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Mathematical Analysis Techniques of Frontal Sinus Morphology with Emphasis on Homo.

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THE ANATOMICAL RECORD 291:1455–1478 (2008)
Mathematical Analysis Techniques of
Frontal Sinus Morphology, with
Emphasis on Homo
Department for Anthropology, University of Vienna, Vienna, Austria
Of all the paranasal sinuses, frontal sinus (FS) morphology, volumes,
outlines, and cross-sectional areas vary most and so their statistical noise
presents particular challenges. To assess and control this statistical noise
requires a suite of mathematical techniques that: model their volume and
cross-sectional area ontogeny, determine the uniqueness and fractal
dimensions of their outlines (useful in forensics), smooth their outlines
via Singular Value Decomposition (SVD), and model their expansion via
percolation cluster models (PCMs). Published data sets of FS outlines,
cross-sectional areas and volumes of Neanderthal and modern crania
(obtained via CT-imaging techniques) are utilized here for application of
these novel mathematical methods, which necessitate a modeling
approach. Results show that: FS noisiness can be explained as cluster
growth, their fractal outlines have properties similar to closed random
walks (Brownian bridges) about predefined curves, and the PCMs can
simulate the emergence of lamellae. The statistical properties derived
from the analysis techniques presented here suggest that an emergence
of the lamellae via PCMs (with pinning and quenching-correlated noise)
resolves the masticatory stress debate by showing that the lamellae are
indeed responses to masticatory stresses, but these are of so low a level
that they cannot be measured with strain gauges. PCMs and Brownian
bridges, defined by local rules, lead to the emergence of macroscopically
observable morphologies. The methodologies presented here contribute to
research in emergence phenomena and are not confined to morphological
analyses of frontal sinuses. Anat Rec, 291:1455–1478, 2008. Ó 2008
Wiley-Liss, Inc.
Key words: frontal sinuses; fractals; singular value decomposition; supraorbital torus; masticatory stress;
Eden model; pinning; sigmoid function; paranasal
sinuses; discrete Fourier transform; stationary
time series; autocorrelation function; Ljung-Box
statistic; Yule-Walker algorithm
*Correspondence to: Hermann Prossinger, Department for
Anthropology, University of Vienna, Althanstrasse 14, A-1091
Vienna, Austria. Fax: 43-1-4277-9547. E-mail: [email protected]
Received 22 April 2008; Accepted 23 April 2008
DOI 10.1002/ar.20783
Published online in Wiley InterScience (www.interscience.wiley.
Historical Précis
Almost contemporaneously with the pioneering work
by Mihalkovićs (1896), Paulli (1900) investigated the
pneumatizations of the crania of all (then) known mammals. He was most emphatic in denying the existence of
one single (functional) role of these pneumatizations.
Rather, he claimed that they resulted from differential
growth rates during the ontogeny of the individual. He
also pointed out that the convention of naming the
sinuses after the bone in which they (predominantly) reside is inaccurate. The frontal sinuses in primates,
although in the os frontalis, are ethmoidally derived, for
example. Despite his claim of no function of the pneumatizations, he does concede that the pneumatization
reduces the bony mass without compromising strength.
(A view that several morphologists do not concur with:
more about this later). A generation after Paulli, Weinert (1925) undertook another comprehensive overview
of the frontal sinuses in a large number of vertebrate
species. In a long two-part article, he made a remarkable morphological analysis of the frontal sinuses, based
on a large data set that encompassed crania from reptiles to primates.
In the primates, certain patterns in the paranasal
sinuses seem to be observed (Anon et al., 1996; Koppe
and Nagai, 1999), predominantly in the maxillary
sinuses (Rae and Koppe, 2000); they are, in the case of
frontal sinuses, tenuous at best (Borovanský, 1936;
Bugyi, 1959). The frontal sinuses of three specimens of
H. heidelbergensis—Broken Hill/Kabwe (Woodward,
1921), Petralona (Kokkoros and Kanellis, 1960), and
Steinheim (Berckheimer, 1933)—are very large (Fig. 1).
It has been speculated (Prossinger et al., 2003) that the
African and European specimens may be different
clades, but with a sample size of two (Europe) and one
(Africa), implications derived from such cladistics would
be questionable indeed. Below, I present statistical evidence for how extremely (statistically) noisy the morphology of frontal sinuses in Homo is. Conventional morphometric assessment (as used by Buckland-Wright,
1990, e.g.), it will be argued, is therefore of limited
value. Alternatively, a suite of mathematical/statistical
analysis techniques that offers progress is presented in
this review.
Reasons for Analyzing Frontal Sinus
1. Morphology assists in cladistic and/or phylogenetic
reconstructions. That the morphology of paranasal
sinuses might be different in different species of
humans seems to have been first expressed by
Schwalbe (1899). Many researchers base their work
on the paradigm that differences in sinus morphology
help defines clades and can be used to reconstruct
phylogenies (Rae and Koppe, 2004).
2. Improved knowledge of ontogeny (Sieglbauer, 1947;
Schmid, 1973; Lanz and Wachsmuth, 1985) may support or reject clinical/epidemiological intervention
strategies (Scuderi et al., 1993). In Homo, the frontal
sinuses communicate with the nasal cavity via very
small openings located between the lamellae of the
ethnoturbinals (Moore, 1981; p. 267). Whenever the
frontal sinuses become infected, treatment is extraordinarily difficult. A simple search with Google Scholar
for ‘‘frontal sinuses’’ yields in excess of 15,900 citations. The overwhelming majority of these refer to
clinical studies and/or reports about intervention
strategies and attendant difficulties. Almost all clinicians report that the large variability of frontal sinus
morphology necessitates interventions that are casespecific. If morphologists cannot succeed in characterizing regularities that supply definitive help to the
clinician, the converse question arises: why is the
morphology of the frontal sinuses so (statistically)
3. Currently, the evidence for the highly variable morphology of the frontal sinuses (Donald et al., 1994)
suggests that the 2D projection outlines are unique to
each individual. They thus, so it is claimed, can be
used in forensic sciences as a fingerprint technique
(Quatrehomme et al., 1996; Nambiar et al., 1999;
Riberiro, 2000; Christensen, 2004a).
4. The assessment of the morphology, especially geometric extent and spatial distribution (Fig. 1), of
the frontal sinuses contribute to the debate as to
whether the mechanical stiffness coupled with
weight decrease is achievable by these structures
(Shea, 1986). As all cranial sinuses are cavities
within bony structures, they do indeed decrease the
weight. Biomechanics researchers are interested in
clarifying the question whether these cavities are
where they are and have the morphology they do,
because putative reasons would explain how they
‘‘lighten’’ the skull without comprising its stiffness/
There is a long, drawn-out debate involving the
issue of stiffness and frontal sinus extent. Homo
erectus and H. heidelbergensis, for example,
have prominent browridges (Schwartz and Tattersall, 2002). Consequently, their frontal
sinuses are very extended (Fig. 1). One group of
researchers interprets these prominent browridges as being a mechanical bar that stiffens
the upper facial cranium (Demes, 1982, 1987;
Russell, 1985; Spencer and Demes, 1993; Prossinger et al., 2000a). The measurements by
Hylander (Hylander et al., 1991a,b; Hylander
and Johnson, 1992; Hylander and Ravosa, 1992)
and colleagues (Moss and Young, 1960; Ross and
Hylander, 1996; Ravosa et al., 2000a,b) have
drawn this view into question, basing their conclusion on the absence of measurable strains
that they attempted to detect in vivo with strain
gauges. They view the frontal sinuses to be the
result of a supraorbital browridge formation
that is an ontogenetic/phylogenetic trait, varying
with species, and not a response to masticatory
stresses. In the previous millennium, Bookstein
et al. (1999) could show that the internal curvature of the frontal bone maintains a constant
curvature over a time span in excess of 2 million
years. They interpreted this as a straightforward result of mechanical pressure exerted by
the growing brain, and therefore argued that
the extended browridges (and, by implication,
the large frontal sinuses) function as (hollow)
Fig. 1. The frontal sinuses of three Homo heidelnergensis specimens: (a) Kabwe/Broken Hill, (b) Petralona, and (c) Steinheim. Yellow:
the frontal sinuses; blue: the sphenoidal sinuses; red: the neuronal
cavity. In all three specimens, the frontal sinuses have an enormous
size. The lateral extent of the frontal sinus of the African specimen is
much smaller than in the European ones. In all specimens, the lamellas can clearly be seen.
stresses during chewing highly fibrous foods.
Note that prominent browridges are observed
in Homo with a large masticatory apparatus
(Fig. 1). A response, disagreeing with this
interpretation was soon published (Ravosa
et al., 2000b). Prossinger et al. (2000a) disagreed, pointing out that the morphological
evidence for phylogenetic browridge formation
is lacking. The absence of evidence for one
argument is not validation for a different one.
