THE SPATIAL STRUCTURES OF PSEUDOPHASE SINGULARITIES IN THE
ANALYTIC SIGNAL REPRESENTATION OF A SPECKLE-LIKE PATTERN
AND THEIR APPLICATION TO BIOLOGICAL KINEMATIC ANALYSIS
Wei Wang (1), Yu Qiao (2), Reika Ishijima (1), Tomoaki Yokozeki (1), Daigo Honda(1),
Akihiro Matsuda (1), Steen G. Hanson (3), and Mitsuo Takeda (1)
1: Department of Information and Communication Engineering, The University of ElectroCommunications, 1-5-1, Chofugaoka, Chofu, Tokyo, 182-8585, Japan
Email: [email protected]
2: Department of Information Management Science, The University of Electro-Communications
3: Optics and Plasma Research Department, Risoe National Laboratory, Technical University of
Denmark, DK-4000 Roskilde, Denmark
ABSTRACT
A new technique for kinematic analysis of biological specimens is proposed that makes use of pseudophase singularities in the
analytic signal generated by a Laguerre-Gauss filter operation applied to a speckle-like random pattern. In addition to the
information about the location and the core structures of the phase singularities, we also detect the spatial structures of a
group of pseudophase singularities. This spatial structure serves as a “constellation” that uniquely characterizes the mutual
position relation between the individual phase singularities, and can thus be used for unique identification of a cluster of phase
singularities. Experimental results of the measurements of angular and linear velocities and a scale change of a swimming fish
are presented along with the kinematic analysis of the fish in motion, which demonstrate the validity of the proposed technique.
Introduction
Measurement of dynamic structures is a key issue in both the field of mechanics and, to a lesser extent, in the field of biology.
The reason for the lack of biological applications being that the structures, themselves, are flexible, and thus are not easily
followed by standard cross-correlation techniques. Let alone, that the entire structure might undergo rotation and maybe even
scaling during movement. Finally, it’s usually of importance to track the dynamical behavior of parts of the sample individually.
This objective has previously been pursued with some success by establishing fix points at the object’s surface, which later are
to be followed. Needless to say, the rather arbitrarily placing of these points will inevitably influence the result. Furthermore,
these methods will usually not possess the needed accuracy. The need for establishing an autonomous method with a high
accuracy for tracking the dynamics of especially biological specimen is obvious.
The concept of singularities in optical fields has during the last two decades been subject for intensive studies [1,2]. These
points in a coherent field are positioned where the intensity vanishes, and thus the phase of the field is undetermined here; as
such, they are considered a nuisance in interferometry. But as the photon noise is zero where the intensity disappears, these
points will carry valuable information as land markers.
On the other hand, ordinary real-valued imagery does not per se have phase singularities, as they are not complex analytical
functions. But based on the real-valued imagery one can construct a unique complex analytical field by deriving the imaginary
part based on a Hilbert transform of the intensity, and subsequently merging the real- and imaginary parts [3]. Unfortunately,
this way of creating an analytical function is asymmetrical with respect to the coordinates, and thus less useful. A rotationally
symmetric transformation (The Riesz transform or Laguerre-Gauss transform) has been introduced and will better serve our
purpose [4,5]. Having converted the incoherent image into an analytical field facilitates the finding of the singularities, which
uniquely depicts the structure under investigation.
In this paper, we will show the benefits of converting a sequence of images of a dynamic structure into a corresponding
sequence of complex analytical fields, from which we only utilize the positions of the singularities. In addition, we assign a
“fingerprint” based on the fine structure of the individual singularity in order for reliably distinguishing the movement of each
singularity over the entire field of view. To further strengthen unambiguous tracking, singularity clusters with specific mutual
spatial structure will be designated.
A series of images of a biological specimen, here a swimming fish, will be transformed and the singularities identified and
tracked during the object’s motion across the entire field of view of the camera. Not only can the position and the velocity be
monitored, but it will be shown that also the rotation and the scaling can be derived. Finally, the tracking of individual parts of
the specimen will be depicted. This article will consist of a theoretical description of the way the analysis is carried out,
followed by the experiments conducted on a living specimen, ending with a conclusion.
Principle
Before explaining the proposed improvement to optical vortex metrology for biological kinematic analysis, we first briefly review
the two-dimensional isotropic analytic signal representation of a speckle pattern.
