The Hartmann test is a well-known technique fortesting large optical components.1 The Hartmanntechnique samples the wave front under analysiswith a screen of uniformly spaced holes situated atthe pupil plane. The rays of light that pass throughthe screen holes are then captured by a photographicplate near the paraxial focus. The uniformly spacedarray of holes at the instrument pupil is distorted atthe photographic plane by the spherical aberrationof the wave front being tested. The screen defor-mations are then proportional to the slope of theaspheric wave front. Traditionally these Hart-manngrams ~the distorted image of the screen at thephotographic plate’s plane! are analyzed by measure-ment of the centroid of the spots generated by thescreen holes in the photographic plate. The devia-tion of these centroids from their uniformly spaced
M. Servin, F. J. Cuevas, and D. Malacara are with Centro deInvestigaciones en Optica A. C., Apartado Postal 1-948, 37150Leon, Gto., Mexico. J. L. Marroquin is with Centro de Investiga-cion en Matematicas A. C., Apartado Postal 402, 36000 Guana-juato, Gto., Mexico.
Received 21 August 1998; revised manuscript received 30 No-vember 1998.
2862 APPLIED OPTICS y Vol. 38, No. 13 y 1 May 1999
positions ~unaberrated positions! are recorded andre proportional to the aspherical aberration slope.1
The centroid coordinates give a two-dimensionalsampled field of wave-front slopes that need integra-tion and interpolation over regions without data.Integration of the wave-front gradient field is nor-mally done by use of the trapezoidal rule.1 The trap-ezoidal rule consists of following several independentintegration paths and averaging their outcome. Inthis way one may approximate a path-independentintegration. When this integration procedure isused, the wave front is known at only the hole’s po-sition. Although this integration technique maygive good wave-front demodulation, the estimation ofthe positions of the holes must be done by hand, andit is a time-consuming and tedious process. Finally,a polynomial or spline wave-front fitting is necessaryto estimate the wave-front values at places otherthan the discrete points where the gradient data arecollected. Another suggestion is to use two-dimensional polynomial fitting.1 A polynomial formfor the wave front is proposed, and it is least-squaresfitted to the slope data in both directions. This poly-nomial must contain every possible type of wave-front aberration; otherwise some unexpected features~especially at the edges! of the wave front may befiltered out. On the other hand, given that the col-lected data are not continuous, that is, they are col-lected in only a discrete set of points that are the
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centroids of the Hartmanngram spots, a high-degreepolynomial may wildy oscillate in the regions whereno data are collected.
We described a newer approach to the problem ofanalyzing a Hartmanngram earlier.2 In this tech-
ique2 we use two robust orthogonal bandpass filterstuned at the dot frequency. In this way we obtaintwo orthogonal carrier-frequency fringe patterns outof a single Hartmanngram. These two fringe pat-terns are then analyzed by the Fourier technique3 orby synchronous interferometry4 to obtain the esti-mated phase gradient. This approach is more ro-bust and direct than that of measuring individualspot centroids.
If the frequency spectrum of the Hartmanngramcontains well-separated different diffracting orders,one may try the synchronous phase-detection meth-od3 or the Fourier transform phase-estimation tech-nique.4 These are probably the best-knownautomatic fringe-detecting methods available in theliterature, so in this paper they are used as referencesto demonstrate the advantages of introducing anewly developed method in Hartmanngram process-ing.
