V Multiple Scattering in HPIV: Use of ODT Analysis
Techniques
J Lobera and J M Coupland
Wolfson School of Mechanical and Manufacturing Engineering
Loughborough University, Ashby Road, Loughborough, Leics. LE11 3TU
J.Lobera@lboro.ac.uk
Abstract. Holographic Particle Image Velocimetry (HPIV) has been used successfully to make
three-dimensional, three-component flow measurements from holographic recordings of
seeded fluid. It is clear that measurements can only be made in regions that contain particles,
but simply adding seeding results in poor quality images that suffer from the effects of multiple
scattering. Optical Diffraction Tomography (ODT) provides a means to reconstruct a 3D map
of refractive index from coherent recordings of scattered fields with different illumination
conditions. In this paper we consider from first principles, how these methods can be used to
extract information from HPIV recordings in the presence of multiple scattering. We discuss
3D reconstruction using the Born approximation and the Born Iteration Methods. Finally we
consider the application of some optimization methods that have been proposed elsewhere as a
means to solve strong scattering problems, but simplified with the use of the a-priori
information concerning the particle shape and refractive index.
1. Introduction
Holographic Particle Image Velocimetry (HPIV) provides a means to make simultaneous three
component, three-dimensional (3C3D) measurements of a seeded fluid flow[1],[2]. Implicit in the
analysis of HPIV recordings is the assumption that light scattered from a laser source is recorded
directly by the hologram such that multiple scattering is negligible. In practice, however, multiple
scattering effects appear to increase background noise, thereby decreasing the SNR and ultimately
limiting the number of velocity vectors that can be retrieved from a given flow field [3]. Optical
Diffraction Tomography (ODT) provides a means to reconstruct a 3D map of refractive index from
coherent recordings of scattered fields with different illumination conditions. In 1969 Wolf proposed a
method applicable to weak scattering objects, based on the first Born Approximation[4], and more
recently the Born Iteration Method (BIM), the distorted Born Iteration Method (dBIM)[5] and other
optimization strategies[6] have been proposed as methods to solve stronger scattering. It is clear that the
ability to identify particle images, even in the presence of partial obscuration and multiple scattering,
would be beneficial to provide detailed velocity measurements from HPIV recordings. It is worth
noting however, that ODT methods generally require significant computational effort and only the
very simplest of the techniques are currently viable in practical HPIV studies. Nevertheless the
methods of ODT are worthy of consideration and in what follows we discuss the techniques’
capability for the case of a simplified problem to reduce the computational overhead.
Consider a holographic reconstruction of two particles, back illuminated with a plane wave such
that the light scattered from one particle illuminates the other and the second particle effectively
obscures the first. To simplify matters, we consider a planar (2D) system with polarization such that
all E-fields (i.e. illuminating and scattered fields) are out of the plane and together are solutions to the
Helmholtz equation (see Section 2). To investigate the performance of this methods we have
developed computational tools to solve the Helmholtz equation for a known refractive index
distribution and illuminating field – the forward problem. The solver runs in Matlab and uses the
Finite Element Method (FEM) with a rectangular mesh. An important aspect of the solver is the use of
an Absorbing Boundary Condition (ABC) or Perfect Matched Layer that surrounds the computational
domain[7]. This type of boundary condition allows outgoing waves to pass freely from the
computational domain without reflections. A rectangular mesh, with a spacing of one tenth of the
wavelength is used.
Figure 1a) and 1b) show the scattered fields that correspond to particles of 0.05m and 1m
respectively, and are separated by distance of 5m. The particles have a refractive index 1.33 and are
assumed to be suspended in air. It can be seen that the smaller particles scatter in a near iso-tropic
manner (i.e. as point sources), while forward scatter dominates for larger particles. Let us suppose that
the cone of forward scattered light is recorded holographically from these systems and reconstructed in
the usual way. Figure 2a) and 2b) show the intensity of the reconstructed images for the case of
0.05m and 1m particles when the recording is made with a large numerical aperture (NA=0.85). For
the case of the 0.05m particles, it can be seen that the recording NA is sufficient to resolve the
particles in the depth direction. For the case of the 1m particles, it can be seen that although the
recording has sufficient depth resolution the images are confused. Furthermore, figure 2c) shows the
intensity of the reconstructed images for the case of 1m particles when the recording is made with a
smaller numerical aperture (NA=0.3) and it can be seen that the two particles appear as one.
a)
b)
Figure 1 Real part of the scattered field form two particles: a) of 0.05m diameter, b) of 1m diameter.
a)
b)
c)
Figure 2 Reconstruction image of two particles: a) 0.05m diameter particles recorded in a NA=0.85
hologram; b) 1m diameter particles recorded in a NA=0.85 hologram; c) 1m diameter particles
recorded in a NA=0.3 hologram.
It is clear from these results that the recording NA plays a very significant role in HPIV recording
and in the past we have made use of relatively large holographic plates, placed close to the object of
interest to optimize this parameter[8]. With a view to digital recording, large NA is only practically
achievable for the case of small scale flows due to space-bandwidth limitations[9], and for this reason
many researchers have considered reconstruction from simultaneous, low NA recordings made from at
least 2 different directions. The recordings can be made incoherently, as in stereo PIV (with a thick
light sheet)[10],[11] or coherently[12]. In the incoherent case all focus information is lost and a small
aperture is used to maintain a reasonable depth of field. In the coherent case some focus information is
retained although for a given sized object, the NA is ultimately limited by the camera resolution. In the
(coherent) digital holographic method reported by Sheng et al.[12], a forward scattering geometry using
two different illuminating waves (at 90 degrees) was used and intensity images were computed from
each hologram. Using our 2D system of 2 particles (Fig. 3a), an image of this kind has been computed
and is shown in figure 3b). For comparison, we have used an NA=0.3 (as in figure 2c) and it is clear
that now the particle positions can be deduced from the brightest peaks in the image. If the particles
are repositioned slightly (turned through 45 degrees as in figure 3c), however, then this type of
reconstruction gives rise to the distribution shown in figure 3d). Here we note that additional “ghost
particles” appear in the reconstruction. Elsinga et al.[11] have shown that the generation of ghost
particles is reduced significantly if recordings at more angles are combined and furthermore enhanced
by a reconstruction and measurement algorithm that they call Multiplicative Algebraic Reconstruction
Tomography (MART). In essence, MART pre-processes each image to decide if, and where, particles
exist and then calculates epipolar lines in a manner similar to classical photogrammetry[13]. The crosscorrelation of the epipolar intersection points identified in a given region of 3D space is then
calculated to estimate particle displacement.
a)
b)
c)
d)
Figure 3 Two in-line holograms with NA=0.3: a) two particles aligned with one observations directions; b)
reconstructed image of a); two particles rotated 45 degrees, d) reconstructed image of c).
Although, some impressive results have been reported, we note that the MART algorithm is highly
non-linear both in the particle identification phase and photogrammetric reconstruction and
consequently it is expected to be intolerant to optical aberrations and background noise. With HPIV,
we have found in the past, that there is significant advantage to collating information in a linear
manner (correlation of the complex amplitude) and postponing the non-linear decision process (in this
case, deciding which is the signal peak)[14]. Although, not always a linear process, ODT provides a
basis to reconstruct the scattering potential (refractive index) from simultaneous recordings of
scattered light with different illumination and viewing directions. In the following we outline the basis
of ODT and discuss its merit for 3D3C digital HPIV.
2. The scattering problem
According to scalar diffraction theory the (complex) amplitude of a monochromatic electric field, E(r),
propagating in a medium of (complex) refractive index, n(r), obeys the Helmholtz equation,
(1)
 2 E(r)  k 02 n 2 (r)E(r)  0
where k0 is the free space wave number defined here such that k0=2 , where  is the free space
wavelength.
The electric field can be written as the superposition of, Er(r), the illuminating field (i.e. that which
would be present if the medium was absent) and, Es(r), the scattered field. Defining the scattering
potential or object function as (r)  k 02 n 2 (r) 1 and noting that  2  k 02 E r (r)  0 we have,
(2)
 2  k 02 E s (r)  (r) E r (r)  E s (r)








