Non-linear iterative phase retrieval based on Frechet derivative

Abstract

Several methods of phase retrieval for in-line phase tomography have already been investigated based on the linearization of the relation between the phase shift induced by the object and the diffracted intensity. In this work, we present a non-linear iterative approach using the Frechet derivative of the intensity recorded at a few number of propagation distances. A Landweber type iterative method with an analytic calculation of the Frechet derivative adjoint is proposed. The inverse problem is regularized with the smoothing L₂ norm of the phase gradient and evaluated for several different implementations. The evaluation of the method was performed using a simple phase map, both with and without noise. Our approach outperforms the linear methods on simulated noisy data up to high noise levels and thanks to the proposed analytical calculation is suited to the processing of large experimental image data sets.

Absorption and phase retrieval in phase contrast imaging with non linear Tikhonov regularization

“…Most existing phase retrieval approaches are based on a linearization of the direct problem valid under some rather restrictive assumptions [1,2,3,4,5]. Recently, a non linear iterative approach based on the Frechet derivative of the intensity has improved the phase retrieval accuracy [6]. Yet, in this approach, the absorption B(X), obtained from the intensity measured for D=0, is assumed to be known.…”

“…[1,2,3,4,5]. Several linear phase retrieval methods have already been investigated [1,2,3,4,5] and improved by non linear approaches recently [6]. Yet, all these approaches are based on a precise knowledge of the absorption.…”

Phase retrieval problems in x-ray physics

“…[29,40] for nonlinear Tikhonov-type regularization, [27] for a Landweber-type method, and, as mentioned above, [47,82] for iteratively regularized Newton-type methods. Moreover, [70] which provides a nice overview of regularization methods in Banach spaces lists a phase retrieval problem as application (but again without considering the special data structure).…”

“…So, motivated by the theory developed there, we solve our problem by a more noise adapted method, the iteratively regularized Newton-type method (IRNM, see Section 3.1): 27) with a Kullback-Leibler like data fidelity functional S (y δ ; y) = KL(y δ + ; y + ), with shift parameter ≥ 0, (1.28) an appropriate penalty term R, and regularization parameters α n > 0. Here T [φ n ] denotes the Fréchet derivative in φ n ∈ X, i.e.…”

“…the IRNM) and also Landweber-type iterations that address the reconstruction of phase information from modulus measurements of its Fourier transform with a Banach or Hilbert norm monomial as data fidelity functional S (y δ ; y) = y − y δ s Y , s > 1, have been suggested in several works, such as [7,13,27,29,44,70]. While most of these approaches are based on a Hilbert space setting, choosing X (and possibly also Y) as a Banach space seems to be more appropriate: Since we are especially interested in the reconstruction of blocky structured biological samples (see e.g.…”