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Distribution of epidermal ridge minutiae.

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AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 77:367-376 (1988)
Distribution of Epidermal Ridge Minutiae
DAVID A. STONEY
Department of Criminal Justice, University of Illinois at Chicago, Chicago,
Illinois 60680
KEY WORDS
Pattern, Fingerprints, Dermatoglyphics
ABSTRACT
The distribution of epidermal ridge minutiae on the distal
portion of male human thumbprints has been characterized. For each of 412
thumbprints, a centrally located focal minutia was chosen and neighboring
minutiae were sampled. Minutiae were considered to be neighbors if there
were no other minutiae appearing in the intervening region defined by the two
minutia events and the ridge system. For each neighbor minutia, the total
ridge distance between the focal and neighbor minutiae was measured. This
distance is the total length of ridges appearing in the intervening region.
The number of neighbor minutiae occurring about the focal minutia was
found to be normally distributed with a mean of 6.42 (n = 412). The distribution of total ridge distances was not adequately described by the negative
exponential distribution, but was well described by a gamma distribution with
a shape parameter of 0.193 and a scale parameter of 5.91 mm. This gamma
distribution reflects a local overdispersion of minutiae.
This study is noteworthy as the first to describe the distribution of epidermal
ridge minutiae within the ridge structure. The results support prior work
based on quadrat sampling and eliminate two possible sources of error.
A possible explanation for the overdispersed distribution lies in the growth
stress model for minutia formation. Minutia formation may alleviate local
growth stress, thereby removing the impetus for formation of additional minutiae in the immediately surrounding region.
Epidermal ridge structures occur on the
volar surfaces of all primate species and on
some marsupials (Montagna and Parakkal,
1974). Two features of the epidermal ridges
are of interest here: 1) the patterns that are
formed by groups of ridges and 2) the irregularities and characteristics of the individual
ridges, known as minutiae. The present work
describes the distribution of epidermal ridge
minutiae observed on the distal tips of male
thumbs.
Locally, epidermal ridges are parallel to
one another, but over larger regions the
ridges flow in groups and form patterns,
known technically as dermatoglyphics, but
familiar to most as “fingerprints.” In addition to general convergence and divergence
of ridges, two fundamental types of pattern
singularities occur: loops, where a group of
ridges turns back upon itself, and triradii,
where ridges flow outward from one point in
three directions. Combinations of these two
0 1985 ALAN R. LISS, INC.
pattern singularities create a variety of local
pattern environments.
Close inspection reveals that individual
epidermal ridges do not flow continuously,
but often branch or terminate within the confines of the general ridge pattern. These discontinuities in individual ridges are
minutiae. Minutiae can take a variety of
forms, but the most common types are ridge
bifurcations and ridge endings (see Fig. 1).
Dermatoglyphics, the epidermal ridge patterns, have been the subject of intensive investigation. Major fields of study have
included their comparative anatomy (Cummins and Midlo, 1943; Newell-Morris, 1979),
genetics (Holt, 1968; Roberts, 1979; Loesch,
19831, anthropology (Steinberg et al., 1975;
Plato et al., 1975; Roberts, 1982), topology
Keceived September 14, 1987; accepted April 4, 1988
368
D.A. STONEY
(Penrose, 1965,1979; Smith, 1979), utility for
medical diagnosis (Schaumann and Alter,
1976), and utility for fingerprint record retrieval (Cowger, 1983).
In spite of this extensive and long-standing
interest in dermatoglyphics, there has been
only minimal study of epidermal ridge minutiae. Systematic studies with a biological focus have been rare (Okajima, 1967a,b, 1970;
Loesch, 1973; Dankmeijer et al., 1980; Liu,
et al., 1982; Loesch, 1983,1984; Okajima and
Usukura, 1984). A number of other studies
by forensic scientists and statisticians are
also of interest (Stoney and Thornton, 1986a),
but the spirit behind these investigations has
been justification of fingerprint comparison
practices rather than of investigation of
minutia variation per se. In all prior work,
minutiae are treated a s point occurrences,
without reference to the ridge structure in
which they are found. The result is that the
distribution of minutiae has never been adequately described within the ridge structure.
