Robust Cortical Reconstruction and Mapping Tools Using Intrinsic - - PowerPoint PPT Presentation

Robust Cortical Reconstruction and Mapping Tools Using Intrinsic Analysis

• f Geometry and Topology

Yonggang Shi Laboratory of Neuro Imaging (LONI) UCLA School of Medicine

Overview

• Automated reconstruction and mapping of

cortical surfaces

Spectrum of the Laplace-Beltrami Operator

• For a surface M, the spectrum of its Laplace-

Beltrami (LB) operator is defined as:

– Property: discrete and isometry invariant

• Numerical computation

– Use the weak form and finite element methods – For triangular meshes, we solve a matrix eigenvalue problem :

• Applications in computer graphics, vision, and

medical imaging (Belkin’03, Reuter’06, Niethammer’07,

Qiu’06, Bruno’07, Shi’08, Lai’09, Lai’10,Reuter’10, …)

• Our focus: use the eigenfunctions to analyze

the underlying domain, i.e., the surface

f f

M

λ = ∆ −

n=1 n=25 n=50 n=75 n=100

β λ β U Q =

Reeb Graph

• Let M be a surface patch or closed surface

and f: M → R be a feature function defined over the shape.

• The Reeb graph R(f) is the quotient space

with its topology defined through the equivalent relation x ∼ y if f(x)=f(y) for ∀x, y ∈ M. (Reeb’46,Jost’01)

– Graph of level sets

• If f is a Morse function, the graph structure

captures the topology of the surface.

Unified Correction of Geometric and Topological Outliers

• Intrinsic Reeb analysis

– Compute Reeb graphs of LB eigenfunctions – Graph analysis to locate outliers – Removal of outliers by incorporating information from tissue classification and geometric regularity

Cortical Reconstruction Workflow

Enhanced Tissue Classification Registration to Atlas Build Evolution Speed Topology and Geometry Correction WM Surface Evolution Geometry Correction GM Surface Evolution Sub-voxel Interpolation Skull-stripped MR Image GM Surface Sub-voxel Interpolation WM Surface

Shi et al., IPMI’11, MICCAI’12 (submitted)

Comparison with FreeSurfer: ADNI data

Our Results FreeSurfer Results

Comparison with FreeSurfer: ICBM data

Our Results FreeSurfer Results

Conformal Metric Optimization for Surface Mapping

• Idea: modify the metrics on surfaces to deform

their embedding and improve surface maps

• Denote the conformal metrics on M as

– All genus zero surfaces are conformally equivalent

• The Laplace-Beltrami operator

+

→ = R M w where wg g : ˆ

n n n g M g M g M

wf f w λ − = ∆ ∆ = ∆ 1

ˆ

Surface Mapping in the Embedding Space

• Two surfaces (M1,w1g1) and (M2,w2g2)
• Discretization

– At each iteration, define the closest point maps – The energy becomes

2 2 2 1 2 2 2 1 2 1 2 2 2 2 1 2 1 1 2 1

) ( ) ( min ) ( d , ) ( ) ( min ) ( d where ) ( ) ( 1 ) ( 1 ) , (

1 1 1 2 2 2 1 2 2 2 1 1 1 2 2 1 2 1

y I x I x y I x I x dM w dM w dM x d S dM x d S w w E

g w M g w M M y g w M g w M M y M M M M

− = − = ∇ + ∇ + + =

∈ ∈

∫ ∫ ∫ ∫

ξ

1 2 2 2 1 1

) ( u , ) ( u BV V AV V = =

Shi et al., MICCAI’11.

Cortical Mapping

• Intrinsic mapping of highly complicated

surfaces

• Application: gyral labeling

Atlas label Left hemisphere Right hemisphere

Group Study: Comparison with FreeSurfer

• 50 NC vs 50 AD

Large Scale Study Results: n=783

Summary

• A fully automated workflow for large scale

cortical reconstruction and mapping

– Reconstructs WM and GM surfaces – Gyral labels – Gray matter thickness

• Comparison with FreeSurfer

– Computationally much faster – More statistical power in detecting group differences