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Optical coherence tomography system optimization
using simulated annealing algorithm
MOHAMMAD-REZA NASIRI-AVANAKI1, S. A. HOJJATOLESLAMI
1, MARIA
PAUN2, SIMON TUOHY
2, ALEXANDER MEADWAY
2, GEORGE DOBRE
2 and
ADRIAN GH. PODOLEANU2
1. Research and Development Centre, School of Biosciences, University of Kent,
Canterbury, Kent, CT2 7PD, UNITED KINGDOM
2. Applied Optics Group (AOG), School of Physical Sciences,
University of Kent, Canterbury, CT2 7NH, UNITED KINGDOM
http://www.kent.ac.uk/physical-sciences/research/aog/
Abstract: - In this study optimized compensation of wavefront aberrations utilizing an electromagnetic
deformable mirror is presented. The mirror is controlled by PCI cards and sound card through simulated
annealing algorithm implemented by using the integration of Visual C++ and MATLAB in MATLAB
environment.
Key-Words: - optimization, simulated annealing algorithm, adaptive optics, Matlab
1 Introduction Optical coherence tomography (OCT) is an
advanced imaging tool, which produces high-
resolution three-dimensional (3D) images [1]. OCT
is a non-invasive imaging modality that applies non-
ionizing safe optical radiation [1]. OCT images are
constructed by measuring the magnitude of the
optical echoes at different transverse positions on
the sample [2]. The optical devices and the surface
of the sample deteriorates the wavefront and
produce aberrations. Wavefront aberrations reduce
the signal to noise ratio and consequently diminish
the contrast of the OCT images. In addition,
aberrations reduce the resolution of the imaging
system. The aberrations are different from sample to
sample, hence it is difficult to correct them by a
static technique. Adaptive optics (AO) has been
introduced to reduce the effect of the aberrations. It
becomes essential especially in the applications
demanding a higher image signal-to-noise ratio and
higher resolution.
An AO system is made up of a wave-front
sensor, such as a Shack-Hartmann, to measure the
wave-front error, a deformable mirror (DM), to
compensate the distortion, and a control loop
method, to find the optimized compensation.
Unfortunately such systems are expensive [3]. On
the other hand, although open-loop wavefront or
beam-shape control can in principle be achieved
using the knowledge of a deformable mirror’s
influence matrix, its nonlinear characteristics make
implementation of a wide range of pre-deformed
shapes problematic without the use of feedback
through a wavefront sensor or other forms of closed-
loop control system. Less expensive closed-loop
methods are required for correcting distortions, such
as the so-called blind optimization or sensor-less
approaches [4-7]. In blind optimization algorithms,
only a DM is required. Such algorithms operate in a
closed-loop iteratively to optimize a single
measurable variable by reducing the wavefront error
obtained by changing the mirror surface shape.
Simulated annealing (SA) introduced in 1982 by
Kirkpatrick et al. [9], is a technique for
combinatorial optimization problems, such as
minimizing multivariate functions [10-11] to
improve the solution to the so-called Travelling
Salesman Problem [12]. SA is favoured due to its
efficiency with less computational complexity since
all other known techniques for obtaining an
optimum solution which require an exponentially
increasing number of steps as the problems become
larger [10]. In comparison with iterative
improvement methods [9], which are captured
simply in local minima, SA improves the result by
occasionally searching in directions that lead to
worse solutions [13]. SA is however a heuristic
optimization technique which does not guarantee in
achieving the optimum value [10]. The concept of
MATHEMATICAL METHODS AND APPLIED COMPUTING
ISSN: 1790-2769 669 ISBN: 978-960-474-124-3
the optimization technique comes from a physical
process of heating to a temperature that permits
many atomic rearrangements, and then slowly
cooling a substance allowing it to come to thermal
equilibrium at each temperature until the material
freezes into an ordered crystalline structure or into a
structure with the lowest energy [14]. In the context
of adaptive-optical control, the simulated annealing
algorithm is well-suited to the task because of its
ability to independently optimize many variables at
once [15].