The present work is a n initial effort to describe this distribution. For each print, a centrally located focal minutia is selected and
the distribution of total ridge distances to
neighboring minutiae is characterized. Total
ridge distance is the total length of ridges
occurring between the two minutiae, as fully
defined below.
MATERIALS AND METHODS
Sample material
Data were collected from inked fingerprints on file at the Contra Costa County
Criminalistics Laboratory in Martinez, California. Right and left thumbprints of adult
males were examined from sequential fingerprint cards in a file sequenced by date of
printing. The cards were screened for clarity
of printing of right and left plain thumbprints a t the bottom of the standard fingerprint card (Cowger, 1983). Plain prints
approximate a planar projection, as opposed
to the cylindrical projection of rolled prints.
Sampling was restricted to male thumbprints in order to avoid substantial discrepancies in finger size.
A total of 412 thumbprints were sampled:
the right and left thumbprints from 206 individuals. The mean age at time of printing
was 26.5 years with a range of 18-56 years
and a standard deviation of 6.5 years. According to the racial classifications on the
cards, there were 147 whites, 56 blacks, and
3 Asians in this sample population. No at-
tempt was made to verify these racial
classifications.
Sample area
The distal tips of digits were chosen as the
sample area for two reasons. First, the geometry of the ridge flow seen in this region
is similar for all dermatoglyphic pattern
types. The ridges conform to the digital outline, flowing in a n arch from one side of the
finger to the other. Comparable ridge geometry within the sampling region is required
in order to use the general ridge flow as a
reference for minutia position. The second
reason for sampling from the distal tip region
was to avoid pattern singularities. These singularities lie proximal to the sampling region on the distal phalange. Minutia density
is known to increase significantly about the
pattern singularities (Kingston, 1964).
Selection of focal minutiae
For each print, a centrally located focal
minutia was chosen using the following procedure (see Figs. 1, 2). A reticle with a circle
of 5-mm radius was placed over the print
while viewing the print under a stereobinocular microscope. The reticle was placed so
that the outline of the circle coincided with
the upper portion of the ridge most closely
corresponding to the circle curvature. This
ridge was designated the 5-mm ridge. While
holding the reticle in place, the center of the
circle was marked with a fine pin prick. This
point was designated the origin. Typically,
the origin was in the area of the print where
the pattern singularities occur.
Epidermal ridges show their highest degree of curvature as they round the apex of
the finger, flattening out as one proceeds laterally. The point of maximum distal extent on
the 5-mm ridge was thus easily noted and
marked with a second pin prick.
A second reticle, marked with angular
measure, was placed with the vertex at the
origin (see Fig. 2). The search for a focal
minutia was conducted beginning on the 5mm ridge. The minutia closest to the point of
maximum distal extent was selected if one
was within 45". If a minutia was not found
within 45", then the search was continued
on successively more distal ridges within the
total 90" arc until a minutia was encountered. Limiting the search to within the 90"
arc ensured that the focal minutia would be
on the distal portion of the finger. Searching
on sequential ridges introduced some varia-
EPIDERMAL R1:DGE MINUTIAE
369
tion in the position of the focal minutia relative to the 5-mm ridge. The intent of
selection, however, was not to define a n invariant point within the pattern, but rather
to choose a centrally located minutia in a
consistent and unbiased manner.
ab-
Data collection
Pattern type. The general pattern type
of each print was noted, following Loesch's
criteria for arches, radial loops, ulnar loops,
and whorls (Loesch, 1983).
Ridge density. After selection of the focal
Cminutia, ridge density was measured along
the direction defined by the origin and the
focal minutia. The distance between the
ridge located five ridges below the focal minutia and that found five ridges above the
%&-+w
*
.
focal minutia were measured (see Fig. 3).
Minutia density. Minutia density was
Fig. 1. Placcmcnt of the reticle on print to be sammeasured by sampling all minutiae in a cirpled. The 5-mm diameter circular reticle is placed so
cular region of 3-mm radius, centered one
that the outline of the circle coincides with the upper
portion of the ridge most closely corresponding to this
ridge directly above the position of maxidegree of curvature. Minutiae labeled (a), (b), and (c) mum distal extent. This position was chosen
represent the three fundamental types of ending ridge,
because the average focal minutia position
fork, and dot, respectively.
was not on the 5-mm ridge, but a t 0.91 ridge
intervals above it.
