Statistical Methods for Automatic Interpretation of Digitally Scanned Fingerprints KV Mardia, AJ ...

29 Pages·2001·3.09 MB·English

Statistical Methods for Automatic
Interpretation of Digitally Scanned Fingerprints
K.V. Mardia, A.J. Baczkowski, X. Feng, T.J. Hainsworth
Department of Statistics, University of Leeds,
Leeds LS2 9JT, U.K.
Internal Report STAT 97/23
December 1997
Statistical Methods for Automatic
Interpretation of Digitally Scanned Fingerprints
K.V. Mardia, A.J. Baczkowski, X. Feng, T.J. Hainsworth
Department of Statistics, University of Leeds, Leeds LS2 9JT, U.K.
Internal Report STAT 97/23
December 1997
Abstract
Orientation flow field is the starting point for feature extraction in various fingerprint
algorithms. Inrecentpapers,Jainandhiscolleagues(Rathaetal.,1995,KaruandJain,
1995, Ratha et al. 1996) introduced a fundamentally new algorithm. Their main moti-
vation for the algorithm has been to work at the problems from a fingerprint matching
point of view. Our aim in contrast has been feature extraction from a dermatoglyphic
point of view. Thus, we are not only interested in classification of the fingerprints but
also other features such as ridge counts (number of ridges between any two points,
say core points and tri-radius) and the average width of ridges in a region of finger-
print (see Loesch, 1983). Our end product is calculation of quantitative features rather
than identification. Again, manual calculations of such quantities is a highly tedious
task. We use some advanced techniques of spatial statistics, directional data analysis
and Bayesian image analysis to obtain the orientation field. The algorithm is used to
extract various features of interest from fingerprints.
1 Introduction
This paper focuses on statistical techniques for extracting structural features from
fingerprint images. We suppose that a fingerprint impression has been taken and is
digitized to give an image, typically containing 400 × 400 pixels, which can be then
1
processed or enhanced in some way to overcome the presence of noise which arises from
an inadequate initial impression or a poor procedure.
To improve fingerprint image quality, directional ridge enhancement is commonly
employed. Fingerprint images can be considered as an oriented texture pattern with
the lines on the image giving the ridges and valleys of the fingerprint. These lines
flow in a locally constant direction; see for example the fingerprint image shown in
Figure 1, see Section §2. This fingerprint is used in this paper for illustrative purposes.
Computing this local direction at each point of the image defines the orientation field
for the image.
In this paper we use the spatial structure of the fingerprint image to provide in-
formation about the orientation field. This spatial structure is summarized using the
semi-variogram, a basic tool of spatial statistics (see, for example, Cressie, 1991).
The sample semi-variogram gives not only the local orientation field but also the
distance between neighbouring lines in the image, and in turn the distance between the
ridges or valleys of the fingerprint.
To smooth the orientation field produced from this semi-variogram approach, we
use techniques of directional statistics (see for example, Mardia, 1972). To further
enhance the orientation field we use a Bayesian framework with a suitable prior for
the directional field. For computation of the posterior mode, we used the iterative
conditional mode procedure of Besag (1986). Most methods for computing orientation
field are based on the variance of groups of pixels, for example, Mehtre (1993) used
the intensity variance in nine directions while Sherlock et al. (1994) projected along 16
directionsandchosethedirectionwithmaximumvariance. Thereareotherapproaches;
for example, Kawagoe and Tojo (1984) divided a fingerprint image into subregions and
estimated the average direction of each subregion by counting micro-patterns of the
image, while Karu and Jain (1996) used a method based on the grey-value sum of
pixels. Our method appears to give better results than that of Karu and Jain (1996).
To extract information about the ridges we use the mean grey level of blocks of
pixels aligned with the orientation field at any location. Other techniques include, for
example, thresholdingto give a binary image, due to Mehtre (1993), but see also Ratha
et al. (1995). The advantage of our method is that it does not require a large window
about each pixel of interest since the semi-variogram provides the necessary distance
2
information. Using a classical thinning algorithm (Pavlidis, 1982, pp.195-209), we thin
the ridges to give a skeleton image with ridges having width of one pixel only. From
this we extract count information about the ridges. Our ridge counting method gives
very low bias and mean square error, superior to those based on using the grey-level of
pixels.
