Phase estimation for Zernike circle polynomials using Phase Locked Loops for investigations of camera aberrations Sonam Singh a,1 , Dinesh Ganotra b,n a University School of Basic & Applied Sciences, GGS Indraprastha University, Dwarka, Delhi 110075, India b Indira Gandhi Institute of Technology, GGS Indraprastha University, Room No. 117B, Civil Block, Kashmere Gate, Delhi 110403, India article info Article history: Received 6 July 2012 Received in revised form 24 September 2012 Accepted 8 October 2012 Available online 9 November 2012 Keywords: Phase Locked Loop (PLL) Zernike circle polynomials Camera aberrations Wavefront fitting abstract This paper proposes a method to investigate aberrations of cameras from phase recovered fringe patterns using first-order Phase Locked Loop (PLL). Images of fringe patterns of different spatial periods are captured using different cameras. First-order PLL is applied to recover phase information from the captured images. Zernike circle polynomials are fitted to the phase recovered and Zernike coefficients are computed using the Gaussian elimination method. Comparison of cameras in terms of aberrations is done using these Zernike coefficients. Phase is also recovered and fitted for fringe patterns captured using the same camera with different lens combinations. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Spatial distortion in the regular fringe patterns captured by an optical system may be attributed to its spherical aberrations. Algo- rithm of Phase Locked Loops in electronics can be applied to recover the spatial phase changes which can be further studied using Zernike polynomials for aberrations. Servin and Rodriguez-Vera [1] were the first to introduce the technique of PLL in fringe profilometry. Methods using PLL have been preferred over the conventional techniques because the problem of phase unwrapping is not encountered while using PLL. Noisy fringe patterns were also demodulated by Servin et al. [2]. Gdeisat et al. [3] extended the demodulation of fringe patterns using digital PLLs from first order to second order. Ochoa et al. [4] compared the results of phase shifting from PLLs. Phase maps generated by the basic PLL algorithm are corrupted by high frequency components added by the phase detector. Kozlowski and Serra [5] modified the PLL algorithm to overcome this problem. Ganotra et al. [6] studied the phase recovery in terms of loop gain coefficients for both first- and second-order PLLs. Zernike polynomials for circular aperture are useful for quan- titatively characterizing aberrated wavefront of an optical system. Noll [7] introduced the orthonormal form of Zernike polynomials and used them to describe the aberrations of an optical wave propagating through atmospheric turbulence. Mahajan [8–10] studied the optical aberrations of systems with circular pupils using orthonormal form of Zernike circle polynomials and Zernike annular polynomials for imaging systems with annular pupils. In the present paper prints of software generated fringe patterns of spatial periods 50, 100 and 150 pixels are taken at 600 dpi. Printed images are captured using a CCTV camera with different lenses and a Prosilica firewire camera. PLL is applied for recovering phase from these captured fringe patterns. Recovered phase is taken to represent the wavefront. The aberrations in the wavefront are estimated by fitting Zernike circle polynomials and their coefficients are deter- mined using the least squares Gaussian elimination method [11]. The remainder of the paper is organized as follows. Section 2 illustrates the use of first-order PLL for recovering phase. Section 3 describes wavefront fitting using Zernike Polynomials and determination of Zernike coefficients. Section 4 presents experi- ments and simulations. Section 5 gives experimental results on real images and Section 6 gives conclusion. 2. Phase estimation using first-order PLL Phase-Locked Loop is a subject in communication engineering and can be applied to recover minor phase change in regularly spaced fringe patterns [4]. A block diagram of a PLL as shown in Fig. 1 contains three basic components: i. A phase detector or a multiplier. ii. A digital filter. iii. A digital-controlled oscillator (DCO). Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/optlaseng Optics and Lasers in Engineering 0143-8166/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlaseng.2012.10.003 n Corresponding author. Tel.: þ91 112 3900 233. E-mail addresses: [email protected] (S. Singh), [email protected] (D. Ganotra). 1 Tel.: þ91 112 3900 233. Optics and Lasers in Engineering 51 (2013) 261–269
Transcript
Optics and Lasers in Engineering 51 (2013) 261–269
Contents lists available at SciVerse ScienceDirect
Phase estimation for Zernike circle polynomials using Phase Locked Loopsfor investigations of camera aberrations