In this paper, we aim to utilize DM and we
propose an effective optimization technique to
provide a simple, compact and affordable adaptive
optic system to compensate aberrations introduced
by distorting optical elements in the optical
coherence tomography.
2 Materials and Methods
2.1 Optical Coherence tomography
system configuration As shown in Figure 1, light is launched from a super
luminescent diode (SLD) with the central
wavelength of λ = 830 nm and with nm17=∆λ .
The light is split by BS1 into the reference and
sample arms by the ratio of 55/45. In the sample
arm, the light is split by another beam splitter, BS2,
with the ratio of 70/30. To compensate for the loss
introduced by the beam splitters, one can increase
the power in the source.
Fig1. Schematic of the optical coherence tomography setup with adaptive optics. SLD: super luminescent laser diode, PD: photodiode,
M: microscope objective, Mirao52d: 52-actuator electromagnetic
deformable mirror, BS1,2: beam splitter, HM1,2: heat mirror, CM1-6: concave mirror, XY: transversal scanners, C: coupler, SIP: sound card
input port, PC: personal computer, EU: electronic unit
Light then goes through a telescope containing
two concave Gold coated mirrors, CM1 (f=20cm,
Ø50.8 mm) and CM2 (f=101.6cm, Ø50.8 mm), in
order to expand the beam diameter. The telescope
increases the beam diameter from 3 mm to 15 mm
so as to cover the whole aperture of the deformable
mirror. The light then is projected onto the
deformable mirror (DM). DM for this study is a
Mirao52d manufactured by Imagine which is an
electromagnetic deformable mirror with 52 actuators
[16]. Each actuator couples a miniature magnet with
a miniature coil attached to the deformable
membrane covered by a silver coated membrane. By
applying a voltage to the coils a magnetic field is
induced in the coils which can attract or repeal the
magnets attached to the membrane [16]. Therefore
by varying the coil voltage one can adjust the
magnetic force and control the local shape of the
mirror so as to correct the aberrations. Reflected
light from the mirror is projected into another
telescope containing two concave Gold coated
mirrors; CM3 (f=76 cm, Ø50.8 mm) and CM4 (f=10
cm, Ø50.8 mm). This is to reduce the beam size on
the deformable mirror to a size that would fit on the
aperture of the XY scanners. The scanners, provided
by Cambridge Technologies, produce the transversal
scans. Another telescope containing two concave
mirrors CM5 (f=20cm, Ø50.8 mm) and CM6
(f=60cm, Ø50.8 mm) have been used to increase
the beam diameter to 6 mm. A hot mirror, HM2 is
used to deflect the beam towards the object.
On the return path, the beam passes through BS2
and on to HM1. The reflected light then is coupled
into the optical fibre. The light in both arms are
joined in a coupler to produce the interference signal
and the resultant light is detected on a Nirvana
photodetector from Newport Corporation. The
electrical signal obtained from the photodetector is
coupled into the computer sound card input port
(SIP) (microphone port) for signal processing and
controlling the shape of the mirror. SIP allows the
photodetector signal to be recorded onto the personal
computer (PC) in real time.
2.2 Signal processing flow The system from a signal processing point of view
includes SLD, optical system excluding the mirror,
detector, SIP, MATLAB as the processing unit
centre, PCI card and the DM. Signal from the SIP is
gathered using MATLAB for data acquisition in real
time. The signal is processed in the optimization
algorithm. The output of the algorithm is a set of 52
values which are sent to the DM. The values are in
the range between -1 and +1 V in order to produce
50 micron summit or depression respectively on the
surface of the mirror, respectively. We send the 52
voltage values to an 8-bit PCI7200. The voltage
MATHEMATICAL METHODS AND APPLIED COMPUTING
ISSN: 1790-2769 670 ISBN: 978-960-474-124-3
signals are amplified in the digital electronics unit,
converted to analogue signal and applied to the
mirror’s actuators to produce a certain shape on the
mirror. When no voltage is applied to the actuators,
the reflecting membrane is an uncontrolled shape.