Here we propose a way for demodulating a Hart-manngram in a direct way by using a simplifiedversion of an earlier proposed regularized phase-tracking ~RPT! technique.5 The RPT technique aspresented herein overcomes all the difficulties of an-alyzing Hartmanngrams mentioned above. That isbecause the RPT technique can measure the trans-verse aberration from a Hartmanngram in a directway, provided that we have knowledge of the ex-pected aberration at the test plane. An additionaladvantage of the RPT technique is that it does notneed any further unwrapping process6 in the esti-mated aberration. That is because the unwrappingprocess is implicit in the RPT technique. The RPTray aberration estimation is computed at every pixelof the framegrabbed Hartmanngram, so least-squares integration of the wave-front slope may beeasily done.7–9
2. Hartmann Test
The Hartmann technique samples the wave front un-der analysis by use of a screen of uniformly spacedholes situated at the pupil plane. A Hartmann platemay be represented by the following mathematicalexpression:
Hs~x1, y1! 5 (n52Ny2
Ny2
(m52Ny2
Ny2
h~x1 2 dn, y1 2 dm!, (1)
here Hs~x1, y1! is the Hartmann screen and h~x1 2dn, y1 2 dm! are the small holes that are uniformlypaced in the Hartmann screen. The point ~x1, y1!ocated at the Hartmann screen plane corresponds tooordinates ~x, y! that are located at the Hart-
manngram plane. The constant d is the spaceamong the holes of the screen. The experimentalsetup to obtain a Hartmanngram for testing a largeprimary mirror is shown in Fig. 1. A screen formed
by regularly spaced holes is positioned in the planewhere the wave front is to be analyzed. For an as-pheric converging wave front, the photographic plateto record the light that has passed through the Hart-mann screen is normally positioned just inside itsparaxial focus. The geometry of the screen as im-aged at the Hartmanngram plate is then distorted bythe wave-front aberrations according to
x 5r 2 L
rx1 1 L
]W~x1, y1!
]x1,
y 5r 2 L
ry1 1 L
]W~x1, y1!
]y1, (2)
where L is the distance from the screen to the pho-tographic plate used to gather the aberration data~see Fig. 1!; the constant r is the radius of curvatureof the wave front under test. Finally, W~x1, y1! is thespheric wave front being tested at the screen plane,nd it is bounded by a pupil P~x1, y1!.As mentioned above, the converging rays of light
passing through the screen holes are captured by aphotographic plate at some distance L from it. Theuniformly spaced array of holes at the instrumentpupil is then distorted at the photographic plane bythe aspherical aberration of the wave front beingtested. The screen deformations are then propor-tional to the slope of the aspheric wave front @see Eqs.~2!#, that is,
H~x, y! 5 H (n52Ny2
Ny2
(m52Ny2
Ny2
hiFr 2 Lr
x1 2 L]W~x1, y1!
]x1
2 dn,r 2 L
ry1 2 L
]W~x1, y1!
]y12 dmGJP~x, y!,
(3)
here H~x, y! is the Hartmanngram obtained at adistance L from the Hartmann screen and P~x, y! isthe pupil that bounded the distorted wave front in theHartmanngram plane. The function hi~x, y! is theimage of the screen hole h~x1, y1! as projected at theHartmanngram plane. Finally, P~x1, y1! is the pupil
Fig. 1. Simplified experimental setup used to obtain an off-axisHartmanngram.
1 May 1999 y Vol. 38, No. 13 y APPLIED OPTICS 2863
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2
of the wave front being tested. As Eq. ~3! shows,only one Hartmanngram is needed to estimate fullythe wave-front sampled gradient. The frequencycontent of the estimated wave front will be limited bythe sampling theorem to the hole’s period d of thecreen.Typically the strongest components for an aspheri-
al converging wave front W~x1, y1! come from therimary spherical aberration. So the first approxi-ation for the testing wave front may be written as
W~x1, y1! 5 B~x12 1 y1
2!2, (4)
where B is the spherical aberration coefficient.hen the approximate geometric transformation suf-
ered by the screen at the Hartmanngram plate ac-ording to Eq. ~3! is
x 5r 2 L
rx1 1 4LB~x1
2 1 y12!2x1,
y 5r 2 L
ry1 1 4LB~x1
2 1 y12!2y1. (5)
As Eqs. ~5! show, the spherical aberration translatesinto a nonlinear spatial transformation.
The Hartmanngram recorded by a photographicplate may be digitized with a framegrabber and avideo CCD camera for further processing within adigital computer. If the distance L in Eqs. ~5! islose to the wave-front radius of curvature r ~as issually the case with this technique!, then the dotattern in the Hartmann plate is strongly curvedoward the optical axis.
