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Computerized restoration of nonhomogeneous deformation of a fossil cranium based on bilateral symmetry.

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Computerized Restoration of Nonhomogeneous
Deformation of a Fossil Cranium Based on Bilateral
Naomichi Ogihara,1* Masato Nakatsukasa,1 Yoshihiko Nakano,2 and Hidemi Ishida3
Laboratory of Physical Anthropology, Department of Zoology, Graduate School of Science, Kyoto University,
Kyoto 606-8502, Japan
Laboratory of Biological Anthropology, Graduate School of Human Sciences, Osaka University, Suita,
Osaka 565-0871, Japan
School of Human Nursing, University of Shiga Prefecture, Hikone, Shiga 522-8533, Japan
Proconsul; thin-plate spline; surface model; computed tomography
We developed a computerized method of
correcting plastic deformation of a fossil skull, based on
bilateral symmetry with respect to the midsagittal plane,
and applied this method to reconstruction of a fossilized
Proconsul heseloni cranium (KNM-RU-7290A). A threedimensional (3D) model of the fossil was generated using
consecutive cross-sectional images retrieved from computed tomography. 3D coordinates of anatomical landmarks that should be located on the midsagittal plane and
pairs of landmarks that should be symmetrical with
respect to this plane were acquired. These landmarks were
then repositioned so that geometrical constraints were satisfied, while translated distances of landmarks were mini-
mized. We adopted a thin-plate spline function to mathematically describe the 3D nonlinear volumetric transformation between acquired and repositioned landmarks. Using
this function, the entire fossil shape was transformed, and
the effect of reversing the deformation could be visualized.
The results indicated that the proposed method was effective in eliminating nonhomogeneous deformation of the fossil skull. The antemortem appearance of the skull cannot
be completely restored by this method alone, due to methodological limitations. However, the presented method has
a role as an adjunct in complementing conventional restoration techniques on account of its objective nature. Am
J Phys Anthropol 130:1–9, 2006. V 2005 Wiley-Liss, Inc.
Unearthed fossils have often undergone plastic deformation due to compaction and diagenesis (Shipman, 1981;
Lyman, 1994). In order to compare morphology among
such specimens, it is important to reconstruct their original antemortem appearance. Conventionally, such reconstructions are manually performed by skilled expert anatomists or paleontologists. For example, the deformed skull
of a female Proconsul heseloni (KNM-RU-7290A) was reconstructed by cutting photographs and plaster casts of
the fossil into pieces and rearranging them manually
while correcting deformation; the reconstruction was then
completed graphically (Davis and Napier, 1963; Walker
et al., 1983). However, such reconstructions might be subjectively influenced, and might not yield reproducible results.
In contrast, a novel computerized reconstruction method
was recently proposed, following advances in computerassisted morphological techniques. For instance, Ponce de
Leon and Zollikofer (1999) attempted to correct taphonomic
deformation of a Neanderthal skull (Le Moustier 1) digitally. Virtual three-dimensional (3D) surface models of fragments of the skull were generated on a computer. The composite skull was then mathematically undeformed so as to
remove its vertical compression by assuming this to be a
homogenous transformation, i.e., a linear transformation
such as simple uniform compression or a shear process.
However, deformation of a fossil skull is generally a nonhomogeneous process (Ramsay and Huber, 1983). Therefore, a
more complex model of deformation is necessary, and this
should be taken into consideration when estimating the
original antemortem appearance of a fossil skull.
In this study, we propose a computerized method to correct nonhomogeneous deformation of a fossil skull. In this
method, the skull is undeformed based on bilateral symmetry as assumed elsewhere (Kohler et al., 2001; Ponce
de Leon and Zollikofer, 1999), but the taphonomic deformation is not modeled as a homogeneous process. We will
first describe the proposed restoration procedure, and then
present the actual application of the method to the Proconsul heseloni cranium for illustration of the procedure.