Sampling of neighbor minutiae. Once a
focal minutia was chosen, positions of neighd
boring minutiae were sampled using the an\
C
gular reticle (see Fig. 4).To be sampled a
minutia had to meet three criteria: 1)appear
within a 45" angle of the focal minutia, 2) be
within a count of six ridges proximally or
e
nine ridges distally, and 3) be a neighbor
f
b
minutia, as defined below.
Minutiae outside the 45" angle, or more
distal than nine ridge intervals, approached
a
the border areas of the print. These areas
could not be sampled on all prints because of
the variable extent of printing. Minutiae appearing more than six ridges proximal to the
focal minutia approached the pattern area
where ridges could not be reasonably approximated a s circular about the origin.
Ridge counting was performed in the direction perpendicular to the ridge flow, beginFig. 2. Selection of the focal minutia. After placement ning at the focal minutia. Each ridge
of the rcticle, the direction showing maximum ridge interval was counted from the focal minutia
curvature (d) defines the point of maximum distal extent to the ridge on which the neighbor minutia
(fJ.Thc focal munutia is selected from within a 90" sector
of this point with vertex at the origin (a).Lines (c) and appeared. Counts to bifurcating ridges were
(e) define the sector limits. A minutia is first sought on made to the nearest branch of the bifurcathe fi-mm ridge. If no minutia is found on the 5-mm tion, and counts to ending ridges were made
ridge, successively more distal ridges are searched until as if the ending ridge extended past the
a focal minutia is found. If two rnunitiae appear on the counting line.
ridge, the one closest to the axis (d) is taken. In this
Only neighboring minutiae were sampled.
example, the minutia at (b) is selected as the focal
Minutiae were defined as neighbors when
minutia.
'
,>L
1
370
D.A. STONEY
examination of fingerprint patterns (Stoney
and Thornton, 1986b). Ambiguities in ridge
counts arise when minutiae are not neighbors, because of the effect of the intervening
minutiae.
For each neighbor minutia, the ridge count
and the angular distance from the focal
b
minutia were recorded. Ridge counts ranged
from six ridges proximal to the focal minutia
to nine ridges distal to the focal minutia.
a
Ridge counts of zero represent minutiae that
appear on the same ridge as the focal minutia. A full discussion of the ridge counting
procedure has been previously described
(Stoney and Thornton, 1986b).
Calculation of total ridge distance. A
single measure of distance between the focal
and neighbor minutiae was obtained by calculating the total length of ridges appearing
in the region between the two minutiae. For
Fig. 3. Measurement of ridge density. Ridge density
is measured along the direction defined by the origin (a) a ridge count of C, there are C + 1 ridges
and the focal minutia (b). The distance (c) is measured
appearing between the two minutiae. The
between the ridge five ridges below the focal minutia
distance
along each of these ridges must be
and the ridge five ridges above the focal minutia.
summed to get the total ridge distance.
The primary data for calculating this distance are the ridge count, the angular distance to the neighbor minutia, the radius of
ridge curvature at the focal minutia, and the
ridge density. Given the ridge density, the
radii of curvature Ri for each ridge are calculated, using Equation (l), where Ci is the
ridge count (distally) from the focal minutia,
Ro is the radius of ridge curvature at the
b
focal minutia, and B is the ridge density.
The total ridge distance is given by sum-a
ming Equation (1)over the integral values
of Ci, ranging from zero to the ridge count C.
(Note that proximal ridge counts give a negative value of Ci since the radius of curvature is decreasing.)
ic
Ri = Ro
+ CiB.