As seen from Eqs. ~5!, the strongest wave-frontberration component of an asphere comes from itspherical aberration term; we suggest using this ab-rration term as prior knowledge in the RPT algo-ithm. Therefore the RPT will detect the remainingberrations, improving in this way its sensitivity andobustness to noise. This sensitivity and noise ro-ustness are improved because the remaining aber-ations have a lower dynamic range and therefore aarrower frequency spectrum.
3. Regularized Phase-Tracking Demodulation of theRay Aberration in the X Direction in a Hartmanngram
A more complete explanation about the RPT tech-nique may be found in Ref. 5. Here we present justa summary of the method as applied to Hart-manngram analysis. The main idea behind the pro-posed method is to estimate the phase of twoorthogonal carrier-frequency fringe patterns out ofthe original Hartmanngram.
From the number of dots in the Hartmanngramand the video framegrabber resolution, one may es-timate the carrier frequency of the dots in radians perpixel at the framegrabbed Hartmanngram. To thislinear phase we must add the strong spherical rayaberration of the Hartmanngram. So in summarywe start our RPT demodulating algorithm with the
864 APPLIED OPTICS y Vol. 38, No. 13 y 1 May 1999
following carrier-frequency fringe pattern as the ini-tial estimation:
where fx~x, y! ~initially set to zero! is the ray aber-ration correction to our prior knowledge of the ex-pected phase. The function P~x, y! is the wave-frontpupil at the testing plane. The constant K 5 4LB@Eqs. ~5!# depends on the expected amount of spher-ical ray aberration, and the spatial frequency v0 isthe average spatial frequency of the dot pattern interms of radians per pixels in the digitized Hart-manngram.
In classical regularization10 one needs to provide aconstraint about the family of solutions that one ex-pects to obtain. A typical constraint in classical reg-ularization is smoothness over the estimatedfunction. This constraint is normally introducedinto the problem as potentials that are formed as thenorm of derivative operators.
As in classical regularization, to regularize thephase-detection problem in fringe analysis2,5 it is nec-essary to find a suitable energy or cost function thatuses at least two terms that contribute to constrainthe estimated phase field. These terms are given bythe following information:
~1! Fidelity between the estimated function and theobservations
~2! Smoothness of the modulated phase field
It is then assumed that the searched phase functionis the one that minimizes the cost function.
In particular in the RPT technique proposed here~as in Ref. 5! we have assumed that locally the dotpattern may be modeled as two cosinusoidal carrier-frequency fringe patterns that are phase modulatedby a plane plus the expected ray spherical aberration@as in Eq. ~6!#. The amplitude of this cosinusoidalfunction must be close in the least-squares sense tothe observed irradiance @statement ~1!#. A phase
lane such as this must adapt itself to every region inhe dot pattern, given that its local frequency changesontinuously in the two-dimensional space. The sec-nd term of the proposed cost function refers to thexpected smoothness and continuity of the estimatedhase @statement ~2!#. Specifically, the proposedost function to be minimized by the estimated phasex~x, y! at site ~x, y! is
U~x, y! 5 (~e,h![@Nx, yùP~ x, y!#
$@H~e, h! 2 cos px~x, y!#2
1 l@fx~e, h! 2 fx~x, y!#2m~e, h!%, (7)
px~x, y! 5 v0 x 1 K~x2 1 y2!x 1 fx~x, y!, (8)
where px~x, y! is the total phase in the x direction.he region Nx,y is a neighborhood region around the
coordinate ~x, y! where the phase fx~x, y! is beingestimated. m~x, y! is an indicator field that equals 1
pt
d
s~mtyt
tl
ovoiar
if the site ~x, y! has already been estimated and 0otherwise ~see below!. Finally, l is the regularizing
arameter that controls ~along with the size of Nx,y!he smoothness of the detected phase.
