Network connectivity via inference over curvature-regularizing line - PowerPoint PPT Presentation

Network connectivity via inference over curvature-regularizing line graphs Asian Conference on Computer Vision Maxwell D. Collins 1 , 2 , Vikas Singh 2 , 1 , Andrew L. Alexander 3 1 Department of Computer Sciences 2 Department of Biostatistics and Medical Informatics 3 Waisman Laboratory for Brain Imaging, Departments of Medical Physics and Psychiatry University of Wisconsin-Madison, Madison, WI mcollins@cs.wisc.edu, vsingh@biostat.wisc.edu, alalexander2@wisc.edu November 10, 2010

White Matter Anatomy ◮ White matter (WM) lies in the brain’s interior. ◮ Large bundles of neurons ◮ Connects the functional areas in the grey matter . ◮ Connectivity may tell us about function or pathology Figure: Coronal slice of a Fractional Anisotropy Our Goal (FA) image, which highlights WM. Map connections in the white matter in vivo from biomedical images.

Diffusion Imaging ◮ Type of Magnetic Resonance Imaging (MRI) ◮ Shows diffusion of water, which will preferentially diffuse along axons. ◮ Each scan measures the diffusion in a given direction. ◮ Can model Orientation Distribution Function (ODF).

Partial Voluming ◮ Diffusion images discretely sample angle. ◮ Multiple fiber orientations may be present in a voxel. ◮ Must recover orientation from context .

Tractography Problem Given an ODF field, trace the local directions to find the full paths of the fibers. Method Classes ◮ Local : Differential equations, Tensorlines ◮ Stochastic Optimization : Spin glass, Gibbs Tracking ◮ Graph-Based

Graph Methods Procedure 1. Construct graph over voxels. ◮ edge corresponds to tract between voxels 2. Weight to model local directional information. 3. Find some optimal subgraph. ◮ i.e. shortest path between given points 4. Extract streamlines or connectivity measures.

Line Graph ◮ Graph of edges incident to a common voxel. ◮ The basic adjacency graph is G = ( V , E ) ◮ The line graph is ( E , L ) for L = { (( ij ) , ( jk )) | ( ij ) ∈ E and ( jk ) ∈ E} ◮ Can interpret as triplets of points. ◮ Shows tract topology.

Proposal Curves Construction ◮ For given orientations, construct Hermite spline ◮ Reduce derivative constraints to orientation constraints by optimizing over magnitude to minimize length and curvature. x i p j ( · ) x k x j C v j

Expected Energy Weight Energy For a given proposal curve C , � 1 K · κ C ( t ) 2 + �C ′ ( t ) � 2 dt , E( C ) = − 1 for curvature κ C and speed C ′ . Weight � � w ijk = E (Curve( x { i , j , k } , ˆ v { i , j , k } )) E v { i , j , k } ∼ p { i , j , k } ˆ Triplets are given low weight if their ODFs align with low-energy proposal curves.

Minimum Cost Flow ◮ User specifies a pair of regions of interest (ROIs) and number of tracts N . ◮ Add “source” and “sink” nodes with edges to members of an ROI. ◮ Replace each edge in L with directed edges in each direction, for directed graph L ± . ◮ Model tries to find lowest-weight edges to carry N units of flow from source to sink. ( S , ( ij )) ijk j S k i source set kji

Minimum Cost Flow � � min w ijk α ijk + λ β j α,β j ∈V ( ijk ) ∈L ± subject to flow constraints � β j ≥ α ijk − 1 ( ijk ) ∈L ± β j ≥ 0 ◮ α ijk : decision variable on whether ( ijk ) is in output tractography ◮ β j : Number of tracts passing through j beyond 1.

Continuation Constraints ◮ Can replace ROI pairs with a single endpoint set M (i.e. WM/GM boundary) where tracts are expected to begin/end. ◮ In line graph setting, can express as continuation constraints. l i l k j l ◮ Recover long tracts by penalizing endpoints outside this set.

Continuation Constraints � min edge selection + µ γ ijk α,β,γ ( ijk ) subject to α ijk ∈ { 0 , 1 } γ ijk ≥ 0 (1) � γ ijk ≥ α ijk − α lij l � γ ijk ≥ α ijk − α jkl , l ◮ For µ > 0, have that γ ijk ≥ 1 iff picking triplet ijk introduces an endpoint. ◮ Relax penality if the corresponding point is in M .

Crossing Results Figure: Comparison of our method with purely local method on a simulated tensor field.

ROI Pairs

Acknowledgements ◮ NIH R21-AG034315: Singh,Collins ◮ NIH MH62015: Alexander ◮ UW ICTR (1UL1RR025011) ◮ UW CIBM (NLM 5T15LM007359) and Morgride Institute for Discovery: Collins ◮ Thanks to Nagesh Adluru for assistance with DTI data.