IOP PUBLISHING JOURNAL OF OPTICS J. Opt. 15 (2013) 075716 (10pp) doi:10.1088/2040-8978/15/7/075716 Least-squares based inverse reconstruction of in-line digital holograms Edwin N Kamau 1 , Nicholas M Burns 2 , Claas Falldorf 1 , Christoph von Kopylow 1 , John Watson 2 and Ralf B Bergmann 1 1 Bremer Institut f ¨ ur Angewandte Strahltechnik, Klagenfurter Str. 2, 28359 Bremen, Germany 2 University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE, UK E-mail: kamau@bias.de Received 27 February 2013, accepted for publication 17 June 2013 Published 2 July 2013 Online at stacks.iop.org/JOpt/15/075716 Abstract We present a least-squares solution for the inverse problem in in-line digital holography which is based on a point source model. We demonstrate that by reformulating the reconstruction problem as an inverse problem and by integrating a contour gradient based auto-focus search algorithm into the reconstruction routine, a more fundamental solution for the inversion of a hologram can be attained. With this approach the inversion can be calculated without any prior knowledge of the object’s shape/size and without imposing any constraints on the imaging system. In a proof-of-concept study we show that our method facilitates a more accurate reconstruction, as compared to conventional methods, and that it facilitates object localization with an accuracy on the order of the optical wavelength. Keywords: diffraction, interference, holography, inverse problems, auto-focus schemes (Some figures may appear in colour only in the online journal) 1. Introduction Digital holographic imaging facilitates capturing the entire wave field scattered or diffracted by a three-dimensional object in a single two-dimensional digital hologram. Both the amplitude and the phase of the object wave field are encoded in the intensity distribution of an interference pattern [1]. This mapping of the 3D object’s features onto a 2D hologram can be referred to as the forward problem in holography. The inverse problem, i.e. the reconstruction of the object from the hologram, is one of the fundamental problems in digital holography. Since diffraction between two parallel planes can be represented as a linear shift-invariant system and therefore numerically implemented with low computational efficiency [2], conventionally applied reconstruction methods are mainly mathematical transformations based on diffraction theory. The reconstruction of the object is in this case achieved through numerical segmentation, whereby a 2D complex amplitude is recovered from the measured hologram for a series of planes across the object volume by means of solving the diffraction integral [3]. Unfortunately, a reconstruction in any given single plane is distorted by wave fields originating from objects located on prior planes leading to formation of defocused images. Furthermore, the reconstruction is prone to errors resulting from noisy measurements, spatial filtering or speckle artifacts. Specifically in particle digital holography and in metrology this might lead to erroneous information extraction. In terms of precision, the achievable depth and lateral resolutions are limited to values of the order of δz = 100λ and δx = 10λ respectively, where λ is the optical wavelength [4]. A thorough summary of the these and other drawbacks of diffraction based methods are presented elsewhere [5, 6]. These drawbacks can be overcome by numerically solving the inverse problem. A number of different solutions have been proposed and implemented for specific applications in digital holography. A regularization method was first investigated by Cetin et al [7] with the aim of suppressing coherent speckle artifacts and preserving important image features. Likewise a penalized-likelihood statistical method specifically for image-plane holography was assessed in [6]. However, these two methods only consider the reconstruction of 2D images and hence do not solve the above mentioned inverse problem. A detailed overview of an inverse approach 1 2040-8978/13/075716+10$33.00 c  2013 IOP Publishing Ltd Printed in the UK & the USA