This study proposes a modified subaperture stitching algorithm, which uses an image sharpening algorithm and particle swarm optimization to improve the stitching accuracy. In subaperture stitching interferometers with high positional accuracy, the highfrequency components of measurements are more important than the lowfrequency components when compensating for position errors using a subaperture stitching algorithm. Thus we use image sharpening algorithms to strengthen the highfrequency components of measurements. When using image sharpening algorithms, subaperture stitching algorithms based on the leastsquares method easily become trapped at locally optimal solutions. However, particle swarm optimization is less likely to become trapped at a locally optimal solution, thus we utilized this method to develop a more robust algorithm. The results of simulations showed that our algorithm compensated for position errors more effectively than the existing algorithm. An experimental comparison with full aperturetesting results demonstrated the validity of the new algorithm.
I. INTRODUCTION
In the development of astronomical optics, space optics
[1

3]
, and inertial confinement fusion, largeaperture optical systems have been widely used. With long production cycle and high cost, largeaperture interferometers cannot satisfy the advanced requirements for testing large optical elements. In 1982, Kim addressed this problem by developing the subaperture stitching method
[4]
. The core of this method is stitching together the measurements obtained from all of the subapertures to generate the fullaperture result. However, location errors are inevitable when using this method, so researchers use the overlapping areas between subapertures to compensate for the location errors
[5

8]
. The location errors include piston, tilt, defocus, clocking, and position errors. Piston, tilt, and defocus errors are easily compensated, and clocking errors are usually quite small, but position errors are not usually compensated well by algorithms. In most cases subaperture stitching interferometers have high positional accuracy, so the highfrequency components of measurements are the key factors when compensating for position errors. Therefore, we use an image sharpening algorithm to strengthen the highfrequency components of measurements. The basis of image sharpening is computing the approximate gradient of a measurement, so that the data may be stripped of many lowfrequency components after image sharpening. However, the lowfrequency components are helpful in avoiding local minima, and a subaperture stitching algorithm based on the leastsquares method can easily become trapped at a local minimum if the image is sharpened. A different optimization algorithm should be used to solve this problem.
Some researchers have used particle swarm optimization to address this problem in image stitching
[9]
. Particle swarm optimization can be more reliable than the leastsquares method for finding the globally optimal solution. Therefore, we utilize particle swarm optimization in our algorithm to calculate the position compensation. In Section II we introduce an existing iterative subaperture stitching algorithm based on the leastsquares method, while in Section III we describe our modified subaperture stitching algorithm, which uses image sharpening and particle swarm optimization. In Section IV we present the results of simulations comparing the performance of the two algorithms, and in Section V we report experimental results that demonstrate the effectiveness of the proposed method. Section VI presents our conclusions.
II. ITERATIVE SUBAPERTURE STITCHING ALGORITHM BASED ON THE LEASTSQUARES METHOD
For convenience, in this article we discuss the testing of only planooptical elements. First, a subaperture (usually the central one) is selected as the reference subaperture. Clearly the other subapertures have tilt and piston relative to the reference subaperture, as shown in
Fig. 1
. Thus it is necessary to compensate for the relative tilt and piston
[5

