Asymptotic Analysis of the SVD for the Truncated Hilbert Transform with Overlap

Rima Alaifari, Michel Defrise, Alexander Katsevich
2015 SIAM Journal on Mathematical Analysis  
The truncated Hilbert transform with overlap H T is an operator that arises in tomographic reconstruction from limited data, more precisely in the method of Differentiated Back-Projection (DBP). Recent work [1] has shown that the singular values of this operator accumulate at both zero and one. To better understand the properties of the operator and, in particular, the ill-posedness of the inverse problem associated with it, it is of interest to know the rates at which the singular values
more » ... ngular values approach zero and one. In this paper, we exploit the property that H T commutes with a second-order differential operator L S and the global asymptotic behavior of its eigenfunctions to find the asymptotics of the singular values and singular functions of H T .
doi:10.1137/140952296 fatcat:souf7emdtjbqtnba4jev7ff2li