a novel riemannian framework for shape analysis of 3d objects
DESCRIPTION
A NOVEL RIEMANNIAN FRAMEWORK FOR SHAPE ANALYSIS OF 3D OBJECTS. S. Kurtek 1 , E. Klassen 2 , Z. Ding 3 , A. Srivastava 1 1 Florida State University Department of Statistics 2 Florida State University Department of Mathematics 3 Vanderbilt University Institute of Imaging Science. - PowerPoint PPT PresentationTRANSCRIPT
S. Kurtek1, E. Klassen2, Z. Ding3, A. Srivastava1
1Florida State University Department of Statistics2Florida State University Department of Mathematics
3Vanderbilt University Institute of Imaging Science
A NOVEL RIEMANNIAN FRAMEWORK FOR A NOVEL RIEMANNIAN FRAMEWORK FOR SHAPE ANALYSIS OF 3D OBJECTSSHAPE ANALYSIS OF 3D OBJECTS
*This research was supported in part by grants from AFOSR, ONR and NSF.
PROBLEM INTRODUCTIONPROBLEM INTRODUCTION
Main Goal: To compare the shapes of these surfaces using a metric that is invariant to
scale, translation, rotation and re-
parameterization.
f1 f2
d(f1,f2) = ?
Consider these two parameterized surfaces:
MOTIVATIONMOTIVATIONMOTIVATIONMOTIVATION
1. Medical Image Analysis
2. Bioinformatics
3. Facial Recognition
4. Geology
5. Image Matching
These do not analyze shapes of parameterized
surfaces directly, which is our goal.
CURRENT METHODSCURRENT METHODS
1. Deformable Templates - Davatzikos et al. 1996; Joshi et al. 1997; Grenander and Miller 1998; Csernansky et al. 2002.
2. Level Set Methods - Malladi et al. 1996.
3. Landmarks, Active Shape Models - Kendall 1985; Cootes et al. 1995; Dryden and Mardia 1998.
4. Iterative Closest Point Algorithm - Besl and McKay 1992; Almhdie et al. 2007.
5. Medial Representation - Siddiqi and Pizer 1992; Bouix et al. 2001; Gorczowski et al. 2010.
PARAMETERIZED SURFACESPARAMETERIZED SURFACESMAIN ISSUEMAIN ISSUE
• S denotes a 2D smooth and differentiable surface.• Define a parameterization of surface S as .
• Let Г be the set of all diffeomorphisms of . The natural action of Г on is on the right by composition .
• In general , is not area preserving and therefore the isometry condition is not satisfied under the metric:
Existing Solutions: 1.Restrict to area preserving re-parameterizations - Gu et al. 2004.2.Fix the parameterization (SPHARM) of all surfaces - Brechbüler et al. 1995; Styner et al. 2006.
Definition: a. Given a differentiable surface f, define as
the “area multiplication factor” of f at s:
where {us, vs} is an orthonormal basis of .
b. Define a q-map, using by
NEW REPRESENTATION OF SURFACESNEW REPRESENTATION OF SURFACES
SHAPE ANALYSIS OF SURFACESSHAPE ANALYSIS OF SURFACES
Achieving the desired invariances:• Remove Directly:
1. Scale, .
2. Translation, .• Remove Using Algebraic Operations:1. Rotation, SO(3): Given , the action of the
rotation group is defined as (O,q)=Oq.2. Re-parameterization, Г: Given , the action of the
re-parameterization group is defined as
• Equivalence Class:
• Shape Space:
• Distance Between Surfaces:
• Distance Between Orbits:
DISTANCE BETWEEN SURFACESDISTANCE BETWEEN SURFACES
Optimization Problem:1. Rotation, SO(3): Procrustes analysis.2. Re-Parameterization, Г: gradient descent.
1. Define the energy as:
where γ0 is fixed and γ is a variable.
2. Define the mapping:
3. The Jacobian of φ(γ) is:
where b is an orthonormal basis of and .4. The directional derivative of E is:
OPTIMIZATION PROBLEM OVER OPTIMIZATION PROBLEM OVER ГГ
• Optimize over 60 elements in the group of symmetries of the dodecahedron.• The largest finite subgroup of SO(3). • Equivalent to placing the North Pole at 60 different positions.