However in this study we assume that the shape of
the mirror with zero volt applied to all the actuators
is flat and applying the positive and negative
voltages results in the membrane going up and
down, respectively. With the variation of the surface
of the mirror one can compensate for all Zernike
modes up to order 4 and also part of Zernike modes
order 6.
As we intended to manage the input and output
signals in a single programming environment, we
converted the port controller code written in VC++
to MATLAB using mex file [17].
2. 3 Our optimization schedule In this study, the cost function is defined as the
difference between the new intensity and the highest
intensity detected up to that point, which we call
Current Maximum Intensity (CMI). The variables
are the 52 components of a vector sent to the
actuators. The vector controls the actuator voltages.
The flowchart of our SA-based algorithm designed
to find the best mirror shape is given in Figure 2.
Fig.2 Flowchart of our simulated annealing algorithm
We use an artificial parameter which we call the
“synthetic temperature” to determine how often the
system switches between local and global search.
The initial value of the synthetic temperature has to
be high enough to let the algorithm jump up to
anywhere in the variable space. We experimentally
considered the initial temperature (melt temperature)
to be 1000 and the final (frozen) temperature to be
10-12
. The algorithm starts by sending a random set
of 52 values as actuators’ voltages between -1 and
+1. A uniformly distributed pseudo-random function
is chosen to generate the random values [11]. The
maximum intensity is set to zero at the beginning.
From now on, the algorithm operates in its ordinary
optimization loop (OOL) while the temperature is
examined to see whether or not it is still less than the
frozen temperature. The shape of the mirror changes
according to the actuator voltages and this results in
a new value of the light intensity detected. If the
intensity is greater than the CMI value, it becomes
the new CMI and also the voltage vector is saved as
the best vector. At each temperature, the OOL is
continued until we reach the situation where in 10
successive iterations the rate of increase of the
intensity becomes less than a threshold value. We
defined an allowed variation at each iteration which
is constant for a determined number of iterations. As
shown in Figure 3-a, the threshold values decrease
by the equation ])4/[*25.02/()001.0()( ttTr += for each
iteration (temperature step decrease), where t is the
iteration step and Tr specifies the allowed variation
at each step. As a result, the thermal equilibrium
condition of the algorithm is obtained in a smaller
range of intensity variation.
(a)
MATHEMATICAL METHODS AND APPLIED COMPUTING
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(b) Fig.3 (a) Threshold intensity decrease profile with iteration number. (b)
Temperature reduction profile during the simulated annealing process
based on successive temperature reductions given by T new = 0.9Told.
The cooling schedule as shown in Figure 3-b is
based on Tnew=0.9Told; in this study, the number of
cooling-down steps based on our choice of initial
and final temperatures is 328 steps. In the OOL in
each iteration, the value of actuators’ voltages are
updated around the best voltage by adding/reducing
a random value in the range of one tenth the
actuator’s voltage that is correspondence to the
CMI. In the OOL the Boltzmann probability
function is compared with a random value in the
interval 0 and 1. If the probability is greater than the
random value we let the algorithm jump to random
points in the variable space 5 times (this value was
obtained experimentally), by changing the voltage
vector in the range of [-1 +1], however
corresponding to the number of iterations, a
coefficient progressively lessened is multiplied to
this range. If the intensity at this stage is higher than
the CMI, it becomes the new CMI and the
corresponding voltage set is recorded and becomes
the best voltage set. After reaching the thermal
equilibrium in each temperature, the system
temperature is decreased. The process terminates
when the temperature becomes lower than the frozen
temperature. In the end, the optimal shape of the
mirror is determined by the best voltage obtained
from the algorithm which leads to producing the
highest obtainable intensity with the optical setup
which accordingly produces a better image quality.
3 Results and discussion The algorithm was tested on a 2.2 GHz Pentium IV
computer with 2.2 GB memory. The progression of
intensity along the simulated annealing algorithm
in finding the more appropriate shape of the mirror
with higher intensity is given in Figure 4.