The impasse may be resolved by a subtle effect
in the percolation cluster model (PCM) with
quenched noise (see later).
The lamella structure of the frontal sinuses
strongly suggests that the extended browridges
are not independent of masticatory stress (Prossinger et al., 2000a). The electronically prepared
images (Recheis et al., 1999; Prossinger et al.,
2000b) shown in Fig. 2 of the interior of the Petralona cranium exemplify the apositional
hypotheses. Its frontal sinus is enormous in
extent. The supraorbital torus formation hypothesis claims that there is no mechanical
necessity for generating such a large sinus. The
masticatory stress hypothesis claims that the
small internal vault alone could not withstand
the forces occurring during chewing. In H. heidelbergensis, the large masticatory apparatus,
so the argument, necessitates the existence of a
second mechanical vault that is considerably
distanced from the first, otherwise the vault
structure would be too weak (Prossinger et al.,
2000a). The large distance between inner and
outer table of the frontal bone generates such a
required double-vault structure, hence, by
implication, the existence of large frontal
sinuses. The masticatory stress hypothesis
thereby ‘‘explains’’ why the frontal sinuses,
which are ethmodially derived, pneumatize the
frontal bone.
The lamellar structure in the frontal sinus
demands a close inspection (Fig. 2). The lamellae are predominantly orientated parallel to the
midsagittal plane, and thus perpendicular to the
occlusal surface of dentition. They are roughly(!)
periodically spaced, yet have no further ascertainable regularity. They are interconnected by
horizontal lamellae, in a way reminiscent of rib
reinforcements in vault architecture. It seems
difficult to claim that such a complicated rib
structure is ‘‘merely’’ a phylogenetic trait of H.
heidelbergensis or Australopithecines with
massive dentition. (The latter also have such
lamellae distribution morphology; Prossinger,
unpublished.) A parsimonious explanation
would be the lamellae distribution morphology
is a reinforcement structure for a ribbed vault
architecture needed to absorb masticatory
stresses. In the Kabwe (Fig. 1a) cranium, the
‘‘vault ribbing’’ is also prominent, even though
the frontal sinuses do not extend as far laterally as in Petralona (Fig. 1b) and Steinheim
(Fig. 1c).
5. The processes active at a cellular level help to clarify
how the sinuses emerge and expand during ontogeny
(Fig. 3) and are then maintained during the remaining lifespan of the individual. These putative processes need not be genetically controlled or indicators
of heritability. Rather, it is conceivable that mechanisms depend on the physiology of the sinus surfaces,
much like the deposition growth during molecular
beam epitaxy of semiconductor materials research,
sedimentation processes in geology, and bacterial colony growth (many examples in Barabasi and Stanley,
1995 and in Vicsek, 2001).
Any Putative Relations Between Function and
Moore (1981) is of the opinion that there is little point
in speculating about the many features that are
observed in frontal sinuses and in looking for an explanation. In part, I concur. One goal of this review is to
show to what extent some of these ‘‘many’’ features are
statistically reliably quantifiable. A suite of methods,
many of which derive from those developed by physicists, statisticians and biologists investigating numerous
biological growth phenomena as presented in Vicsek
(2001), can be used for (and have successfully been
applied to) the analysis of frontal sinus morphology. As
will become clear in the descriptions of the Methods,
none are restricted to analyzing the morphological features of frontal sinuses. Some are most suitable for
many morphological analyses of cross sections and outlines.
Further research into the morphological assessment of
anatomical structures needs to take the many novel statistical techniques presented here into account. These
methods, evidently successful (as manifest by the robustness of the estimators), point toward explanations that
involve an uncommon perception of sinus ontogeny. This
latter claim is at odds with Moore: a suitable suite of
analysis techniques may point to physiologically satisfactory explanations.
Frontal Sinus Volumes
One data set consisted of five H. sapiens crania from
the Department of Anatomy Collection, University of
Vienna and an Avar cemetary (Lippert, 1969) ranging
from 2 years of age to adulthood (Fig. 4) and the
other of four H. neanderthalensis crania from 3 years of
age to adulthood (Fig. 5; Gorjanovic-Kramberger, 1899;
Klaatsch and Hauser, 1908; Blanc, 1939; Bartucz et al.,
1940). Each cranium was CT-scanned, and the method
of flood-filling the cavity in the scan (Prossinger et al.,
2003, 2005) was used to estimate frontal sinus volumes.
Fig. 2. The frontal sinus of the Petralona fossil in two mutually perpendicular cross sections; (a) longitudinal and (b) coronal. These
images are electronically prepared from a CT-scan. In (a), the internal
table of the frontal bone has been stippled a dark gray. It has been
electronically removed in (b) so that the lamella structure can be
Frontal Sinus Cross-Sectional Areas
A data set published by Szilvàssy (1982), namely the
cross-sectional areas of the frontal sinuses of young H.
sapiens (105 boys and 87 girls, aged 3–11 years), is combined with of 50 adult male and 50 adult female frontal
sinus cross sections published in 1973 (Szilvàssy, 1973).
Fig. 3. An illustration of the pneumatization process leading to the
frontal sinus morphology in Homo sapiens. The smallest cranium is of
a 2-year-old girl, the medium-sized of a 15-year-old adolescent, and
the largest a 35-year-old woman. All imaged frontal sinuses have the
openings to the ethnoturbinals in common. The growth of the cranium
thus accompanies the corresponding expansion of the frontal sinus
lobes. This ontogenetic process is one of the modeling tasks explored
in this review article.
This sample of 292 individuals is ethnically homogeneous. Graph 1 shows the distribution of the cross sections
by sex.
Frontal Sinus Outlines
In the course of his thesis work, Kritscher (1980)
made roentgenograms of many crania in the Caldwell
projection. From the roentgenograms, he traced the outlines, which were orientated so that the midsagittal
plane of each cranium is perpendicular to the top border
of the page. These tracings have been scanned; of these,
25 of Chinese (Fig. 6) are used herein.
Analyzing Morphology Mathematically
This review limits itself to presenting the various
methods that assess morphology via quantifiable morphological features. As the vast literature (Vlcek, 1967;
Tillier, 1977) has shown, classical approaches to frontal
sinus morphologies, describing overall/summative features, like dimensions, angles, etc. or solely verbal
assessments, invariably demonstrates the absence of
any generalizable patterns (Strek et al., 1992). This
review, therefore, concentrates on methods of morphological analysis that depart from classical descriptive
Fig. 4. Images from CT-scans of H. sapiens crania at four different
ages: (a) 2 years old, (b, c) 8/9 years old, (d) 11 years old, and (e)
adult. Details in Table 1.
approaches. ‘‘Traditional’’ morphological assessments,
such as those inventorizing height, width, surface area,
and arithmetically derived variables (such as ratios),
allow at best a limited description of frontal sinus morphology. More often than not, noisiness will render such
assessments as inconclusive. Mathematical modeling
techniques, on the other hand, promise to overcome the
limitations of traditional ones.
Frontal Sinus Volume Ontogeny
as Sigmoids
Frontal sinuses do not grow; rather, they expand
(Maresh, 1940; Fairbanks, 1990; Weiglein, 1999). It is, of
course, unreasonable to monitor the expansion in a single individual: repeated CT-scans would be unethical
(because of the X-ray radiation exposure). Instead, one
samples the ontogeny by using CT-scans of crania of
deceased individuals and assumes that this sample is (in
some way) representative for the frontal sinus volumes
of contemporaneous individuals whose sinuses we cannot measure. Figure 3 shows a graphical representation
of how, as the cranium grows, the frontal sinuses
expand. This expansion need not be the same mathematical function as the increase in braincase volume or
dimensions of the facial cranium. Nonetheless, many
Fig. 5. Images from CT-scans of H. neanderthalensis crania at
three/four different ages: (a) 3 (?) years old, (b) 15 (?) years old, (c)
adult, and (d) adult. Details in Table 1.
geometric aspects of the frontal sinuses (volumes, crosssection areas, and outlines) can best be described with a
sigmoid function
VðtÞ ¼ V1
1 þ aert
West et al. (2001) have shown that many ontogenetic
trajectories derived from biological principles are sigmoid functions. If so, one challenge is to see how well
the parameters V1 , a, and r can be estimated from the
data. Based on the assumption that the ontogeny of
frontal sinuses is an expansion that is most rapid during
adolescence, and then asymptotically approaches the
(constant) adult volume, a sigmoid regression analysis is
Modeling Statistical Noisiness of Frontal
Sinus Cross-Section Ontogeny
Morphological characterization of the frontal sinuses
of Homo is fraught with difficulties. Attempts at a verbal
description are usually unsuccessful, because the features detected are rarely (if ever) found in another specimen.
Because the data sets of frontal cross sections are so
noisy, a more refined estimation technique is neces-
Graph 1. Distribution of the total frontal sinus cross-sectional areas
for 155 males and 137 females from an Eastern Austrian population.
