Abstract
In this paper, we present a novel dictionary learning framework for data lying on the manifold of square root densities and apply it to the reconstruction of diffusion propagator (DP) fields given a multi-shell diffusion MRI data set. Unlike most of the existing dictionary learning algorithms which rely on the assumption that the data points are vectors in some Euclidean space, our dictionary learning algorithm is designed to incorporate the intrinsic geometric structure of manifolds and performs better than traditional dictionary learning approaches when applied to data lying on the manifold of square root densities. Non-negativity as well as smoothness across the whole field of the reconstructed DPs is guaranteed in our approach. We demonstrate the advantage of our approach by comparing it with an existing dictionary based reconstruction method on synthetic and real multi-shell MRI data.
This research was in part funded by the NIH grant NS066340 to Baba C. Vemuri, and the following grants AFOSR FA9550-12-1-0304, ONR N000141210862, NSF CCF-1018149 to Alireza Entezari.
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References
Basser, P., Mattiello, J., Lebihan, D.: Estimation of the effective self-diffusion tensor from the nmr spin echo. Journal of Magnetic Resonance (1994)
Callaghan, P.T.: Principles of nuclear magnetic resonance microscopy. Oxford University Press (1991)
Ozarslan, E., Shepherd, T.M., Vemuri, B.C., Blackband, S.J., Mareci, T.H.: Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT). Neuroimage (2006)
Jian, B., Vemuri, B.C., Ozarslan, E., Carney, P.R., Mareci, T.H.: A novel tensor distribution model for the diffusion-weighted MR signal. NeuroImage (2007)
Descoteaux, M., Deriche, R., Bihan, D.L., Mangin, J., Poupon, C.: Multiple q-shell diffusion propagator imaging. MIA (2011)
Assemlal, H., Tschumperle, D., Brun, L., Siddiqi, K.: Recent advances in diffusion MRI modeling: Angular and radial reconstruction. MIA (2011)
Aharon, M., Elad, M., Bruckstein, A.: K-svd: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing (2006)
Fletcher, P., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing (2007)
Sra, S., Cherian, A.: Generalized dictionary learning for symmetric positive definite matrices with application to nearest neighbor retrieval. In: Gunopulos, D., Hofmann, T., Malerba, D., Vazirgiannis, M. (eds.) ECML PKDD 2011, Part III. LNCS (LNAI), vol. 6913, pp. 318–332. Springer, Heidelberg (2011)
Caruyer, E., Deriche, R.: Diffusion MRI signal reconstruction with continuity constraint and optimal regularization. MIA (2012)
Tuch, D.S.: Q-ball imaging. MRM (2004)
Wedeen, V.J., Hagmann, P., Tseng, W.Y., Reese, T.G., Weisskoff, R.M.: Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. MRM (2005)
Pickalov, V., Basser, P.: 3D tomographic reconstruction of the average propagator from MRI data. In: ISBI (2006)
Wu, Y., Alexander, A.: Hybrid diffusion imaging. NeuroImage (2007)
Ye, W., Portony, S., Entezari, A., Blackband, S.J., Vemuri, B.C.: An efficient interlaced multi-shell sampling scheme for reconstruction of diffusion propagators. IEEE TIP (2012)
Bilgic, B., Setsompop, K., Cohen-Adad, J., Wedeen, V., Wald, L.L., Adalsteinsson, E.: Accelerated diffusion spectrum imaging with compressed sensing using adaptive dictionaries. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012, Part III. LNCS, vol. 7512, pp. 1–9. Springer, Heidelberg (2012)
Merlet, S., Caruyer, E., Deriche, R.: Parametric dictionary learning for modeling EAP and ODF in diffusion MRI. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012, Part III. LNCS, vol. 7512, pp. 10–17. Springer, Heidelberg (2012)
Ye, W., Vemuri, B.C., Entezari, A.: An over-complete dictionary based reguralized reconstruction of a field of ensemble average propagators. In: ISBI (2012)
Schwab, E., Afsari, B., Vidal, R.: Estimation of non-negative ODFs using the eigenvalue distribution of spherical functions. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012, Part II. LNCS, vol. 7511, pp. 322–330. Springer, Heidelberg (2012)
Cheng, J., Jiang, T., Deriche, R.: Nonnegative definite EAP and ODF estimation via a unified multi-shell HARDI reconstruction. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) MICCAI 2012, Part II. LNCS, vol. 7511, pp. 313–321. Springer, Heidelberg (2012)
Spivak, M.: A comprehensive introduction to differential geometry. Publish or Perish, Berkeley (1979)
Cetingul, H.E., Vidal, R.: Sparse Riemannian manifold clustering for HARDI segmentation. In: ISBI (2011)
Absil, P., Mahony, R., Sepulchre, R.: Optimization algorithms on matrix manifolds. Princeton University Press (2008)
Rao, C.R.: Information and accuracy attainable in the estimation of statitical parameters. Bull. Calcutta Math. Soc. (1945)
Srivastava, A., Jermyn, I., Joshi, S.: Riemannian analysis of probability density functions with applications in vision. In: CVPR (2007)
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Sun, J. et al. (2013). Dictionary Learning on the Manifold of Square Root Densities and Application to Reconstruction of Diffusion Propagator Fields. In: Gee, J.C., Joshi, S., Pohl, K.M., Wells, W.M., Zöllei, L. (eds) Information Processing in Medical Imaging. IPMI 2013. Lecture Notes in Computer Science, vol 7917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38868-2_52
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DOI: https://doi.org/10.1007/978-3-642-38868-2_52
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