INITIALIZATION OF GRADIENT SEARCHINITIALIZATION OF GRADIENT SEARCH
f1 Energy Minimizer
f2 Cost Function
BRAIN STRUCTURE SURFACESBRAIN STRUCTURE SURFACESTWO LEFT PUTAMENSTWO LEFT PUTAMENS
BRAIN STRUCTURE SURFACESBRAIN STRUCTURE SURFACESTWO LEFT PUTAMENSTWO LEFT PUTAMENS
f1 f2 O*(f2 ◦ γ*)
Energyat each iteration
0 10 20 30 40 50 600.02
0.022
0.024
0.026
0.028
0.03
0.032
||γ*(s)-s||
d([q1],[q2])==0.0207
BRAIN STRUCTURE SURFACESBRAIN STRUCTURE SURFACESLEFT PUTAMEN AND LEFT THALAMUSLEFT PUTAMEN AND LEFT THALAMUS
BRAIN STRUCTURE SURFACESBRAIN STRUCTURE SURFACESLEFT PUTAMEN AND LEFT THALAMUSLEFT PUTAMEN AND LEFT THALAMUS
f1 f2 O*(f2 ◦ γ*)
Energyat each iteration
0 20 40 60 80
0.08
0.09
0.1
0.11
0.12
0.13
d([q1],[q2])==0.0790
||γ*(s)-s||
• T1 weighted brain magnetic resonance images of young adults of ages between 13 and 17 recruited from the Detroit Fetal Alcohol and Drug Exposure Cohort.• Left and right brain structures (total of 11) for 34 subjects, 19 with ADHD and 15 healthy. • Leave-one-out nearest neighbor classification scheme.
ADHD STUDYADHD STUDYSINGLE STRUCTURE CLASSIFICATIONSINGLE STRUCTURE CLASSIFICATION
ADHD STUDYADHD STUDYMULTIPLE STRUCTURE CLASSIFICATIONMULTIPLE STRUCTURE CLASSIFICATION
• Combined weighted single structure distances to maximize the ADHD classification rate.• Using our method, the combination of left putamen, left pallidus and right pallidus distances provided a 91.2% classification rate.• Other methods:
1. Harmonic – 85.3%2. ICP – 88.2%3. SPHARM-PDM – 85.3%
• So far we have presented the framework and results for shape analysis of closed surfaces only.• The extension to quadrilateral (D=[0,1]2) and hemispherical (D=unit disk) surfaces is straightforward.
EXTENSION TO OTHER TYPES OF SURFACESEXTENSION TO OTHER TYPES OF SURFACES
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
QUADRILATERAL SURFACESQUADRILATERAL SURFACESIMAGE MATCHINGIMAGE MATCHING
QUADRILATERAL SURFACESQUADRILATERAL SURFACESIMAGE MATCHINGIMAGE MATCHING
f1 f2 O*(f2 ◦ γ*)
γ*Energy
at each iteration
0 20 40 60 80 100 1200.04
0.06
0.08
0.1
0.12
0.14
0.16
I1 I2
|I1-I2| |I1-O*(I2 ◦ γ*)|
d([q1],[q2])==0.0567
QUADRILATERAL SURFACESQUADRILATERAL SURFACESIMAGE MATCHINGIMAGE MATCHING
QUADRILATERAL SURFACESQUADRILATERAL SURFACESIMAGE MATCHINGIMAGE MATCHING
f1 f2 O*(f2 ◦ γ*)
γ*Energy
at each iteration
I1 I2
|I1-I2| |I1-O*(I2 ◦ γ*)|
d([q1],[q2])==0.0953
HEMISPHERICAL SURFACESHEMISPHERICAL SURFACESCROPPED FACESCROPPED FACES
HEMISPHERICAL SURFACESHEMISPHERICAL SURFACESCROPPED FACESCROPPED FACES
f1 f2 O*(f2 ◦ γ*)
Energyat each iteration
d([q1],[q2])==0.0288
||γ*(s)-s||
OPTIMAL PATHS BETWEEN SURFACESOPTIMAL PATHS BETWEEN SURFACESOPTIMAL PATHS BETWEEN SURFACESOPTIMAL PATHS BETWEEN SURFACES
Without Re-Parameterization
WithRe-Parameterization
• We computed certain optimal paths between two toy shapes with and without re-parameterization.• The displayed paths are not geodesic with respect to our metric but under a metric described by Kilian et al. 2007.
M. Kilian, N. Mitra, and H. Pottman. “Geometric Modeling in Shape Space.”, in ACM Transactions on Graphics, vol. 26, no. 3, 2007, 1-8.
CONCLUSION AND FUTURE WORKCONCLUSION AND FUTURE WORKCONCLUSION AND FUTURE WORKCONCLUSION AND FUTURE WORK
• Shape analysis of 3D objects is very important in many scientific fields.• We have proposed a novel approach for the analysis of 3D objects, which is invariant to rigid motion, scaling and most importantly re-parameterization.• This results in a proper metric on the space of q-maps.• In the future, we would like to be able to show geodesics between surfaces.• We would also like to apply this methodology to more data sets.
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