Sonam Singh a,1, Dinesh Ganotra b,n
a University School of Basic & Applied Sciences, GGS Indraprastha University, Dwarka, Delhi 110075, Indiab Indira Gandhi Institute of Technology, GGS Indraprastha University, Room No. 117B, Civil Block, Kashmere Gate, Delhi 110403, India
a r t i c l e i n f o
Article history:
Received 6 July 2012
Received in revised form
24 September 2012
Accepted 8 October 2012Available online 9 November 2012
This paper proposes a method to investigate aberrations of cameras from phase recovered fringe
patterns using first-order Phase Locked Loop (PLL). Images of fringe patterns of different spatial periods
are captured using different cameras. First-order PLL is applied to recover phase information from the
captured images. Zernike circle polynomials are fitted to the phase recovered and Zernike coefficients
are computed using the Gaussian elimination method. Comparison of cameras in terms of aberrations is
done using these Zernike coefficients. Phase is also recovered and fitted for fringe patterns captured
using the same camera with different lens combinations.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Spatial distortion in the regular fringe patterns captured by anoptical system may be attributed to its spherical aberrations. Algo-rithm of Phase Locked Loops in electronics can be applied to recoverthe spatial phase changes which can be further studied using Zernikepolynomials for aberrations. Servin and Rodriguez-Vera [1] were thefirst to introduce the technique of PLL in fringe profilometry. Methodsusing PLL have been preferred over the conventional techniquesbecause the problem of phase unwrapping is not encountered whileusing PLL. Noisy fringe patterns were also demodulated by Servinet al. [2]. Gdeisat et al. [3] extended the demodulation of fringepatterns using digital PLLs from first order to second order. Ochoaet al. [4] compared the results of phase shifting from PLLs. Phasemaps generated by the basic PLL algorithm are corrupted by highfrequency components added by the phase detector. Kozlowski andSerra [5] modified the PLL algorithm to overcome this problem.Ganotra et al. [6] studied the phase recovery in terms of loop gaincoefficients for both first- and second-order PLLs.
Zernike polynomials for circular aperture are useful for quan-titatively characterizing aberrated wavefront of an optical system.Noll [7] introduced the orthonormal form of Zernike polynomialsand used them to describe the aberrations of an optical wavepropagating through atmospheric turbulence. Mahajan [8–10]
ll rights reserved.
. Singh),
studied the optical aberrations of systems with circular pupilsusing orthonormal form of Zernike circle polynomials and Zernikeannular polynomials for imaging systems with annular pupils.
In the present paper prints of software generated fringe patternsof spatial periods 50, 100 and 150 pixels are taken at 600 dpi. Printedimages are captured using a CCTV camera with different lenses and aProsilica firewire camera. PLL is applied for recovering phase fromthese captured fringe patterns. Recovered phase is taken to representthe wavefront. The aberrations in the wavefront are estimated byfitting Zernike circle polynomials and their coefficients are deter-mined using the least squares Gaussian elimination method [11].
The remainder of the paper is organized as follows. Section 2illustrates the use of first-order PLL for recovering phase. Section3 describes wavefront fitting using Zernike Polynomials anddetermination of Zernike coefficients. Section 4 presents experi-ments and simulations. Section 5 gives experimental results onreal images and Section 6 gives conclusion.
2. Phase estimation using first-order PLL
Phase-Locked Loop is a subject in communication engineeringand can be applied to recover minor phase change in regularlyspaced fringe patterns [4]. A block diagram of a PLL as shown inFig. 1 contains three basic components:
i.
A phase detector or a multiplier. ii. A digital filter.
Fig. 1. Block diagram of digital Phase Locked Loop.
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The basic principle of the PLL is to follow the phase changes of theinput signal by varying the value to the DCO signal so that the phaseerror between the input signal and the DCO signal is minimized. Thephase detector is used to compare the phase of a discrete input signalwith the phase of the digital-controlled oscillator; difference betweenthese two signals is measured at the output of the phase detector.The digital filter filters this output and the filtered output is appliedto the DCO to change the frequency of the DCO in a direction thatreduces the phase difference between the input signal and the DCO.The combination of multiplier and digital filter (low pass filter) iscalled the phase comparator of the PLL.