To demodulate the remaining aberration in the yirection, one needs to replace px~x, y! in Eq. ~7! above
with
py~x, y! 5 v0 y 1 K~x2 1 y2!y 1 fy~x, y!, (9)
where py~x, y! is the total phase in the y direction.By using Eqs. ~7! and ~8! and following the demodu-lating strategy proposed below, one is able to demod-ulate the wave-front gradient along the x direction @oralong the y direction by using Eq. ~9!#. In Eq. ~7! wehave assumed that the Hartmanngram’s irradianceH~x, y! is stretched to fit within the range ~0, 1!.
The first term in Eq. ~7! attempts to keep the localfringe model in phase with the observed dot patternin a least-squares sense within the neighborhood Nx,y@statement ~1!#. The second term enforces the as-umption of smoothness and continuity @statement2!# by using only previously demodulated pixelsarked by m~x, y!. One can see that the second
erm will make a small contribution to the cost U~x,! for only smooth estimated phase fields. Note alsohat the local phase error fx~x, y! @or fy~x, y!# is
adapted simultaneously to the observed data~through its cosinusoidal model! and to the phase-error values that have already been estimated. Thisis done in order to find the smoothest phase compat-ible with the observed dot’s irradiance. Note that inthe cost function U~x, y! one is adapting horizontalplanes. This is a simplified version of the RPT al-gorithm presented in Ref. 5, in which one adaptsplanes with arbitrary orientation.
To demodulate the dot pattern given by Eq. ~3! weneed to find the minimum of the cost function U~x, y!with respect to the field fx~x, y! at site ~x, y!. Toestimate the phase error fx~x, y! @or fy~x, y!#, wepropose to follow the sequential crystal-growing ~CG!demodulating algorithm described below, optimizingfor each U~x, y! by adapting the function fx~x, y! ateach site ~x, y! in P~x, y!. The CG algorithm gives usour first global phase estimation. After the firstglobal CG iteration is performed, we may improve theestimated phase error by performing additionalglobal iterations in P~x, y!, taking as the initial con-dition the last estimated phase. Given that our firstglobal iteration ~given by the CG algorithm! is close tothe actual modulating phase, few ~typically less than4! additional iterations are needed to reach the sta-tionary point of U~x, y!.
The first CG iteration @the one that gives the firstphase estimation on P~x, y!# is performed as follows:To start, the indicator function m is set to zero @m~x,y! 5 0 in P~x, y!#. Then we choose a seed or startingpoint ~x0, y0! inside P~x, y! to begin the demodulationof the Hartmanngram’s dot pattern. The functionU~x0, y0! is then optimized with respect to fx~x0, y0!and the visited site is marked as detected, i.e., we setm~x0, y0! 5 1. Once the seed pixel is demodulated,
he sequential phase demodulation proceeds as fol-ows:
~1! Choose the ~x, y! pixel inside P~x, y! ~randomlyor with a prescribed scanning order!.
~2! ● If m~x, y! 5 1 return to the first statement.● If m~x, y! 5 0, then test whether m~x9, y9! 5
1 for any adjacent pixel ~x9, y9!.● If no adjacent pixel has already been esti-
mated, return to the first statement.● If m~x9, y9! 5 1 for an adjacent pixel, take
fx~x9, y9! as the initial condition to minimizeU~x, y! with respect to fx~x, y!.
~3! Set m~x, y! 5 1.~4! Return to the first statement until all the pixels
in P~x, y! are estimated.
The term CG was given to this algorithm because ofits similarity to a CG process. In a CG process newmolecules are added to the bulk in that particularorientation that minimizes the local crystal energy@in our case U~x, y!#, given the geometric orientationof the adjacent and previously positioned molecules@marked by m~x, y!#.
To optimize U~x, y! at site ~x, y! with respect tofx~x, y! we have used a simple gradient descent as
fxk11~x, y! 5 fx
k~x, y! 2 t]U~x, y!