(1)
Statistical methods
Standard deviations are given as the
Fig. 4. Selection of neighbor minutiae. Neighbor
minutiae (circled) are selected from within a 90" sector square root of the adjusted sample variance.
centered on the focal minutia (d) with vertex at the Chi-square tests were used to test hypotheses
origin (a). Lines (bl and (e) define the sector limits. of homogeneity and goodness of fit. Equality
Neighbor minutiae are sampled where the ridge count
from the focal minutia is within the range of six ridges of variance was tested using the F-ratio, and
the Z statistic was used to test for equality of
proximally to nine ridges distally.
means. Parameters for the gamma distribution were estimated by the matching mono other minutiae appeared in the region ments method (Hastings and Peacock, 1975),
defined by the ridge flow and the perpendic- and expected frequencies for the gamma disular ridge counts from each minutia to the tribution were calculated by linear interpoother. Pairs of neighboring minutiae are the lation from Pearson's tables of the incomplete
fundamental unit of variation in the forensic gamma function (Pearson, 1965).
EPIDERMAL RIDGE MINUTIAE
371
Distribution of Number of Neighbor Minutiae
100 7
,
I
0
2
i
6
i
I
,
1 0 1 2 1 4
NUMBER OF NEIGHBOR MINUTIAE
5
Fig. 5. The distribution of the number of neighbor
minutiae per print. The line follows the expected values
for a normal distribution with mean 6.42 and standard
deviation 1.76.
RESULTS
minutiae per square mm for the mean. Given
the mean ridge density, the estimated density of minutiae o n the ridges is 0.103 minutiae per mm. The standard deviation,
calculated using the individual ridge densities for each print, was 0.0429 minutiae per
mm.
Variation of density with pattern type is of
possible concern. In the present sampling,
however, identical mean density values were
obtained for the 225 loops and for the 174
whorl patterns (0.222 minutiae per square
mm). There was also no significant variation
in minutia density between the right and left
hands, with mean values of 0.217 and 0.228
minutiae per square mm, respectively
(equality of means accepted a t 5% significance with nl and n2 of 206 and a Z value of
0.12, P = ,941.
Frequencies of pattern types
Although the dermatoglyphic patterns are
not the principal focus of this study, the representatives of the sample with respect to
pattern type is of interest. This sample of 412
thumbprints contained 223 ulnar loops, 174
whorls, 13 arches, and 2 radial loops. This
distribution of pattern types does not differ
significantly from the expected distribution,
given the racial composition of the sample
(Stoney and Thornton, 1987).
Ridge density
The mean ridge density for the 412 thumbprints was 0.463 mm per ridge. Traditionally, ridge density has been expressed in
ridges per cm. This value is 21.6 ridges per
cm with a standard deviation of 2.82 ridges
per cm.
Number of neighbor minutiae per focal
minutia
Minutia density
The distribution of the number of neighbor
For each of the 412 thumbprints, the num- minutiae per print appearing in the sample
ber of minutiae occurring in the circular region is shown in Figure 5. A total of 2,645
sampling region of 3-mm radius was re- neighbor minutiae were sampled from about
corded. The mean number of minutiae was the 412 focal minutiae. The distribution was
6.29, with a standard deviation of 2.61 minu- found to be normal with a mean of 6.42 and
tiae. The area of the circular sampling region a standard deviation of 1.76 (normal approxwas 28.3 square mm, giving a density esti- imation accepted a t 5% significance with a
mate of 0.223 minutiae per square mm with chi-square value of 4.63 and 6 degrees of
a 95% confidence interval of 0.215-0.230 freedom [dfl,P = .59).
372
D.A. STONEY
Frequency of Neighbor Minutiae
by Total Ridge Distance
200-.-
Frequency
I
.
-
i
~
~~
-Gamma Prediction
---- Exponential Prediction
150
0
Observed Frequency
100
ti 0
0
0
5
6
10
15
Total Ridge Distance (mm)
20
25
Fig. 6. Observed TRDs of neighbor minutiae and expected frequencies under the assumptions of 1) the negative exponential distribution and 2) the gamma
distribution. Although the two distributions are close for
large TRD, the negative exponential distribution pre-
dicts unacceptably high frequencies for the shorter distances and is rejected at high significance ( P < < ,0011.
The gamma distribution gives a n acceptable fit, demonstrating that minutiae are somewhat overdispersed ( P =
.37).
Minutia distribution
A curve following expected values for the
negative exponential distribution, given the
mean value of 7.05 mm, is presented in Figure 6. The negative exponential distribution
is rejected at high significance (goodness-offit test rejected at 0.1% significance with a
chi-square value of 311 and 60 df, P < <
.001).
