Analysis of Seidel Aberration by Use of the Discrete Wavelet Transform

Analysis of Seidel aberration by use of thediscrete wavelet transform

Rong-Seng Chang, Jin-Yi Sheu, and Ching-Huang Lin

Seidel aberration coefficients can be expressed by Zernike coefficients. The least-squares matrix-inversion method of determining Zernike coefficients from a sampled wave front with measurement noisehas been found to be numerically unstable. We present a method of estimating the Seidel aberrationcoefficients by using a two-dimensional discrete wavelet transform. This method is applied to analyzethe wave front of an optical system, and we obtain not only more-accurate Seidel aberration coefficients,but we also speed the computation. Three simulated wave fronts are fitted, and simulation results areshown for spherical aberration, coma, astigmatism, and defocus. © 2002 Optical Society of America

OCIS codes: 100.2650, 100.5070, 100.7410.

1. Introduction

A polynomial representation of the optical wave frontis essential in the analysis of interferometric testdata and optical system performance. Zernike poly-nomials1,2 are ideal for fitting the measured datapoints in an interferogram wave front to a two-dimensional polynomial; some Zernike polynomialsare related to the classical aberrations. Numericalresults of estimating the Zernike coefficients by theGram–Schmidt method indicate that they are prac-tically identical to those obtained by the least-squares matrix-inversion �LSMI� method.3,4 Eitherof these two approaches would become numericallyunstable from a sampled wave front with measure-ment noise.3

Recently, the discrete wavelet transform �DWT�has been introduced in image processing.5–7 TheDWT8,9 provides an orthogonal multiresolution archi-tecture of input images and therefore is a powerfultool for image processing.

Our purpose in this paper is to estimate the Seidelaberration coefficients from a sampled wave front byusing the DWT method over an x–y data uniformsampling. To improve numerical accuracy perfor-mance, it is desirable to minimize the approximationerror. In other words, this problem is based on max-

The authors are with the Institute of Optical Sciences, NationalCentral University, Chung-Li, 320, Taiwan. R. S. Chang’s e-mailaddress is [email protected].

Received 14 November 2001.0003-6935�02�132408-06$15.00�0© 2002 Optical Society of America

2408 APPLIED OPTICS � Vol. 41, No. 13 � 1 May 2002

imizing the projection of the wave front on low-frequency space.10 In this way, we have a uniquesolution of the unknown wave-aberration �Seidel ab-erration� coefficients by solving simultaneous linearequations.

The Seidel aberration coefficients obtained fromthe Zernike coefficients are given in Section 2. InSection 3 we use the DWT technique to estimate theSeidel aberration coefficients. A comparison be-tween the LSMI method of determining the Zernikecoefficients and the DWT method of determining theSeidel aberration coefficients is given in Section 4, inwhich computer-simulated testing data are used.The final results and conclusions are discussed inSection 5.

2. Seidel Aberration Coefficients Computed with theZernike Polynomials

In many cases a measured wave front can be de-scribed by a few coefficients that multiply the termsof a well-chosen polynomial, such as Seidel polyno-mials; it can be expressed as

W� x, y� � �s�0

5

ws Us� x, y� � w0 x � w1 y � w2� x2 � y2�

� w3� x2 � 3y2� � w4 y� x2 � y2�

� w5� x2 � y2�2, (1)

where �x, y� are the Cartesian coordinates of anypoint in the exit pupil plane and the coefficients wscorrespond to Seidel aberrations and are given inwavelength units. The aberration function W�x, y�is referred to as the Gaussian image point.

For wave-front sensing problems, discrete samplesare usually taken over the pupil area, such as posi-tions of fringes from an interferogram. The totalnumber of measurements is limited. Here it is as-sumed that the number of measurements may beincreased to as many as desired for data analysis, andthe effect of measurement noise on the determinationof Zernike coefficients is considered. In this section,we consider the data fitting to the measured points ina wave front by using a monomial representation ofZernike polynomials, as shown in Table 1.11

Let the provided optical wave front W�xi, yi� beexpressed in terms of Zernike polynomials as

W� xi, yi� � �j�1

N

ajZj� xi, yi�, i � 1, 2, . . . , M,

(2)

where N � 15 is the number of expansion terms �i.e.,number of degrees of freedom�, M is the number ofmeasurements, and W�xi, yi� contains random mea-surement noise. Equation �2� may be written con-veniently in a matrix form:

W � Za. (3)