]fx~x, y!, (10)
where t is the step size, which is ;0.01, and k is theiteration number. It is important to remark that thetwo-dimensional RPT technique gives the estimatedphase error fx~x, y! already unwrapped, so no addi-tional phase-unwrapping process is required.
The first CG global phase estimation is usuallyclose to the actual modulating phase; if needed, onemay perform additional global iterations over U~x, y!to improve the phase-estimation process. To movecloser to the optimal values of fx~x, y!, one may againuse the gradient descent system @Eq. ~10!#, but nowne uses as the initial state the previously estimatedalues at the same site ~do not use the detected phasef a neighborhood site as the initial state as was donen the first CG iteration!. In practice only one or twodditional ~non-CG! global iterations are needed toeach a stable minimum of U~x, y! at every site in P~x,
y!. Note that the indicator function m~x, y! in theadditional global iterations is equal to 1 in P~x, y!;therefore one may scan the lattice in any desired waywhenever all the sites are visited.
The same procedure followed to estimate the phaseerror fx~x, y! is applied to estimate the phase errorfy~x, y!. This time our initial estimation is thecarrier-frequency fringe pattern along the y direction,whose phase is given by Eq. ~9!.
4. Experimental Results
A. Regularized Phase-Tracking Phase Estimation
The RPT demodulating scheme was applied to theexperimentally obtained Hartmanngram shown in
1 May 1999 y Vol. 38, No. 13 y APPLIED OPTICS 2865
2
Fig. 2. This Hartmanngram was taken off axis, asshown in Fig. 1, and was digitized with a resolution of160 3 160 pixels each of 256 gray levels. The pri-mary mirror being tested had a curvature radius of280.6 cm, a conic constant of 21.0133, and a diameterof 64 cm. Given the experimental arrangementused, the off-axis aberration must be taken into ac-count in the detected ray aberration.
The initial fringe-pattern estimation is shown inFig. 3. We obtained the phase of this fringe patternby using as the prior aberration model a pure spher-ical ray aberration ~the theoretical model contains
Fig. 2. Framegrabbed Hartmanngram obtained from a primarymirror that has a curvature radius of 280.6 cm, a conic constant of21.0133, and a diameter of 64 cm.
Fig. 3. Fringe pattern used as prior phase model for the Hart-manngram aberration. The RPT technique herein presented willthen estimate the phase difference between this fringe pattern andthe digitized Hartmanngram shown in Fig. 2.
866 APPLIED OPTICS y Vol. 38, No. 13 y 1 May 1999
other aberration terms not considered in the simpli-fied aberration model!. The RPT system will thenestimate the phase error fx~x, y! or fy~x, y! betweenthe prior ray aberration model and the actual rayaberration of the framegrabbed Hartmanngram.Then the actual ray aberration Tx~x, y! and Ty~x, y! inthe x and the y directions, respectively, will be givenby
Tx~x, y! 5 @K~x2 1 y2!x 1 fx~x, y!#P~x, y!,
Ty~x, y! 5 @K~x2 1 y2!y 1 fy~x, y!#P~x, y!. (11)
The detected ray aberration error fx~x, y! in the xdirection is given in gray levels in Fig. 4. To obtainthe estimated aberration shown in Fig. 4 it was nec-essary to calculate the first CG global iteration plustwo additional ~non-CG! global iterations. The totalcomputing time required for obtaining this estima-tion was ;90 s on a 90-MHz Pentium PC.
Figure 5~a! shows the fringe pattern I1~x, y! of theprior assumed reference
The actual Hartmanngram dot pattern was superim-posed on this fringe pattern @Eq. ~12!# to make cleartheir initial phase differences. On the other hand,Fig. 5~b! shows the fringe pattern I2~x, y! given by
I2~x, y! 5 $1 1 cos@v0 x 1 K~x2 1 y2!x
1 fx~x, y!#%P~x, y!. (13)
Again the Hartmanngram dot pattern was superim-posed on this fringe pattern @Eq. ~13!# to verify thatthe obtained phase clearly corresponds to the ex-pected Hartmanngram dot phase.