Let I ( x , y ) be the original intensity distribution of the speckle pattern, and let its Fourier spectrum be ℑ( f x , f y ) . We can relate
I ( x, y ) to its isotropic analytic signal I ( x, y ) through a Laguerre-Gauss (L-G) filter. Thus, our definition of a 2-D isotropic
analytic signal representation for the speckle pattern is
I ( x, y ) =
+∞ +∞
∫ ∫ LG ( f
−∞ −∞
x
, f y ) ⋅ ℑ( f x , f y ) exp ⎡⎣ j 2π ( f x x + f y y ) ⎤⎦ df x df y
(1)
where LG ( f x , f y ) is a Laguerre-Gauss filter in the frequency domain defined as follows:
LG ( f x , f y ) = ( f x + jf y ) exp ⎡⎣ − ( f x2 + f y2 ) ω 2 ⎤⎦ = ρ exp( − ρ 2 ω 2 ) exp( j β ) .
where ρ =
(2)
f x2 + f y2 , β = arctan( f y f x ) are the polar coordinates in the spatial frequency domain. In addition to the advantage
of spatial isotropy common to the Riesz transform stemming from the spiral phase function with the unique property of a
signum function along any section through the origin [4], the Laguerre-Gauss transform has the favorable characteristics to
automatically exclude any DC component of the speckle pattern [5]. It can also readily be seen from Eq.(2) that the doughnutlike amplitude for the L-G function serves as a band-pass filter by suppressing the high spatial frequency components that
create unstable phase singularities. Therefore, the density of phase singularities in a analytic signal can be controlled by
choosing a proper bandwidth of the Laguerre–Gauss function, ω , to adjust the average speckle size. After straightforward
algebra, we find that
(3)
I ( x, y ) = I ( x, y ) exp[ jθ ( x, y )] = I ( x, y ) ∗ LG( x, y ) ,
where ∗ denotes the convolution operation, and LG( x, y ) is again a Laguerre-Gauss function in the spatial signal domain,
LG( x, y ) = F
−1
{LG ( f x , f y )} = ( jπ 2ω 4 )( x + jy ) exp[ −π 2ω 2 ( x 2 + y 2 )]
= ( jπ 2ω 4 )[ r exp( −π 2 r 2ω 2 ) exp( jα )]
where, F
−1
,
(4)
is inverse Fourier transform, and r = x 2 + y 2 , α = arctan( y x ) are the spatial polar coordinates defined as usual.
The phase θ ( x , y ) of the analytic signal of a speckle pattern is referred to as the pseudophase to distinguish it from the true
phase of the optical field responsible for the speckle pattern. Although it is not the true phase of the complex optical field, the
pseudophase does provide useful information about the object, as will be shown below.
Just as a random speckle-like intensity pattern imprints marks on a coherently illuminated object surface, randomly distributed
phase singularities in the pseudophase map of the speckle-like pattern imprint unique marks on the object surface. In our
previous investigation [4], we have proposed a technique for nano-metric displacement measurement, which makes use of the
information about the locations of phase singularities before and after the displacement. To do this, we need to be able to
identify the corresponding phase singularities between the pre- and post-displacement pseudophase maps. If the
displacement is known to be small a priori, we can restrict our search only to the closest neighbor phase singularities of the
same topological charge. However, when the displacement is large and/or non-uniform and no information is given a priori, we
cannot uniquely identify the corresponding phase singularities. This has been the drawback of our previous technique, though
it has the advantage of having a high resolution.
To solve this problem, we have already proposed the use of additional information about the core structure of the phase
singularities [5]. Similarly to optical vortices in random laser speckle fields [6, 7], the changes of the pseudophase around the
phase singularities are non-uniform, and the typical core structure around the phase singularity is strongly anisotropic. Figure 1
shows an example of the reconstructed amplitude contours and pseudophase structure of the analytic signal representation of
a speckle pattern in the neighborhood of a pseudophase singularity. The pseudophase singularity is located at the center of
the elliptical contours of the amplitude, which is the intersection of the zero crossings of the real and imaginary parts of an
analytic signal I ; i.e. the pseudophase has a characteristic feature of a 2π helical structure. We note that the eccentricity of
the contour ellipse e and the zero crossing angle θ RI between the real and imaginary parts, shown in Fig. 1(a), are invariant
to the in-plane rigid-body motion of the object involving translation and rotation, and we can use these two geometric
parameters to describe the local properties of the phase singularities. In addition, each phase singularity has its own
G
topological charge q and vorticity defined by Ω ≡ ∇ Re[ I ( x, y )] × ∇ Im[ I ( x, y )] , which we also assume to be invariant during
{
} {
}
an in-plane rigid-body displacement involving translation and rotation. Just as no fingers have exactly the same fingerprint
pattern, no phase singularities have exactly the same local properties with identical eccentricity e , zero-crossing angle θ RI ,
G
topological charge q , and vorticity Ω . It is this uniqueness of the core structure that enables the correct identification and the
tracking of the complicated movements of phase singularities.
