Computerized restoration of nonhomogeneous deformation of a fossil cranium based on bilateral symmetry.код для вставкиСкачать
AMERICAN JOURNAL OF PHYSICAL ANTHROPOLOGY 130:1–9 (2006) Computerized Restoration of Nonhomogeneous Deformation of a Fossil Cranium Based on Bilateral Symmetry Naomichi Ogihara,1* Masato Nakatsukasa,1 Yoshihiko Nakano,2 and Hidemi Ishida3 1 Laboratory of Physical Anthropology, Department of Zoology, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan 2 Laboratory of Biological Anthropology, Graduate School of Human Sciences, Osaka University, Suita, Osaka 565-0871, Japan 3 School of Human Nursing, University of Shiga Prefecture, Hikone, Shiga 522-8533, Japan KEY WORDS Proconsul; thin-plate spline; surface model; computed tomography ABSTRACT 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 V WILEY-LISS, INC. C MATERIALS AND METHODS 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: email@example.com Received 14 March 2005; accepted 1 June 2005. DOI 10.1002/ajpa.20332 Published online 13 December 2005 in Wiley InterScience (www.interscience.wiley.com). 2 N. OGIHARA ET AL. 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- 3 COMPUTERIZED RESTORATION OF DEFORMED CRANIUM TABLE I. Anatomical landmarks used for the reversal of deformation Midsagittal landmarks expression Median posterior border of palate Prosthion Nasospinale 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 (anterior) Maxilla nasal crest (posterior) Median palatine suture (posterior) Median palatine suture (anterior) Incisive foramen Orale Intermaxillary suture Rhinion 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 Opisthocranion Bilateral landmarks expression 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; ð1Þ 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 : ð2Þ 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 : ð3Þ 4 N. OGIHARA ET AL. 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 trans0 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 0 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 b ^ j a k Þ2 ðbj ak Þ2 ¼ ðb ^ 0 a k Þ2 ðb0j ak Þ2 ¼ ðb j ^ j ak1 Þ2 ðbj ak1 Þ2 ¼ ðb ^ 0 ak1 Þ2 ðb0 ak1 Þ2 ¼ ðb j ðbj ð4Þ j b0j Þ2 ^j b ^ 0 Þ2 ¼ ðb j 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 : ðb j j ð5Þ Then, these paired landmarks were further modiﬁed to be bilaterally symmetrical with respect to the skull’s 0 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 ð6Þ n¼0 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 5 COMPUTERIZED RESTORATION OF DEFORMED CRANIUM ^j b ^ 0 Þ2 : ðbj b0j Þ2 ¼ ðb j V ¼ PW þ QA 6 6 6 Uðr2;1 Þ 6 P¼6 6 6 6 4 ð7Þ 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 j j ð8Þ 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 rp;q Uðr1;2 Þ Uðr2;2 Þ . 2 .. 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: T UðrK;1 Þ UðrK;2 Þ ¼ xp xq Uðr1;K Þ 3 7 7 Uðr2;K Þ 7 7 7 7 7 7 5 ð10Þ 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 ð11Þ Uðrp;q Þ ¼ r2p;q lnðr2p;q Þ ð12Þ 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 2 6 6 6 Q¼6 6 4 1 xT1 7 xT2 7 7 ; V ¼ ½x1 .. 7 7 . 5 1 xKT QT W ¼ 0 ð13Þ that minimizes the magnitude of the nonlinear transformation component. The simultaneous equations VT 0 ¼ P QT Q 0 W A ð14Þ 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 3 1 .. . 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 x2 L 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 6 N. OGIHARA ET AL. 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. RESULTS 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 parallelepiped. 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. COMPUTERIZED RESTORATION OF DEFORMED CRANIUM 7 Fig. 5. 3D surface of fossil skull before (A) and after (B) reconstruction. 3D grid enables spatial deformation to be visualized. DISCUSSION 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 8 N. OGIHARA ET AL. 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 b COMPUTERIZED RESTORATION OF DEFORMED CRANIUM 9 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. ACKNOWLEDGMENTS 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. 0 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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