The Representation of Fingerprint Minutiae as Defects in a ...dieter/courses/Math... · The...

The Representation of Fingerprint Minutiae as Defects in a Pattern-Formation System

Jonathan Alfson, David Hjelmstad, Lucas Malin, Dominick Ortiz, Wacey Teller

Main Idea

•  Main strengths of pattern formation models lie in Level 1 detail !  Some Level 2-type defects are observed, but they are not distributed on

the fingerprint in a way that is statistically consistent with real fingerprints

•  Thus the ability to insert defects into patterns would allow for the generation of statistically realistic fingerprints

•  We present a mathematical characterization of defects, a basic proof of concept via tests on simple pattern systems, and relate back to the end goal of generating fingerprints.

Ending Ridge Bifurcation Dot

Three Types of Level 2 Characteristics

Citation: Eric Ray, “An Overview of Fingerprints for MAT 451”

Newell-Whitehead-Segel (NWS) Equation

•  Nonlinear Partial Differential Equation •  Relates Amplitude to spatial components x and y as well as to time t •  Useful for describing defects – i.e. points where the amplitudes are

zero •  Can be analytically solved by making certain assumptions

Assumptions

•  Solution to amplitude is a function of phase •  Thus, the NWS equation applies to phase •  The steady-state solution for the phase will result in stable patterns

in time

Field near defects Assuming we are near the core of the defect, we get the following:

Field far from defects Assuming we are far from the core of the defect, we get the following:

Growing a Defect

•  Place a structure such as the previous two functions into a field of rolls

•  Run a reaction-diffusion model with the above as initial conditions •  Examine result to see what kind of defect has grown

‘erf’ function initial conditions

Step size N=59

‘erf’ function result

Step size N=59

Result

•  There appears to be a defect, most likely a bifurcation with some noise

•  Resolution is too low to allow for proper growth/calculation of defect

Determination of steady state roll pattern

u0 = sin2(3! y)

Observation: Steady state is reached in about .015 seconds

v0 = cos2(3! y)

Steady state pattern

•  This state is used as the basis for the initial conditions in the experiments that follow

Case 1 Enforce a short line of zeros in center for .005 seconds then release

u: v:

Case 1 After the release:

v: u:

Case 1 Resulting steady state

u:

Case 1 Resulting steady state

v:

Case 2 Enforce zero along line from center to one edge for .005 s

u:

"  Ridge Ending

Case 2 Enforce zero along line from center to one edge for .005 s

v:

"  Bifurcation

Case 3 Same as Case 2, but hold for less time (.0025 s)

u:

"  Illustrates the importance of timing: By reducing the time held at zero, the discrepancy between the amplitudes before and after the defect occurs is reduced.

Case 3 Same as 2, but enforce for less time (.0025 s)

v:

Case 4 !but don’t go too far

"  Holding at zero for too short of a time results in ‘healing’ of the defect

Case 4 !but don’t go too far

Case 5 Use a rectangle as wide as the ridges instead of just a line

u: v:

"  Produces results similar to those of Case 3 when held for a very short time. Holding longer produces an enlarged middle ridge in the steady state

Bifurcation on a larger domain

Bifurcation on a larger domain Top view with simple filter applied

Ridge ending on a larger domain

Ridge ending on a larger domain With simple filter applied

Dot?

"  Much more difficult to produce by these methods

Defects of a more Natural Look

•  To give the finger prints a more natural look. A controlled randomness was added to the initial grid

•  The randomness was taking the set grid and taking some set percentage of the values than adding it back on.

Comparison

Non-randomized Bifurcation With randomization

Initial grid

Without randomization With 50% randomization

Conclusion

•  The “growth” of defects in the steady state of a pattern-formation system has been shown to be possible

•  To implement in the generation of fingerprints, one would need to have the probability distribution of all minutiae. An algorithm could then be developed to distribute the defects randomly, but according to the probabilities via a system of weights.

Acknowledgements

•  Bruno Welfert !  Use of Schnakenberg pattern formation code

•  Keith Hjelmstad

!  Assistance with numerical methods in second version of code