G
Ω
e
θRI
Figure 1. Core structure around a phase singularity with zero crossings of real and imaginary parts of complex signal
representation for a speckle pattern. (a) Amplitude contours and zero crossing lines; (b) Pseudophase structure.
When observed with focus on the object surface, the displacement of each pseudophase singularity can be directly related to
the local displacement of the object surface. After identifying corresponding phase singularities before and after displacement
making use of their core structures as fingerprints, we can estimate the local displacement of an object by tracing the
movements of the pseudophase singularities. Thus, the in-plane displacement of an object can be estimated from the
coordinate change (∆x, ∆y ) for each pseudophase singularity within the probing area.
In addition to the information about the location and the core structures of the phase singularities, we also detect the spatial
structures of a group of the pseduophase singularities. As shown in Fig. 2, this spatial structure can serve as a “constellation”
that uniquely characterizes the mutual position between the individual pseudophase singularities. After identifying the
corresponding pseudophase singularities of the speckle pattern after displacement through their core structures, we can
decompose a complicated movement of the object into translation, rotation and scaling, and therefore make the kinematic
analysis.
y
e
G
Ω
θ RI
e′
G
Ω′ ( xi′, yi′)
θ RI′
( xi , yi )
( x ', y ')
(x, y)
δ
x
Figure 2. Schematic diagram for the constellation of pseudophase singularities. (a) Before movement; (b) After movement.
Let ( xi , yi ) , ( xi′, yi′) be the coordinates of the i -th pseudophase singularity in one constellation before and after movement,
respectively. Based on the movement decomposition analysis, the coordinate relation for the object before displacement can
be written as
x 'i − x ' = a ( xi − x ) cos δ + a ( yi − y )sin δ ,
(5)
y 'i − y ' = − a ( xi − x )sin δ + a ( yi − y ) cos δ ,
(6)
where a is scale factor, δ is rotation angle, ( x = ∑ i xi N , y = ∑ i yi N ) and ( x ′ = ∑ i xi′ N , y ′ = ∑ i yi′ N ) are the coordinates
of the gravity centers before and after movement, respectively, and N is the total number of pseudophase singularities in this
constellation. Therefore, the figure-of-merit for best matching of pseudophase singularities has been chosen as
{
}
E = ∑ i [ x 'i − x '− C ( xi − x ) − S ( yi − y )] + [ y 'i − y '+ S ( xi − x ) − C ( yi − y )] ,
2
2
(4)
where C = a cos δ and S = a sin δ . On the basis of the least square fitting routine, the condition for E to be a minimum is that
∂E / ∂C = 0 and ∂E / ∂S = 0 . After straightforward algebra, we find that
∑ (x
- x y′ ) ∑ (x
C = ∑ i ( x i′x i + y ′i y i )
S = ∑ i ( x i′y i
i
i
)
).
i
2
i
+ y i2 ,
(6)
i
2
i
+ y i2
(8)
where x i′ = xi′ − x ′ , x i = xi − x , y i′ = yi′ − y ′ and y i = yi − y . From the definition for S and C , we have
a = S2 + C2 ,
δ = arctan( S C ) ,
(9)
(10)
∆x = x '− x ,
(11)
∆y = y '− y .
(12)
Thus a complicated movement of an object can be decomposed into a sum of a translation ( ∆x, ∆y ) , a deformation a and a
rotation δ by tracing the movements of the pseudophase singularities.
Experiments
Experiments have been conducted to demonstrate the validity of the proposed principle for biological kinematic analysis. In the
experiments, the biological sample is a swimming fugu fish with a speckle-like pattern on its body surface. The fugu is imaged
by a COSMICAR television lens ( f = 12.5 mm) onto the sensor plane of a high-speed camera, FASTCAM-NET 500/1000/Max
(PHOTRON). Under the illumination given by a halogen lamp, sequences of the fugu swimming have been taken at a frame
rate of 30 frames per second. From the recorded images, we generated an isotropic analytic signal by Laguerre-Gauss filtering,
and retrieved the pseudophase information. In the experiment, we adjusted the average speckle size and controlled the
density of the pseudophase singularities carefully by choosing a proper bandwidth of the Laguerre-Gauss filter ω in Eq. 2, so
that the pseudophase singularities within the fugu body are clearly separated with a large average separation distance. After
identifying the corresponding pseudophase singularities inside the fugu by using their core structures as the fingerprints, and
determining their coordinates for every two consecutive pictures, we conducted the biological kinematic analysis by the
proposed optical vortex metrology.