C 2005
Outline of restoration procedure
If a skull is intact, we can assume bilateral skeletal
symmetry, as shown in Figure 1: anatomical landmarks
on the median sagittal plane should lie on the same
plane, and each bilateral landmark pair should be symGrant sponsor: Japan’s Ministry of Education, Culture, Sports,
Science and Technology; Grant number: Grant in Aid for Scientific
Research 15770158; Grant sponsor: Japan Society for the Promotion
of Science; Grant number: Grant for Biodiversity Research of the
21st Century COE A14.
*Correspondence to: Naomichi Ogihara, Laboratory of Physical
Anthropology, Department of Zoology, Graduate School of Science,
Kyoto University, Kitashirakawa-oiwakecho, Sakyo-ku, Kyoto 6068502, Japan. E-mail:
Received 14 March 2005; accepted 1 June 2005.
DOI 10.1002/ajpa.20332
Published online 13 December 2005 in Wiley InterScience
Fig. 1. Outline of restoration procedure. A: Intact skull. B: Deformed skull. C,D: Reversal of deformation process. Landmarks
which should be on midsagittal plane were repositioned on this plane (C). Pairs of landmarks which should be symmetrical with
respect to plane were then repositioned so that vector connecting each pair intersected midsagittal plane perpendicularly at its midpoint (D).
metrical to each other with respect to this plane. On the
other hand, if the skull is subjected to taphonomic deformation, such geometrical constraints are no longer maintained. Therefore, 3D coordinates of these median sagittal and bilateral landmarks are first modified so that
these geometrical constraints are satisfied: landmarks
which should be on the midsagittal plane are restored to
this plane (Fig. 1C), and then pairs of landmarks that
should be symmetrical with respect to this plane are
repositioned so that the vector connecting each pair intersects the plane perpendicularly at its midpoint (Fig. 1D).
Subsequently, the entire shape of the skull is transformed
according to the displacements of the landmarks, using an
interpolation function.
then transferred to commercial medical image-processing
software (Analyze 4.0, BRI, Mayo Clinic) for segmentation,
and the 3D surface of the scanned fossil was represented
by a triangular mesh model. The original occipital fragment fossil, which was later found to be continuous with
the main skull (Walker et al., 1983), was unavailable for
analysis. Instead, we scanned a cast of this fragment with
a pQCT scanner (Norland-Stratec, Germany) at the Laboratory of Physical Anthropology, Kyoto University (Kyoto,
Japan), and constructed a 3D surface model of this fragment in a similar manner. The model of this fragment
was joined in continuation with that of the main skull,
using commercial software (RapidForm2002, Inus Technology, Korea). Figure 2 shows the mesh model of the skull
created in this study. The surface model contained
279,579 vertices and 559,212 polygons.
Acquisition of 3D morphometrical data from
the skull
Extraction of landmark coordinates
Three-dimensional morphometrical data from a Proconsul heseloni cranium (KNM-RU-7290A) was acquired with
a CT scanner (Siemens Somatron AR SP) at the Diagnostic Centre, Nairobi, Kenya. In terms of scanning parameters, tube voltage, exposure, and slice thickness were set
to 130 kV, 158 mAs, and 1 mm, respectively. Two hundred
and ninety-three consecutive cross-sectional images were
reconstructed at 0.5-mm intervals, using an SP90 kernel
and the extended CT scale (Spoor et al., 2000). XY pixel
size was 0.195 mm. The two-dimensional images were
The 3D coordinates of midsagittal and bilateral landmarks were extracted by pointing to their locations on the
3D surface model, using a virtual probe incorporated in the
software (RapidForm2002). To help locate the landmarks,
we superimposed a color map of surface curvature on the
model. Table 1 provides the list of anatomical landmarks
recorded in this study. While some are commonly used morphometric landmarks or points that can easily be defined
based on morphological characteristics such as inflections or
ridges of the surface, others are quasilandmarks. For exam-
TABLE I. Anatomical landmarks used for the reversal of
Midsagittal landmarks
Median posterior
border of palate
Fig. 2. Virtual replica of skull of Proconsul heseloni (KNMRU-7290A), incorporating its occipital fragment (7290C, arrow).
