Sharp adaptation for spherical inverse problems with applications to medical imaging

  • Authors:

  • Ja-Yong Koo;Peter T. Kim

  • Affiliations:

  • Department of Statistics, Korea University, Anam-Dong Sungbuk-Ku, Seoul 136-701, Korea;Department of Mathematics and Statistics, University of Guelph, Guelph, Ont., Canada N1G 2W1

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2008

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Abstract

This paper examines the estimation of an indirect signal embedded in white noise for the spherical case. It is found that the sharp minimax bound is determined by the degree to which the indirect signal is embedded in the linear operator. Thus, when the linear operator has polynomial decay, recovery of the signal is polynomial, whereas if the linear operator has exponential decay, recovery of the signal is logarithmic. The constants are determined for these classes as well. Adaptive sharp estimation is also carried out. In the polynomial case a blockwise shrinkage estimator is needed while in the exponential case, a straight projection estimator will suffice. The framework of this paper include applications to medical imaging, in particular, to cone beam image reconstruction and to diffusion magnetic resonance imaging. Discussion of these applications are included.