Extracting Proximity for Brain Graph Voxel Classification

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slides from our invited talk at the 5th Panhellenic Conference on Biomedical Technology, Athens, Greece

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Extracting Proximity for Brain Graph Voxel Classification

  1. 1. Extracting Proximity for Brain Graph Voxel Classification N. Sismanis 1 D. L. Sussman 2 J. T. Vogelstein 3 W. Gray 4 R. J. Vogelstein 4E. Perlman 5 D. Mhembere 5 S. Ryman 6 R. Jung 6 R. Burns 2 C. E. Priebe 2 N. Pitsianis 1,7 X. Sun 7 1 Electrical and Computer Engineering Department, Aristotle University, Thessaloniki, Greece 2 Applied Mathematics and Statistics Department, Johns Hopkins University, Baltimore MD, USA 3 Statistical Science and Mathematics Department, Duke University, Durham NC, USA 4 Johns Hopkins University, Applied Physics Laboratory, Laurel MD, USA 5 HHMI Janelia Farm Research Park, Ashburn VA, USA 6 Neurosurgery Department, University of New Mexico, Albuquerque NM, USA 7 Computer Science Department, Duke University, Durham NC, USA 5 April 2013 5th Panhellenic Conference of Biomedical Technology, Athens, Greece
  2. 2. Brain Maps & Connectomes ⋄ Connectome: The totality of neuron connections in a nervous system ⋄ Connectomics: The science concerned with assembling, analyzing connectomesNeural Activity, Association, or Difference Scales of Neuron Systemsbetween ◦ anatomical regions C.elegans 102 neurons ◦ individual neurons fruit fly 102 × 103 ◦ physical properties and mental behaviors [J. Vogelstein et al., Scient. Rep., 2011] mouse 102 × 103 × 103 ◦ spatial cortical regions and functionality [R. Desikan et al., NeuroImage, 2006] ◦ gender human 102 × 103 × 103 × 103 [J. Vogelstein et al., IEEE Trans. Pattern Anal. and Mach. Intell, 2012] N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 2 / 16
  3. 3. Brain Graph Generation from ImagingEstimations of brain graphs or connectomes obtained from 3D MRI scans 1 1 G. R. Gray et al., IEEE PULSE, 2012 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 3 / 16
  4. 4. Brain Graph Analysis : ClassificationVoxel-Vertex Graph Voxel-to-Region Classification ⊲ voxels smallest distinguishable partition in a 3D image ⊲ Voxels in gray matter: classified via adaptive image registration ⊲ connections in PDD chromatic code: Interior ↔ Posterior: Green ⊲ Voxels in white matter: highly uncertain, to be labeled by connection, as- Superior ↔ Inferior: Blue sociation and inference Left ↔ Right: Red ⊲ http://www.humanconnectomeproject.org ⊲ http://www.humanconnectomeproject.org N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 4 / 16
  5. 5. Computational Challenges in Classification ⊲ A huge number of vertices (potentially 100 billion) ⊲ A large of classes (70) Desikan et al., NeuroImage, 2006 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 5 / 16
  6. 6. Computational Challenges in Classification ⊲ A huge number of vertices (potentially 100 billion) ⊲ A large of classes (70) Desikan et al., NeuroImage, 2006 ⊲ Noisy data ⊲ Partially available connectivity and labels ⊲ Complex geometry ⊲ Individual variation N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 5 / 16
  7. 7. Recent Advances in Voxel-to-Region Classification A Magnetic Resonance Connectome Automated Pipeline 2 B Spectral Embedding of Graphs 3, using also efficient SVD package C Universally Consistent Latent Position Estimation and Vertex Classification for Random Dot Product Graphs 4 2 Grey et al. IEEE PULSE, 2012 3 Rohe et al. Annals of Statist. 2011 4 Sussman, et al. in preprint, 2012 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 6 / 16
  8. 8. Spectral Embedding of a Brain GraphSpectral embedding : a graph placed in an Euclideanspace with d chosen singular-vectors of adjacency orLaplacian matrix as the axes A2 = U Σ2 U ⊤ truncated to A2 = Ud Σ2 Ud d d ⊤ ⋄ Σ, U : singular-value, singular-vector matrices 0.6 ⋄ Rd by Ud as a feature space : 0.4 0.2 0 encoding connections, revealing latent info. −0.2 −0.4 −0.6 −0.8 ⋄ each vertex coded with a d-vector −1 −1.2 −1.4 −1.5 ⋄ each edge associated with a pair of d-vectors −1 −0.5 0 0.8 1 1.2 0.5 0.2 0.4 0.6 0 ⋄ a metric established for similarity/dissimilarity, −0.6 −0.4 −0.2 −0.8 a critical connection to standard classification methods Data and figure source: http://www.openconnectomeproject.org/ N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 7 / 16
  9. 9. Proximity Analysis in an Embedding Vector Space : Status and Gaps - Proximity analysis so far limited to the sequential use of k-NN search in a low-dimension embedding space - Highly efficient, robust k-NN search all at once is needed, especially for large data sets in relatively high-dimensional space All-k-NN : Among an ensemble of N points in a d-dimensional Euclidean space, locate for each and every point its k nearest neighbors, according to a distance metric ◦ Exact search methods prohibitively expensive ◦ Resort to approximate methods (statistical, numerical or both) N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 8 / 16
