Imaging Systems Laboratory: Research on Digital Holography

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Objective

We develop computational algorithms  —  mostly inverse imaging  —  for sectional image reconstruction and resolution enhancement in digital holography.

Introduction

Digital holography is about the acquisition of holographic image data using a digital sensor, and the subsequent processing to reconstruct individual images. One important use is in three-dimensional microscopy. In fact, holography was first invented for microscopy, and making the recording digital has the benefit of enormous processing power in the recovery of images at specific sections, a process commonly called sectioning or sectional image reconstruction.

We have worked primarily with a specific form of digital holography called optical scanning holography [1]. The system diagram is shown in Figure 1.

optical scanning holography architecture

Fig 1: The optical scanning holography system. [1]

Using k_x and k_y to denote the transverse spatial frequency coordinates, we can derive the optical transfer function (OTF) to be

 mathcal{H}(k_x,k_y; z) = {rm exp} left{ -j frac{z}{2k_0}(k_x^2+k_y^2) right},

where k_0 is the wave number. The free-space spatial impulse response is then

 h(x,y; z) = -j frac{k_0}{2pi z} {rm exp} left{jfrac{k_0}{2z}(x^2+y^2) right}.

For an object with complex amplitude f(x,y,z), the complex hologram is given by

 g(x,y) = int_{-infty}^infty left(| f(x,y,z)|^2 * h(x,y; z) right) , {rm d}z .

The goal of sectioning is to recover |f(x,y,z)| from g(x,y) at specific depths (i.e. different values of z).

Inverse imaging

The conventional method for optical sectioning suffers primarily from defocus noise, i.e., appearance of images arising from sections other than the one we want to reconstruct. We develop an inverse imaging approach that can significantly suppress such noise. If we discretize the imaging equation above and converts the 2D images into vectors using raster scan, we have an equation of the form

 vec{g} = sum_{i=1}^N H_i vec{f}_i = H vec{f}.

vec{g} is from the observed hologram, and H is known if we know where the sections are. We can therefore derive vec{f} from various inverse imaging techniques.

  • Our first method, using (the simplest) Tikhonov regularization, is reported in [2].

  • An improvement, which using total variation regularization to preserve the edges better, is reported in [3].

Examples

(1) Biological sample (fluorescent beads)

We experimentally capture a hologram of fluorescent beads (DukeR0200, 2mum in diameter, excitation around 542 nm, emission around 612 nm). The beads mainly assemble at the top and bottom surfaces. We then reconstruct the two sections. These are shown in Figure 2. In the reconstructed sections we can clearly see the individual beads.

volume view      first section second section

Fig 2: [left] Volume view; [right] Two sections reconstructed by the inverse imaging method [1].

(2) Blind sectional image reconstruction

We derive a technique that can estimate where the sections of interest are in the hologram. Such information is then used in the inverse imaging algorithm. Details of our edge-based blind section identification technique can be found in [4]. A representative example is given in Figure 3.

recon by convention method      recon by inverse imaging

Fig 3: [left] Two reconstructed sectional images by the convolution method; [right] by the inverse imaging method. [4]

(3) Multiple sectional image reconstruction

One advantage of the inverse imaging method is that it can be applied on objects with multiple sections. An example with five sections is given in Figure 4. Here, the locations of the five sections are first estimated by the blind reconstruction technique [4] before using the information in the inverse imaging algorithm.

recon with five sections

Fig 4: [top] Locations of the five sections; [bottom] Sectional image reconstruction. [4]

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Selected references

  1. Edmund Y. Lam, Xin Zhang, Huy Vo, Ting-Chung Poon, and Guy Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Applied Optics, Vol. 48, no. 34, pp. H105–H112, December 2009.
    (pdf copy)  (bibtex entry)

This paper reviews the state-of-the-art of optical scanning holography sectional image reconstruction. It also demonstrates the capability of 3D microscopy with a biological sample (fluorescent beads). Both the paper and the dataset are freely available at OSA Optics Infobase.

  1. Xin Zhang, Edmund Y. Lam, and Ting-Chung Poon, “Reconstruction of sectional images in holography using inverse imaging,” Optics Express, vol. 16, no. 22, pp. 17215–17226, October 2008. Also published in The Virtual Journal for Biomedical Optics, vol. 3, no. 12, December 2008.
    (pdf copy)  (bibtex entry)

This paper presents for the first time an inverse imaging method to reconstruct sectional images in optical scanning holography. We demonstrate the superiority of this method compared to other existing techniques. The defocus noise has been suppress significantly and the reconstructed images show the sectional intensity clearly.

  1. Xin Zhang and Edmund Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” Journal of the Optical Society of America A, vol. 27, no. 7, pp. 1630–1637, July 2010. Also published in The Virtual Journal for Biomedical Optics, vol. 5, no. 11, August 2010.
    (pdf copy)  (bibtex entry)

This is the state-of-the-art sectional image reconstruction method that uses total variation regularization to achieve sharp edges in the resulting sectional images.

  1. Xin Zhang, Edmund Y. Lam, Taegeun Kim, You Seok Kim and Ting-Chung Poon, “Blind sectional image reconstruction for optical scanning holography,” Optics Letters, Vol. 34, no. 20, pp. 3098–3100, October 2009.
    (pdf copy)  (bibtex entry)

We develop an automated algorithm to identify the sections of interest that should be reconstructed. This is complementary to the inverse imaging technique in the actual reconstruction.

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Current members

External Collaborators and former members