The basic iterative equation in discrete form for the first-orderPLL [5] is given
where fðxÞ is the phase of the fringe pattern, x is the sampleindex, t is the closed loop gain, IðxÞ is the intensity and f is thespatial frequency. As it takes certain iterations before the phasesettles to the correct values the above iteration is carried in bothforward and backward directions through the same line acrossthe image of captured fringes to recover the incorrect phase. Inbackward scanning the last term calculated by Eq. (1) is con-sidered as the initial condition and is given
The value of t should be less than one for stability considerationand good noise rejection. Low values of t act as a low pass filterremoving the need of an additional filter.
In our experiment f is the spatial frequency fixed for softwaregenerated images and varies when these images are capturedusing different cameras. It also changes as the distance betweenthe camera and the fringe pattern being captured is varied. IðxÞ isthe normalized intensity. fðxÞ is the phase recovered in Cartesiancoordinates. We have normalized the recovered phase and con-verted it to polar coordinates fðr,yÞ for fitting Zernike polyno-mials as described in Section 3.
3. Wavefront fitting using Zernike circle polynomials anddetermination of Zernike coefficients
A wavefront fðr,yÞ can be expanded into the linear combina-tion of Zernike polynomials Zjðr,yÞ as
f r,yð Þ ¼X1j ¼ 1
ajZj r,yð Þ, ð3Þ
where aj is the Zernike coefficient and Zjðr,yÞ denotes the jthZernike polynomial which is a set of orthogonal polynomialsdefined on a unit circle given by eq. (4).
Zeven j r,yð Þ ¼ ½2 nþ1ð Þ�1=2 Rmn ðrÞcosmy
Zodd j r,yð Þ ¼ ½2 nþ1ð Þ�1=2 Rmn ðrÞsinmy
)ma0
Z j r,yð Þ ¼ ½ nþ1ð Þ�1=2 Rmn ðrÞ m¼ 0
9>>>=>>>;
ð4Þ
where the radial polynomial Rmn ðrÞ is given as
Rmn rð Þ ¼
Xn�mð Þ=2
s ¼ 0
�1ð Þs n�sð Þ!
s!½ nþmð Þ=2�s�!½ n�mð Þ=2�s�!rn�2s, ð5Þ
where 0orr1 and 0oyr2p. The indices n and m are the radialdegree and the azimuthal frequency respectively and shouldsatisfy mrn and n�m¼even, j is a function of n and m calledmode ordering number.
The Gaussian elimination method [11] utilizes matrix leftdivision and computes ‘a’ as
a¼ Z \f r,yð Þ, ð6Þ
which yields results similar to the matrix inversion method.Table 1 gives the list of different modes of Zernike coefficients
and the corresponding aberration names.
4. Experiment and simulations
Experiments and simulations were done to recover phasechanges of fringe patterns of different spatial periods captured usingdifferent cameras and lenses using PLL. Recovered phase was fittedusing Zernike circle polynomials and Zernike coefficients werefurther computed to estimate the aberrations associated with theoptical system. ‘‘Proposed algorithm’’ is summarized in the form of apseudo code below. It was also verified experimentally.
1.
Generate fringe patterns of spatial periods 50, 100 and 150pixels.
2.
Read the captured image. Determine the spatial period (p) ofthe fringe pattern. The number of pixels between two succes-sive maxima corresponds to the spatial period such thatf ¼ 1=p: Choose the suitable value of closed loop gain,t.
3.
for y¼1 to y¼ image_lengthfor x¼1 to x¼ image_width
f xþ1ð Þ ¼fðxÞ�t½I xþ1ð Þ�IðxÞ�
�sin½2pf xþfðxÞ� Forward scanning
for x¼ image_width to x¼1
fðxÞ ¼f xþ1ð Þ�t½I xþ1ð Þ�IðxÞ�
�sin½2pf xþf xþ1ð Þ� Backward scanning
4.
