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

Robust Cortical Reconstruction and Mapping Tools Using Intrinsic Analysis of 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: n=100 n=50 n=75 n=25 n=1 − ∆ = λ f f M – Property: discrete and isometry invariant • Numerical computation – Use the weak form and finite element methods – For triangular meshes, we solve a matrix eigenvalue β = λ β Q U 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

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 Build Topology and Classification Skull-stripped Evolution Geometry MR Image Speed Correction Registration to Atlas WM WM Sub-voxel Surface Surface Interpolation Evolution GM Geometry GM Sub-voxel Surface Correction Surface Interpolation Evolution 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 + = → ˆ g wg where w : M R – All genus zero surfaces are conformally equivalent • The Laplace-Beltrami operator 1 ∆ ˆ = ∆ g g M M w ∆ = − λ g f wf M n n n

Surface Mapping in the Embedding Space • Two surfaces ( M 1 , w 1 g 1 ) and ( M 2 , w 2 g 2 ) 1 1 ∫ ∫ ∫ 2 ∫ 2 = + + ξ ∇ + ∇ 2 2 E ( w , w ) d ( x ) dM d ( x ) dM ( w dM w dM ) 1 2 1 1 2 2 1 1 2 2 S S M M M M 1 2 1 2 1 2 = − = − w g w g w g w g where d ( x ) min I ( x ) I ( y ) , d ( x ) min I ( x ) I ( y ) 1 1 2 2 2 2 1 1 1 M M 2 M M ∈ 2 ∈ 2 y M 1 2 y M 2 1 2 1 • Discretization – At each iteration, define the closest point maps = = u ( V ) AV , u ( V ) BV 1 1 2 2 2 1 – The energy becomes Shi et al., MICCAI’11.

Cortical Mapping • Intrinsic mapping of highly complicated surfaces • Application: gyral labeling Left hemisphere Atlas label 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