The orientation field can also be used to segment the fingerprint image into regions
reflectingthequalityoftheoriginalimage. Wedothisusingamovingwindowtoassess
the local pixel grey value variance where the window size depends upon the distance
between local ridges estimated from the semi-variogram and does not need to be set
manually as in the procedure of Ratha et al. (1995).
Wefurtheruseourorientationfieldforclassificationofthefingerprintsbyextracting
feature points such as core and tri-radius points from the fingerprint. This application
is supplemented by incorporating Karu and Jain’s (1996) method to detect the “flow
pattern” of the orientation field. The method involves the use of the Poincare´ index,
see Rosen (1970) and Kawagoe and Tojo (1984).
2 Orientation field obtained from semi-variogram
Suppose that a fingerprint image is made up of a m×n rectangular array of pixels.
For a pixel located at (i,j) let its grey level value be x . A fingerprint image is then
ij
described by the array x = {x : i = 1,···,m and j = 1,···,n}, where i and j are
ij
row and column labels respectively.
For each pixel (i,j), we obtain the sample semi-variogram along each of 16 equi-
spaced directions making angle α with the horizontal axis, where α = πd/16 for
d d
d = 0,···,15. To ensure that the calculated sample semi-variogram reflects the local
spatial structure about each pixel we only include a maximum of (2r +1) pixel grey
level values centred on (i,j) in the semi-variogram. Typically we used r ≈ 30. The
sample semi-variogram at lag h and direction d centred on pixel (i,j) is given by,
1 r−h
g (h;d) = (y −y )2,
ij 2(2r+1−h) (cid:88) k k+h
k=−r
where y denotes the grey level value at a distance k from (i,j) along line d. Thus
k
3
for d = 0, y = x . For an arbitrary direction d, y is assigned the grey level value
k ik k
of the pixel at the corresponding position. To ensure that g (h;d) is averaged over a
ij
sufficient number of terms we only consider lags h up to a maximum of 40 in this work.
Figure 1 shows a blurred fingerprint which will be used for illustrative purposes
throughout this paper. Figure 2 shows the semi-variogram functions plotted against
lag h for a selection of directions d for a typical pixel (i,j) located in the fingerprint
image of Figure 1.
INSERT FIGURE 1.
INSERT FIGURE 2.
Itisobservedthatasthedirectiondapproachesthedirectionorthogonaltotheridge
direction, the semi-variogram plots become more cyclical and the average distances
between any two neighbouring local minima become smaller.
This observation is easily explained. Suppose that the ridges can be represented by
parallellinesacommondistanceH apart. Foradirectionmakingangleθ withtheridge
lines, the distance between intersections of neighbouring ridges is H cosecθ which is
1
minimized for θ = π. At lags h ≈ H cosecθ the semi-variogram will be approximately
2
1
zero, and at lags h ≈ H cosecθ the semi-variogram will be averaged over many pairs
2
of pixels with one on a ridge and one in a valley which gives a large semi-variogram
value.
The algorithm for obtaining both the local direction of the orientation field and the
distances between ridgesfrom thesemi-variogramfunctionis summarized as follows for
pixel (i,j).
(1) Find the local minima of the semi-variograms in each direction d. These can be
found from the sample semi-variogram g (h;d) by comparison with the adjacent
ij
values g (h + 1;d) and g (h − 1;d). If n (d) minima are found, let H (k,d)
ij ij ij ij
denote the distance between the (k−1)th and kth minima, k = 2,···,n (d). To
ij
ensure that a local minima of the semi-variogram has been found we also check
g (h;d) with the values g (h±2;d) and g (h±3;d).
ij ij ij
Furthermore,ifwefindacasewithH (k,d) < 5thenwereplacethetwoapparent
ij
minima by one at their average location; this is done to avoid spurious minima
4
since the minimum distance between the ridges in the fingerprint images we have
used is greater than 5 pixels.
(2) Calculate the mean distance H¯ (d) between the pairs of neighbouring local min-
ij
ima for each direction d. If only 0 or 1 local minima were found then H¯ (d) is
ij
not defined.
(3) If there is some direction d(cid:48) giving a unique minimum mean, H¯ (d(cid:48)), then this
ij
direction is perpendicular to the local ridge orientation and H¯ (d(cid:48)) gives the dis-
ij
tancebetweenridgesintheneighbourhoodof(i,j). Ifthereareseveraldirections,
d ∈ D say, giving the same mean value H¯ (d) for all d ∈ D then proceed to step
ij
(4).