Points are landmarks listed in Table 1. Axes show skull’s local
coordinate frame.
ple, although no points on the intermaxillary suture are
regarded as formal anatomical landmarks, some of these
points were used as landmarks in the present study, since
only insufficient commonly used midsagittal landmarks
could be obtained due to the condition of the fossil skull.
Similarly, each of a bilateral pair of points along the temporal lines was considered a landmark, the locations of which
were determined so that the distance along the surface
between each of the points and the jugale on one side was
equal to that on the other side.
The skull’s local coordinate frame was defined as follows: the x-axis was an axis through the median posterior border of the palate and prosthion; the z-axis was
perpendicular to the plane inclusive of the x-axis and
the nasospinale; and the y-axis was the cross product of
these two axes. The origin of the coordinate frame was
set at the median posterior border of the palate. The
plane containing the x- and y-axes was the midsagittal
plane of the skull.
All extracted coordinates were then transformed to
this new coordinate frame. Figure 3 shows the distribution charts of landmarks projected on the midsagittal
and transverse planes. These charts illustrate that the
above-mentioned geometrical constraints that must be
satisfied in the antemortem state were violated due to
the taphonomic deformation of this fossil skull.
Modification of landmark coordinates
Let ai denote the 3D coordinate of the ith midsagittal
landmark extracted and transferred to the skull coordinate frame, and ai denote the same landmark coordinate
after modification. In addition, N is the total number of
midsagittal landmarks registered: 24 in this study. In
order for landmarks to lie on the midsagittal plane after
modification, ai must satisfy the following equation:
Maxilla nasal crest
Maxilla nasal crest
Median palatine suture
Median palatine suture
Incisive foramen
Intermaxillary suture
Internasal suture
Internasal suture
(inside) #1
Internasal suture
(inside) #2
Pt on groove for superior
sagittal sinus #1
Pt on groove for superior
sagittal sinus #2
Pt on groove for superior
sagittal sinus #3
Pt on groove for superior
sagittal sinus #4
Pt on groove for superior
sagittal sinus #5
Pt on groove for superior
sagittal sinus #6
Pt on groove for superior
sagittal sinus #7
Pt on groove for superior
sagittal sinus #8
Pt on groove for superior
sagittal sinus #9
Bilateral landmarks
Inferior pt of maxillary
alveolar process between 11–12
Inferior pt of maxillary alveolar
process between 12–C
Inferior pt of maxillary alveolar
process between P3–P4
Inferior pt of maxillary alveolar
process between P4–M1
Inferior pt of maxillary alveolar
process between M1–M2
Inferior pt of maxillary alveolar
process between M2–M3
Lateral edge of nasal cavity
Piriform aperture
Zygomaticoalveolar crest
Frontal process of maxilla
Pt on temporal line #1 (jugale)
Pt on temporal line #2
Pt on temporal line #3
Pt on temporal line #4
Pt on temporal line #5
Pt ¼ point.
ai n ¼ 0;
where n is the vector perpendicular to the midsagittal
plane. However, this geometrical relationship is not sufficient for determining the location of ai. Hence, another
geometrical constraint was introduced:
ðai ai1 Þ2 ¼ ðai ai1 Þ2 :
This equation indicates that the distance between the
two adjacent midsagittal landmarks i and i 1 is not
varied, assuming that during the deformation process,
the distance is not subjected to shrinking or stretching.
While satisfying these constraint Equations (1, 2), we
determined the new location of the ith landmark so as to
minimize the magnitude of the change in position, i.e.,
the displacement vector ai ai by solving the following
optimization problem:
ðai ai Þ2 ! min :
Fig. 3. Distribution charts of landmarks, projected on (A) midsagittal, (B) transverse, and (C) frontal planes. Bilateral landmark
pairs are connected by a line to indicate orthogonality with coordinate planes.
Modified locations of all the midsagittal landmarks were
recursively calculated from i ¼ 4 to N.
According to this mapping that makes the midsagittal
landmarks coplanar, bilateral landmarks were
formed by the following method. Let bj and bj be the 3D
coordinates of the 0right and left bilateral pair of jth land^ j denote these landmarks after this
^ j and b
marks, b
landmark nearest
transformation, ak is the midsagittal
to the midpoint between bj and bj, and M is the total
number of pairs of bilateral landmarks: 15 in this study.