A more generalized distribution, known as
the gamma distribution, is suitable for describing a wide variety of interevent distributions (Jewell, 1960). Estimates of the
gamma parameters were made by the matching moments method (Hastings and Peacock,
1975), using s2/x for the scale parameter b,
and (xIsl2 for the shape parameter c, where x
is the sample mean and s is the unadjusted
sample variance. The estimated scale and
shape parameters were 5.91 and 0.193 mm,
respectively. Using these parameters, expected frequencies for the gamma distribution were determined using the incomplete
gamma function as shown in Equation (3)
and tabulated by Pearson (1965).
The distribution of the total ridge distance
(TRD) for the 2,645 sampled minutiae is
shown in Figure 6. The mean TRD was 7.05
mm with a standard deviation of 6.45 mm.
The distribution was found to show no significant differences between the right. and left
hands and between whorl and loop patterns
(see Figs. 7, 8). (Tests for homogeneity accepted at 5% significance. For right and left
hands, sample sizes were 206 each, with a
chi-square value of 28.3 and 26 df, P = .35.
For whorl and loop patterns, sample sizes
were 174 and 225, respectively, with a chisquare value of 22.9 and 27 df, P = .69.)
Insufficient numbers of arch and radial loop
patterns were present in this sample set to
extend the test of homogeneity to these pattern types.
If minutiae are positioned randomly with
respect to TRD, minutia occurrence can be
treated as a Poisson process and a negative
exponential function should describe the distribution of distances. With a mean value of
y, the frequencies of the negative exponential
distribution are given by Equation (2).
P(X1 < x < X2) = exp(-X2ly)
- exp(-Xl/y).
(2)
373
EPIDERMAL RIDGE MINUTIAE
Comparison of Minutia TRD Frequencies
between Right and Left Hands
200
150
Frequency
I
_______--
E B RIGHT (N=1336)
~
_
I
LEFT (N=1309)
~
_
t---
_
_
_
~
7
TRD (mm)
Fig. 7. Observed TRD frequencies on the right and
left hands. The distrlbutions are not statistlcally different ( P = ,351.
Comparison of Minutia TRD Frequencies
between Loop and Whorl Patterns
Frequency
F Z WHORLS (N=1139)
I
LOOPS (N=1414)
~~~
8
TAD (mm)
Fig. 8. Observed TRD frequencies for whorl and ulnar
loop patterns. The distributions are not statistically different ( P = .69).
A curve following the expected values for the
DISCUSSION
Ridge density
gamma distribution is presented in Figure 6.
Observed and predicted values are shown i n
The ridge density of 0.463 mm per ridge
Table 1. The agreement is very good (goodness-of-fit test accepted at 5% significance obtained in the present sampling is in excelwith a chi-square value of 57.8 and 55 df, lent agreement with prior studies (Cummins
et al., 1941; Kingston, 1964) as fully disP = .37).
