Reconstruction of Holograms

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    Reconstruction of sectional images inholography using inverse imaging

    Xin Zhang, 1 Edmund Y. Lam, 1, and Ting-Chung Poon 21 Imaging Systems Laboratory,

    Department of Electrical and Electronic Engineering,The University of Hong Kong, Pokfulam Road, Hong Kong

    2 Bradley Department of Electrical and Computer Engineering,Virginia Polytechnic Institute and State University,

    Blacksburg, Virginia 24061, USA

    elam@eee.hku.hk

    Abstract: This paper discusses the reconstruction of sectional imagesfrom a hologram generated by optical scanning holography. We present a

    mathematical model for the holographic image capture, which facilitatesthe use of inverse imaging techniques to recover individual sections. Thisframework is much more exible than existing work, in the sense that it canhandle objects with multiple sections, and possibly corrupted with whiteGaussian noise. Simulation results show that the algorithm is capable of recovering a prescribed section while suppressing the other ones as defocusnoise. The proposed algorithm is applicable to on-axis holograms acquiredby conventional holography as well as phase-shifting holography.

    2008 Optical Society of America

    OCIS codes: (090.1760) Computer holography; (180.6900) Three-dimensional microscopy;(100.3190) Inverse problems; (100.3020) Image reconstruction-restoration; (110.1758) Com-putational imaging.

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    space Filtering with Wigner Distribution Functions, Applied Optics 47, 164175 (2008).

    #97622 - $15.00 USD Received 19 Jun 2008; revised 3 Sep 2008; accepted 6 Oct 2008; published 13 Oct 2008(C) 2008 OSA 27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 17215

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    13. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Opticsand Signal Processing , 1st ed. (Wiley, Chichester, 2001).

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    1. IntroductionOptical scanning holography (OSH) is a technique which records the holographic informa-tion of a three-dimensional (3-D) object on a two-dimensional (2-D) hologram by lateral scan-ning [1]. Nowadays, it covers a great number of application areas, such as holographic uores-cence microscopy, remote sensing and biological microscopy [2, 3, 4, 5, 6]. The rst systemwas designed by Poon in 1985 [7]. It is a two-pupil system, in which two coherent beams il-luminate a point pupil and a uniform pupil, respectively [7]. They are then combined to takea raster scan over an object, and the holographic information is captured to form electronicholograms [8].

    Using optical scanning, multiple sections of an object are recorded on a 2-D hologram [9].Generally, the hologram is complex-valued and contains information from all sections. Thetechnique is advantageous as the hologram contains volume information of the 3-D object be-

    ing scanned. However, the challenge is with the reconstruction of individual sections from thehologram. The major reason lies in the suppression of defocus noise [10]. From a hologramcontaining two or more sectional images to be reconstructed, when one intends to retrieve asection, the other sections will manifest as defocus images, which we treat as noise with regardto the in-focus section. The more sections a hologram contains, the more serious the defocusnoise appears. Therefore, the suppression of defocus noise is mandatory in the reconstructionof sectional images from a hologram.

    The conventional method to obtain sectional images in OSH involves computing the con-volution of the hologram with the conjugate impulse response at the focused position. Thisconvolution operation extracts the focus image but the image is contaminated with interfer-ences from other defocused sectional images. Two recent methods were developed particularlyto reduce the visibility of the defocus images. Kim [11] proposed to reconstruct sectional im-ages by means of a Wiener lter, which is an optimal lter, in a statistical sense, widely used inimage restoration. The method was shown to outperform the conventional technique describedabove. However, the Wiener lter is optimal provided we can get accurate measurement of thepower spectral density. When the intensity distribution of a section is quite similar to the other,the estimator is biased greatly from the true noise, so that results would be less meaningful. Inaddition, it seems to work only with two sections, and with virtually no noise in the imagingsystem. More recently, Wigner distribution function (WDF) is also used in the reconstructionof sectional images [12]. The method takes advantage of the relationship between fractional

    #97622 - $15.00 USD Received 19 Jun 2008; revised 3 Sep 2008; accepted 6 Oct 2008; published 13 Oct 2008(C) 2008 OSA 27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 17216

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    Fourier transform and WDF to distinguish the information of sectional images [13]. Yet, for animage the Wigner distribution is a 4-D function, and the image reconstruction is thus compu-tationally intensive. Furthermore, both methods described above have only been demonstratedfor two sections. With more sections, the interference among those sections in a hologram islarger, resulting in more pronounced defocus noise.

    In this paper, our goal is to design a exible image reconstruction technique that can obtainsectional images from OSH. Our method requires only a moderate computational load typicalof image reconstruction techniques, can handle multiple sections, and incorporates noise in theimaging model. We achieve this by formulating the imaging using a matrix approach, resultingin an inverse imaging problem for the sectioning of the hologram. To cope with the ill-posednature of the imaging, regularization is used, together with an iterative approach using conjugategradient (CG) for the successive computations.

    This paper is structured as follows. In Section 2, we describe the basic principles of the OSHsystem and introduce the mathematical model we use for the imaging. A hologram contain-ing two sections is described as an example. In Section 3, the regularization technique andimplementation using iterative techniques are presented. Section 4 shows the results of twoexperiments on two- and three-sectional images resulting from holograms. Finally, Section 5concludes this paper.

    2. Imaging Model

    2.1. Principles of Optical Scanning Holography

    Optical Scanning Holography (OSH) records the holographic information of a 3-D object byoptical heterodynescanning. Its schematic is shown in Fig. 1, which was rst analyzed by Poonin 1985 [7]. In the system, two coherent beams of different temporal frequencies + and pass through two pupils p1 and p2 respectively, and