Light gray: adults; dark gray: 3–11 years old.
sary. The ontogeny of the frontal sinus cross section
can also be modeled as a sigmoid function (Prossinger,
AðtÞ ¼ K
1 þ aert
with K the asymptotic value of the cross-sectional area,
r a measure of the rate of expansion at the point of
inflection, and a a parameter related to the age at which
maximum expansion occurs via
t0 ¼
ln r
As evidenced in the data sets published by Szilvassy
(1973, 1982) and shown in Graph 2, noisiness needs to
be controlled. To do so, note that, at each subadult age
Fig. 6. The outlines of 25 H. sapiens frontal sinuses in Caldwell
projection obtained by tracing roentgenograms made by Kritscher
(1980); they are roughly ordered by increasing cross-sectional area.
The right outline no. 11 (marked by the symbol ) is used in many of
the following figures and graphs. All tracings have been made with the
Frankfort Horizontal parallel to the top of the page.
group, the data set consists of more than one crosssectional area. There are nk such areas for each age
cohort k (k 5 1. . .9 for the ages t1. . .t9 5 3. . .9 years),
both for the boys and for the girls. Let Ak denote the
mean area of the Ajk areas of the j (j 5 1. . .nk) individuals in each age cohort k. Prossinger and Bookstein
(2003) presented the rather involved reasoning that
implies that frontal sinus cross sections at each
age cohort are well approximated by a lognormal
Because, to a good approximation,
varðln AÞ 1
one can fit the sigmoid with common lognormal variance
by fitting the observed data to a curve with the cost
1 X
Wk ¼ 2
Ajk Aðtk Þ
Ak j¼1
for each (child) cohort k with weightings A2
for the
residuals of each age cohort. The cost criterion for the
adult cross-sectional areas is
Wadult ¼
Aadult m¼1
ðAm Aðtm ÞÞ2
(where tm (m 5 1. . .50) are the ages at death of the
adults in the data set) because, as will be shown, the
variation of age of the individual hardly influences the
fitted sigmoid. The total cost function to be minimized,
pooled for each sex, is therefore
Wk þ Wadult >
>! min :
Graph 2. The ontogeny of total frontal sinus cross-sectional areas
for 155 males and 137 females from the distribution shown in Graph
1. The sigmoids are estimated according to the methods explained in
the text. Note that the dots for the 3–11 years old do not express their
multiplicity; more than one individual can have the same total frontal
sinus cross-sectional area. Refer to Graph 1.
Fourier Descriptors of Frontal Sinus Outlines
In the following three sections, various outline analysis methods are presented. First, the method of obtaining the outline data set(s) of frontal sinuses is exemplified on Kritscher’s (Kritscher, 1980) tracings of roentgenograms of 25 Chinese skulls (Fig. 6). The details are
exemplified with one particular right outline (no. 11,
marked with the symbol ).
The center of mass of the flood-filled outline (Fig. 7a)
is the origin of the coordinate system in the subsequent
calculation steps. The outline has been digitized, so it is
effectively a polygon. In the next algorithmic step, points
on the polygon that are nearest to integer multiples of
some angle (2-degree angle in this example) are selected
(Fig. 7b). A linear interpolation determines the arrowhead of the vector at 2-degree angle (the tail being the
origin; see Fig. 8).
Each outline j is therefore characterized by 180 vectors. The k (k 5 0. . .179) norms rjk of the vectors are a
function rj 5 rj(a) of the angle—one function for each
individual j (j 5 1. . .25). The differences between the
norm at each angle and the circle with a radius equal to
Fig. 8. An illustration of how the 2-degree angle-polygon derived
from a digitized outline is constructed. The points of the digitized outline (300 dpi resolution) are dots and circles. Gray arrows point from
the center of mass to points on the outline that straddle integral multiple of 2-degree angle; the corresponding dots are drawn as small
circles. The vectors that are integral multiples of 2-degree angle are
drawn in black. Their tips do not (in general) point to integral multiples
of 2-degree angle; rather, these tips are found through triangulation
(the error due to linearization of the digitized outline is exceedingly
small because of the smallness of the angles).
Fig. 7. Digitized outline of the right frontal sinus of Chinese specimen no. 11 from Fig. 6. (a) The frontal sinus in cross section as a
disk, with the center of mass marked as a dot. (b) The radial vectors,
in 2-degree angle spacings from the center of mass to points interpolated on the digitized outline, as clarified in Fig. 8. In this, and in all
other outlines, the 0-degree angle vector is the horizontal one pointing
to the left.
the mean norm (Fig. 9) estimate a periodic function
(Graph 3).
One can fit harmonics to this function by using the
discrete Fourier transform (Press et al., 1992). For every
individual j,
rj ðaÞ ¼
1 X
1 X
Amj eixm a ¼
Amj ðcos xm a i sin xm aÞ
180 m¼0
180 m¼0
where Amj is amplitude of the mth harmonic pwith
(angular) frequency xm ¼ m 2p32
360 ¼ m 90 (and i ¼ 1).
There are at most 89 such harmonics (Nyquist theorem)
and the 0th harmonic is the mean of the function estimation. For the outline in this example, the Fourier
spectrum is shown in Graph 10. Note that in this spectrum, the second harmonic has a very large amplitude
because the outline is periodic in 180-degree angle. The
second largest harmonic is the third harmonic, corresponding to a periodicity of 90-degree angle. The Fourier
spectrum shows that a few harmonics are markedly
larger than all the others. A cursory inspection seems to
indicate that the other harmonics constitute the noise
spectrum. The inverse of the Fourier transform using
only the large amplitude harmonics (those with Am 0.1 in this example) shows a smooth outline (Graph 3)
and the others the noisy fluctuations around the
smoothed outline. Further results of this method of analysis will be discussed later.
Using only the harmonics with the largest amplitudes
is part of a standard toolkit in function smoothing. A
graph of the Fourier spectrum of all 25 outlines (Graph
11) demonstrates one special application to frontal sinus
outlines. First, observe that less than 20 harmonics
(around 15) capture the salient features of the outlines,
and the remaining ones describe each outline’s noisiness
(see a more detailed argument later). Second, observe
that there is no pattern in the 20 first harmonics for the
25 individuals. The Fourier spectrum of each individual
is unique (a generalization derived from the 50 outlines
of 25 individuals). Forensic scientists use this ‘‘fingerprint’’ as an identification methodology (Nambiar et al.,
1999; Christensen, 2004b). Not all forensic scientists use
the full outline in their investigations; some use an outline arc (details can be found in Christensen, 2004b).
Using the full outline as described here has a major
advantage; it is a method free of coordinate orientation
artifacts. Further research shows that the uniqueness of
the ‘‘outline fingerprint’’ converges to a stable value at
around 2-degree angle, hence the choice of this angle in
this article. Because there are two outlines (a left and a
right one) per individual, the probability of all the Fourier harmonics ‘‘fingerprints’’ in two different crania
being identical is then halved.
Graph 3. The result of smoothing the 2-degree angle-polygon
deviations from mean curves as shown in Fig. 9. The 2-degree anglepolygons are defined as shown in Fig. 7b; (a) the deviations as
defined in Fig. 9a. (b) the deviations as defined in Fig. 9b. The small
circles are the actual deviations, the black curves are the superposition of the first five harmonics (those with the largest amplitudes; refer
to Graph 10). The residuals are graphed as noise. The abscissa for
the noise curves has not been drawn through the origin of the graph.
The noise graphs have been shifted for clarity only, they fluctuate
about an axis actually passing through the origin. Note the difference
in scale of the ordinates. The noise curves are not iid, as explained in
the text.
Fig. 9. Deviations of a 2-degree angle-polygonized outline from
two different curves. (a) The curve is a circle with radius equal to the
mean distance of the outline from its center of mass. (b) The curve is
the first SVD smoothed outline (shown in Fig. 10a).
SVD as a Method of Smoothing Noisy Outlines
The sample of the 25 Chinese outlines (Fig. 6) are
mathematically 25 radial vector sets that form a (180 3
25) matrix
> .
r180 1
r180 j
r1 25
r180 25
In A, each column vector (r1j . . . . . . rij . . . . . . r180j)T
j 5 1. . . 25 consists of the norms of the 2-degree anglevectors, scaled for Centroid Size (Slice, 2005). We calculate the singular value decomposition (SVD) of A (Leon,
1998) namely,
A ¼ :... ~
> 1
9 >
...; >
: :: ~
:: ;
(There are 25 singular values rj because 25 individuals
were used for constructing the decomposition.) The matrix A1 ¼ r1~
vT1 can be obtained by setting all singular
values to zero except r1. This matrix A1 is the first
approximation of A with respect to the Frobenius norm
(Leon, 1998). The column vectors of this matrix A1 are
the 25 outlines that are the closest common fit for the
25 individuals. In fact, they are practically identical, as
can be seen in Fig. 10a. Therefore, SVD is a powerful
smoothing technique that produces a mean outline of
the 25 (fractal) frontal sinus outlines. One investigation
approach is to compare these mean outlines for the left
versus the right lobes to assess any possible asymmetries, either by sex or by population.