The terms on the right-hand-side of equation 2 can be identified as source terms and can be
integrated to give

(3)
s
r
s
3
E (r ' ) 




G 0 (r 'r )(r ) E (r )  E (r ) dr
where G 0 (r)  e jk0 r r is the free-space Green’s Function. It is noted that a holographic microscope (or
any far-field optical instrument) is only capable of measuring the propagating part of the scattered
field that is observed at a distance from any inhomogeneity. In this, r’>>r and the Green’s Function
becomes,
(4)

exp jk 0 r'r  e jk0 r '
r.r' 
  A exp  jk.r 
exp  - jk 0

r' 

where A is a complex constant and k is the wave vector defined as k  k 0 r' r' . We can therefore write
G 0 (r'-r) 

r'r
r'
our observation as, Em(k), given by,
E m (k )  A





exp(  jk .r )(r ) E r (r )  E s (r ) dr 3
(5)
Equation 5 can be recognized as the three-dimensional Fourier Transform of the source terms of
equation 2 and is general in the sense that it is based only on the assumption of scalar diffraction
theory. Although Em(k) appears to be the full spectrum of the source terms, it is noted however, that
the Fourier transform is only evaluated over a limited portion of k-space, the (Ewald) sphere, defined
by k  2 /  . Consequently, the problem is generally under determined and the scattering potential
can only be estimated subject to additional properties (for example minimum variance) or certain
assumptions as follows.
3. First Born approximation
Reconstruction under the assumption of weak scattering was first considered by Wolf 8 in 1969. Let us
assume that we use digital holography to make a finite set of measurements E mij, of the
monochromatic plane wave components identified by the wave vectors, kj ,that are scattered by a
system which is illuminated by a set of plane waves identified by ki . Further assume that in each case
the illuminating field, Er(r) is assumed to be significantly stronger than the scattered field, Es(r), such
that from equation 5 we have,