Two examples of the recorded images for the swimming fugu at the different instants time are represented in Figure 2(a) and
(b). The image in Fig. 2(b) is delayed by 3.3s from that in Fig. 2(a). The corresponding amplitude distributions of the generated
analytic signals of the recorded images are shown in Fig. 2(c) and (d), where the locations for the positive and negative
pseudophase singularities have been indicated by different gray scale points, respectively. In all of these figures, we can
observe that the fugu has experienced a large rotation during the recording period, and the pseudophase singularities have
similar structures transformed with respect to each other by a certain amount that corresponds to the complicated movements
of the fugu. As expected, most of the pseudophase singularities can find their correct counterparts, and constitute a stable
constellation with specific mutual spatial structure. The coordinate differences between the corresponding pseudophase
singularities inside the constellation gave a good estimation for the swimming fugus, and could be used for biological kinematic
analysis as far as we use only matching phase singularities. However, we also found that some pseudophase singularities
failed to find their correct counterparts. This phenomenon may be attributed to the following: Because of the change in the
illumination condition and the body distortion during the fugu’s swimming, the recorded images begin to change their shapes;
in other words, speckle decorrelation increases when the time interval between two recorded images is large. This gives rise to
the creation or annihilation of singularities in the pseudophase map. Meanwhile, the local shape change in the recorded
images also introduces the distortions of the core structures of the phase singularities. These newly created or annihilated
pseudophase singularities definitely have no counterparts in the pseudophase map. Furthermore, the distortion of the core
structure for the pseudophase singularities enhances the difficulty in finding the correct counterparts for calculation of their
coordinate difference.
(a)
(b)
(c)
(d)
Figure 2. Recorded images for the swimming fugu at different instants of time and the generated analytic signals with
pseudophase singularities inserted.
(a)
(b)
(c)
(d)
Figure 3. The trajectory of the pseudophase singularities inside the fugu’s body
After identifying corresponding pseudophase singularities for each pair of consecutive images making use of their core
structures as fingerprints, we can trace the movement of the swimming fugu through its trajectory as shown in Fig. 4, and
obtain the configuration of singularity constellation within the fugu body by eliminating those pseudophase singularities that
failed to find their correct counterparts. From the coordinate information for each pseudophase singularity in the constellation,
the complicated movement of the swimming fugu can be decomposed into a translation, a scaling and a rotation based on
Eqs.(9)-(12). Meanwhile, we can also obtain the fugu’s dynamic information about the linear and angular velocities from the
calculated displacement in translation and rotation.
Figure 4. Movement decomposition based on the pseudophase singularities constellation
Figure 4 shows the movement decomposition for the swimming fugu based on the constellation of pseudophase singularities.
The linear velocity for translation has a typical periodical structure with an average frequency around 5 hertz. This rhythmic
swimming behavior was produced by the fin of fugu to provide the internal force through striking water, and resulted in the
forward-and-back movements. Meanwhile, a large rotation and a slight depth change inside the water can also be observed for
the fugu. Figure 4 serves as an experimental demonstration of the validity of the proposed technique for biological kinematic
analysis.
Conclusions
We have proposed a technique for biological kinematic analysis based on the improved Optical Vortex Metrology, which
makes use of the spatial structures of the constellation for the pseudophase singularities in the analytic signal generated by
Laguerre-Gauss filtering. This singularity constellation uniquely characterizes the mutual position relation between the
individual pseudophase singularities, and can be used for the purpose of unique identification of a cluster of phase
singularities. Experimental results for angular and linear velocity measurement, and scaling changes for the fish kinematic
analysis are presented, which demonstrate the validity of the proposed technique. Furthermore, the proposed method for
decomposing the biological movement establishes a new relationship between singular optics and biological dynamics, which
appears to have been regarded as being based on mutually independent disciplines, and thus introduces new opportunities to
explore other potential application of the phase singularities in biological phenomena.
Acknowledgments
Part of this work was supported by Grant-in-Aid of JSPS B (2) No. 18360034, Grant-in-Aid of JSPS Fellow 15.52421, and by
st
The 21 Century Center of Excellence (COE) Program on “Innovation of Coherent Optical Science” granted to The University
of Electro-Communications.
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