If one calculates A2 ¼ r1~
vT1 þ r2~
vT2 by setting
rj 5 0 (for j 5 3. . . 25), then one obtains a better approximation of A (in the Frobenius norm sense). In this case,
the 25 column vectors rTj of A2 represent a very interesting approximation of these 25 frontal sinus outlines, as
can be seen in Fig. 10b. One can take advantage of a remarkable feature: Each first approximation outline must
intersect the second approximation outline in four
points; however, these four intersection points of all 25
outlines are remarkably close together; they are (almost)
common to all outlines (Fig. 10b). The four points
obtained this way can be used to define surrogate landmarks. A discussion of the biological implications can be
found in Prossinger (2005). Again, the distribution of
such surrogate landmarks (and their number) in comparisons of left/right lobes, by sex, in a sample population is a novel approach to morphological assessment.
One can introduce a further methodological advancement. Rather than look at the differences in radial vectors with the unit circle, investigate the differences
against the first and second SVD-smoothed outlines
(shown for the 1st SVD-smoothed outline in Fig. 9b).
Not the uniqueness (in the fingerprint sense), but rather
the patterns in the harmonics relative to the common
form of a population (as morphologists—rather than forensic scientists—are wont to do) is the essence of the
summary morphological description. In particular, the
amplitudes of the higher harmonics reveal common features, because the overall periodicity of the common
form is already parceled out. Note that, in the example
drawn in Fig. 9b, deviations are most pronounced on
one side of the lobe, whereas in the deviation from a
circle (Fig. 9a), the deviations are large and quite periodically distributed.
Analyzing Frontal Sinus Outlines as Fractals
and Random Walks
The residuals remaining (after approximating the outline with only the largest amplitude harmonics) ‘‘look’’
like noise (Graph 3). Alas, they are not. The autocorrelation function for many lags (Brockwell and Davis, 2002)
reveals that there are a large number of periodicities
remaining (not shown). In fact, the time series of the
residuals (in both cases) after parceling out the first four
harmonics is not even stationary. Despite the seemingly
reassuring apparent ‘‘randomness’’ of the residuals’ time
series (Graph 3), it must be stressed that the first four
harmonics reveal considerable morphology, yet there is
apparently more information to be extracted. Why is
this so? The answer lies in the fact that the outlines
have fractal properties (Prossinger, 2004, 2005).
Because of the high resolution of the digitizing process, the sum of the distances between the outline pixels
is a good measure of the circumference U of the outline.
The number of pixels enclosed by the outline is its crosssectional area A (in pixel units; the dpi scale factor of
the scan can be used to convert to physical cross-sectional area in cm2). For smooth outlines, A U2, so the
graph A(U) should be parabolic. For objects with fractional dimension (fractals), the exponent is not an integer; the area increases less rapidly with circumference
(because of the self-similar jaggedness of the outline).
Each frontal sinus outline 2-degree angle-polygon is a
sequence r(j) (j 5 1 . . . n) of radii. We are interested how
the radial component r(j) covaries with its neighbor F
(0 < F 2p) steps away. We look at the differences
DrU ðjÞ ¼ rðj þ UÞ rðjÞ:
The periodicity r(j 1 n) 5 r(j) ensures that DrF(j) is
defined Vj. We estimate the variance of these differences
for a given F, viz,
Fig. 10. Two outcomes of SVD smoothing method for the 2-degree
angle-polygonized right outlines of the 25 Chinese specimens from
Fig. 6. (a) The 25 outlines using only the first singular value. The 25
outlines are essentially identical. The outline is the best (in the Frobenius sense) representation of a common form. (b) The 25 outlines
using the first and second singular values only (details in the text). The
variation among the outlines is in distinct regions; all 25 outlines have
four points in common.
VarðDrU Þ ¼ hDr2 iU ¼
1 X
ðDrU ðjÞ hDriÞ2
n 1 j¼1
where hDri denotes the (arithmetic) mean over all DrF(j)
(observing that this average is independent of F). The
standard deviation C(F) 5 hDr2iU is a function of F. If
C(F) varies as C(F) FF, then the graph of ln C(F) 5 ln
(hDr2iU ) versus ln (F) should be a straight line with
slope F. If F is integer, then the curve is smooth (a onedimensional geometric object); if not, then F is a measure of the fractal dimension of the outline (Baumann
et al., 1997). Geometric outlines that derive from biologi-
Fig. 12. An image of the changes in the perimeter list generated
by the Percolation Cluster Model (PCM) algorithm with P0 5 0.72 after
80 iterations. Sites are drawn as squares or circles. The starting perimeter list is the straight, horizontal set of sites at the base of the
drawing. After 80 iterations, the cluster consists of 80 sites (white) and
the perimeter list consists of 25 sites (gray). (a) After an additional 25
iterations, 18 more sites (drawn as circles) have been generated; they
are adjacent to sites from the perimeter list one iteration step earlier.
Not all iterations generate an addition to the perimeter list, because P0
5 0.72. In one case, a perimeter site has been added at the recently
added perimeter site (identifiable by two stacked circles). (b) The cluster (96 sites drawn as white squares) and the perimeter list (25 sites
drawn in gray) after the iterations shown earlier (in a) just before the
next iteration. Note that the perimeter site (square) at k 5 5 is isolated
after only 80 iterations.
cal morphologies are statistical fractals; they do not
have a fixed F for all F, but only over some range. For
outlines to be considered statistical fractals, then the linear region should be over some reasonably large interval
Generating Frontal Sinus Outlines as Fractals
via PCMs
Fig. 11. Three samples of Brownian bridges around (a) a circle, (b)
an ellipse, and (c) two crossed ellipses. (Brownian bridges are random
walks that return to the starting point.) Each drawn sample is one of
the 609 Brownian bridges generated from 10,000 random walks. The
circles are the 180 positions of steps around the curves.
This method derives from predictions of two spreading
models: the Eden model (Eden, 1961) and its generalization, the PCM (Gaylord and Wellin, 1995). In both models, a perimeter list is defined on a lattice, and a seed
site is chosen randomly (from the perimeter list). A site
that is adjacent to (but not an element of) the perimeter
list is added to the perimeter list. This process is iterated. In the PCM, a predefined probability P0 determines whether the site adjacent to the randomly chosen
perimeter site is added to the perimeter list. In both
cases, the resulting union of all perimeter lists (after
many iterations) is called a cluster, and its final perimeter list is sometimes called the outline.
These two remarkably simple algorithms can generate
a wealth of different clusters. Depending on the geometry of the original perimeter list, the clusters show unexpected geometries. If the PCM is used (note that the
Eden model is the PCM with P0 5 1), then changing the
predefined value of P0 not only changes the geometry
(primarily the fractal dimension) of the cluster but also
introduces other biologically relevant features, especially
in the geometrical properties of the outline, as will be
TABLE 1. The specimens used for estimating the sigmoids V(t) that model ontogenetic expansion
of the frontal sinuses, as shown in Graph 4
Homo neanderthalensis
Le Moustier I
Krapina C
Guattari I (Monte Cerceo)
Homo sapiens
Age (years)
Frontal sinus volume (mL)
Bartucz et al. (1940)
Klaatsch and Hauser (1909)
Gorjanovic-Kramberger (1899)
Blanc (1939)
Dept. Anat. Collection, University of Vienna
Avar Cemetery (Lippert, 1969)
Avar Cemetery (Lippert, 1969)
Dept. Anat. Coll., University of Vienna
Inst. Anthrop. Collection, University of Vienna
The sex and the ages at death of the specimens are estimated. Details can be found in the references. The ages of the
adults are simulated to estimate uncertainties (see Graphs 5 and 6).
TABLE 2. The parameter estimates in the sigmoids V(t) 5 V‘ (1 1 ae2rt)21 for
Homo neanderthalensis and H. sapiens
Sigmoid parameter estimates
Homo neanderthalensis
Homo sapiens
V1 (mL)
r (year21)
Range in
9.223 < V1 < 9.375
11.50 < t0 < 12.03
The age t0 when maximum frontal sinus expansion occurs is related to the parameter r via t0 5 a21 ln r . The uncertainties
in the parameters are exemplified by simulating a range of ages and volumes for the input data (see Graphs 5 and 6) and
their distributions are shown in Graph 7.