Figure 6 shows the ray error aberration in the ydirection fy~x, y!. The resulting fringe patterns,
Fig. 4. Estimated phase difference between the prior phase modelshown in Fig. 3 and the actual phase of the digitized Hart-manngram shown in Fig. 2 along the x direction.
similar to those shown in Figs. 5~a! and 5~b!, areshown in Figs. 7~a! and 7~b!.
B. Comparison with the Synchronous and the FourierPhase Estimations
In this section we use the well-known synchronousand Fourier phase-demodulation techniques to esti-mate the modulating ray aberration of the Hart-
Fig. 5. ~a! Fringe pattern modulated by the expected phase model~spherical ray aberration!. We have superimposed the digitizedHartmanngram on this fringe pattern to see the initial phasemismatch. ~b! Fringe pattern modulated by the expected phasemodel plus the estimated phase error shown in Fig. 4. We havesuperimposed the digitized Hartmanngram on this fringe patternto see clearly the spatial phase match of the Hartmanngram dotsalong the x direction.
manngram shown in Fig. 2. This is done to comparethe RPT technique presented above with these twowell-known techniques that may also be used to solvethe same problem. The Fourier spectrum of theHartmanngram is shown in Fig. 8. As seen fromthis figure the lower diffraction orders seem to beseparated well enough that the synchronous andFourier phase-estimation techniques can be tried.
First we show the results of using the Fourier tech-nique. As seen in Fig. 8, we have isolated themarked diffraction order to the left of the spectrum.The ellipse shown represents approximately the foot-print of the isolating quadrature filter used. Theamplitude response of the filter used was Gaussian,and its bandwidth was determined to obtain the bestphase reconstruction. After isolating this diffrac-tion order, we performed an inverse Fourier trans-form of the signal and obtained its phase. Figure 9shows a cosine function phase modulated by the de-tected ray aberration
IF~x, y! 5 1 1 cos@TFx~x, y!#, (14)
where TFx~x, y! is the ~carrier plus the ray aberration!detected with the Fourier transform. From Fig. 9 itcan be seen that the retrieved ray aberration is notreliable near the central stop nor at the edges of thepupil.
Now we show the performance of the synchronousmethod applied to the Hartmanngram shown in Fig.2. As usual, we have multiplied the Hartmann ir-radiance by the sine and the cosine of the referencecarrier. Then we have used a 3 3 3 convolutionaverager seven times to obtain two in-quadraturelow-pass frequency signals used to estimate the mod-ulating ray aberration. The ray aberration in the x
Fig. 6. Estimated phase difference between the prior phase modeland the actual phase of the digitized Hartmanngram shown in Fig.2 along the y direction.
1 May 1999 y Vol. 38, No. 13 y APPLIED OPTICS 2867
2
direction, TSx~x, y!, estimated with the synchronoustechnique, is then given by
TSx~x, y! 5 arctanHLPF@H~x, y! sin~v0 x!#
LPF@H~x, y! cos~v0 x!#J , (15)
where H~x, y! represents the Hartmanngram irradi-ance, v0 is the mean spatial frequency of the Hart-
Fig. 7. ~a! Fringe pattern modulated by the expected phase model~spherical ray aberration!. We have superimposed the digitizedHartmanngram on this fringe pattern to see the initial phasemismatch. ~b! Fringe pattern modulated by the expected phasemodel plus the estimated phase error shown in Fig. 6. We havesuperimposed the digitized Hartmanngram on this fringe patternto compare its spatial phase with the phase of the Hartmanngramdots along the y direction.
868 APPLIED OPTICS y Vol. 38, No. 13 y 1 May 1999
manngram spots along the x direction, and the LPF@.#operator is the low-pass-filter operation performed.