Fig.4 Output intensity amplitude profile recorded from audio card
during 328 iterations (or 7405 number of voltage changes) in our
simulated annealing algorithm. The minimum and maximum intensities are specified on the graph with pink arrows. The green
starts show the improved intensity obtained from the ordinary
optimization loop (OOL) and the red starts show the improved intensity obtained from the Boltzmann optimization loop (BOL).
As shown in Figure 4, the output intensity of the
system is increased from 0.0024 mA to 0.0626 mA
which shows a 26-fold increase. We can not claim
that the algorithm is able to enhance the DOCI by
the order of 26 times for instance, but we can say
that if the optical system presents a static aberration,
the proposed algorithm is able to find an appropriate
shape of the mirror to correct and compensate for the
aberration such that the system achieves the highest
possible intensity and the highest image quality
accordingly.
As seen in Figure 4, consideration the
application of the Boltzmann probability condition
resulted in large variations of the intensity. In this
loop the algorithm looks for a better intensity value
throughout the variable space by changing the
voltage set values (randomly in the interval between
-1 and +1). The number of improved intensities
indicated by red stars in Figure 4 shows the
significance of the use of Boltzmann probability
scheme. As the algorithm reaches to the lower
temperature the probability condition is fulfilled
fewer times and the OOL operates without going
through the Boltzmann loop. We examined our
algorithm for a similar setup and different samples
to make sure that the algorithm works correctly.
The mirror is hexagonal and the actuators are
placed in the centre of square pixel. The shape of the
mirror with the numbering of the actuators are given
in Figure 5.
MATHEMATICAL METHODS AND APPLIED COMPUTING
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Fig.5 Mirao52d mirror schematic. The numbers show the index of the
actuators.
We simulated the surface of the deformable
mirror simplistically for 12 sets of voltages by using
the actuator’s voltage value and an interpolation
technique. The surfaces display a shaded surface
based on a combination of ambient, diffuse, and
specular lighting models [11]. For datagrid we used
built-in Matlab function with v4 method [18]. The
surfaces are demonstrated in Figure 6. This
simulation performed on the voltage sets obtained
from the experiment that its results have been given
in Figure 4.
Fig.6 Surface curvature of the mirror demonstrated at 60o azimuthally
with 60o elevation which result the different value of Imax during our
simulated annealing algorithm. The z-axis in the graph indicates the voltage range applied on the mirror.
As seen, the surface of the mirror is changed
quite a lot during the optimization process. However
towards the end of the algorithm where the system
reaches to a very low energy the mirror surface is
just slightly changed.
In some SA applications, due to time limitations,
it is not possible to obtain the optimum answer in the
allocated time [19]. Further research is aimed at
investigating the optimum run time on our OCT
system, which will allow faster optimization.
4 Conclusion In summary, we have demonstrated an efficient
method for optimization of an optical coherence
tomography system using an electromagnetic
deformable mirror in conjunction with our simulated
annealing algorithm (Figure 2). We configured an
adaptive optics OCT setup (Figure 1) and examined
the optimization algorithm successfully. We used a
simple and inexpensive computer interface (audio
card for receiving and PCI7200 card for sending
signals) in single programming environment,
Matlab. We showed that the algorithm is quite
promising in finding the global maximum intensity
(Figure 4 and 6) and that is reproducible by running
it several times on different samples. We showed
that with this configuration, a compact, inexpensive
and simple optimization approach is achievable.
We conclude that while this technique is
unlikely to find the optimum solution, it can often
find a very good solution. We have plans to
investigate the use of this approach to remove the
effect of aberrations on the images obtained from
OCT. This will enable deeper exploration of
specimens.
References:
[1] E. A. S. D. Huang, C. P. Lin, J. S. Schuman, W.
