Investigation on Compressed Wavefront Sensing in Freeform Surface
Eddy Chow Mun Tik
, Xin Wang
, Ningqun Guo
, Ching SeongTan
and Kuew Wai Chew
School of Engineering, Monash University Malaysia, Jalan Lagoon Selatan, Bandar Sunway, Subang Jaya, Malaysia
Faculty of Engineering, Multimedia University, Cyberjaya, Malaysia
Faculty of Engineering and Science, University Tunku Abdul Rahman, Kampar, Malasyia
Shack-Hartmann Wavefront Sensing, Compressed Sensing, Freeform Surface Measurement.
Abstract: In this paper, conventional modal wavefront reconstruction is compared with compressed wavefront sensing
to reconstruct freeform surface profiles using the Shack-Hartmann wavefront sensor. The modal wavefront
reconstruction represents the phase or the wavefront in the Zernike domain. The compressed wavefront
sensing method based on the sparse Zernike representation (SPARZER) represents the phase slopes in the
Zernike domain. The effectiveness of compressed wavefront sensing in freeform surface profile
measurements is investigated.
The Shack-Hartmann wavefront sensor has been
popular due to its simplicity in measuring the shape
of a wavefront. Since the start of its development in
the late 1960s for the use of improving images
captured from ground telescopes (Platt & Shack,
2001), its applications expanded to measurement of
aberrations of the eye and optical component
characterization among others (Schwiegerling &
Neal, 2005). There have been a great amount of
research to improve the accuracy of the Shack-
Hartmann wavefront sensor. This includes new
centroid detection algorithms (Yin, et al., 2009), de-
noising centroid images (Basden, et al., 2015), use of
new basis functions (Lundstrom & Unsbo, 2004), and
new wavefront reconstruction algorithms (Rostami,
et al., 2012).
Compressive sensing meanwhile is a great
optimization technique to recover sparse signals even
when the sampling rate is lower than required by the
Shannon-Nyquist sampling theorem (Donoho, 2006).
This is done by solving underdetermined linear
systems where the signal is sparse in a particular
domain. Another requirement is the incoherence
between the sampling and the representing domains
e.g. the time and frequency domains (Candes &
Romberg, 2005). The linear equations are solved
using l1 minimization, which does not have an
analytical solution. It is solved using iterative
numerical methods such as linear programming etc.
(Candes, et al., 2006)
Application of compressed sensing on the Shack-
Hartmann wavefront sensor is possible because there
exist a sparse representation of the projected
wavefront. A popular representation of the wavefront
is in the Zernike domain (Noll, 1976). Early work has
shown different implementations of compressive
sensing in Shack-Hartmann wavefront sensors. This
includes the representation of phase slopes in the
Zernike domain (Polans, et al., 2014), and defining
the sensing domain as the Dirac comb (Hosseini &
Michailovich, 2009).
The use of Shack-Hartmann wavefront sensing on
free-form surfaces presents some challenges due to
the nature of freeform surfaces themselves. Due to
large slopes or curvature of freeform surfaces, the
focal spot on the image sensor could be distorted
(Guo, et al., 2013). This causes the centroid detection
algorithm to be inaccurate, and thus producing
inaccurate phase slope measurements. Compressive
wavefront sensing has shown to reconstruct
wavefronts accurately even with noisy measurements
(Polans, et al., 2014).
Tik, E., Wang, X., Guo, N., SeongTan, C. and Chew, K.
Investigation on Compressed Wavefront Sensing in Freeform Surface Measurements.
DOI: 10.5220/0005743901520157
In Proceedings of the 4th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2016), pages 154-159
ISBN: 978-989-758-174-8
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
2.1 Modal Wavefront Reconstruction
The classic modal reconstruction of Shack-Hartmann
wavefront sensors uses the least squares optimization
to solve the linear equations which approximate the
wavefront to a summation of some decomposed
polynomials (Dai, 1994).
Where a
is the ith coefficient, and Z
is the ith
polynomial. The commonly used polynomials are the
Zernike polynomials and Fourier transforms.
The information obtained from the Shack-
Hartmann wavefront sensor meanwhile are the phase
slopes in the x and y direction of phase Φ, which can
be approximated to the distance of a focal spot to its
reference divided by the focal length of the lenslets.
For a Shack-Hartmann wavefront sensor with
lenslets of n×n grid, the number of phase slope
measurements will be 2n
. In matrix form, the above
equations can be represented as
Where F is the column matrix of phase slope
measurements, E is matrix of the partial differentials
of polynomials Z in the x and y direction, and A is the
matrix of coefficients of Z.
The elements in matrix E, are the average Zernike
derivatives over the corresponding sub-apertures of
the Shack-Hartmann wavefront sensor.
The least squares solution to the above equation
would be
2.2 Compressed Wavefront Sensing
The method used for compressed wavefront sensing
is the sparse Zernike representation (SPARZER).