Figure 10 shows a cosine function phase modulatedby the detected ray aberration with the synchronoustechnique, that is,
IS~x, y! 5 1 1 cos@v0 x 1 TSx~x, y!#, (16)
where TSx~x, y! is the ray aberration detected withthe synchronous technique and v0 is the mean spatialfrequency of the Hartmanngram spots along the xdirection. In Fig. 10 we can also see that the re-trieved phase is not reliable near the central stop norat the edges of the pupil.
Fig. 8. Fourier ~frequency! spectrum of the Hartmanngramshown in Fig. 2. We may see that the diffraction orders are stillseparable, so linear phase-detection techniques may be used.
Fig. 9. Cosine of the estimated modulating phase obtained fromthe Hartmanngram in Fig. 2 with the synchronous technique.
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Comparing the estimated ray aberration obtainedby using the RPT technique ~Fig. 5! with the resultsbtained by using the Fourier technique ~Fig. 9! andhe synchronous technique ~Fig. 10!, we can clearlyee that by using the proposed RPT technique webtain more reliable results.
5. Conclusions
We have presented a direct way to analyze Hart-manngrams by using a regularized phase-tracking~RPT! technique. This technique may use the ana-ytical form of the strongest aberration of the testedave front to improve its phase-demodulation capa-ilities. The strongest aberration in testing as-heres comes from the spherical aberration, whichay be used as prior knowledge. Then the RPT
echnique will detect the remaining aberrations orhe actual phase error between the aberration modelnd the one actually recorded over the Hart-anngram.An additional advantage of using the RPT tech-
nique is the fact that this technique does not need aseparate unwrapping process, given that it is implicit
Fig. 10. Cosine of the estimated modulating phase obtained fromthe Hartmanngram in Fig. 2 with the Fourier technique.
within the RPT method. The RPT technique givesthe estimation of the ray aberration at every pixelover the pupil of the framegrabbed Hartmanngram.This is a good feature, given that the phase gradientmay be easily integrated with a least-squares inte-gration technique.7–9
We have also compared the RPT technique with thetwo best-known methods for automatic fringe pro-cessing, namely the synchronous and the Fouriertechniques. We have demodulated the given Hart-manngram by using these two techniques, and wehave found that the proposed RPT technique givesbetter phase-estimation results.
We acknowledge the financial support for this re-search of Consejo Nacional de Ciencia y Tecnologia~CONACYT! under grants 0580P-E and 4424-A.
References1. J. Ghozeil, “Hartmann and other screen tests,” in Optical Shop
Testing, D. Malacara, ed. ~Wiley, New York, 1992!, pp. 367–396.
2. M. Servin, D. Malacara, J. L. Marroquin, and F. J. Cuevas,“New technique for ray aberration detection in Hart-manngrams based on regularized bandpass filters,” Opt. Eng.35, 1677–1683 ~1996!.
3. K. H. Womack, “Interferometric phase measurement usingspatial synchronous detection,” Opt. Eng. 23, 391–395 ~1984!.
4. M. Takeda, H. Ina, and S. Kobayashi, “Fourier transformmethods of fringe-pattern analysis for computer-based topog-raphy and interferometry,” J. Opt. Soc. Am. 72, 156–160~1982!.
5. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulationof a single interferogram by use of a regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 ~1997!.
6. D. C. Ghiglia and L. A. Romero, “Robust two-dimensionalweighted and unweighted phase unwrapping that uses fasttransforms and iterative methods,” J. Opt. Soc. Am. A 4, 107–117 ~1994!.
7. B. R. Hunt, “Matrix formulation of the reconstruction problemof phase values from phase differences,” J. Opt. Soc. Am. 69,393–399 ~1979!.
8. R. H. Hudgin, “Wave-front reconstruction for compensated im-aging,” J. Opt. Soc. Am. 67, 375–378 ~1977!.
9. R. J. Noll, “Phase estimates from slope-type wave-front sen-sors,” J. Opt. Soc. Am. 68, 139–140 ~1978!.
10. A. N. Tikhonov, “Solution of incorrectly formulated problemsand the regularization method,” Sov. Math. Dokl. 4, 1035–1038 ~1963!.
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