In general, Eq. �3� represents a determined systemof equations, wherein there are more equations �M�than unknown �N�. These expansion coefficientsmay be computed by the LSMI method3; that is, wetry to minimize

� � �i�1

M

�ajZj� xi, yi� � Wi �2. (4)

The resultant equation may be written in the familiarform,

ZTZa � ZTW, (5)

and the desired coefficients may be obtained by directinversion,

a � �ZTZ��1ZTW. (6)

The Seidel aberration coefficients are determinedby following relations:

tilt in x � w0 � a2 � 2a8, (7)

tilt in y � w1 � a3 � 2a9, (8)

defocus � w2 � 2a5 � 6a13 � �a42 � a6

2�1�2, (9)

astigmatism � w3 � �4a42 � 4a9

2�1�2, (10)

coma � w4 � �9a82 � 9a9

2�1�2, (11)

spherical � w5 � 6a13, (12)

where a2, a3, a4, a5, a6, a8, a9, and a13 are the Zernikecoefficients for tilt in the x axis, tilt in the y axis,astigmatism with the axis at �45°, focus shift, astig-matism with the axis at 0° or 90°, third-order comaalong the x axis, third-order coma along the y axis,and third-order spherical aberration, respectively.

3. Seidel Aberration Coefficients Computed by theDiscrete Wavelet Transform

The DWT provides a powerful tool for image process-ing. An image can be decomposed into a sequence ofdifferent spatial-resolution images through the DWT.

Let �Vj�j�Z be a separable multiresolution approx-imation of L2�R2�. Let ��x, y� � ��x��y� be theassociated two-dimensional scaling function, and let�x� be the one-dimensional wavelet associated withthe scaling function ��x�. Then the three waveletsare 1�x, y� � ��x��y�, 2�x, y� � �x��y�, and 3�x,y� � �x��y�.

Let �j,k,l�x, y� � 2j��2jx � k��(2j y� l)(k,l)�Z2 be anorthonormal basis of Vj, let j,k,l

1(x, y) � 2j�(2j x �k)�2jy � l �, j,k,l

2�x, y� � 2j�2jx � k��2jy � l �, andlet j,k,l

3�x, y� � 2j�2jx � k��2jy � l � be an orthonor-mal basis of Wj.

The relation

Vj � Vj�1 � Wj�1 (13)

Table 1. Zernike Polynomials Un,m up to Fourth Degree

n m n � 2m jZernike

Polynomial Monomial Representation Meaning

0 0 0 1 1 1 Constant1 0 1 2 sin � x Tilt in x direction

1 �1 3 cos � y Tilt in y direction2 0 2 4 2 sin 2� 2xy Astigmatism with axis at �45°

1 0 5 22 � 1 2x2 � 2y2 � 1 Focus shift2 �2 6 2 cos 2� y2 � x2 Astigmatism with axis at 0° or 90°

3 0 3 7 3 sin 3� 3xy2 � x3

1 1 8 �33 � 2�sin � �2x � 3xy2 � 3x3 Third-order coma along x axis2 �1 9 �33 � 2�cos � �2y � 3x2y � 3y3 Third-order coma along y axis3 �3 10 3 cos 3� y3 � 3x2y

4 0 4 11 4 sin 4� 4y3x � 4x3y1 2 12 �44 � 32�sin 2� �6xy � 8xy3 � 8x3y2 0 13 64 � 62 � 1 1 � 6y2 � 6x2 � 6�x2 � y2�2 Third-order spherical aberration3 2 14 �44 � 32�cos 2� �3y2 � 3x2 � 4y4 � 4x4

4 �4 15 4 cos 4� 4y4 � 6x2y2 � 4x4

1 May 2002 � Vol. 41, No. 13 � APPLIED OPTICS 2409

implies the Mallat transform

Vj � Wj�1 � Wj�2 � · · · � W1 � V0. (14)

Let Aj be the wavelet multiresolution analysis ofthe original wave front W�x, y� in the spatial resolu-tion 2j� j is the index of spatial resolution, 0 � j � J�.Aj can be decomposed into approximate and detailedsignals in multiresolution analysis space.12

According to the Mallat algorithm,12,13 the decom-position of Aj can be performed by

Aj�1�m, n� � �k,l�Z

h�k�h�l � Aj�2m � k, 2n � l �,

(15)

Dj�11�m, n� � �

k,l�Zh�k�g�l � Aj�2m � k, 2n � l �,

(16)

Dj�12�m, n� � �

k,l�Zg�k�h�l � Aj�2m � k, 2n � l �,

(17)

Dj�13�m, n� � �

k,l�Zg�k�g�l � Aj�2m � k, 2n � l �.