In order to preserve the spatial relationship among the
jth bilateral landmark pair, the nearest midsagittal landmark ak, and ak1 before and after the transformation,
^ j0 must satisfy the following equations:
^ j and b
^ j a k Þ2
ðbj ak Þ2 ¼ ðb
^ 0 a k Þ2
ðb0j ak Þ2 ¼ ðb
^ j ak1 Þ2
ðbj ak1 Þ2 ¼ ðb
^ 0 ak1 Þ2
ðb0 ak1 Þ2 ¼ ðb
ðbj ð4Þ
b0j Þ2
^j b
^ 0 Þ2
¼ ðb
While satisfying these constraint equations, we computed the new location of the jth pair of landmarks so as
to minimize changes in their position (the sum of the
magnitude of the two displacement vectors) by solving
the following optimization problem:
^ j bj Þ2 þ ðb
^ 0 b0 Þ2 ! min :
Then, these paired landmarks were further modified to
be bilaterally symmetrical with
respect to the skull’s
midsagittal plane. Let bj and bj be the 3D coordinates of
the modified jth pair of landmarks. In order for these to
be bilaterally symmetrical, the following geometrical conditions must be satisfied:
ðbj b0j Þ a2 ¼ 0
ðbj b0j Þ a3 ¼ 0
ððbj b0j Þ=2Þ
The first and second equations of (6) represent the condition that the vector connecting the paired landmarks
is perpendicular to the midsagittal plane: a2 and a3 are
the position vectors of the prosthion and nasospinale,
which define the midsagittal plane. The third equation
represents the condition that the vector intersects the
^j b
^ 0 Þ2 :
ðbj b0j Þ2 ¼ ðb
V ¼ PW þ QA
6 Uðr2;1 Þ
6 6
While satisfying these constraint Equations (6, 7), we
determined the new locations of the jth paired landmarks by solving the following optimization problem:
^ j Þ2 þ ðb0 b
^ 0 Þ2 ! min :
ðbj b
Modified locations of all M pairs of landmarks were individually calculated. Therefore, the landmark transformation process was twofold: all landmarks were first transformed by the mapping that makes the midsagittal landmarks coplanar, and then the bilateral landmarks were
transformed to their new locations.
Uðr1;1 Þ
Uðr1;2 Þ
Uðr2;2 Þ
midsagittal plane perpendicularly at its midpoint. These
are the conditions necessary for the paired landmarks to
be bilaterally symmetrical. Additionally, as the distance
between paired landmarks was assumed to be equal before and after the deformation in this study, the following condition was added:
UðrK;1 Þ UðrK;2 Þ
¼ xp xq Uðr1;K Þ
Uðr2;K Þ 7
UðrK;K Þ
where W and A are the (K 3 3) and (4 3 3) coefficient
matrices, and rp,q is the Euclidean distance between the
pth and qth landmarks. In this matrix, U(rp,q) is the
basis function. The first term on the right side of Equation (10) (i.e., PW) represents a nonlinear transformation, whereas the second term (i.e., QA) represents a linear (affine) transformation.
The basis function used for the TPS function here is
described as
Uðrp;q Þ ¼ rp;q
Uðrp;q Þ ¼ r2p;q lnðr2p;q Þ
instead of
3D shape transformation
In order to restore the shape of the skull, not only the
displacements of the landmarks, but also all intermediate vertices describing the skull topography, must be
estimated from the correspondence of landmark coordinates before and after modification. In this study, we
attempt to describe this nonhomogenous mapping between the two sets of landmarks by a thin-plate spline
(TPS) function. The TPS function describes mapping
that is consistent with the assigned correspondence of
these landmarks (Bookstein, 1989, 1991). This function
is often utilized to quantify morphological change (cf.
Ponce de Leon and Zollikofer, 2001), but here it is solely
used to perform 3D interpolation of the intermediate vertices. The calculation procedure is described briefly
below; readers are referred to Bookstein (1989, 1991) for
a more complete description of TPS function and its
application to 3D morphometrics.