In Fig. 12 a simple model—very much simpler than
the one used in the frontal sinus outline investigations
presented in this article—depicts the outcome of the
PCM algorithm. The original perimeter list is a set of
sites along a straight line, and the top squares (shaded
gray) form the ‘‘surface’’ (i.e., the perimeter; note that it
need not be connected) that is generated after a (finite)
number of iterations. The circles above the perimeter
are the sites that have been chosen for the ‘‘growth’’ of
the cluster. However, the figure does not show the temporal sequence of when the sites had been chosen. It
only shows that repeated calls to the perimeter sites
have not resulted in every site being chosen to define
the subsequent ‘‘surface’’ or outline. It often happens
that a site can be selected before all other nonvisited
sites have been visited (exemplified in Fig. 12 by some
circles being stacked). Furthermore, in the PCM, randomly chosen perimeter sites need not be added to the
perimeter list to form the new ‘‘surface’’ of the cluster;
they will only be added with a probability P0.
Frontal Sinus Volume Ontogeny
Sigmoid regression can quantify the volumetric expansion of H. sapiens and H. neanderthalensis frontal
sinuses. Data and regression parameters are listed in
Tables 1 and 2, and Graph 4 shows the estimated sigmoids.
Frontal sinus volumes are very difficult to obtain
(Prossinger et al., 2000b). Using four (H. neanderthalensis) and five (H. sapiens) data points gives a seemingly
reliable parameter estimate (Table 2). However, it is
well known that the frontal sinuses are statistically
extremely noisy (see later), so one should be suspicious
of the apparently successful modeling. The analysis presented here does not, therefore, imply that the asymptotic frontal sinus volume V1 in H. neanderthalensis is
smaller than that of H. sapiens. Quite the contrary, the
observation of the large Neanderthaler browridge (Tattersall, 1995; Schwartz and Tattersall, 2002) leads one
to expect a different morphological outcome. Rather, this
example shows the sensitivity of the parameter estimation to the data by addressing the assumptions made in
the published literature. There is a long, involved debate
about the age-at-death of the LeMoustier I adolescent
(Thompson and Nelson, 2005). Graphs 5 and 6 show
how simulating a variation in the ‘‘raw’’ data influences
the parameter estimates. All numerical simulations
show that the parameter estimates are markedly sensitive to age estimates of the time of maximal growth rate
(mathematically: the point of inflection)—around the age
of death of the Le Moustier I individual. Because of technical sophistication, the uncertainties in volumes determined by the flood-filling algorithms applied to CT-scans
(Prossinger et al., 2000b) are less than 1%, so the ageat-death estimates are what drives the uncertainty in
the parameter estimates V1, a, and r. However, summary histograms (Graph 7) show that these parameter
estimates are quite robust. Graph 7a shows that the asymptotic volume derived from the Neanderthal varies by
less than 61 mL in a total of 24 simulated uncertainties;
Graph 7b shows that the putative age-at-death of the
adult does not influence the estimation of sigmoid parameters, and the uncertainty of the age-at-death of one
adolescent individual results in a fluctuation of the age
at maximal expansion by 61/4 year, less than the published uncertainty in the estimated age-at-death of Le
Graph 4. Sigmoid regressions of Homo neanderthalensis (large
circles) and H. sapiens (small circles) modeling volume ontogeny of
the frontal sinuses illustrated in Figs. 4 and 5.
Moustier I (Thompson and Nelson, 2005). Sigmoid functions are not only model biological processes very well
but also they supply robust estimators.
Frontal Sinus Cross-Sectional Area Ontogeny
Applying the methods of controlling for noise to the
Szilvassy data sets, one obtains the following sigmoid
For the males, the regression is
A# ðtÞ ¼ 12:299
; t0# ¼ 12:53 years;
1 þ 37:15e0:2885t
and for the females it is
A$ ðtÞ ¼ 10:451
; t0$ ¼ 9:51 years
1 þ 129:18e0:5110t
Both fitted sigmoids are shown in Graph 2. The asymptotic areas for the males (K# 5 12.30 cm2) and the
females (K$ 5 10.45 cm2) are close to the averages
(12.32 and 10.31 cm2) as published by Szilvassy (1981).
The logarithm of the interpolating sigmoid and the logarithms of the cross-sectional areas are graphed in Graph
8. The fits are very good, despite the noisiness of the
The significance of the difference in models is tested
with a log-likelihood test: test the hypothesis H1 (interpolating the two sexes separately with a total of eight
parameters K#, K$, a#, a$, r#, r$, r#2 , and r$2) against the
hypothesis H0 (the complete data set is to be interpolated with the four parameters Kall, aall, rall, and r2all ) by
calculating the statistic
k ¼ 2 ln>
:LH0 LH1 >
where L is the likelihood of H, the product of the probabilities of the observations over each datum of the dataset. k has approximately a v2-distribution with 4 (5 8–4)
degrees of freedom. In the complete data set, ln LH0 5
Graph 5. Fluctuations in regression sigmoids due to variation of
age-at-death as input: Homo neanderthalensis. (a) Varying the estimated age-at-death of the adult specimens. (b) Varying the estimated
age-at-death of the adolescent specimen. (c) Varying the estimated
ages at death of the adults and the adolescent.
Graph 6. Fluctuations in regression sigmoids due to variation of age-at-death as input: Homo sapiens. (a) Varying the estimated age-at-death of the adult. (b) Varying the estimated age-at-death of the adolescent and the adult. (c) Varying the estimated volume of the 8/9 years old and the estimated age-atdeath of the adult. (d) Varying the estimated age-at-death of the 8/9 years old.
2358.571 and ln LH1 5 2348.571, k 5 19.30 (P < 0.001,
highly significant; P 0.001 at v24 5 18.5).
Is there a significant laterally asymmetry of the crosssections? This tests the hypothesis: is it to be expected
that the left lobes to have a larger/smaller cross-sectional area? A paired two-tailed t test shows that there
is a difference only at the 6.1% significance level
(females) and 7.4% significance level (males). Although
the total cross-sectional areas are lognormally distributed, the left and right lobes are not (Graph 9).
Fractality of Frontal Sinus Outlines
Graph 12 shows the relation A(U) obtained from the
25 left and right Chinese frontal sinus outlines in the
functional form A 5 z1 Uz (left: z 5 1.897 6 0.071; right:
z 5 1.864 6 0.059). The exponent is significantly different for 2 (first successful outcome of a test for fractality)
and that the fractal ‘‘growth’’ is statistically the same for
both the left and right lobes. The graphs in Graph 13
superimpose the fractal dimensions of the 25 Chinese
left and 25 Chinese right frontal sinus outlines. The fractal dimensions so obtained are consistent and no trend
separating left from right outlines can be ascertained.
The 2-degree angle-polygon outlining the right sinus
lobe of Crapina C (Gorjanovic-Kronberger, 1899; Wolpoff,
1999; Schwartz and Tattersall, 2002), together with the
arrows that define this polygon, is shown in Fig. 13
(compare with Fig. 5c). The log–log function shows that
the outline scales self-similarly over a scale factor of e2
7.38 (both for right and left lobe outlines; corresponding to an angle range 28–288)—a scale factor that is very
Graph 7. Distribution of the estimators of the sigmoid regression
parameters. (a) The distribution of the estimates of the asymptotic
frontal sinus volume obtained with the variations shown in Graph 5.
Light gray: variation in Graph 5a; medium gray: variation shown in
Graph 5b; dark gray: variation shown in Graph 5c. (b) The distribution
of the age of maximal frontal sinus expansion obtained with the variations shown in Graph 6. Light gray: variation in Graph 6a; medium
gray: variation shown in Graph 6b; dark gray: variation shown in
Graph 6c; black: variation shown in Graph 6d.
Graph 8. The demonstration of the regression quality of sigmoids
when using weighted residuals for lognormally distributed data points.
The logarithms of the total frontal sinus cross-sectional areas are
graphed, together with the logarithms of the regressions sigmoids that
estimate the frontal sinus ontogeny. The fits are remarkably good,
especially for such statistically very noisy data.
large for a biological object! The fractal dimension estimate of the right frontal sinus lobe outline is 0.660 6
0.011 (r2adjusted 5 0.9997); for the left one (not shown) the
fractal dimension estimate is even better: 0.6197 6
0.0063 (r2adjusted 5 0.9997).