(18)

Here, Aj�1, Dj�11, Dj�1

2, and Dj�13 are the wavelet

coefficients in spatial resolution 2j�1 �subband cod-ing14,15�, m and n are the pixel coordinates, Z is theinteger set, and h�l � and g�l � are a pair of quadraturemirror filters.16

The reconstruction can be made from Aj�1, Dj�11,

Dj�12, and Dj�1

3, which can be written as

Aj�m, n� � 4 �k,l�Z

h�k�h�l �Aj�1�m � k�2, n � l�2�

� 4 �k,l�Z

h�k�g�l �Dj�11�m � k�2, n � l�2�

� 4 �k,l�Z

g�k�h�l �Dj�12�m � k�2, n � l�2�

� 4 �k,l�Z

g�k�g�l �Dj�13�m � k�2, n � l�2�,

(19)

where h�l � and g�l � are also a pair of quadraturemirror filters, and

h�l � � h��l �, g�l � � g��l �, g�l �

� ��1�1�lh�1 � l �. (20)

Because any physically measurable signal is pro-cessed in a specific resolution, the original signal canbe approached by Aj of the maximum spatial resolu-tion 2J, namely,

Aj�m, n� � W�m, n�. (21)

Thus when Eqs. �15�–�20� and approximation �21� areused, the original wave front W�m, n� can be decom-posed and reconstructed.

Let the provided optical wave front be expressed by

Seidel polynomials; the approximation signal of originalwave front at resolution 2j can be expressed by Aj as

Aj � �k,l�Z

W� x, y�, �j,k,l� x, y��

� �s�0

5

ws �k,l�Z

Us� x, y�, �j,k,l� x, y��

� �s�0

5

ws Ajs, (22)

where Ajs � ¥k,l�Z Us�x, y�, �j,k,l�x, y��.

Equation �22� may be written conveniently in amatrix form:

Aj � Ajsw. (23)

From Eq. �23�, the Seidel aberration coefficients canbe obtained by direct inversion:

w � �� Ajs�TAj

s��1� Ajs�TAj. (24)

The measured data of the wave front W�x, y� haveoptical random noise that makes it difficult to eval-uate the wave-aberration coefficients from them di-rectly. In our approach an algebraic expressionW�x, y� is fitted to the set of wave-front measure-ments by least-squares error criteria. The new formof wave front can be written as

W� x, y� � �s�0

5

ws Us� x, y�. (25)

If ei is the difference between wave-front measure-ments, its fitted value

ei � W� xi, yi� � W� xi, yi�, (26)

and the sum of squares of the errors is denoted by

E � �i

�ei�2. (27)

In this paper, a Haar wavelet was used in theDWT. To determine accurate Seidel aberration co-efficients, the signal must be projected on a low-frequency space �suitable spatial resolution� forminimizing E.

4. Computer Simulation

The performance of the new wave-front estimatetechnique was tested by computer simulation. Thesimulation was also repeated for the LSMI methodfor the purpose of comparison.

To test the method described in Section 3, we pro-vided the three test wave fronts:

W1� x, y� � �3.2� x2 � y2� � 0.9� x2 � 3y2� � 1.1y� x2

� y2� � 3.1� x2 � y2�2, (28)

W2� x, y� � �4.05� x2 � y2� � 1.35� x2 � 3y2�

� 3.25� x2 � y2�2, (29)

W3� x, y� � 0.15y � 1.55� x2 � y2� � 3.55y� x2 � y2�

� 2.05� x2 � y2�2. (30)


The coefficients correspond to Seidel aberrations andare given in wavelength units. In the simulations,the Gaussian white noise is added to the calculatedmeasurement as a simulation of the measurementnoise; the added Gaussian white noise is generatedrandomly. For the test, the data are uniformly sam-pled over a 65 � 65 grid �4225 points in the unitsquare�, and the noise of the mean and the standarddeviations are preset as 0 and 0.19, respectively.The sampling data are fitted to the Seidel polynomi-als.

For the examples shown here, the wave aberrationrepresented by the Zernike coefficients is estimatedby the LSMI method and the wave aberration repre-sented by the Seidel aberration coefficients is esti-mated by the DWT method. Using Eqs. �6�–�12�, wecan compute w0 �tilt in the x axis�, w1 �tilt in the yaxis�, w2 �defocus�, w3 �astigmatism�, w4 �coma�, andw5 �spherical� by Zernike coefficients. From Eqs.�24�–�27�, the Seidel aberration coefficients can beestimated by the DWT method �with a suitable spa-tial resolution chosen for minimized E�.