Let x1, x2 , . . . , xK represent K landmark coordinates
before the modification, and x1, x2 , . . . , xK represent these
landmark coordinates after modification. K is the total
number of points: K ¼ (N þ 2M), i.e., 54 in this study.
Using matrix representation, these can be written as
xT2 7
; V ¼ ½x1
.. 7
. 5
QT W ¼ 0
that minimizes the magnitude of the nonlinear transformation component. The simultaneous equations
can now be solved for a unique set of W and A. Therefore, the mapping function from the original to the modified coordinates is defined as
sTp ¼ Uðrp;1 Þ Uðrp;2 Þ Uðrp;K Þ W þ ð1 sTp Þ A ð15Þ
because the former is 3D but the latter is 2D (Meinguet,
1979, 1984).
Equation (10) must be solved for W and A in order to
define the mapping function. However, the solution is
indeterminate because the number of unknown parameters in W and A is greater than the number of equations.
Hence Equation (10) is solved, given the following additional condition
xK ð9Þ
where Q is a (K 3 4) and V is a (3 3 K) matrix. Using
the TPS function, correspondence of the landmarks between before and after modification (the mapping from
Q to V) can be described by a linear combination of basis
functions as
where sp and sp are the coordinates of the pth intermediate vertex before and after modification. The equation
approximates the displacement vector field defined by
the assigned correspondence of landmarks, and can be
used to estimate a displacement vector at each of the
intermediate vertices between landmarks.
Outline of computational algorithm
The proposed method therefore consists of the following four steps: 1) extraction of midsagittal and bilateral
Fig. 4. Displacement vector field of landmarks, projected on (A) midsagittal, (B) transverse, and (C) frontal planes.
landmarks, 2) transformation of all landmarks by the
mapping that makes midsagittal landmarks coplanar
(Equations 1–5), 3) transformation of bilateral landmarks (Equations 6–8), and 4) 3D shape transformation
of the entire cranium (Equations 9–15).
A software program to calculate and visualize the 3D
shape transformation was written using C language and
the Open GL graphic library. Inputs to the software were
1) the surface model data from the original fossil, and 2)
the sets of landmark coordinates before and after modification, Q and V. The optimization problems for obtaining V
were solved with Microsoft Excel Solver, using the quasiNewtonian method. The 3D shape transformation software
then calculated and displayed the modified shape of the
fossil. The modified surface model can be dumped to a file
in STL format, for use in other software for further analyses or in stereolithography to produce a physical replica of
the result. This 3D shape transformation software is available from the corresponding author (N.O.) on request.
Figure 4 shows the displacement vector field of landmarks projected on the midsagittal and transverse
planes. Displacement vectors join the original and modi-
fied positions of the landmarks. As illustrated, landmark
coordinates were correctly shifted so as to satisfy the
geometrical constraints of bilateral skeletal symmetry,
by solving the optimization problems.
Figure 5 compares the 3D surface of the fossil skull
before and after the modification. Following the displacement vector field in Figure 4, the entire shape of the
skull was successfully undeformed. The 3D grid displays
the spatial deformation caused by the 3D transformation
between the original and the modified shape, as defined
in Equation (15). The deformation of the grid implies
that the modification of the fossil skull involved heterogeneous shearing in all three directions. This deformation is a very complicated nonlinear transformation,
unlike the deformation from a rectangular prism to a
Figure 6 shows the projected views of the restored Proconsul cranium. The quality of bilateral symmetry of the
restored cranium depends on the number of bilateral
landmark pairs considered for the 3D transformation
using the TPS function. Therefore, as you increase the
number of bilateral landmark pairs, the quality of the
reconstruction becomes higher. The volume of the
restored cranium model was increased by 4.4%, whereas
its surface area was decreased by 1.8% from the outset.
Fig. 5. 3D surface of fossil skull before (A) and after (B) reconstruction. 3D grid enables spatial deformation to be visualized.