  10. 10. All-k-NN Search : Exact Methods Prohibitively ExpensiveQuadratic Scaling in N by na´ve use of one-to-all k-NN search for each point ı O(d · N 2 ) C. Elegans : d · 1002 Fruit Fly : d · 1002 · 106 Mouse : d · 1002 · 106 · 106 Human : d · 1002 · 106 · 106 · 106Exponential Growth with d (dimension curse) by spatial partition/binning 5 , limited tolow-dimension spectral embedding ≤ O(d N 2 ) when d < log N O(2d N) > O(d N 2 ) otherwise 5 P. B. Callahan et al., Jurn. ACM, 1995; J.Sankaranarayanan et al., Comput. Graph. 2007 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 9 / 16
  11. 11. All-k-NN Search : Status of Approximate Methods ≻ By randomized projections for locality sensitive hashing , O(λ N d 2 ) + O(k d λ N) 6 λ: number of hash tables ≻ randomized kd-trees, O(N log N) + O(α d N) 7 α: number of tree nodes traversed ≻ Hierarchical k-means, O(N log N) + O(α d N) 8Shortcomings low dimension assumption limited in parallel execution poor data locality 6 Indyk and Motwani, STOC, 1998; M. Trad et al., ICMR, 2012 7 Silpa-Anan and Hartley, CVPR, 2008 8 Fukunaga and Narendra, IEEE Trans. Comput. 1975 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 10 / 16
  12. 12. AkNN-RARE: All-k-NN Search with RAndom REflectionsWe developed a fast and robust algorithm for All-k-NN search O(d h N log (N)) + O(k h d 2 N) ≻ use h random distance-preserving coordinate transforms with Householder reflections ≻ sort along each axis, in parallel ≻ merge kNN among all axesAdvantages ⋄ defying the dimension curse : superlinear in N, quadratic in d, a small number h sufficient for desired accuracy ⋄ simple data structure, regardless geometric, relational structures ⋄ high parallel potential, at multiple levels ⋄ high degree of data locality, hashing free ⋄ simple program structures, hassle free N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 11 / 16
  13. 13. Experimental Results with AkNN-RARE ≻ Data collected at the Mind Research Network (MRN), New Mexico ≻ Data labeled via adaptive image registration and inference techniques ≻ Data size : about a half million voxel-vertices ≻ Performance evaluation of AkNN-RARE - Accuracy metric : RECALL - Comparison with FLANN, a popular package for kNN search N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 12 / 16
  14. 14. Accuracy and Efficiency of AkNN-RARE ◦ Total arithmetic complexity AkNN−RARE recall k=4 k=8 O(d h N log N) + O(d 2 k h N) k=16 1 0.9 ◦ R ECALL precision: percent of cor- 0.8 rect kNN found 0.7 ◦ High recall precision with only h = recall 0.6 0.5 10 transformations 0.4 0.3 ◦ Reflection transformations can be 0.2 7 executed concurrently 6.5 6 8 10 ◦ Problem size shown N = 500, 000 6 5.5 4 5 2 ◦ Enabling high-dim. space embed- log2 embedding dimension 0 number of reflections ding 32 ≤ d ≤ 128 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 13 / 16
  15. 15. Comparison with FLANNFLANN : based on randomized kd-trees, widely used for kNN search 9 0.035 FLANN d=32 FLANN d=64 FLANN d=128 0.03 AkNN−RARE d=32 AkNN−RARE d=64 AkNN−RARE d=128 0.025 ≻ Cost : actual number of pairwise dis- percentage of points searched 0.02 tances calculated 0.015 ≻ At a higher level of recall pre- cision, AkNN-RARE incurs much 0.01 lower cost 0.005 0 0.4 0.5 0.6 0.7 0.8 0.9 1 recall precision 9 Muja and Lowe, VISSAPP, 2009 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 14 / 16
  16. 16. Recap: Extracting Proximity for Brain Graph Voxel ClassificationDraw upon recent advance A Magnetic Resonance Connectome Automated Pipeline B Spectral Embedding of Graphs, using also efficient SVD package C Universally Consistent Latent Position Estimation and Vertex Classification for Random Dot Product GraphsWe developed D a fast, robust algorithm, enabling proximity extraction - at increasingly larger scale toward 100 billion - in sufficiently high-dim. info.-encoding space - on high accuracy demand - utilizing highly parallel computing architectures N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 15 / 16
  17. 17. Acknowledgments ≻ The authors at AUTh acknowledge the support of Marie Curie International Reintegration Program, EU ≻ J. Vogelstein acknowledges the support of Research Program in Applied Neuroscience and the London Institute for Mathematical Science Subcontract on HDTRA1 − 11 − 1 − 0048 and NIH RO1ES017436 ≻ R. Jung and S. Ryman acknowledge the John Templeton Foundation-Grant #22156: The Neuroscience of Scientific Creativity ≻ J. T. Vogelstein, R. J. Vogelstein and W. Gray acknowledge the Research Program on Applied Neuroscience NIH/NINDS 5R01NS056307 N. Sismanis (AUTH) Extracting Proximity for Brain Graph Voxel Classification 5 April 2013 16 / 16

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