Total Variation-Based Reconstruction and Phase Retrieval for Diffraction Tomography with an Arbitrarily Moving Object

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Total Variation-Based

Reconstruction and Phase Retrieval

for Diﬀraction Tomography with an

Arbitrarily Moving Object

Robert Beinert∗

beinert@math.tu-berlin.de

Michael Quellmalz∗

quellmalz@math.tu-berlin.de

We consider the imaging problem of the reconstruction of a three-dimen-

sional object via optical diﬀraction tomography under the assumptions of

the Born approximation. Our focus lies in the situation that a rigid object

performs an irregular, time-dependent rotation under acoustical or optical

forces. In this study, we compare reconstruction algorithm in case i) that

two-dimensional images of the complex-valued wave are known, or ii) that

only the intensity (absolute value) of these images can be measured, which

is the case in many practical setups. The latter phase-retrieval problem can

be solved by an all-at-once approach based utilizing a hybrid input-output

scheme with TV regularization.

1 Introduction

We are interested in the tomographic reconstruction of a three-dimensional (3D) rigid

object, which is illuminated from various directions. In optical diﬀraction tomography,

the wavelength of the imaging wave, visible light with wavelength hundreds of nanome-

ters, is in a similar order of magnitude as features of the µm-sized object we want to

reconstruct. This setup substantially diﬀers from the X-ray computerized tomography,

where the wavelength is much shorter and therefore the light can be assumed to propa-

gate along straight lines.

∗TU Berlin, Institute of Mathematics, MA 4-3, Straße des 17. Juni 136, 10623 Berlin, Germany.

arXiv:2210.03495v2 [math.NA] 15 Nov 2022

incident ﬁeld uinc

measurement plane x3“rM

Figure 1: Experimental setup with measurement plane located at x3“rM, see [3].

Formally, in optical diﬀraction tomography we want to recover an object or, more specif-

ically, its scattering potential

f:R3ÑR,

which has compact support. At each time step tP r0, T s, the object undergoes a rotation

described by a rotation matrix RtPSOp3q, such that its scattering potential becomes

fpRt¨q. Here, the rotation is not restricted to some direction, we only require that Rt

depends continuously diﬀerentiably on t, which is the case for objects in acoustical traps,

cf. [1]. We further assume that the rotation Rtis known beforehand; for the detection

of motion from the tomographic data we refer to [2]. The object is illuminated by an

incident, plane wave

uincpxq:“eik0x3,x“ px1, x2, x3q P R3,

of wave number k0ą0, which induces a scattered wave ut:R3ÑC. We measure the

resulting total ﬁeld

utot

tpxq “ utpxq ` uincpxq,xPR3,

at a ﬁxed measurement plane x3“rM, see Fig. 1.

Under the Born approximation, the scattered wave utis a solution of the partial diﬀer-

ential equation

´p∆`k2

0qut“ put`uincqfpRt¨q,

see [4]. We assume the Born approximation to be valid, which is the case for small,

mildly scattering objects, see [5, § 3.3]. Then the relationship between the measured

ﬁeld utand the desired function fcan be expressed via the Fourier diﬀraction theorem

[6,7] by

ieiκrM

F1,2rutspk1, k2, rMq “ Frfs pRtpk1, k2, κ ´k0qq , k2

1`k2

2ăk2

0,(1)

where

κpk1, k2q:“bk2

0´k2

1´k2

2, k2

1`k2

2ăk2

Here the 3D Fourier transform Fand the partial Fourier transform F1,2in the ﬁrst two

coordinates are given by

Frfspkq:“ p2πq´3

2żR3

fpxqe´ix¨kdx

and

F1,2rfspk1, x3q:“ p2πq´1żR2

fpx1, x3qe´ix1¨k1dx1

for kPR3and k1PR2. The left-hand side of (1) is fully determined by the measure-

ments utot

tp¨,¨, rMq, and the right-hand side provides non-uniform samples of the Fourier

transform Frfs, evaluated on the union of semispheres which contain the origin and are

rotated by Rt. Therefore, the reconstruction problem in diﬀraction tomography can be

seen as a problem of inverting the 3D Fourier transform Fgiven non-uniformly sampled

data in the so-called k-space. If these samples have a positive Lebesgue measure, the

scattering potential is uniquely deﬁned by the measurements.

Theorem 1.1 (Unique inversion, [3, Thm. 3.2]) Let fPL1pR3qhave a compact

support in txPR3:}x}2ďrMu, and let the Lebesgue measure of

Y:“ tRtpk1, k2, κ ´k0q:k2

1`k2

2ďk2

0, t P r0, T su Ă R3(2)

be positive. Then fis uniquely determined by utot

tp¨,¨, rMq,tP r0, T s.

In § 2of this paper, we consider algorithms for the tomographic reconstruction of ffrom

utot

tp¨,¨, rMq,tP r0, T s. Since in practice one can often measure only intensities |utot

t|,

cf. [8], in § 3we will incorporate phase retrieval methods to retrieve f. In contrast to

[3], we focus here on the numerical studies for the case of an arbitrarily moving rotation

axis.

2 Reconstruction methods

We consider three diﬀerent reconstruction methods for the situation that we know the

complex-valued ﬁeld utp¨,¨, rMq. First, the ﬁltered backpropagation is based on a trunca-

tion and discretization of the inverse 3D Fourier transform,

fbppxq “ żY

Frfspyqeix¨ydy,xPR3,

where Yis the set where the data Ffis available in k-space. We state the following

ﬁltered backpropagation formula for a moving rotation axis, the special case for constant

axis is well established [9].

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摘要：

TotalVariation-BasedReconstructionandPhaseRetrievalforDiractionTomographywithanArbitrarilyMovingObjectRobertBeinert*beinert@math.tu-berlin.deMichaelQuellmalz*quellmalz@math.tu-berlin.deWeconsidertheimagingproblemofthereconstructionofathree-dimen-sionalobjectviaopticaldiractiontomographyundertheass...

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Total Variation-Based Reconstruction and Phase Retrieval for Diffraction Tomography with an Arbitrarily Moving Object

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