This study presents the first computerized restoration
method of a nonhomogeneously deformed fossil skull
based on bilateral skeletal symmetry. The method was
then applied to a deformed Proconsul heseloni skull and
was demonstrated to successfully eliminate the asymmetrical component of the nonhomogenous deformation,
as shown in Figure 6.
However, it must be noted that the method relies on
the following assumption: that the distance between
each of the two adjacent landmarks on the midsagittal
plane and the distance between each of the paired bilateral landmarks are not changed by the modification, as
expressed in Equations (2) and (7). This assumption does
not hold if a skull is stretched or compressed during fossilization, and these distances are changed drastically
from those in the antemortem condition, as illustrated in
Fig. 6. Projected views of restored Proconsul cranium.
Figure 7A. However, a deformed cranium has usually
suffered from some bilateral compression or shear in
reality, such that the distance between bilateral landmarks is changed. Therefore, if there is a certain portion
in a fossil cranium that is undistorted during deformation, it is better to use this relationship, instead of Equations (6–8), for restoration of the region. Supposing that
^ j is a bilateral landmark that is not distorted with
is currently impossible to reproduce the antemortem
appearance of a fossil with complete accuracy, and the
insights of experienced anthropologists are indispensable
in reconstructing and interpreting its original shape.
Despite these limitations, the proposed computerized
method has certain advantages. For instance, it could
successfully reverse the asymmetrical component of the
deformation, while eliminating some of the subjectivity
that may occur in conventional restoration due to differences in anthropologists’ techniques. Moreover, the restoration process is reproducible as long as the landmarks
used for the calculations are recorded. The effect of the
transformation can be visualized, and the time required
for restoration could be considerably reduced. The
present method therefore has a role as an adjunct, complementing conventional restoration techniques.
Fig. 7. Schematic drawings of deformed skulls that cannot
be restored by present method. A: Skull is compressed in lateral
direction, and distances between bilateral landmarks are drastically changed. B: Skull is symmetrically deformed in antero-posterior direction. C, D: Intact skulls.
respect to the midsagittal plane, then bj and bj should
^ j and its mirror-image inversion,
be determined as b
respectively. Yet if a fossil cranium is entirely deformed,
this is not applicable.
Furthermore, the landmark coordinates were calculated according to the minimization principal as expressed in Equations (3) and (8), following the assumption that deformation was minimal. However, this
assumption does not hold unless the deformation is mild
or moderate. Another assumption that we made implicitly is size invariance of the cranium during the period
before and after deformation. In a precise sense, the volume and surface area of the cranium model are subjected to change because no volumetric and planimetric
constraints are introduced here, but the proposed
method largely depends on this assumption. However,
size distortion of the fossil may be present from the outset during fossilization.
In addition, if the skull was deformed in the direction
parallel to its midsagittal plane (as in Fig. 7B), the proposed method would not be applicable since the skull,
although deformed, would remain symmetrical with
respect to this plane. Comparisons of a midsagittal cross
section of the modified skull with that of other living
hominoids suggest that the fossil was actually deformed
in the sagittal plane; however, such deformation cannot
be treated by the present algorithm. These methodological limitations mean that the undeformed shape of the
Proconsul skull was not completely accurate.
If it were possible to record all the properties of a fossilized bone and the geological materials surrounding it
(such as material properties and distributions), as well as
the change in the force field over the fossilization period,
it might be possible to calculate the antemortem appearance of a skull in a retrograde manner directly from its
deformed condition. Nevertheless, this would be very difficult, and the problem remains that the temporal history
of the force field, i.e., the taphonomic history, cannot be
quantitatively estimated from fossil findings. Therefore, it
The authors sincerely thank Emma Mbua (National
Museums of Kenya) for permission to scan the specimen,
and Bablu Sokhi (Diagnostic Centre) and Katsutoshi
Murata (Siemens-Asahi Medical Technologies) for technical assistance with CT scanning. This study was supported by a MEXT Grant in Aid for Scientific Research
(15770158) and a JSPS Grant for Biodiversity Research
of the 21st Century COE (A14) to Kyoto University.
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