } q(ξ θ) = A α=1 Pr(θ, ξ α+1, H p, A, B) b (B\B Pr(θ, ξ α) α (α, b), H p, A, B)

19 A better estimate for LR We want to choose p 0 so that p 0 Pr(A, B, ξ θ, H p )π(θ) (1 p 0 )Pr(A, B H d )q(ξ θ)q(θ) over a large portion of the state space (θ, ξ) This is very difficult, so we tune p 0 to ensure good mixing based on our previous samples Letting l(θ, ξ, A, B) = Pr(θ, ξ, A, B H p )/Pr(θ, ξ, A, B H d ), [ E θ,ξ A,B {1 p 0 + p 0 /l(θ, ξ, A, B)} 1] LR = [ E θ,ξ A,B {p 0 + (1 p 0 )l(θ, ξ, A, B)} 1]. Can show that replacing the expectations with sample averages leads to an valid estimator of LR even if we change p 0 each iteration! p0 n n m=n p0 n = {1 + 1/l(θm, ξ m, A, B)} 1 n m=n 100 {1 + l(θm, ξ m, A, B)} 1

20 Gibbs sampler Algorithm 1 Gibbs sampler for joint posterior of (θ, ξ, H) Require: θ 0, ξ 0 set to some initial value. Set H 0 = H p. for n = 1,..., N do if H = H p then (qa n, qn B ) Sample ( q A, q B A, B, τ n 1 A, τ n 1 B, σ n 1, ψ n 1, ξ n 1, H n 1) (τa n, τ B n) Sample ( τ A, τ B A, B, qa n, qn B, σn 1, ψ n 1, ξ n 1, H n 1) σ n Sample ( σ A, B, qa n, qn B, τ A n, τ B n, ψn 1, ξ n 1, H n 1) ψ n Sample ( ψ A, B, qa n, qn B, τ A n, τ B n, σn, ξ n 1, H n 1) ξ n ξ n 1 for j = 1,..., n A do do repeatedly to reduce autocorrelation ξ n Sample ( ξ A, B, qa n, qn B, τ A n, τ B n, σn, ψ n, ξ n, H n 1) end for end if p0 n adaptive value to increase mixing in H H n Sample (H A, B, qa n, qn B, τ A n, τ B n, σn, ψ n, ξ n ) end for Not quite reversible jump: we don t change states under H d This approach provides better model mixing when the proposal distributions q(θ, ξ) are far from the posterior distributions

21 ξ sampler Basic Metropolis Hastings algorithm like (Green and Mardia, 2006) has accept rates less than 10 5 infeasibly slow Instead we sample directly by reducing the adjacent states with an auxiliary variable α which takes values uniformly on A There are B + 1 states adjacent to any (α, ξ), obtained by matching α to each b B or leaving α unmatched α b α b α α b α b (a) Add (b) Swap b (c) Remove (d) Swap a (e) Swap Figure: Matches ξ which are adjacent to (α, ξ 0 ).

22 NIST-FBI dataset NIST-FBI dataset of 258 fingerprint/fingermark pairs All images have their minutiae labelled by expert examiners Split into three subsets (good, bad, and ugly) based on fingermark quality Figure: Example fingermarks from Garris and McCabe (2000).

23 Simulated dataset Generated 258 fingerprint/fingermark pairs based on model assumptions Split into three sets (good, bad, and ugly) in order of decreasing B Computed all pairwise likelihood ratios Computed LRs are almost entirely determined by B

24 MCMC behavior

25 MCMC behavior

26 Results: simulated data 0.15 G B U Histogram of the log 10 -likelihood ratios for good, bad and ugly simulated fingermarks. Inset ROC curve has false

27 Results: NIST-FBI data 0.15 G B U Histogram of the log 10 -likelihood ratios for good, bad and ugly NIST-FBI fingermarks. Inset ROC curve has false

28 Better model for latent minutiae The intensity for the latent finger MPPP is inaccurate, since most minutiae occur in areas of high minutia curvature Figure: Minutia density over various fingerprints from Chen and Jain (2009)

29 Better distortion model Basic model assumes observed minutiae vary from true minutiae by a rigid motion and iid noise Actually, nearby minutiae have spatially correlated distortions Account for this using a smoothing thin plate spline model, which leads to a Gaussian process for the distortions Figure: Example smoothing thin plate spline from Chui and Rangarajan (2000)

30 Conclusion We have developed a simple model to quantify the strength of evidence for forensic fingerprints Better latent distributions and distortion models will increase discrimination between true and false matches Must manage trade off between model complexity and computational efficiency Computed likelihood ratios should be calibrated against ground-truth database

31 Conclusion Chen, Y. and A. K. Jain (2009). Beyond minutiae: A fingerprint individuality model with pattern, ridge and pore features. Chui, H. and A. Rangarajan (2000). A new algorithm for non-rigid point matching. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, pp Garris, M. and R. McCabe (2000). NIST special database 27: Fingerprint minutiae from latent and matching tenprint images. Technical report, NIST, Gaithersburg, MD, USA. Green, P. J. and K. V. Mardia (2006). Bayesian alignment using hierarchical models, with applications in protein bioinformatics. Biometrika 93(2), Neumann, C., I. W. Evett, and J. E. Skerrett (2012). Quantifying the weight of evidence from a forensic fingerprint comparison: a new paradigm (with discussion). Journal of the Royal Statistical Society: Series A 175(2),