The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
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We consider the phase retrieval problem in 3-D holotomography for strongly absorbing objects. Holotomography combines phase retrieval from Fresnel diffraction patterns with tomographic reconstruction to reconstruct the 3-D refractive index distribution. The main interest is the increase in sensitivity of up to three orders of magnitude compared to standard, absorption based tomography. Most existing algorithms are based upon linearization of the forward problem. This is motivated by the large problem size, since it yields computationally efficient solutions. Here, the mixed approach is used, which allows for both strong absorption and long propagation distances. Previous implementations have shown promising results, but in practice often suffer from strong low frequency artifacts. To address this problem, we introduce a homogeneous object assumption through a regularizing term based upon the absorption image. This allows the homogeneous object assumption to be introduced only in the low frequency range. The proportionality constant between absorption and refractive index is assumed to be known. The regularizing parameter is found using the standard L-curve technique. The benefits of our approach are illustrated using data measured at the European Synchrotron Radiation Facility. Low frequency noise in the reconstruction is alleviated, but the result is only quantitative in the areas of the sample where the homogeneous object assumption is fulfilled.