374
D.A. STONEY
TABLE 1. Total ridge distance (TRD) frequencies
Interval
(mm)
Observed
frequency
0-0.5
0.5-1.0
1.O- 1.5
1.5-2.0
2.0-2.5
2.5-3.0
3.0-3.5
3.5-4.0
4.0-4.5
4.5-5.0
5.0-5.5
5.5-6.0
6.0-6.5
6.5-7.0
7.0-7.5
7.5-8.0
8.0-8.5
8.5-9.0
9.0-9.5
9.5-10.0
10.0-10.5
10.5-11.0
11.0-11.5
11.5-12.0
12.0-12.5
12.5-13.0
13.0-13.5
13.5-14.0
14.0-14.5
14.5-15.0
15.0-15.5
15.5-16.0
16.0-16.5
16.5-17.0
17.0-17.5
17.5-18.0
18.0-18.5
18.5-19.0
19.0-19.5
19.5-20.0
20.0-20.5
20.5-21.0
21.0-21.5
21.5-22.0
22.0-22.5
22.5-23.0
23.0-23.5
23.5-24.0
24.0-24.5
24.5-25.5
25.5-26.5
26.5-27.5
27.5-28.5
28.5-30.0
30.0-31.5
31.5-33.5
33.5-36.0
36.0
114
132
144
112
147
124
126
154
99
117
95
91
88
82
86
73
70
54
64
55
53
39
45
31
41
33
25
25
29
24
21
27
16
15
Frequency
predicted by
gamma distribution
125.3
139.3
144.5
143.0
137.7
131.0
124.9
118.6
111.9
103.5
97.5
91.7
86.2
79.4
73.8
69.2
64.7
60.0
55.0
51.4
48.0
44.8
40.8
37.9
35.3
32.9
30.3
27.7
25.9
24.1
22.4
20.3
18.8
17.5
16.3
14.9
13.7
12.7
11.8
11.0
9.9
9.2
8.6
8.0
7.3
6.7
6.2
5.8
5.3
9.3
8.3
6.7
5.8
7.1
5.6
5.6
5.6
9.5
study is comparable to other reported values
that range from 0.181 to 0.295 minutiae per
square mm (Galton, 1892; Amy, 1946; Kingston, 1964; Osterburg et al., 1977;Dankmeijer
et al., 1980). These mean values are not sufficiently close to be regarded as repetitive
samples from the same population. The differences are probably attributable to the different sampling regions within the
fingerprints and to differences in which fingers and pattern types were sampled. Full
discussion of these factors is presented elsewhere (Stoney and Thornton, 1987).
Minutia distribution
The distribution of TRDs among neighboring minutiae has been shown to be well described by a gamma distribution. The
particular distribution parameters will not
be valid in other regions of thumbprints, or
on other fingers, because the parameters will
be affected by variations in minutia density.
Acceptable description by the gamma distribution, however, is significant because it
demonstrates that, given any particular minutia, there is a decreased probability of encountering neighboring minutiae in the
immediate vicinity. Minutiae are thus overdispersed or more regularly spaced than
would be expected from a random (Poisson)
process.
11
Note that for large TRD both the gamma
22
and the negative exponential distributions
16
8
follow the observed frequencies fairly well.
12
Pielou (1969)has commented on this similar10
ity and on the difficulty that can occur in
5
distinguishing a gamma distribution from a
6
11
random one. In the present data, the large
6
sample size allows sufficiently narrow inter10
vals
so that the distinction is quite clear.
2
A complementary demonstration of the
5
4
regularity of minutia spacing results from
3
comparison of the mean TRDs predicted from
11
1) the minutia density estimate and 2) the
6
estimated parameter of the negative expo6
4
nential distribution. Under the hypothesis of
6
a random minutia distribution, these two
6
mean
distances are estimates of the same
5
population mean and their values should
7
12
+
therefore be within sampling error of one
another. In the case of a n overdispersed distribution, the estimated parameter of the
cussed elsewhere (Stoney and Thornton, negative exponential distribution will be
1987).
lower than the value predicted by the density
(Pielou, 1974).
Minutia density
For the present data, the density gives a
The mean minutia density of 0.223 minu- mean distance of 9.70 mm with a variance of
tiae per square mm obtained in the present 6.62 mm (n = 412).The negative exponential
EPIDERMAL RIDGE MINUTIAE
distribution parameter gives a mean distance of 7.05 mm and a variance of 6.46 mm
(n = 2,645). The assumption of equal variance is justified, and the hypothesis of equal
means is rejected with high significance
(equal variance accepted a t 5% significance
with a n F value of 1.05 with 411 and 2,644
df, P = .29. The test of equal means is rejected a t 0.1% significance with a n F value
of 1.38 with 411 and 2,644 df, P < < .001).
The mean distance predicted by the assumption of random spacing is thus significantly
lower than the observed mean, indicating a n
overdispersed distribution.
Two areas of possible bias in the sampling
method need to be considered in their relationship to the findings: the method for selection of focal minutiae and the limited extent
of the sample region.
Focal minutiae were selected after defining
the point of maximum distal extent on the
ridge with curvature most closely approximating a 5-mm radius. The minutia closest
to the point of maximum distal extent on the
5-mm ridge was selected if one was found
within 45". If no minutia was found on this
ridge, then the search continued on successively more distal ridges within the total 90"
arc until a minutia was encountered.
