Deterministic regularization of three-dimensional
optical diffraction tomography
Yongjin Sung* and Ramachandra R. Dasari
G. R. Harrison Spectroscopy Laboratory, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
*Corresponding author:
Received January 11, 2011; revised May 8, 2011; accepted May 24, 2011;
posted May 26, 2011 (Doc. ID 140886); published July 6, 2011
In this paper, we discuss a deterministic regularization algorithm to handle the missing cone problem of three-
dimensional optical diffraction tomography (ODT). The missing cone problem arises in most practical applica-
tions of ODT and is responsible for elongation of the reconstructed shape and underestimation of the value of the
refractive index. By applying positivity and piecewise-smoothness constraints in an iterative reconstruction
framework, we effectively suppress the missing cone artifact and recover sharp edges rounded out by the mis-
sing cone, and we significantly improve the accuracy of the predictions of the refractive index. We also show
the noise-handling capability of our algorithm in the reconstruction process. © 2011 Optical Society of America
OCIS codes: 100.3010, 100.3190, 120.3180, 180.0180, 170.3880.
1. INTRODUCTION
Diffraction tomography is an inverse scattering technique
used to reconstruct a three-dimensional (3D) object from a
series of two-dimensional (2D) measurements of the scat-
tering field. The principle of diffraction tomography was
proposed by Wolf in 1969 [1]. Based on the first Born approx-
imation, he suggested a way to map 2D scattering fields onto a
3D frequency spectrum of the scattering potential. The Four-
ier diffraction theorem was later extended by Devaney [2]to
adopt the first Rytov approximation. Recently, we have ap-
plied Fourier mapping based on the first Rytov approximation
to the imaging of live biological samples [3]. This Fourier map-
ping is conceptually straightforward and fast, but the mapping
of the 2D scattering field onto 3D rectangular coordinates re-
quires an interpolation and induces an artifact. The filtered
backpropagation method [4] resolves this problem, and is
known to provide a better quality [5]. However, the filtered
backpropagation is valid only within the Born approximation,
and its 3D application is quite involved. On the other hand,
iterative reconstruction can overcome these limitations by
iteratively updating a trial solution based on a scattering mod-
el and a priori knowledge about a sample. Iterative recon-
struction has been applied to projection tomography that
neglects diffraction effects [6–8] and then to diffraction tomo-
graphy [9,10], which Zunino et al. used to find the magnetiza-
tion distribution in a buried layer from measurements of the
magnetic field at the surface [9]. Bronstein et al. applied it to
suppress noise in broadband ultrasound tomography [10]. For
the optical regime, Belkebir and Sentenac presented an itera-
tive method for retrieving the map of permittivity in transmis-
sion and total internal reflection microscopy [11], and Maire
et al. applied the theory to highly scattering samples in a
reflection configuration [12]. Belkebir et al. presented a full-
vectorial nonlinearinversionschemetoretrievethe3Dmapof
permittivity in total internal reflection microscopy [13]. The
main benefits of iterative reconstruction come from the reg-
ularizing effect of the iterative process and the capability of
incorporating various pieces of information about a sample,
such as maximum energy, piecewise smoothness, etc.
In this paper, we apply iterative reconstruction to 3D opti-
cal diffraction tomography (ODT) and focus on the recovery
of the missing cone. The missing cone originates from incom-
plete angularcoverage of the incident beam and arises inmost
practical applications of diffraction tomography. The missing
cone leads to elongation of the reconstructed shape along the
optical axis and to underestimation of the refractive index [2].
It is worthwhile to note that rotating-sample tomography also
suffers from the missing cone problem when the sample is ro-
tated around one axis [14]. Iterative reconstruction without
additional constraints has been called ART (algebraic recon-
struction technique). Ladas and Devaney [15] showed that
ART is capable of reconstructing a sample from a limited
amount of data only insofar as the data are noise-free [5].
A recent overview of optical diffraction tomography princi-
ples can be found in Haeberlé et al. [16].
2. ITERATIVE RECONSTRUCTION OF
OPTICAL DIFFRACTION TOMOGRAPHY
In the real experiment, a collimated laser beam is delivered to
the sample plane using a telescopic system, and the incident
beam onto a sample can be approximated to a plane wave,
since only a small region near the center of the wave front
is used [3,17]. The interaction of the plane wave with a weakly
scattering sample can be reasonably well described by the
scalar theory [18]. A 3D object can be represented by the scat-
tering potential [1],