Extraction of Fingerprint Ridges from Pore Data
Bryony Hill with Elke Thönnes & Wilfrid Kendall
University of Warwick
① Overview of Fingerprints
③ Averaging Tensors and the Log-Euclidean Mean
The main components of fingerprints are ridgelines, sweat pores and
minutiae (ridge endings and bifurcations) at the local level,
and singularities at the
global level.
④ Results

To create a gradient field (undirected vector field) we need to be able to calculate a tensor at any
position – interpolated from the sparse set of tensors at each pore. It's convenient to smooth the
tensor field at the same time.
In order to interpolate we have to find a suitable definition of the mean of two tensors.
minutia
There are a number of possible means for tensors, including:
sweat pore
●
singularity (loop)
Euclidean mean - averaging
tensors element by element
M=
ridgeline
●

This doesn't give very good results as the
determinant of M may be bigger than the
determinants of both A and B.

With this mean we lose some of the information
given by the tensors: the measure of directional
dependence called anisotropy.
② Outline of Problem
●
Fingerprint
Filtering methods can be used to
extract most of the pores from the
fingerprint.
Pores
Fingerprint used here is a section from print a00205 from the NIST Special Database 30 (Dual
resolution images from paired fingerprint cards)
At each pore an inertia tensor
can be calculated [1] which
indicates the prominent
direction of neighbouring pores.
dist  A ,B = ∥ log A −1/2 B A −1/2  ∥
●
The log-Euclidean mean [2] is
another geometric mean, but here
multiplication is defined by
M = exp


1
[log A logB ]
2
The general formula for the mean is
M = exp
Despite the generated gradient field matching closely with the underlying
fingerprint in most central areas, there are parts which differ greatly - often
due to a large number of missing pores or areas where the fingerprint ridges
have high curvature.
The geometric mean has better theoretical
properties but computationally it takes a long time
to calculate.
It would be interesting to see how the log-Euclidean mean affects the
anisotropy of tensors, and potentially use this information to improve the
gradient field. Other areas of further work include identifying a set of
streamlines to represent the ridgelines and finding methods to compare
gradient fields.
∑ w
i
log A i 
The log-Euclidean mean has most of the good
theoretical properties of the geometric mean but is
a lot easier to calculate.
Using the isomorphism between positive semi-definite matrices
(tensors) and general
symmetric matrices,
3
ℝ
we can convert all
tensor problems to
symmetric matrix
problems.
X = log A 

where the w i are weights, typically
dependent on some distance
Above: arrows denote the directions
of the eigenvectors of each tensor –
lengths are proportional to the
corresponding eigenvalues
⑤ Known Issues and Further Work
to get M = A 1/ 2 A −1/ 2 BA −1/ 2 1/2 A 1/ 2
so the mean of two tensors
becomes
Inertia tensors are symmetric
matrices with non-negative
eigenvalues: compare
covariance matrices.
How can we use the
information given by the
tensors calculated at the pores
to find the ridgeline pattern?
The geometric mean [3] uses the
distance function
A∗B = exp log AlogB
Tensors
principal
eigenvector of the
mean tensor.
1 A 11 B 11 A 12 B 12
2 A 12 B 12 A 22 B 22
Averaging the direction of the
principal eigenvectors
A = exp X 
Cone of tensors

I calculated a tensor at each point x on the
−d i , x 2
fingerprint using the log-Euclidean mean with weights w i = exp
2h
corresponding to
the tensor at the
Left:
i-th pore where
Gradient field
d  i , x is the
generated by logdistance between
Euclidean mean
x and the i-th pore
(h=30). Green lines
and h is the
indicate the
direction of the
scaling factor.
Where
A=
and
X=




A 11 A 12
A 12 A 22
X 11 X 12
X 12 X 22
⑥ References
[1] Jionglong Su, PhD Thesis, Warwick, 2008.
[2] Arsigny et al., Log-Euclidean Metrics for Fast and Simple Calculus on
Diffusion Tensors, Magnetic Resonance in Medicine, Vol. 56, No. 2.
(August 2006), pp. 411-421.
[3] Bhatia, Positive Definite Matrices, Princeton, 2007.
[4] Maltoni et al., Handbook of Fingerprint Recognition, Springer, 2003.