7]
:
Measurement of the ith subaperture in terms of the tilt and piston.
where
z_{i}
(
x
,
y
) is the measured value of the
i
th subaperture,
z_{i}
*(
x
,
y
) is the value after compensation,
a_{i}
and
b_{i}
are the coefficients of the relative tilt for the
i
th subaperture in the
x
and
y
directions respectively, and
c_{i}
is the coefficient of the relevant piston for the
i
th subaperture.
The subaperture stitching interferometer possesses six types of mechanical location errors, and the other errors must also be compensated. For the
x
position error (shown in
Fig. 2(a)
),
y
position error (shown in
Fig. 2(b)
), and clocking error (shown in
Fig. 2(c)
) of the
i
th subaperture, the value after compensation is (using linear approximation)
[8]
:
Mechanical location errors of the ith subaperture: (a) xposition error, (b) yposition error, (c) clocking error.
where
θ
is the angle of the
i
th subaperture.
Assuming that there are
N
subapertures, where the reference subaperture is number 1, the subapertures (except number 1) have to be compensated in terms of their tilt, piston,
x
position error,
y
position error, and clocking error to minimize the difference in the subaperture overlapping area between adjacent subapertures. The leastsquares method is used
[8]
, as follows.
Since linear approximation is used to solve the nonlinear problem described above, an iterative algorithm can be used
[8]
: alternately use overlapping subaperture areas to calculate compensation values, and use compensation values to calculate overlapping aperture areas.
III. MODIFIED SUBAPERTURE STITCHING ALGORITHM USING IMAGE SHARPENING AND PARTICLE SWARM OPTIMIZATION
Before discussing our algorithm, we provide brief descriptions of image sharpening and particle swarm optimization.
Image sharpening is a technique that reinforces the highfrequency components of images, and there are many different image sharpening algorithms
[10
,
11]
. We use an image sharpening algorithm based on the Sobel operator
[12]
, a famous gradient operator which is widely used in image processing.
Reference 13 states the following: “Particle swarm optimization is a computational intelligencebased technique that is not largely affected by the size and nonlinearity of the problem, and can converge to the optimal solution in many problems where most analytical methods fail to converge.”
In our algorithm we select the position value indicated by the instrument as the initial candidate solution, and the range of allowable errors is used as the search space for particle swarm optimization.
A flowchart of the process used to calculate the fitness of each particle is shown in
Fig. 3
. The leastsquares method is used to compensate for the tilt and piston errors
[5
,
6]
, and the fitness is the average residual error in the overlapping areas. After clarifying the variables and fitness function, we use the particle swarm optimization algorithm to calculate the positions of the subapertures. After the positions of the subapertures have been determined, the stitching result can be calculated using the algorithm described by Otsubo et al.
[6]
.
Flowchart showing the process used to calculate the fitness of each particle.
IV. SIMULATION
In the simulation, a measurement (301 × 301 pixels, as shown in
Fig. 4
) was divided into seven subapertures (
Fig. 5
), which were then stitched together using both the existing algorithm and our proposed algorithm.
Measurement results. (unit: wavelength)
Layout of the subapertures (blue) used to analyze a large aperture objective (red).
After dividing the measured image into subapertures, we added random noise ([−0.0002, 0.0002] wavelength) and the piston, tilt, and position errors to each subaperture. The position errors were random lengths between −Width/2 and Width/2, where Width was a parameter used in the simulation.
To evaluate the performance of the two algorithms with increasing values of position errors, the simulation results are compared in
Table 1
, where the initial error is the variance in the initial position errors. In the table the results for the existing and proposed algorithms show the variance in the position errors based on simulations using the respective algorithms. The existing algorithm could not compensate for position errors when Width was less than one pixel, whereas the proposed method performed quite well.
Simulation results. All units are the length of one pixel
Simulation results. All units are the length of one pixel
V. EXPERIMENT
Using the layout of subapertures shown in
Fig. 6
, a flat surface (caliber = 150 mm) was measured with a subaperture stitching interferometer (caliber = 100 mm). The measurement results obtained with a fullaperture using our algorithm are shown in
Fig. 7(a)
. For the measurement results obtained using an interferometer with a caliber of 150 mm (shown in
Fig. 7(b)
), the root mean squared error was 0.006372 λ and the residual error image is shown in
Fig. 8
.
Layout of the subapertures.
Experimental results: (a) stitched using our algorithm, (b) fullaperture measurement. (unit: wavelength)
Residual error. (unit: wavelength)
VI. CONCLUSION
In this study we proposed a modified subaperture stitching algorithm, which uses image sharpening algorithms and particle swarm optimization to improve the stitching accuracy for subaperture stitching interferometers with high positional accuracy. The results of simulations demonstrated that the proposed algorithm performs better than the existing method. A comparison with experimental fullaperture testing results demonstrated the validity of the proposed algorithm.
Li X. L.
,
Xu M.
,
Ren X. D.
,
Pei Y. T.
2012
An optical design of offaxis fourmirroranastigmatic telescope for remote sensing
J. Opt. Soc. Korea
http://dx.doi.org/10.3807/JOSK.2012.16.3.243
16
243 
246
Li X. L.
,
Xu M.
,
Pei Y. T.
2012
Optical design of an offaxis fivemirroranastigmatic telescope for near infared remote sensing
J. Opt. Soc. Korea
http://dx.doi.org/10.3807/JOSK.2012.16.4.343
16
343 
348
Jin H.
,
Lim J.
,
Kim Y.
,
Kim S.
2013
Optical design of a reflecting telescope for cubesat
J. Opt. Soc. Korea
http://dx.doi.org/10.3807/JOSK.2013.17.6.533
17
533 
537
Kim C. J.
1982
Polynomial fit of interferograms
Appl. Opt.
http://dx.doi.org/10.1364/AO.21.004521
21
4521 
4525
Otsubo M.
,
Okada K.
,
Tsujiuchi J.
1992
Measurement of large plane surface shape with interferometric aperture synthesis
Proc. SPIE
1720
444 
447
Otsubo M.
,
Okada K.
,
Tsujiuchi J.
1994
Measurement of large plane surface shapes by connecting smallaperture
Opt. Eng.
http://dx.doi.org/10.1117/12.152248
33
608 
613
Sjödahl M.
,
Oreb B. F.
2002
Stitching interferometric measurement data for inspection of large optical components
Opt. Eng.
http://dx.doi.org/10.1117/1.1430727
41
403 
408
Golini D.
,
Forbes G.
,
Murphy P.
2005
Method for selfcalibrated subaperture stitching for surface figure measurement, US Patent:6956657B
Zhang Y.
,
Zhou H.
2013
Image stitching based on particle swarm and maximum mutual information algorithm
Journal of Multimedia
8
580 
587
Gui Z.
,
Liu Y.
2011
An image sharpening algorithm based on fuzzy logic
OptikInternational Journal for Light and Electron Optics
http://dx.doi.org/10.1016/j.ijleo.2010.05.010
122
697 
702
Zeng J.
2006
An image sharpening algorithm based on edge detection
Modern Electronics Technique
12
033 
Qu Y. D.
,
Cui C. S.
,
Chen S. B.
,
Li J.Q.
2005
A fast subpixel edge detection method using SobelZernike moments operator
Image and Vision Computing
http://dx.doi.org/10.1016/j.imavis.2004.07.003
23
11 
17
del Valle Y.
,
Venayagamoorthy G. K.
,
Mohagheghi S.
,
Hernandez J.C.
,
Harley R. G.
2008
Particle swarm optimization: Basic concepts variants and application in power systems
IEEE Transactions on Evolutionary Computation
http://dx.doi.org/10.1109/TEVC.2008.930279
12
2 