In nearest-neighbor studies of the spatial
distribution of organisms, Pielou (1974) has
noted that selecting the individual nearest to
a randomly selected point does not constitute
a selection of a random individual. The distinction is of no consequence if the distribution is, in fact, random, but there is a biased
selection of comparatively isolated individuals under other circumstances. A similar
source of bias in the present work must be
considered.
Two key features set the present work
apart from standard nearest-neighbor organism sampling methods, and both act to reduce the indicated bias. First, the focal
minutia was not sought on the basis of minimum TRD, but rather along individual
ridges. This means that minutiae present on
the other ridges may well show a shorter
TRD to the point of maximum distal extent
than does the focal minutia. Second, the sampling is not limited to the one nearest-neighbor. All neighbors without intervening
minutiae were sampled (an average of 6.7
per focal minutia). This is a much better approximation of a total sampling of interevent
distances than is single-distance sampling.
In any case, the initial selection of a n event
closest to a randomly selected point is a n
unbiased random selection under the hypoth-
375
esis of a random distribution, and therefore
rejection of this null hypothesis is sound.
A second possible source of bias in the sampling is the limited extent of the sampling
region. As noted above, the extent of the
region was limited in order to ensure that
uniform sampling would be possible for all
prints and that within the sampling region
the course of the ridges could be reasonably
approximated by circular arcs. The effect of
this limitation is that on each ridge there is
an upper bound on the TRD that can be sampled. Minutiae appearing outside the 45"
range on either side of the ridge were not
sampled. On any given ridge, therefore, distances could not exceed rl4. Thus, based on
an average focal radius of 5.42 mm, distances
greater than 4.3 mm on the focal ridge fall
outside the sampling region. On the focal
ridge, this distance is equivalent to the TRD.
With a ridge count of +1, assuming the average ridge density, distances of 4.6 mm
along the ridge, or 8.9 mm TRD, fall outside
the sampling region. Maximum TRDs increase with ridge count, and the probability
of encountering neighbor minutiae falling
outside the sampling region becomes very
small for ridge counts greater than three.
Bias introduced by restriction of the sampling region is thus limited to the few ridges
about the focal minutia, and the bias is distributed over a relatively wide range of
TRDs. The direction of this bias is in favor of
shorter TRDs.
Kingston's work (1964) is in qualitative
agreement with this finding. Kingston ignored the ridge structure itself and performed quadrat sampling of minutiae. His
major findings were 1)that there was a regular decrease in the variancelmean ratio with
decreasing quadrat size and 2) that among
quadrats containing exactly two minutiae the
probability of the two minutiae appearing in
the same quarter of the quadrat regularly
decreased with decreasing quadrat size.
Kingston showed that similar results would
follow if about each minutia there was a region in which the occurrence of other minutiae was excluded.
Kingston suspected two contributing
sources for the overdispersal of minutiae. The
first was the ridge structure itself. Minutiae
appearing on different ridges were necessarily spaced a minimum of one ridge interval
apart. Kingston's second suspected source of
overdispersion was the way in which minutiae were defined. Kingston had defined certain closely spaced minutia pairs as single,
compound events. This practice served in a n
376
D.A. STONEY
artificial way to reduce the incidence of
closely spaced minutiae. In the present study
neither of these two explanations for the observed overdispersion is tenable. The use of
ridge distances eliminates the argument that
the ridge structure is responsible, and
throughout this study closely spaced minutiae were strictly defined as two events.
Osterburg et al. (1977) and Sclove (1979,
1980) also studied minutia distribution without reference to the ridge structure, but their
model was not based on interminutia distances and cannot be reasonably compared
to the present study.
ACKNOWLEDGMENTS
The author wishes to thank John E. Murdock of the Contra Costa County Criminalistics Laboratory for access to the fingerprint
data described in this study. This project was
supported in part by grant 82-IJ-CX-0023
from the National Institute of Justice, U.S.
Department of Justice. Points of view or
opinions stated in this document are those of
the authors and do not necessarily represent
the official position or policies of the U.S.
Department of Justice.
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