Perelman School of Medicine at the University of Pennsylvania

Section for Biomedical Image Analysis (SBIA)

Manifold-based DTI Statistics


The development of methods of statistical analysis is challenging as it requires the solution of many mathematical and technical issues, imposed by the high dimensionality of DTI data with its complex and non-linear underlying manifold structure (as seen in Fig. 1), as the tensors are restricted to lie on a non-linear sub-manifold of the six-dimensional Euclidean space. We are in the process of developing and validating methods of statistical analysis based on:
  1. Manifold learning techniques such as Isomap that determine the underlying manifold structure of DTI measurements, and then perform statistical analysis on the estimated manifolds after flattening. Averaging, interpolation and related operations can also be performed on the manifold. Fig. 2 shows such an isomap-based average of spatially normalized DTI datasets. Details of the framework can be found in [1,3].
  2. Kernel-based techniques in which we “kernelize” the tensors to a higher dimensional linear space in which we perform statistics. Details of the method can be found in [2,3]

 

Fig.1. Manifold structure of tensors. The grey surface represents the non-linear manifold fitted through the tensors represented as ellipses. The green line represents the Euclidean distance between tensors treated as elements of R6 and the red line represents the geodesic distance along the manifold that will be used for all tensor manipulations.

Fig.2. Color map of Isomap based average of 10 DTI datasets

We are in the process of applying these methods to group-based studies in schizophrenia and Alzheimer’s disease. [8]

 

People
Publications
  1. Ragini Verma, Parmeshwar Khurd and Christos Davatzikos "On Analyzing Diffusion Tensor Images by Identifying Manifold Structure using Isomaps", IEEE Transactions on Medical Imaging, 26(6): 772-778, 2007.
  2. Parmeshwar Khurd, Ragini Verma and Christos Davatzikos Kernel-based Manifold Learning for Statistical Analysis of Diffusion Tensor Images Information Processing in Medical Imaging (IPMI), 2007
  3. Ragini Verma, Parmeshwar Khurd and Christos Davatzikos: "On Analyzing Diffusion Tensor Images by Identifying Manifold Structure using Isomaps" IEEE Transactions on Medical Imaging 26(6): 772-778, June 2007.
  4. Peng Wang and Ragini Verma, “On Classifying Disease-induced Patterns in the Brain using Diffusion Tensor Images”, MICCAI, New York, September 6-10, 2008.
  5. Ofer Pasternak, Ragini Verma, Nir Sochen, and Peter J. Basser, “On What Manifold Do Difusion Tensors Live?” Workshop on Manifolds in Medical Imaging: Metrics, Learning and Beyond in MICCAI, New York, September 6-10, 2008.
  6. Lindsay Walker, Jinzhong Yang, Xiaoying Wu, Ragini Verma, Carlo Pierpaoli, “Regional distribution of outliers of diffusion MRI of the human brain”, ISMRM, Toronto, 3-9 May 2008.
  7. Ragini Verma, Parmeshwar Khurd, James Loughead, Raquel Gur, Ruben Gur, Christos Davatzikos, “Manifold based morphometry applied to schizophrenia”, International Symposium on Biomedical Imaging (ISBI), Paris, France, May 14-17, 2008.
  8. P. Nucifora, R. Verma, S.-K. Lee and E. Melhem, “Diffusion MR imaging and  tractography: exploring brain microstructure and connectivity”, Radiology, 245(2): 367-384, November 2007.
  9. Parmeshwar Khurd, Ragini Verma and Christos Davatzikos: “Kernel-based Manifold Learning for Statistical Analysis of Diffusion Tensor Images", Information Processing in Medical Imaging (IPMI) July 2007.
  10. Parmeshwar Khurd, Sajjad Baloch, Ruben Gur, Christos Davatzikos, Ragini Verma: “Manifold Learning Techniques in Image Analysis of High-dimensional Diffusion Tensor Magnetic Resonance Images” Workshop on Component Analysis Methods for Classification, Clustering, Modeling, and Estimation Problems in Computer Vision, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2007.
  11. Peng Wang, Christian Kohler, Ragini Verma: “Estimating cluster overlap on Manifolds and its Application to Neuropsychiatric Disorders”Workshop on Component Analysis Methods for Classification, Clustering, Modeling, and Estimation Problems in Computer Vision, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2007.
  12. Paolo Nucifora, Elias Melhem, Ruben Gur, Ragini Verma: "Tract-specific effects of age and sex on human white matter demonstrated with quantitative MR diffusion tractography", ISMRM, Berlin, May 2007.
  13. Ragini Verma and Christos Davatzikos: "Creating Large Scale population Atlases using Diffusion Tensor Images", ISMRM, Seattle, May 2006.
  14. Ragini Verma and Christos Davatzikos: "Manifold Based Analysis of Diffusion Tensor Images using Isomaps" International Symposium on Biomedical Imaging (ISBI) April 2006.
  15. Parmeshwar Khurd, Ragini Verma and Christos Davatzikos: "On Characterizing and Analyzing Diffusion Tensor Images by Learning their Underlying Manifold Structure" MMBIA: IEEE Computer Society Workshop on Mathematical Methods in Biomedical Image Analysis 2006.
  16. Paolo Nucifora, Ragini Verma, Elias Melhem, Ruben Gur: "Diffusion tensor tractography demonstrates asymmetry in arcuate fiber density", 13th Scientific meeting and exhibition, ISMRM, May 2005.
  17. Paolo Nucifora, Ragini Verma, Elias Melhem, Raquel Gur, Ruben Gur: "Sex-related differences in the human uncinate fasciculus demonstrated with diffusion tensor tractography"  ISMRM Workshop on methods for quantitative diffusion of human brain, Canada, March 2005.
  18. Davatzikos, C. and R. Verma. : "Constructing statistical brain atlases from diffusion tensor fields" ISMRM Workshop on methods for quantitative diffusion of human brain, Lake Louise, Alberta, Canada, March 2005.