Normalize the phase obtained and change the coordinatesfrom Cartesian to polar to fit it to Zernike circle polynomials.
f r,yð Þ ¼X1j ¼ 1
ajZj r,yð Þ
5.
Compute first 70 modes of Zernike coefficients using thematrix left division method
a¼ Z \f r,yð Þ
(Table 1 gives the values of aberrations corresponding todifferent modes of Zernike coefficients.)
6.
Generate the fringe pattern of different spatial periods cor-rupted with additive white Gaussian noise and repeat thesteps from 2 to 5.
4.1. Application of proposed algorithm on known aberrated system
In order to validate the proposed algorithm, phase was recoveredand fitted for a known aberrated system. Straight and 901r rotatedfringe patterns of spatial period 100 pixels were first generatedusing software. Fig. 2(a) and (b) shows such images. Softwaregenerated straight fringe pattern of spatial period 100 pixels was
Table 1Values of different modes of Zernike coefficients and corresponding aberrations.
n m No. Polynomial Name
0 0 0 1 Piston
1 1 1 rcosy0 x tilt
2 rsiny0 y tilt
0 3 2r2�1 Defocus
2 2 4 r2cos2y09>=>; Astigmatism plus defocus5 r2sin2y0
Fig. 2. Software generated (a) straight and (b) 901 rotated fringe patterns of spatial period 100 pixels.
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further tilted in y-direction using projective transformation tointroduce tilt aberration (Fig. 3(a)). Phase was recovered for thisimage using PLL and was fitted using Zernike circle polynomials.Fig. 3(b) shows the corresponding bar plot of computed Zernikecoefficients. The plot shows a significant magnitude of third mode ofZernike coefficients which corresponds to y-tilt aberration (Table 1).The results obtained is in correspondence with the theory thusvalidating the proposed algorithm of recovering phase using PLL andfitting the recovered phase to Zernike circle polynomials for esti-mating the aberrations of an optical system.
4.2. Phase estimation using different cameras
Experiments and simulations were also performed for recover-ing phase changes of fringe patterns with different spatial periodsusing PLL. Images of fringe patterns of different spatial periods(50, 100 and 150 pixels) are generated using software. Prints ofthese software generated images are taken at 600 dpi. Printedimages are captured using different cameras and with differentlenses. In our camera–lens system the in-built circular aperture isused. The iris of the lens is not adjusted. The electronics of thecamera lens is such that it adjusts the overall intensity of theimage. The image center and lens’ optical center were matched
Fig. 3. (a) Software generated fringe pattern of spatial period 100 pixels tilted
in y-direction and (b) bar plot of magnitude of Zernike coefficients computed for
y-tilted image shown in (a).
before capturing the image in order to reduce assembly errors.Image captured is rectangular in shape and to apply Zernike circlepolynomials, circle of maximum diameter is extracted from therectangular fringe pattern which is kept constant throughout theexperiment. Fig. 2(a) and (b) shows software generated straightand 901 rotated fringe patterns of spatial period 100 pixelsrespectively. Fig. 4(a)–(c) shows the images of straight fringepatterns of spatial periods (50, 100, and 150 pixels) respectivelycaptured by a CCTV camera (sensor size 1/3 in. and 12 mm lens).Note that the spatial periods of the fringes in the captured imageswill not be now same as 50, 100 and 150 pixels as they will nowdepend upon the x and y resolutions of the camera pixels.Similarly, Fig. 4(d)–(f) shows the 901 rotated set of imagescaptured by the CCTV camera. To detect the phase changesforward and backward scanning techniques are used. Phasechange was recovered for observed images using PLL(Eqs. (1) and (2)) described in Section 2. These equations of thePLL also require the spatial period to be known. Standard imagesare generated using software with the spatial period alreadyknown. However due to varying magnifications the spatial peri-ods of the captured fringes are different. The spatial period for theimages captured using camera is determined by counting thenumber of pixels between the successive maxima of the intensitypattern observed (Fig. 5).
Simulations were also repeated for different values of loopgain coefficients (t¼0.2, 0.3 and 0.4). The corresponding resultsare discussed in Section 5.