Results of our analysis of the three test wave frontsare listed in Tables 2–4, including the Seidel aberra-tion coefficients of the preset wave front and the es-timated results of the two methods. As indicated inTables 2–4, before the noise was added, the wavefronts estimated with the two methods are shownwith high accuracy. The consistent results identifythe correctness of the new technique. After thenoise was introduced, the reduced coefficients had aslight difference from the preset ones for the newtechnique; the errors of the reduced coefficients andthe rms value of the wave front estimated from the

Fig. 1. Contour of the test wave front W1 estimated �a� withoutnoise, �b� with noise by the DWT method, and �c� with noise by theLSMI method.


Table 2. Results of Computer Simulation

PresetAberrationCoefficients

Free from Noise Noise Added

DWT LSMI DWT LSMI

w0 � 0 0 0 0.010 0.001w1 � 0 0 0 0.014 0.007w2 � �3.2 �3.2 �3.2 �3.245 �3.844w3 � 0.9 0.9 0.9 0.932 0.896w4 � 1.1 1.1 1.1 1.050 1.078w5 � 3.1 3.1 3.1 2.813 4.137

rms � 0.137 0.137 0.137 0.141 0.148




DWT LSMI DWT LSMI

w0 � 0 0 0 0.012 0.001w1 � 0 0 0 0.016 0.007w2 � �4.05 �4.05 �4.05 �4.111 �4.691w3 � 1.35 1.35 1.35 1.383 1.345w4 � 0 0 0 0.055 0.036w5 � 3.25 3.25 3.25 3.022 4.287

rms � 0.154 0.154 0.154 0.157 0.161




DWT LSMI DWT LSMI

w0 � 0 0 0 0.010 0.001w1 � 0.15 0.15 0.15 0.164 0.157w2 � �1.55 �1.55 �1.55 �1.595 �2.359w3 � 0 0 0 0.032 0.079w4 � 3.55 3.55 3.55 3.450 3.528w5 � 2.05 2.05 2.05 1.763 3.087

rms � 0.325 0.325 0.325 0.327 0.334


DWT method are less than those of the LSMI method.The contours of the original wave front and the wavefront reconstructed by the two methods are shown inFigs. 1–3. Figures 1�a�, 2�a�, and 3�a� describe thetrue topographies of the three test wave fronts. Thecontour maps of the test wave fronts in Figs. 1�b�,2�b�, and 3�b� were calculated with the DWT method,whereas the contour maps of the test wave fronts in

Figs. 1�c�, 2�c�, and 3�c� were found with the LSMItechnique. The contour maps found with the DWTmethod agree quite well with maps computed withthe original test wave front. On the meridional sec-tion �x � 0� and the transverse section �y � 0�, theoriginal three test wave fronts and the wave frontsreconstructed by the two methods are shown in Fig. 4;it is evident that the DWT method provides betterresults than the LSMI method does.

We implemented this new technique in our simu-lations to estimate the Seidel aberration coefficientsand found that with this implementation the calcu-lation of the fitted wave front �65 � 65 data points�took only 3 s on a personal computer, whereas withthe LSMI method it took �40 s.

In summary, these simulations indicate that theperformance of the new technique is better than thatof the LSMI method; not only are the more accurateSeidel aberration coefficients obtained but also thecomputation is speeded up. In this simulation, allresults were evaluated with the MATLAB system.17

5. Conclusion

In this paper a new technique for the direct evalua-tion of Seidel aberrations of optical system has beenintroduced. This method evaluates the wave frontwith a suitable filter �suitable spatial resolution� toobtain unique values for the Seidel aberration coeffi-cients; it provides not only more-accurate coefficientsbut also speeds the computation. The DWT methodalso provides better a rms value than the LSMImethod does. Using the rms value, we numericallydemonstrated that the DWT method can do a betterjob than the LSMI method.

This method may be applied in other areas, such asinterferogram reduction, aberration extraction in theHartmann test, adaptive optics, phase retrieval, andinterferometry.

This research was supported by the National Sci-ence Council, Taiwan, under project grants NSC88-2215-E-008-008 and NSC89-2215-E-008-010, and bythe Kwang Wu Institute of Technology.

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Analysis of Seidel Aberration by Use of the Discrete Wavelet Transform