If H¯ (d) is not defined for all directions d then the orientation of the local ridges
ij
cannot be determined. In practice this does not occur, there being at least one
direction d for which H¯ (d) exists so that a minimum can be found.
ij
(4) Assess the cyclical behaviour of the semi-variogram function for each direction
d ∈ D by evaluating for each such direction d the sample variance c (d) of the
ij
distances H (k,d).
ij
Select the direction d(cid:48) which minimizes c (d) for d ∈ D. This direction gives the
ij
direction perpendicular to the local ridge direction. As before H¯ (d(cid:48)) gives the
ij
local distance between neighbouring ridges.
(cid:48)
Given this orthogonal direction d it is trivial to obtain the local direction d of
ij
the orientation field at pixel (i,j).
3 Smoothing the orientation field
The directions obtained from the semi-variogram approach are usually quite noisy and
thedirectionsforsomepixelscannotbeestimated. Thereforetheyneedtobesmoothed
and averaged in a local neighbourhood.
A method which is similar to that of Karu and Jain (1995) and Wilson et al. (1993)
is used but using a proper mean from the statistical theory of directional data (see for
example, Mardia, 1972). For a pixel located at (i,j) let the computed orientation field
have local direction d with corresponding angle α , α ∈ [0,π).
ij ij ij
5
To smooth the directional image, we present the direction α as vector. Since the
ij
angles α and π + α are not distinguished, this vector is not directed. We would
ij ij
describe α as axial data, see Mardia (1972, p1). The directed vector for direction α
ij ij
is γ = (cos2α ,sin2α ), where cos2α and sin2α are the Cartesian co-ordinates
ij ij ij ij ij
of the directed vector γ . Karu and Jain (1996) seem to have used this approach.
ij
The directional image can now be smoothed by averaging the two components in
γ separately. We used a 3×3 mean box filter (see Pavlidis, 1982, Section 3.4). The
ij
smoothed directional vector becomes,
1 9 1 9
γ¯ = (cid:32) cos2α , sin2α (cid:33),
ij 9 (cid:88) k 9 (cid:88) k
k=1 k=1
where α denotes the directions of the pixels in the 3×3 box centred at (i,j).
k
The smoothed direction α¯ can then be found from γ¯ = (cos2α¯ ,sin2α¯ ), see
ij ij ij ij
Mardia (1972, p26). The general angle α¯ is discretized to give a smoothed direction
ij
d¯ taking sixteen equi-spaced possible values. Repeating these calculations for every
ij
pixel gives a set of smoothed directions d for a fingerprint image, which we refer to as
the smoothed orientation field.
An alternative method for obtaining the orientation field is the grey-value sum
method of Karu and Jain (1995) and Wilson et al. (1993), following the work of Stock
andSwonger(1969). Thismethodusesa9×9maskcentredonthepixelofinterestand
adds up the grey level value of selected pixels in eight different directions to obtain the
orientationdirection. Wefoundthatthisproceduredoesnotcomparewellinpracticeto
the semi-variogram method. Furthermore it does not readily give the distance between
adjacent ridges. Also, as discussed in Section 4, our method can be improved further
which is especially important for noisy fingerprint images as shown in Figure 3.
Example: For the fingerprint image shown in Figure 1 the smoothed orientation
field is obtained using the semi-variogram method and the method of Karu and Jain.
These are shown superimposed on the original fingerprint image in Figures 3a and 3b
respectively.
INSERT FIGURE 3.
It can be seen that the semi-variogram method gives a more accurate orientation
field than the method of Karu and Jain. In clear areas, such as in the top part of the
6
image, both methods satisfactorily give the directions of the ridge flows. In blurred
areas, such as the middle part of the image, the directions obtained from the semi-
variogram method more precisely express the ridge flows than the procedure of Karu
and Jain.
In blurred regions the semi-variogram is superior because it extracts information
about the orientation field over a wider region than the 9×9 mask of Karu and Jain.
Also, note that the letter “C”, used for identification purposes, goes through the same
directional process and it is not surprising to see degradation by both methods.
Figure 4a shows a small area of the fingerprint image containing two sections of
ridges. The distance between the ridges is approximately 11 pixels.
INSERT FIGURE 4.
Figure 4b shows the estimated ridge spacing obtained from the semi-variogram
method for the same image. It can be seen that at all pixels the correct distance is
obtained. A 3×3 mean filter is used to smooth these distances and yields a value of
11 in nearly all the pixels in Figure 4.