Frontal Sinus Outlines as Random Walks
Another way of assessing the distribution of the fractal dimension of outlines is to compare those with the
properties of a closed random walk. Consider the following analogy: a drunkard, while attempting to follow a
closed path, lurches randomly from side to side. Each
lurch to the side is 61 step while he moves forward by
2-degree angle (in this simulation). A further condition
is that the drunkard return exactly to the position he
started from: the ‘‘lurching drunk’’ has performed a
closed random walk (a Brownian bridge). One simulates
closed random walks by using a (suitably chosen) step
In detail: from 10,000 random sequences rjk (k 5
.181) of 11s and 21s, collect those j for which
k51 rjk 5 0 — a total of 609 Brownian bridges were
found in this simulation. Using these bridges and adding
steps of width sf generated 609 random walks ranj and
random walk outline points ranjk 5 rk 1 sf rjk (k 5
1. . .180; j 5 1. . .609) were obtained. If rk 5 const, one
simulates a lurching drunk trying to follow a circle.
rk ¼
1 þ e cosðk 2 Þ
one simulates a random walk along an ellipse with eccentricity e and parameter P. Another case is the sum of
two ‘‘crossed’’ ellipses, viz.,
rk ¼
1 þ e1 cosðk 2 Þ 1 þ e2 cosðk 2 þ XÞ
(two eccentricities e1 and e2, two parameters P1 and P2,
and an angle X between the two major axes). Figure 11
shows the three closed paths chosen for this analysis, together with one such random walk for each.
Graph 15 shows the distribution of the fractal dimensions of these 3 3 609 Brownian bridges. Notice, how
the fractal dimension distribution of the Brownian
bridges about the ‘‘crossed’’ ellipses straddles the fractal
dimensions of the 25 left and 25 right Chinese frontal
sinus outlines. Also, notice that the distribution of fractal dimensions of the Chinese outlines exhibits no left/
right asymmetry.
Graph 11. The discrete Fourier transform spectra of the deviations
of the 2-degree angle-polygons from a mean circle (as defined in Fig.
9a) of all the 25 right Chinese frontal sinus outlines shown in Fig. 6.
The pattern of large amplitudes is unique for every individual; the harmonics beyond 20 contribute to noise (for details, refer to the text).
Graph 9. The distribution of left and right frontal sinus cross-sectional areas of 155 males and 137 females from the population shown
in Graph 1. Light gray: right lobes; dark gray: left lobes. A paired t test
shows that there is no significant asymmetry in the cross-sectional
area distribution.
Graph 12. The relation between circumference and enclosed area
of the 25 left and 25 right Chinese frontal sinus outlines shown in Fig.
6. Stars: right frontal sinus outlines; squares: left frontal sinus outlines.
Note that there are two regression functions graphed: Aleft U1.897
and Aright U1.864, but they are so close that they cannot be graphically resolved. The two exponents are the fractal dimensions of the
left/right outlines.
Fractal Features of Frontal Sinus Outlines
Generated by PCMs
Graph 10. The spectra of the discrete Fourier transforms of the
deviations from the mean curves as shown in Fig. 9. Main graph: for
the deviations as defined in Fig. 9a. Inset: the deviations as defined in
Fig. 9b. Note the difference in amplitudes in the two deviation conditions. There are 2 3 89 harmonics; only half are graphed, as the other
half is symmetric. The 0th harmonic is the average of the fluctuations,
which has been parceled out before calculating the discrete Fourier
The resulting outlines generated by the PCM are fractals (Czirok, 2001; Prossinger, 2005). Graph 16 shows
the outlines of a cluster after 25,000 and then 75,000
iterations with two different perimeter list geometries,
both with P0 5 0.75. As is to be expected, the cluster
becomes larger as the number of iterations increases
and the number of sites that define the perimeter length
also becomes larger. Note that the ‘‘length’’ of the perimeter is defined as the number of lattice sites in the perimeter list after the last iteration, not the length (in a
geometric sense). In Graph 16b, there are perimeter
sites that are ‘‘inside’’ the cluster. One way of finding
the geometric perimeter, i.e., the biologically relevant
Graph 13. The distribution of the fractal dimensions of the 25 left
and 25 right Chinese frontal sinus outlines shown in Fig. 6. The individual numbers have been reordered, left and right separately to demonstrate the distribution of these fractal dimensions.
one, using PCM, is demonstrated later in the case of the
Crapina C outlines.
Because of the limitation of (randomly) choosing a site
in the perimeter list, and not any arbitrary lattice site,
the elements of perimeter list are correlated. One feature of the growth is the pinning effect (Barabasi and
Stanley, 1995): some elements of the perimeter list are
not chosen frequently enough (because of chance) and
this part of the perimeter lags behind as the perimeter
moves ‘‘outward.’’ (An example is the k 5 5 site in Fig.
12) Whenever pinning sites are close, the entire
‘‘surface’’ lags behind, as shown at P1 and at P2 in
Graph 16b. Computer simulations have revealed that
such pinning effects become more pronounced when the
modeling of physical and biological systems become
more realistic. In bacteria growth modeling, for example,
the pinning occurs predominantly at any inhomogeneity
of the agar gel; in some physical systems, pinning is
enhanced because of small irregularities in the density
of the substrate or because of disorders in a medium or
dislocations in a crystal. Consider the example of the
propagation of a liquid through a paper towel (Barabasi
and Stanley, 1995, p. 119–122). The lag in flow because
of inhomogeneities of the paper results in a quenched
noise phenomenon. The pinning due to quenched noise
is not to be confused with limiting the expansion of the
cluster due to a barrier (Graph 17). A barrier is straightforward to incorporate in PCMs: it is a topology of
absent lattice sites. The quenched noise effect enhances
pinning, whereas the existence of a barrier does not. In
Graph 17b, the deep groove in the perimeter is due to
the lag as the perimeter sites ‘‘try’’ to ‘‘catch up’’ after
having successfully surrounded the barrier. At P1 and P2
in Graph 16b, the lags are due to the pinning effect—
there is no barrier at these lags.
The ‘‘growth’’ of the cluster can be used to model the
expansion of sinuses. In other words, pneumatization as
a PCM is modeled with some P0. Cranial bones grow by
periosteal intramembranous ossification (Martin et al.,
1998). Using PCM as a method of morphological assessment is justified by noting that the shaping of the pneumatization occurs via osteonal remodeling: osteoblasts
deposit bony material and, therefore, build up the bone,
whereas osteoclasts sculpt it by removing bony material.
In the cranium, bones have respond to strains/stresses
differently than do long bones (Rawlinson et al., 2000):
location-dependent patterning has traditionally been
ascribed to physiology, but Skerry (2000) claims this
issue is unresolved. This unresolved issue is addressed
by looking at the outcomes of algorithmically simulations of osteonal remodeling as a PCM process. The bony
Fig. 13. A simulation of the ontogeny of the two Crapina C frontal
sinus outlines using a PCM with P0 5 0.55 and a total of 175,000 iterations. The fractals generated by the PCM are shown in Graph 18.
The outermost fractal is splined once onto the left frontal sinus outline,
once onto the right one. The previous (ontogenetically: ‘‘earlier’’) fractals are also splined, with a scale factor proportional to the ratio of
their length to the length of the outermost fractal. The black dots are
the corners of the 2-degree-angle-polygon (shown as arrow tips in
Graph 14), the contours of the digitized outlines derived from Fig. 5c
are drawn in mauve. Color-coding of the fractals is the same as in
Graph 18.
material is the lattice, the growth of the perimeter list is
the expansion of the sinus(es) because of the osteoclasts
removing bony material with a higher probability than
the osteblasts depositing it. The difference between deposition and removal probability must satisfy the condition 0 < P0 < 1. Therefore, inhomogeneities in the bone
(though not necessarily the composition of the bony material!) result in quenched noise sculpting. In the context
of frontal sinus morphology, the most important sources
of inhomogeneity are the slight, repeated compressions
of bony material during mastication, as the resulting
compression waves nonuniformly compress the frontal
bone. As has been noted earlier, the inhomogeneities
that produce pinning via the quenched noise phenomenon can be (indeed: are) very slight. As the osteoclasts
remove bone, the pinning results in (locally) delaying/
halting the outcomes of their activity: wherever the lattice sites are denser, more new sites must be added to
the perimeter list (at constant P0). In other words, the
apparent slowing of osteoclastic activity outcomes and
the attendant removal of less bony material is modeled
by inhomogeneities (primarily a higher density of lattice
sites) in the lattice in the PCM, and the slowing of sinus
expansion is due to masticatory stresses. The pinning
sites are therefore the observed bone lamellae in the
sinuses (Fig. 2). The emergence of lamellae in the pneumatizations is modeled as pinning and quenched noise
phenomena in PCMs.
One can show how this effect comes about in the generation of the lamellar morphology of the frontal sinuses
in Crapina C (Fig. 5c). First, generate a cluster using a
PCM with P0 5 0.55. ‘‘Snapshots’’ of the perimeter list
after 75,000, 145,000, and 175,000 iterations (with a
suitably complicated initial perimeter list geometry,
details can be requested from the author) are shown in
Graph 18. We note how the pinning sites become more
Graph 15. The distribution of the fractal dimensions of three sets
of 609 Brownian bridges around curves shown in Fig. 11, and the
fractal dimensions of the 25 left and 25 right Chinese frontal sinus outlines shown in Fig. 6. White bars: Brownian bridges about a circle;
light gray bars: Brownian bridges about an ellipse; black: Brownian
bridges about crossed ellipses (see Fig. 11c). Inset: the distribution of
the left (labeled L) and the right (labeled R) fractal dimensions of the
25 left and 25 right Chinese outlines, and the fractal dimensions of the
Brownian bridges about crossed ellipses (Fig. 11c). The left/right fractal dimensions straddle the Brownian bridges about the crossed ellipses very well. Note the scale of the ordinates.
scaling factor for the spline is the ratio of the length of
each perimeter list to the longest one. This procedure
not only models lamellae emergence but also reconstructs the ontogeny of Crapina C (Fig. 13).