SPARZER is a method proposed by James Polans
where the phase slopes itself are represented using
Zernike polynomials (Polans, et al., 2014). Due to the
condition where the phase map has to be continuously
differentiable in the Zernike space, implementation of
this technique is simpler. First, the phase slopes are
represented in the Zernike orthonormal basis.
and c
are the coefficients in the Zernike
domain while Z is the matrix transforming the slope
information into the Zernike domain across the entire
phase map. The amount of phase slope information is
then compressed by randomly selecting a set
percentage of the slope data. Then, using this limited
amount of information, SPARZER reconstructs the
sparse signal in the Zernike domain using the
In equation 7, c is the matrix of coefficients in the
Zernike domain, b is the phase slope measurements,
and is the sparse sampling operator.
While in his paper, Polans uses randomised
samples from a set of slope data from high lenslet
density (HLD) array, this investigation uses samples
similar to the shape of Shack-Hartmann lenslet arrays
with lower density to reconstruct the signal.
Five different lenslet sizes are used in this simulation.
The number of lenslets for each case are 317, 197,
149, 113 and 81 respectively. The simulated reference
focal spot image for the highest and lowest number of
lenslets are shown in Figure 1.
Figure 1: Reference focal spot images.
Investigation on Compressed Wavefront Sensing in Freeform Surface Measurements
Figure 2: from left: a) test freeform wavefront b) x-slope of wavefront and c) y-slope of wavefront.
The simulation is run using a test freeform
wavefront shown in Figure 2. The test wavefront has
various peaks and valleys and a region of large phase
slope. This would be able to test out and compare both
reconstruction techniques.
The conventional modal wavefront reconstruction
reconstructs the test wavefront relatively well for all
cases. For the case of 317 lenslets, it can be seen that
the reconstructed wavefront retains the overall profile
of the test wavefront from Figure 3. The maximum
error of the reconstructed wavefront is 0.13 microns.
Figure 4 shows the overall absolute error across the
entire phase map.
Figure 3: Reconstructed wavefront using conventional
modal reconstruction (317 lenslet).
Figure 4: Absolute error (in microns).
It can be seen that the highest amount of error is
in the peak region near both ends of the wavefront,
and the lowest valley on the right. The wavefront was
reconstructed using 36 Zernike polynomials. Due to
the limited amount of higher order polynomials, and
the edge being the highest point of the wavefront, the
peak could not be resolved accurately. This occurs for
all the different number of lenslet used. This shows
the limitation in representing phase in the Zernike
domain. Low order Zernike polynomials are very
smooth and are unstable at the outer regions (Dalal, et
al., 2001). Thus they are not very suitable for
freeform wavefronts.
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
Figure 5: x and y slopes from the reconstructed wavefront.
Figure 6: Absolute error of x and y slopes.
The x and y slopes calculated from the
reconstructed wavefront are shown in Figure 5 and
their error in Figure 6. Again, it can be seen that the
edges have a higher error which is caused by the
inaccuracies in the edges of the reconstructed
wavefront. Discounting the peak at the edges, the
mean error for the x and y slopes are 2.575×10
SPARZER meanwhile reconstructs the slopes of
the wavefront. The reconstructed slopes using 317
lenslets are shown in Figure 7 and 8, while the
absolute error of the reconstructed slopes are in
Figure 9. Similar to the slope errors from the modal
reconstruction, the largest errors are at the edges
where the slopes are larger in value. Discounting the
peak values at the edges, the mean error for x and y
slopes reconstructed are 2.6346×10
and 2.042×10
Table 1 shows the mean error of x and y slopes
obtained using compressed sensing and modal
reconstruction. From the table, it can be seen that
compressed sensing has a lower mean error when
compared to modal reconstruction for all cases.
Figure 7: x-slope reconstructed with SPARZER.
Figure 8: y-slope reconstructed with SPARZER.
Investigation on Compressed Wavefront Sensing in Freeform Surface Measurements
Figure 9: Absolute error of x and y slopes reconstructed with SPARZER.
Table 1: Mean error of wavefront slopes.
mean error for compressed
mean error for modal
x-slope y-slope x-slope y-slope
Besides that, the mean error for compressed
sensing when using the lowest number of lenslet is
lower than the mean error for all cases using modal
reconstruction. The error from compressed sensing
follows the trend where the higher the number of
samples, the lower the error. Meanwhile for modal
reconstruction, the error stays relatively constant.
However, while the mean errors are lower, the
slopes reconstructed using 81 lenslets do not match
well with the original slopes visually. Shown in
Figure 10 are the slopes reconstructed using 81
Figure 10: x-slope reconstructed with SPARZER.
Figure 11: y-slope reconstructed with SPARZER.
The profile at the edges deviate the most when
compared to the original slopes of the wavefront. This
is again due to the limitations of Zernike polynomials
in freeform surface measurements.
It can be seen that at lower lenslet resolutions, there
are large deviations in the edges of the slopes
reconstructed by compressed sensing. This is caused
by the inability of Zernike polynomials to accurately
represent freeform surface profiles. Similarly, for
modal wavefront reconstruction, the error is also
highest at the edges of the wavefront. However, it is
PHOTOPTICS 2016 - 4th International Conference on Photonics, Optics and Laser Technology
clear that compressed sensing yields a lower error
when compared to modal reconstruction for all lenslet
The authors gratefully acknowledge the support and
funding from Monash University Malaysia and
Ministry of Higher Education, Malaysia under the
Grant No: FRGS/1/2013/SG02/MUSM/02/1.
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Investigation on Compressed Wavefront Sensing in Freeform Surface Measurements