4 Bayesian framework for improving the orientation field
To further enhance the orientation field we use a Bayesian framework with a suitable
prior for the directional field. The computational work involved for the maximum a
priori (MAP) solution is intense since the images are usually very large-scale data.
The iterated conditional modes (ICM) method of Besag (1986) is used to reduce the
computational burden.
Suppose the observed fingerprint image x is a realization of a random matrix, X =
{X : i = 1,···,m and j = 1,···,n}. Let an arbitrary orientation field be denoted by
ij
d = {d : i = 1,···,m and j = 1,···,n},
ij
whered isanintegerbetween0and15indicatingthecorrespondingdirectionforpixel
ij
(i,j). This can be interpreted as a realization of a random matrix,
D = {D : i = 1,···,m and j = 1,···,n},
ij
7
where D assigns a direction to pixel (i,j). We make the following two standard
ij
assumptions.
Assumption 1. Given any orientation field d, the random variables X are con-
ij
ditionally independent and each X has the same known conditional density function
ij
f(x |d ). Thus, the conditional likelihood of the observed fingerprint image x given d
ij ij
is,
m n
l(x|d) = f(x |d ).
(cid:89) (cid:89) ij ij
i=1j=1
Assumption 2. The true orientation field d is a realization of a locally dependent
Markov random field with specified distribution p(d).
The conditional density f(x |d ) is taken to be Gaussian with mean µ and vari-
ij ij ij
ance σ2, where the variance is estimated by the sample variance s2 of the nine grey
ij ij
levels for pixels in the 9×9 window centred on (i,j). The mean µ is estimated by
ij
x¯ (d ), the grey level mean of those pixels within the 9×9 window which lie along
ij ij
direction d .
ij
The prior distribution p(d) is taken to be from the Ising model (see Besag, 1986),
m n
p(d) ∝ exp{βn(d )},
(cid:89) (cid:89) ij
i=1j=1
where n(d ) is the number of the pixels located in the surrounding 3×3 block and
ij
having the same direction as pixel (i,j). The motivation for this is that we expect the
directions of neighbouring pixels in an image to be similar. The parameter β is taken
to be positive to encourage the directional similarity of neighbouring pixels.
We have used β = 1.5 which seems to work well. Taking β = 0 gives the maximum
likelihood estimate of the orientation field which gives the direction with pixels having
mean grey level closest to the corresponding observed pixel. As β → ∞, the ICM
provides a smoothing solution acting on the initial orientation field.
Theposteriordensityp(d|x)oftheorientationfieldd,giventheobservedfingerprint
image x is, from Bayes’s theorem,
p(d|x) = l(x|d) p(d).
8
We would estimate the orientation field using the value dˆ which maximizes this
posterior density. Since the fingerprint image size is large this imposes considerable
computational demands. To overcome this problem we use the iterative conditional
mode procedure of Besag (1986).
If d˜ denotes a provisional estimate of the orientation field at all locations excluding
pixel(i,j)thenwecanestimated bymaximizingp(d |x,d˜)withrespecttod ,where,
ij ij ij
p(d |x,d˜) ∝ f(x |d )p(d |d˜).
ij ij ij ij
On taking logarithms it can be seen that this is achieved by minimizing,
{x −x¯ (d )}2/2s2 −βn(d ).
ij ij ij ij ij
When applied to each pixel in turn, this procedure defines a single cycle of an
iterative algorithm for estimating the orientation field. As an initial estimate d˜ we use
the results of the semi-variogram method. To obtain the estimated orientation field dˆ
we apply the algorithm for a fixed number of cycles or until convergence occurs. In
practice convergence is rapid with few changes occurring after about six cycles.
For the smoothed orientation field of Figure 3a the resultant orientation field after
using the ICM method is shown in Figure 5, which shows the orientation field superim-
posed on the original fingerprint image. Comparison of Figure 3a and Figure 5 shows
a clear improvement from using ICM.
INSERT FIGURE 5.
To show the improvement clearly, three regions in Figure 3a and the corresponding
regions in Figure 5 are enlarged and shown in Figure 6.
INSERT FIGURE 6.
The small squares and triangles in Figures 5 and 6 show the core and delta points
for the fingerprint patterns. Their recognition from the orientation field is discussed in
detail in Section 7. We note here that the positions of these points in the ICM output
orientation field are more accurate than using the original orientation field.
9