The recovered phase was fitted using Zernike polynomials (Eq.(5)) and first 64 Zernike coefficients were computed using theGaussian elimination method (Eq. (6)). Aberrations of the cameraswere studied by comparing the magnitudes of Zernike coeffi-cients. Cameras with higher magnitudes of Zernike coefficientsare considered to have more aberrations and those with lowervalues of Zernike coefficients are considered to have lesseraberrations.
4.3. Phase estimation using the same camera and different lens
combinations
Phase changes were also estimated by capturing fringe pat-terns using the same camera and different lens combinations.Images of fringe patterns having spatial period 100 were capturedusing the CCTV camera with lenses of different focal lengths(8 mm, 16 mm and 25 mm). Fig. 6(a)–(c) shows these images andFig. 7 gives the comparison of magnitudes of Zernike coefficientscalculated for these images. The magnitudes of Zernike coeffi-cients are maximum for the camera–lens system having mini-mum focal length 8 mm (dot-dashed line) and minimum for theone having maximum focal length 25 mm (dashed line). Thesmaller the focal length the more the aberrations is a logicalconclusion which is evident from our observations as well.
4.4. Effect of additive white Gaussian noise
In order to check the efficiency of the proposed algorithm wehave corrupted our software generated fringe pattern of spatialperiod 100 pixels with additive white Gaussian noise of mean, m¼0and different values of standard deviation s¼ 0:1, 0:2 and 0:3ð Þ.Fig. 8(a)–(c) shows these noise corrupted images captured using theCCTV camera with lens of focal length 12 mm. The same sets ofiterative equations of PLL were applied to demodulate these noisyfringe patterns. Phases recovered were fitted using Zernike poly-nomials and corresponding Zernike coefficients were estimated.Fig. 9 shows the plot of magnitude of Zernike coefficients for theseimages. It is evident from the plot that the phase is successfullyrecovered for the noisy fringe patterns and the magnitudes of only
Fig. 4. Images of (a)–(c) straight fringe patterns and (d)–(f) 901 rotated fringe patterns of spatial periods 50, 100 and 150 pixels captured using CCTV camera.
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lower modes of Zernike coefficients increase with the increase in thevalue of standard deviation of additive white Gaussian noise. Themagnitude of lower modes of Zernike coefficients are minimum forstandard deviation s¼0.1 (dot-dashed line) and maximum forstandard deviation s¼0.3 (dashed line). Values of the higher modesare random for different degrees of additive white Gaussian noise.
5. Experimental results and discussion
Proposed algorithm of recovering phase using PLL and Zernikepolynomial fitting on the phase recovered was implemented on aknown aberrated system (tilted in y-direction). Fig. 3(b) shows the
corresponding bar plot of Zernike coefficients which is in agreementwith the theory thus validating the proposed algorithm.
Phase changes of fringe patterns of spatial periods 50, 100 and150 pixels were also recovered using PLL. These reference imageswere first generated using software and their prints were capturedusing different cameras and lenses. The captured fringe patternimage pixels were introduced to the PLL line by line and each linewas scanned twice (both forward and backward). The recoveredphase map for fringe pattern of spatial period 50 pixels is shown inFig. 10. Phase recovered was fitted using Zernike circle polynomialsand Zernike coefficients were further computed.
Fig. 11(a) and (b) shows the plot of Zernike coefficients forstraight fringe patterns of spatial periods (50, 100 and 150 pixels)demodulated using PLL and captured using the CCTV camera and
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firewire camera respectively. Fig. 11(c) and (d) shows the similarplots of Zernike coefficients for 901 rotated fringe patterns.Different line pointers correspond to different fringe patterns.
Fig. 12 compares the magnitudes of Zernike coefficients forstraight and 901 rotated fringe pattern of spatial period 100 pixelscaptured using the CCTV camera. As the plots in Figs. 11(a)–(d)and 12 are almost overlapping, it means that the magnitudes ofZernike coefficients for each individual camera is same. It isexpected that the spherical aberrations of the camera shouldnot change with the changes in images being captured or withchanges in their orientation. The change in orientation is notchanging the magnitudes of the coefficients which confirms that
Fig. 6. Fringe patterns of spatial period 100 pixels captured using CCT
Fig. 5. Plot of one of the rows of the image captured by CCTV camera for
determining spatial period.
the phase is successfully recovered using PLL and Zernike coeffi-cients are a property of the camera which remains consistent asexpected.
