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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