G. Stinson, W. Chang, M. R. Hee, T. Flotte, K.
Gregory, C. A. Puliafito, and J. G. Fujimoto.,
"Optical coherence tomography," Science 254,
1178-1181 (1991)
[2] Podoleanu, A. G.: Optical Coherence
Tomography. Br. J. Radiol., 78 (2005) 976-988
[3] Plett, M. L., Barbier, P. R., Rush, D. W.:
Compact Adaptive Optical System Based on Blind
Optimization and a Micromachined Membrane
Deformable Mirror. Appl. Opt., 40 (2001) 327-330
[4] Booth, M.: Wave Front Sensor-Less Adaptive
Optics: A Model-Based Approach using Sphere
Packings. Optics Express, 14 (2006) 1339-1352
[5] Grisan, E., Frassetto, F., Da Deppo, V. et al.: No
Wavefront Sensor Adaptive Optics System for
Compensation of Primary Aberrations by Software
Analysis of a Point Source Image. 1. Methods. Appl.
Opt., 46 (2007) 6434-6441
[6] Zommer, S., & Ribak, E.: Adaptive Optics
almost without Wave Front Sensing, 2006
Conference on Visual Optics, Ed. W F Harris,
MATHEMATICAL METHODS AND APPLIED COMPUTING
ISSN: 1790-2769 673 ISBN: 978-960-474-124-3
Mopani Camp, Kruger National Park, South Africa,
August 5-10 (2006).
[7] Toledo-Suárez, C. D., Valenzuela-Rendón, M.,
Terashima-Marín, H. et al.: On the Relativity in the
Assessment of Blind Optimization Algorithms and
the Problem-Algorithm Coevolution. (2007) 1436-
1443
[8] Paterson, C., Munro, I., Dainty, J.: A Low Cost
Adaptive Optics System using a Membrane Mirror.
Optics Express, 6 (2000) 175-185
[9] Kirkpatric, S., Gelatt, C., Vecchi, M.:
Optimization by Simulated Annealing. Science, 220
(1983) 671-680
[10] Yang, P., Liu, Y., Yang, W. et al.: Adaptive
Mode Optimization of a Continuous-Wave Solid-
State Laser using an Intracavity Piezoelectric
Deformable Mirror. Opt. Commun., 278 (2007) 377-
381
[11] MathWorks, M.: The Language of Technical
Computing. At: http://www.mathworks.com, last
accessed April, (2009)
[12] Lawler, E. L., Rinnooy-Kan, A., Lenstra, J. K.
et al.: The traveling salesman problem: A guided
tour of combinatorial optimization. John Wiley &
Sons Inc (1985)
[13] Rutenbar, R.: Simulated Annealing
Algorithms: An Overview. IEEE Circuits Devices
Mag., 5 (1989) 19-26
[14] Fang, M.: Layout Optimization for Point-to-
Multi-Point Wireless Optical Networks Via
Simulated Annealing & Genetic Algorithm. Master
Project, University of Bridgeport, Fall, (2000)
[15] El-Agmy, R., Bulte, H., Greenaway, A. et al.:
Adaptive Beam Profile Control using a Simulated
Annealing Algorithm. Optics Express, 13 (2005)
6085-6091
[16] Gambra, E., Sawides, L., Dorronsoro, C. et al.:
Development, Calibration and Performance of an
Electromagnetic Mirror Based Adaptive Optics
System for Visual Optics. (2008) 322–328
[17] Carr, Roger. "Simulated Annealing." From
MathWorld-A Wolfram Web Resource, created by
Eric W. Weisstein.
http://mathworld.wolfram.com/SimulatedAnnealing.
html
[18] Sandwell,D.T.:Biharmonic Spline Interpolation
of GEOS-3 and SEASAT Altimeter Data. Geophys.
Res. Lett., 14 (1987) 139–142
[19] Rutenbar, R.: Simulated Annealing
Algorithms: An Overview. IEEE Circuits Devices
Mag., 5 (1989) 19-26
MATHEMATICAL METHODS AND APPLIED COMPUTING
ISSN: 1790-2769 674 ISBN: 978-960-474-124-3