2d shape analysis of mandible outlines in fossil …
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2D SHAPE ANALYSIS OF MANDIBLE OUTLINES IN FOSSIL HOMININS UTILIZING
SYNTHESIS TECHNIQUES FROM PLANAR MECHANISM DESIGN
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Bingjue Li1, Andrew P. Murray2, David H. Myszka2, Gérard Subsol3
1 JiangSu Key Lab. For Design and Manufacture of Micro-Nano Biomedical Instruments, School of mechanical Engineering, Southeast University, Nanjing, China
2 DIMLab, Department of Mechanical and Aerospace Engineering, Univ. of Dayton, USA3 Research-Team ICAR, LIRMM, Univ. Montpellier, CNRS, France
June 25, 2019
International Symposium on Biological Shape Analysis
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Outline-based Morphometrics
In general based on Fourier descriptors which is a powerful methodology:
• Is EFF well adapted to the shape?20 harmonics gives 4x20=80 coefficients to explain the positions of 84 points (i.e. 2x84=168 scalar values)...
• EEF parameters are not straightforward to analyze as they are not directly related to local geometrical features.
• May be sensitive to the definition of the origin point.
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A New Method
• New method based on mechanical considerations…..
• Planar shape changing rigid-body mechanisms
• Mechanism: revolute joints (pivot) / prismatic joints (glissière) which parameters are easy to understand.
B. Li, A.P. Murray, D.H. Myszka, G. Subsol. "Synthesizing Planar Rigid-Body Chains for Morphometric Applications". ASME International Design Engineering TechnicalConferences & Computers and Information in Engineering Conference, Charlotte (U.S.A.), August 2016.
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The Goal
Revolute joint
C-segment (links containing prismatic joints)
M-segment
Fused connection
Chain of rigid bodies
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General Procedure
Li, B., Murray, A., Myszka, D., and Subsol, G., “Synthesizing planarrigid-body chains for morphometric applications”, submitted to the2016 ASME International Design and Engineering TechnicalConferences, Charlotte, NC, Aug. 21-24, 2016
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Design Profiles
Can be generated from:• Mathematical functions• Point coordinates• Line drawings
j = 1
j = 2
j = 3
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Target Profiles
1. Compute the arc length of each design profile
2. Determine the number of pieces should be in each target profile
3. Desired piece length of target profile
2000 pieces
2322 pieces
2253 piecesj = 1
j = 2
j = 3
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Specify Chain Structure
Segment vector, 𝐕 = 𝐶 𝑀 𝐶 𝑀 𝑀
Connection vector, 𝐖 = 𝑅 𝐹 𝑅 𝑅
Minimum number of pieces in a segment, 𝛼
j = 1
j = 2
j = 3
C-segment
M-segment
Revolute joint
Fused connection
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Initial Segment Matrices (SM)
V = [ C M C M M ]
𝑆𝑀 =359506346
596596596
337512603
343343343
365365365
𝑗 = 1𝑗 = 2𝑗 = 3
j = 1
j = 2
j = 3
Segmentation points
359
596
337
343365
346
596
603
343
365
506 596
512
343
365
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Generating Segments
C-Segments
Average radius (reciprocal of curvature)
Segment arc length
Average piece length
ҧ𝑟𝑒
ҧ𝑠𝑒
𝛿
(0, ҧ𝑟𝑒)
(0,0)
M-segment
C-segment
𝜃𝑗,𝑒
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Generate Segments
V = [ C M C M M ]
𝑆𝑀 =359506346
596596596
337512603
343343343
365365365
𝑗 = 1𝑗 = 2𝑗 = 3
Segment to be elongatedSegment to be shorted
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Error Evaluation
C M C M M
𝐸𝑀 =0.350.460.81
1.001.193.23
1.961.194.31
0.270.780.55
0.521.270.76
Overall mean error,
ത𝐸 = 1.24
Mean error of each M-segment:
ത𝐸2 = 1.81, ത𝐸4 = 0.53, ത𝐸5 = 0.86
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SM Optimization
C M C M M
𝑆𝑀0 =359 596 337 343 365506 596 512 343 365346 596 603 343 365
𝑆𝑀1 =360 595 335 344 366507 595 510 344 366346 595 602 344 366
⋮
𝑆𝑀𝑓 =326 420 376 522 356660 420 364 522 356509 420 446 522 356
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Assembling with Revolute Joints
𝐱 = 𝑓𝑚𝑖𝑛𝑐𝑜𝑛 𝑓, 𝐱0, 𝐜
𝐱, the final chain configuration• Starting coordinates and orientation• Angle of each revolute joint• Length of each C-segment (centric angle of the prismatic joint)
𝐱0, the initial configuration after segments are fused and aligned with the profiles
𝑓, average matching error of all points
𝐜, nonlinear constraints
• Adjustment made to the length of each C-segment• Endpoint locations for closed and fixed-end profiles• Orientation of the end links for closed-F and fixed-end
profiles
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Shifting Endpoints for Closed Profiles
ത𝐸 = 7.0414
ത𝐸 = 6.3331 (improved by 10%)
Before shifting endpoints
After shifting endpoints
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Growth Segment Generation
Growth ratio, 𝑟 =𝑚2
𝑚1
Profile 1𝑚1, ҧ𝑠
Profile 2𝑚2, ҧ𝑠
Profile 1’𝑚1, 𝑟 ҧ𝑠
Profile 1’’𝑚2, ҧ𝑠
G-segment 1’
𝑚2, 1
𝑟ҧ𝑠
Arc length: 𝑚1 ҧ𝑠 = 𝑚21
𝑟ҧ𝑠
G-segment 1𝑚1, ҧ𝑠
G-seg 2
Profile 2
Profile 1’’
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Head Circumference
Fused connection between endpointsV = [C M C M C M C M C M C]W = [F R F R F F R F R F]10 parameters to characterize shapes (4 R joints & 6 C segments)Max diameter = 877.17, mean matching error = 6.90 (0.79% of max diameter)
C1 C2 C3 C4 C5 C6 R1 R2 R3 R4
Profile1 0.44 0.49 0.38 0.28 0.17 0.46 -0.27 -0.09 -0.08 -0.11
Profile 2 0.96 0.37 0.70 0.16 0.28 0.50 0.21 0.03 -0.05 0.09
Profile 3 0.68 0.73 0.43 0.62 0.46 0.38 0.16 0.26 0.02 0.09
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The Cochleae
V = [M M M M C M M C M M M]W = [R R … R R] (10 revolute joints)12 parameters to characterize shapes (10 R joints + 2 C segments)
C
M
Average profile width = 798.68Average profile length = 976.40 Mean matching error = 6.91
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The Mandible
33 parameters: 23 R joints + 10 G segments
Discriminant analysis
24 segments [M M G M M G M M G M G G M G G M M G M M G M M G ] + 23 R joints
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1. How to define the chain? (segment and connection vectors)
anatomical hypotheses / accuracy.
2. Multivariate analysis of the C segments and the R joints.
Future Work
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Application to Growth or Evolution
V = [G G G C G G G C ... G G G C G G G] (8 C segments and 27 G segments)W = [R R … R R] (35 revolute joints)
Average profile width = 165.50Average profile length = 171.20ത𝐸 = 1.09
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