Graph 14. The 2-degree angle-polygon approximating the right
frontal sinus outline of Crapina C (Fig. 5c) and the log–log graph of the
standard deviation C(F) as defined in the text. The circles are the
points used for estimated the linear regression. The self-similarity of
the outline is a statistical fractal (a self-similar object) from 28 to 288.
The variables C(F) and F are explained in the text.
pronounced as the perimeter becomes longer: these are
the predicted lags. We then generate 2-degree angle-polygons of the observed outlines (one left and one right) of
Crapina C, with the method demonstrated in ‘‘Frontal
Sinus Outlines as Fractals and Random Walks’’ (earlier).
The left polygon is then splined onto the digitized left
outline of Crapina C, likewise the right outline, along
with the fractals (with lengths 1,706, 2,268, 2,549). Each
Attempting to estimate the ontogeny of frontal sinus
volumes via a sigmoid function is fraught with difficulty:
the sample size is often too small but never too large.
Because only a few well-behaved points ensure a good
sigmoid fit, readers should be warned of the false reassurance in Graph 1. The case of the very noisy cross-sectional areas of the outlines show most dramatically how
challenging it is to properly estimate a biologically
meaningful sigmoid. However, the sigmoids modeling
the volume ontogeny do reveal several things. First,
they show how sigmoids behave with variation in measurement error—they are quite robust, in fact, for biologically relevant estimators. Second, there is a debate
of the age-at-death of Le Moustier I and the sex of the
individual because of the observed gracility of the cranium (Thompson and Nelson, 2005); one can justify why
the debate is crucial in this context. Only the volumes
near the point of inflection drive the parameter estimates of the sigmoid function. Accurate age-at-death
determination of adults is not very important for estimating the parameters. Intriguing is the consistency of
the frontal sinus data despite their paucity. Although
Neanderthals have large browridges, their ontogeny
seems to indicate that the frontal sinuses are much
smaller than those of H. sapiens. In the debate about
‘‘accelerated ontogeny’’ of Neanderthals, the sigmoids
indicate that their frontal sinus ontogeny may be chronically comparable (in some statistical measure) with H.
Graph 16. Two clusters generated by the Percolations Cluster
Model (PCM) algorithm after (a) 25,000 and (b) 75,000 iterations with a
predefined probability P0 5 0.75. The initial perimeter list consisted of
two straight strings, perpendicular to each other.
SVD smoothing has revealed the common underlying
form of the 25 Chinese outlines. The smoothed shape
(Fig. 10b) is closer to that generated by two crossed
ellipses than by a single ellipse (Fig. 11). The observed
distribution of fractal dimensions of Chinese outlines is
closer to the fractal dimensions of random walks around
crossed ellipses than around a single ellipse; the fractal
dimensions of these random walks comfortingly confirm
the underlying common form detected by SVD smoothing. Random walks Si are not iid-noise (Brockwell and
Davis, 2002) while their first differences Xi 5 Si11 2 Si
are. Brownian bridges are a subset of random walks, but
their noise properties are more involved (Bookstein, personal communication). Details of the noise properties of
Graph 17. Two clusters generated by the PCM algorithm after (a)
45,000 (b) 175,000 iterations with a predefined probability P0 5 0.55
in the presence of a barrier. In both cases, the initial perimeter list
consisted of a straight string of sites, roughly along the first median of
the coordinate system. For illustration purposes, the two barriers have
different geometries; in both cases, after sufficiently many iterations,
the barrier will be engulfed by the cluster (one not shown).
Brownian properties on SVD-smoothed frontal sinus outlines have not yet been resolved and are a topic of
ongoing investigations.
The noisiness of the frontal sinuses is one reason why
this article reviews so many different analysis techniques. The noisy signals do contain biologically relevant
information, but it cannot be extracted all with one
assessment methodology. The sigmoids found by Prossinger and Bookstein (2003) do reveal biologically consistent, even interesting, relevant insights. The histo-
Graph 18. Three clusters generated by the PCM algorithm after
75,000, 145,000, and 175,000 iterations with a predefined probability
P0 5 0.55. The initial perimeter list consisted of a straight string of
sites, roughly along the first median of the coordinate system.
grams of left and right cross-sectional areas indicate
that they are not well approximated by a lognormal distribution, whereas their sum (the total cross-sectional
area) is. One implication of this is: the observed lognormality of the total cross-sectional area of both lobes together, yet its absence for each lobe population points
out that septum morphology also exhibits noisiness.
Rarely is the septum close to being planar, and it is
also rarely orientated parallel to the midsagittal
plane. In projection, therefore, the parts of the outlines of the individual lobes near the midsagittal plane
are even noisier than the other parts. In Fig. 4e, the
septum is not detectable in the projection. The frontal
sinus lobe projections in Figs. 5b,c mask some of the
septum; its morphology can only be assessed in 3D CTscans.
The clusters generated by PCMs that model the putatively earlier outlines of the Crapina C lobes when the
individual was younger must have been separated by a
smaller distance than is rendered in Fig. 13. Further
studies of how the lobes move apart—in all species of
Homo—are needed. Such studies will also contribute to
our knowledge of the ontogeny of the septum, its morphology, and further details of the pneumatization process in the frontal bone.
In general, morphologists try to bridge the gap
between two diverging challenges: to find the unique
morphology of an individual specimen and to extract
general features inherent in the specimens of a population. Frontal sinuses present, in this regard, probably
(although this probability has yet to be calculated!) the
most extreme challenge. The noisiness ensures that each
individual’s frontal sinus outlines are unique enough to
warrant using them for identification purposes.
Although human digit fingerprints and iris patterns are
similarly unique, this uniqueness of the frontal sinus
outlines is frustrating for the researcher attempting to
extract biological generalities in the face of extreme
noisiness (pun intended!). The methods presented here
(many of which had been developed and first applied to
the identification of cell outline morphologies and extraction of statistical descriptors of bacteria growth colony
outlines) show how this challenge can be met. The
ontogeny of the frontal bone seems to have wider biological implications than the patterns of the iris.
How many Fourier components need to be retained for
the identification of an individual? Answer: retain all
those frequencies that do not constitute iid noise. Those
Fourier components that contribute to iid noise can be
detected with a suite of tests (Brockwell and Davis,
2002): (1) the maximum and minimum values of the
sample autocorrelation function, (2) the sample value of
the Ljung-Box statistic, (3) the sample value of the turning-point statistic, (4) the sample value of the difference
sign statistic, (5) the rank test (Kendall and Stuart,
1976) for the existence of a trend, and (6) testing for the
possibility to fit an autoregressive model with the YuleWalker algorithm. These tests have been carried out in
the case of the 25 Chinese outlines, but the details are
too involved to be presented here. As a rule of thumb,
some 20 harmonics need to be retained.
The successful characterization of general biological
principles that lead to highly individualized pneumatization morphologies is, as is to be expected, unorthodox.
This review shows that a suite of statistical techniques
does exist and that they can successfully assess morphology, but only by modeling the pneumatization (i.e.,
its cellular process). It is interesting that a reductionist
modeling approach can extract biologically relevant macromorphological features that conventional morphometrics methods fail to find. The PCMs presented here offer
an added bonus: they may resolve the masticatory stress
Chewing results in compression waves (albeit of very
small amplitude) propagating through the facial skull,
including the outer frontal bone. Bone modeling
(osteonal remodeling) is a competition between osteoblast and osteoclast activity, which is best modeled by
PCM. The resulting pinning sites are stochastically distributed along the perimeter. Wherever they form, however, they lag more and more behind as the perimeter
list lengthens. The lamellae (the set of pinning sites)
therefore have an observable morphology (Fig. 2b). They
apparently form rib-reinforced architecture. The ‘‘masticatory stress debate’’ (Prossinger et al., 2000a; Ravosa
et al. 2000b) has heretofore been inconclusive, because
the observed large frontal sinuses imply a mechanical
response, yet the stresses could not be detected in vivo
(Hylander and Ravosa, 1992). The PCMs show that
lamellae emerge because of mechanical stresses and
strains, but their amplitudes are so small that they can
hardly be detected with strain gauges applied to the
browridges and faces of small primates. And, to boot,
the strain gauges were not attached to the lamellae. A
careful morphological analysis, which necessitates modeling of fractals, resolves this debate. The lamellae,
then, are the responses to masticatory stress, not the
large frontal sinuses per se.
