Network connectivity via inference over curvature-regularizing line graphs

Transcription

1 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

2 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 Our Goal Map connections in the white matter in vivo from biomedical images. Figure: Coronal slice of a Fractional Anisotropy (FA) image, which highlights WM.

3 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).

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

5 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

6 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.

7 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.

8 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 C p j ( ) x j v j x k

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

10 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 S i j k source set kji

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

12 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. i l l j k l Recover long tracts by penalizing endpoints outside this set.

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

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

15 ROI Pairs

16 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.