(6)
m
3
E ij  A


exp( j(k i  k j ).r )(r ) dr
In this case the measured field values can be recognised as an incomplete set of Fourier
coefficients. Accordingly, an estimate of the scattering potential under the assumption of weak

scattering, (r) , is given by,

1
(7)
m
( r ) 
A
E
ij
exp( j(k i  k j ).r)
ij
It can be seen that according to the first Born approximation, the object function is a coherent sum
of periodic variations in refractive index that are analogous to Bragg gratings formed in a thick
(volume) hologram. Returning to the two particle problem of figure 3c), an estimate of the refractive
index distribution obtained by applying equation 7 to the FEM data is shown in figure 4a). This result
can be compared directly with the image shown in figure 3d) and a significant contrast enhancement
can be seen. The improvement can be attributed to the fact that the reconstruction according to the
Born approximation is a coherent addition of the information obtained from different views whereas
the intensity based reconstruction of figure 3d) is incoherent. We also remark that the Born
reconstruction is linear and presents no extra computational burden compared to the incoherent
method. Nevertheless it can be seen that the two problematic ghost images remain.
3. Born expansion method
Rather than neglect the scattered field, Es(r), in equation 5 Chew and Wang[5] have considered iterative
solutions. In this paper we consider the distorted Born Iteration method (dBIM) that essentially use the
best estimate of the refractive index, n, as the background object. If we define Er as the solution of
 2 E r (r)  k 02 n  2 (r )E r (r )  0 , we can obtain a equation similar to equation 2:
(8)
 2  k 2 n 2 (r ) E s (r)   (r) E r (r)  E s (r)

0 

1

1

1

Let us remark that in an integral notation the Green function would be not longer a free-space
Green function, and that in each iteration, we obtain the increment of the scattering potential
 1 (r ) . It is clear that if we start from a background that is air, the solution after the first iteration is
the first Born approximation (Fig. 4a).
The main drawback of this method appears to be stability and there is not guarantee of
convergence. For the case considered before of two particles of water (n=1.33) in air the image
obtained is clearly wrong after 3 iterations (Fig. 4b). A similar but weak scattering case of particles
with a refractive index n=1.05 is shown in figure 4c) and it can be clearly seen that the contrast has
increased. This limitation makes the method unpractical in most cases of interest in HPIV.
4. Optimization method
Finally we turn our attention to reconstruction methods that minimise the error defined by
J ( ) 

i
E im  E ith  
E im
2
(9)
2
where Eith() is the scattered field obtained by a forward solver. This is a non-linear optimization
problem that can be solved by the Conjugated Gradient Method[6]. This method can be applied to
strong scattering problems, but the computational cost make unpractical for PIV. In PIV the object is a
disperse medium, composed by particles of a known diameter and refractive index value in most of the
cases. The use of this a priori information can simplify and reduce significantly the computing task.
Returning to the solution obtained from the application of the first Born approximation to the 2
particle problem (Fig 4a), if the 4 possible particle locations are identified we can test whether each is
likely to be a ghost particle. In essence, we can test whether removal of particles reduces the
discrepancy between measured and predicted results and therefore reduces the error defined by
equation 9. In this case, we find the error from our first guess (all 4 particles present) is 0.2989. If we
test each particle in turn we find that removal of the bottom-left (ghost) particle has the most
significant effect on the error and reduces the value to 0.1244. Removal of top-right the (ghost)
particle reduces the error to 0.0065 and note that no further reduction is possible. We conclude that we
have the particle distribution shown in figure 4d) and this appears consistent with figure 3c) We note
however, that even in this noiseless case the error is not exactly zero and on close examination note
that the position of the particles in 4d) are shifted of 0.05m, showing that our initial estimates of
particle position were inaccurate. Clearly further dithering of the particle positions could be
implemented to decrease the error further.
a)
b)
c)
d)
Figure 4 Reconstruction image of two particles using: a) 1st Born approximation; b) 3rd iteration of dBIM
for particles of n=1.33; c) 3rd iteration for particles of n=1.05; d) optimization method.
5. Conclusions
In this paper we have considered the application of the methods of Optical Diffraction Tomography
(ODT) to the analysis of HPIV recordings. We used the case of 2 closely spaced particles modeled in
2D to illustrate the effect of multiple scattering and have used the Finite Element Method to compute
the scattered fields. From typical 2 view geometries, ODT based on the first Born approximation was
found to provide a subjectively better reconstructions than intensity based methods, however,
erroneous ghost particles remain in the image. The ghost particles diminished using iterative ODT
methods based on the Born expansion, but were found to be unstable unless refractive index changes
were unrealistically small. Finally, the problem was formulated as an optimization problem with apriori knowledge of particle shape and, in this noiseless case, it was shown that there is sufficient
information to eliminate the ghost particles. Further work is necessary to assess the practical feasibility
of this approach.
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