On the other hand, the mechanical strains/stresses,
although responsible for the emergence of the lamellae,
do not explain why the pneumatization occurs in the
first place. The pneumatization itself must be due to
phylogenetic/ontogenetic processes (Leicher, 1928). The
pneumatization of the diploë between inner and outer
table of the frontal bone is described as a difference in
growth rates of the two growth fields (Enlow, 1975; Liebermann, 2000). What happens within the diploë needs
to be ‘‘described’’ (more precisely: mathematically modeled) as well. It is therefore reasonable—even necessary—to analyze frontal sinus morphology using PCMs.
Such analyses should give insight into the emergence of
phylogenetic trees, or at least clades.
Morphological assessment using PCMs also explains
another observed feature: why the lamellae distribution
in a population is (statistically) noisy yet has an
observed quasi-regularity (Figs. 1 and 2) in each single
individual. The PCM algorithm shows that the pinning
sites occur stochastically. It is their lagging that is
enhanced by mechanical (in the case of frontal sinuses!)
feedback mechanisms, the quenched noise effect. No two
lamellae patterns can be expected to be the same, the
probability (the product of two very small probabilities)
for that is miniscule. Using PCM algorithms, therefore,
shows why the projection of frontal sinus outlines can be
used as ‘‘fingerprints’’ unique to an individual. In projection, the lamellae generate the (stochastically emerging)
outlines. PCMs supply a justification for the statistics of
fractal forms.
PCMs do not predict that the volume distributions are
as noisy as the outlines. If pneumatization is primarily
phylogenetically driven, then sigmoids are good models
for volume ontogeny. The outlines, as projections of
the 3D-fractals, become noisy and the highly variable
outline morphology generates the uniqueness of the
In this review, Moore’s skepticism can therefore be
addressed: although we cannot derive a function from
the traditional assessment of frontal sinus morphology,
attempts at deriving statistical estimators that are applicable to noisy data sets point to physiological and phenomenological insights (including insights as to the
‘‘roles’’ and ‘‘functions’’ of the frontal sinuses) that had
heretofore remained inaccessible.
This article reviews a suite of analysis techniques, not
an inventory of conventional descriptive morphological
features of specimens or populations. The presented
techniques show that frontal sinuses are not only in-
triguing objects of study but also remain elusive when
conventional techniques, which have been successful for
other features of the skeleton, are applied.
Novacek (1993) cautiously claims that sinuses may
merely be spaces between struts and pillars in the cranium or that there are many possible functions. The
analysis techniques presented here have more to offer
than the highly verbose morphological assessments so
ubiquitous in the literature. Three sequence analysis
methods (Fourier descriptors, random walks, and fractal
dimension determination) and, fourth, the PCMs provide
morphological analysis novelty. Each of these four can
be employed to focus on possible answers (plural!) to the
elusive question of the frontal sinuses’ role(s). Indeed,
the techniques presented here allow one to ascertain
whether there could be one, several, or no role of the
observed morphology of these pneumatizations. Here, it
has been shown, for example, how the pinning effects in
PCMs can explain the emergence of lamellae and their
(statistical) noisiness—both in one single individual and
also within groups.
Why are models of the morphology of frontal sinuses
needed, rather than conventional, verbose descriptors?
More specifically: need one model in order to statistically
describe? The rationale for theory is the necessity of logical consistency, not simplification or ad hoc narratives.
Simplification is most parsimoniously achieved with
models; they should be designed to be consistent with
theory (albeit not an ad hoc one!) and demonstrate the
salient features observed. It is properly constituted
theory that defines what salient is. Ad hoc theories,
alas, are at odds with the definition of salient, as they
cannot identify—let alone specify—saliency. Whenever,
in any scientific undertaking, models are successful, their
parameters constitute the sine-qua-non information extractable from the observed natural phenomena. Describing roughness with simple summary statistics, for example, is not very appropriate (because roughness is statistically too noisy). Describing how roughness varies (better:
changes) by finding the numerical values of the statistical
estimators is an endeavor more appropriate and more rigorous for fractal morphologies, including anatomical ones
(Graph 15). The intent of modeling fractal dimensions of
frontal sinus outlines did not a priori include supplying a
resolution of the masticatory stress debate. Pinning and
quenched noise effects are salient features that are not
added to the models of pneumatization in an ad hoc manner; they are emergent properties.
The noisiness of frontal sinus morphology has necessitated a shift in how anatomists approach difficult-toresolve morphology questions. The methodology needed
to assess morphology now demands model building; otherwise, the statistical variances do not converge. Fractals do not have converging variances; so repeated measurements of ever-larger data sets are not worthwhile.
Only parameters of an underlying model can be estimated by applying the model to the data—to boot, it
must be a model not derived from an ad hoc theory.
Sinuses are cavities; the outlines are projections of
their lateral extent. A scale change that is the same in
all directions will change the morphology of self-affine
objects. Scale changes that differ in different directions
do not change the morphology of interfaces (Barabasi
and Stanley, 1995). The latter changes behave like fractals. The projection of the fractal interface bone/pneuma-
tization will produce a fractal outline. But an inherent
limitation of outline analysis remains: it misses out on
processes that take place in the direction perpendicular
to the projection plane. Although morphological features
detected in the analysis of outlines are not misleading,
one would like to know what happens in the (third)
dimension perpendicular to the projection. Even though,
in the case of frontal sinuses, where cavity dimensions
perpendicular to the Caldwell projection are not large,
analyses incorporating this dimension promise to reveal
a considerable wealth of further information—information (not only morphological) that doubtless is highly desirable in its own right.
The frontal sinuses are cavities whose enveloping surfaces follow the curvature of the inner and outer tables
of the frontal bone. Envelope morphology needs to be
investigated. Statistical analysis of envelope morphology
necessitates, however, 3D-morphometrics.
The septum not only has fractal surfaces (these are
the fractals of the pneumatizations) but also a highly
irregular orientation distribution within the frontal
bones of a population. Assessment of septum morphology
and orientation distribution is only possible in 3D.
The perimeter generated by a PCM is rougher than
the surfaces observed in biological specimens (Figs. 4
and 5). This is because perimeter and cluster growth in
PCMs generate uncorrelated surface structures. The
introduction of correlated noise (Barabasi and Stanley,
1995) smoothes the surface to some degree and—most
importantly—removes the isolated perimeter sites
within the cluster. In the case of pneumatizations,
PCMs with correlated noise are only meaningful in 3D,
because the finite-distance correlations are not restricted
to correlations in the outlines’ plane.
For all these reasons, the motto for the next round of
morphological assessment(s) must henceforth be: 3D
analysis techniques are needed!
Many researchers and colleagues have given (invaluable) assistance during the development and tailoring of
the mathematical methods presented here: their (often
critical) advice was not restricted to mathematics and
statistics; quite the contrary, many were morphological
practitioners whose criticisms helped hone the statistical/mathematical approaches by enhancing the understanding of the biological aspects of frontal sinuses. The
author thanks (in alphabetical order): Leslie Aiello
(Wenner-Gren Foundation for Anthropological Research,
New York, USA), Fred Bookstein (Department of Statistics, University of Washington, Seattle, USA), Phillip
Gunz (Department of Human Evolution, Max Planck
Institute for Evolutionary Anthropology, Leipzig, Germany), Thomas Koppe (Institute of Anatomy, Ernst Moritz Arndt University, Greifswald, Germany), Les Marcus
(y), Roberto Macchiarelli (Université Poitiers, France),
Philipp Mitteröcker (Konrad Lorenz Institute for Evolution and Cognition Research, Altenburg, Austria), Gerd
Müller (Department of Theoretical Biology, University of
Vienna, Austria), Todd Rae (Department of Anthropology, University of Durham, UK), Wolfgang Recheis (Universitätsklinik für Radiodiagnostik, University of Innsbruck, Austria), Silvia Scherbaum (Library for the Biological Sciences, University of Vienna, Austria), Dennis
Slice (Department of Anthropology, University of Vienna,
Austria), Chris Stringer (Department of Palaeontology,
Natural History Museum, London, UK), Ian Tattersall
(Division of Anthropology, American Museum of Natural
History, New York, USA), Maria Teschler-Nicola
(Department of Anthropology, Natural History Museum,
Vienna, Austria), Philipp Tobias (Department of Anatomy, University of Witwatersrand, Republic of South
Africa), and Lothar Wicke (Vienna, Austria).
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The PCM algorithm has been programmed in MathematicaTM by Gaylord and Wellin (1995). Most researchers investigating biological fractals program their own
software, as I did. The sigmoid regressions, maximumlikelihood estimations, and the determinations of A(U)
are available on request.
Software used for flood-filling the CT-scans of frontal
sinuses is available in standard medical software packages.
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