Experiments were also done using the CCTV camera withlenses of different focal lengths (8 mm, 16 mm and 25 mm).Fig. 7 shows the comparison of Zernike coefficients obtained forthese images. It is evident from the graph that smaller the focallength of the lens used the more the aberrations and greater thefocal length the lesser the aberrations.
V and lenses of focal lengths (a) 8 mm, (b) 16 mm and (c) 25 mm.
Fig. 7. Comparison of Zernike coefficients computed by using phase recovery from
PLL for images shown in Fig. 5(a)–(c). Dot-dashed line denotes the camera–lens
system of focal length 8 mm, dotted line denotes the camera–lens system of focal
length 16 mm and dashed line denotes the fringe camera–lens system of focal
length 25 mm.
Fig. 8. Fringe patterns of spatial period 100 pixels corrupted with additive white Gaussian noise of mean m¼0 and standard deviation (a) s¼0.1, (b) s¼0.2 and (c)
s¼0.13.
Fig. 9. Comparison of magnitudes of Zernike coefficients for images corrupted
with different values of additive white Gaussian noise.
Fig. 10. Recovered phase map for fringe pattern of spatial period 50 pixels
captured using CCTV camera and demodulated using PLL.
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Fig. 13 shows the comparison of Zernike coefficients for fringepattern of spatial period 100 by varying the value of closed loopgain (t¼0.2, 0.3 and 0.4).
Experiments were also done with the addition of whiteGaussian noise. The magnitude of lower modes of Zernikecoefficients increases with the increase in the standard deviationof the Gaussian noise. Values of the higher modes are random fordifferent degrees of additive white Gaussian noise (Fig. 9).
6. Conclusion
Fringe pattern demodulation with first-order Phase LockedLoop is extended for computing Zernike coefficients and deter-mining the aberrations of cameras used for capturing the fringepatterns. Experimental results on fringe patterns of differentspatial periods demonstrate that the proposed method of fittingphase recovery using PLL has successfully been able to estimatethe Zernike coefficients. The magnitude of the Zernike coefficientsremains same by changing the orientation of the images captured.
Fig. 13. Comparison of Zernike coefficients computed for fringe patterns (cap-
tured using CCTV camera) of spatial period 100 pixels by varying closed loop gain
QUOTE in PLL. Dot-dashed line denotes t¼0.2, dotted line denotes t¼0.3, and
dashed line denotes t¼0.4.
Fig. 12. Comparison of Zernike coefficients for straight and 901 rotated fringe
patterns of spatial period 100 pixels captured and demodulated using CCTV
camera and PLL. Dot-dashed line denotes straight fringe pattern and dotted line
denotes 901 rotated fringe pattern.
Fig. 11. Plots of Zernike coefficients for straight fringe patterns of spatial periods 50, 100 and 150 pixels demodulated using PLL and captured using (a) CCTV camera and
(b) firewire camera; 901 rotated fringe patterns of spatial periods 50, 100 and 150 pixels demodulated using PLL and captured using (c) CCTV camera and (d) firewire
camera. Dot-dashed line denotes fringe pattern of spatial period 50 pixels. Dotted line denotes fringe pattern of spatial period 100 pixels. Dashed line denotes fringe
pattern of spatial period 150 pixels.
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Keeping the imaging sensor (camera) same and changing thelenses we have found that smaller focal length lenses give moreaberrations and vice-versa. Variation of loop gain coefficient isalso studied experimentally. Aberrations remain same for differ-ent values of loop gain coefficient in first-order PLL (t¼0.2,
0.3 and 0.4). Effect of additive white Gaussian noise on phaserecovery and computation of Zernike coefficients were alsostudied. The proposed algorithm has also worked successfullyon higher order Zernike coefficients for noisy images as well.
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Acknowledgment
The author acknowledges University Grants Commission andDepartment of Science and Technology, the Government of Indiafor their financial support. The author also acknowledges GuruGobind Singh Indraprastha University.
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