Estimation of Non-negative ODFs Using the Eigenvalue Distribution of Spherical Functions
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- Schwab E., Afsari B., Vidal R. (2012) Estimation of Non-negative ODFs Using the Eigenvalue Distribution of Spherical Functions. In: Ayache N., Delingette H., Golland P., Mori K. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2012. MICCAI 2012. Lecture Notes in Computer Science, vol 7511. Springer, Berlin, Heidelberg
Current methods in high angular resolution diffusion imaging (HARDI) estimate the probability density function of water diffusion as a continuous-valued orientation distribution function (ODF) on the sphere. However, such methods could produce an ODF with negative values, because they enforce non-negativity only at finitely many directions. In this paper, we propose to enforce non-negativity on the continuous domain by enforcing the positive semi-definiteness of Toeplitz-like matrices constructed from the spherical harmonic representation of the ODF. We study the distribution of the eigenvalues of these matrices and use it to derive an iterative semi-definite program that enforces non-negativity on the continuous domain. We illustrate the performance of our method and compare it to the state-of-the-art with experiments on synthetic and real data.
Keywordsdiffusion imaging orientation distribution functions spherical harmonics Toeplitz matrices eigenvalue distribution theorem
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