abel inversion of deflectometric data: comparison of accuracy and noise propagation of existing...

Abel inversion of deflectometric data: comparisonof accuracy and noise propagation

of existing techniques

Pankaj S. Kolhe and Ajay K. Agrawal*Department of Mechanical Engineering, University of Alabama, Tuscaloosa, Alabama 35487, USA

*Corresponding author: [email protected]

Received 6 April 2009; revised 1 June 2009; accepted 15 June 2009;posted 17 June 2009 (Doc. ID 109732); published 1 July 2009

Abel inverse integral to obtain local field distributions from path-integrated measurements in an axi-symmetric medium is an ill-posed problem with the integrant diverging at the lower integration limit.Existing methods to evaluate this integral can be broadly categorized as numerical integration techni-ques, semianalytical techniques, and least-squares whole-curve-fit techniques. In this study, Simpson’s1=3rd rule (a numerical integration technique), one-point and two-point formulas (semianalyticaltechniques), and the Guass–Hermite product polynomial method (a least-squares whole-curve-fit tech-nique) are compared for accuracy and error propagation in Abel inversion of deflectometric data. For dataacquired at equally spaced radial intervals, the deconvolved field can be expressed as a linear combina-tion (weighted sum) of measured data. This approach permits use of the uncertainty analysis principle tocompute error propagation by the integration algorithm. Least-squares curve-fit techniques should beavoided because of poor inversion accuracy with large propagation of measurement error. The two-pointformula is recommended to achieve high inversion accuracy with minimum error propagation. © 2009Optical Society of America

OCIS codes: 000.2190, 100.6950, 120.5820, 280.1740, 280.2490.

1. Introduction

Nonintrusive diagnostics typically consist of localtechniques that measure a quantity at a point, plane,or volume, and line-of-sight techniques measuring achordally integrated quantity. The latter techniquesrequire a deconvolution approach to reconstruct thelocal field from measurements of the integratedquantity. For an axisymmetric medium, Abel inver-sion [1] formulas are used to reconstruct the fieldvariable from its one-dimensional projections. Paststudies have generally focused on Abel formulasfor phase measurements, given as

PðxÞ ¼ 2ZRx

f ðrÞrffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − x2

p dr; ð1Þ

The Abel inversion integral transform of Eq. (1) is gi-ven as

f ðrÞ ¼ −1π

ZRr

dPðxÞdx

dxffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 − r2

p : ð2Þ

Equations (1) and (2) find applications in radio as-tronomy [2], atomic and molecular scattering [3],measurements by spectroscopy (flame/plasma) [4–8],photoelectron imaging experiments [9], combustionand fluid-flow measurements by interferometry[10–12], x-ray radiography [13], optical-fiber refrac-tive-index measurements [14,15], and spray-dropletstudies [16]. The integral in Eq. (2) requires differen-tiation of measured data, which can amplify the mea-surement error in PðxÞ.

In the present study, we are dealing with Abel for-mulas for deflectometric measurements. For exam-ple, the transverse deflection angle of a ray in aschlieren setup is given by Eq. (3):

0003-6935/09/203894-09$15.00/0© 2009 Optical Society of America

3894 APPLIED OPTICS / Vol. 48, No. 20 / 10 July 2009

ΘðxÞ ¼ 2xZRx

∂δ∂r

drffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − x2

p : ð3Þ

The field variable is the normalized refractive-indexdifference [δ ¼ ðη=η0 − 1Þ) where η is the local refrac-tive index and η0 is the refractive index of the sur-rounding] obtained from an Abel inverse integraltransform of Eq. (3), given as

δðrÞ ¼ −1π

ZRr

ΘðxÞ dxffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 − r2

p : ð4Þ

Equations (3) and (4) are used with measurementsobtained by moiré deflectometry [17,18], laserspeckle photography [19], and rainbow schlieren de-flectometry [20–25], among others. The integral inEq. (4) is improper with the integrant diverging atthe lower integration limit. Moreover, deflection an-gle measurements will inevitably contain experi-mental uncertainties or noise. Thus, the procedureto evaluate the integral in Eq. (4) is important fromperspectives of accuracy and propagation of mea-surement uncertainties even though the computa-tional effort is insignificant. The primary objectiveof the present work is to compare different methodsproposed in the literature so that the best approachto evaluate the integral in Eq. (4) can be identified.Existing algorithms to evaluate local field distribu-

tion using either Eq. (2) or Eq. (4) can be catego-rized as:1. numerical integration techniques using piece-

wise curve fitting of the diverging integrant in theAbel integral transform (namely, trapezoidal inte-gration, Simpson’s 1=3rd rule, quadrature methods,etc.), [6,26–28].2. data approximation techniques,

a. semianalytical techniques using piecewise ap-proximation (namely, Lagrangian polynomials,Taylor series expansion) of measured data in eachsegment evaluated analytically [29–33], orb. least squares approximation of entire data set

or whole-field curve-fit methods (namely, use ofFourier series, orthogonal polynomials, Hermitepolynomials, etc.) coupled with analytical integrationof the curve-fit function [5,12,14,34].

Most of the studies in each category are similar intheme and they typically present a specific techniquewithout a comparative account of different methods.Cremers and Birkebak [4] compared different techni-ques based on piecewise approximation of data.Minerbo and Levy [10] presented a method to ac-count for error/uncertainty in the inversion processand used it to systematically compare differentpolynomials for data approximation. Dasch [30] com-pared semianalytical techniques to integrate theAbel inverse transform in Eq. (2) and showed, for thefirst time, that all of the techniques could be cast intolinear operator form in Eq. (5):

f ðriÞ ¼1Δr

·X∞j¼i

Dij · Pj ð5Þ

Dasch [30] used Eq. (5) as a benchmark to compareinversion accuracy and noise propagation behavior ofseveral semianalytical techniques. Past studies dealwith phase measurements and, to date, a quantita-tive assessment of techniques to evaluate Abel inte-gral with deflectometric measurements [Eq. (4)] hasnot been conducted to the best of our knowledge.

Past investigators have employed two differentmethods to account for noise propagation. In the firstmethod, input data are infiltrated with noise to de-termine its propagation during the inversion process[35]. The second method is the typical uncertaintyanalysis, determining how a known uncertainty inmeasured data [given in terms of standard deviationor root mean square (RMS)] propagates into uncer-tainty in a local or inverted quantity [36,37]. How-ever, none of these investigators compared errorpropagation for inversion techniques of differentcategories listed above (namely, numerical integra-tion, semianalytical integration, and least-squaresapproximation).

In this study, four different techniques to evaluatethe Abel inversion integral in Eq. (4) are presented.It is shown that each of these techniques can be ex-pressed in a linear operator form similar to that em-ployed by Dasch [30]. This feature allows comparisonof different techniques using the standard errorpropagation analysis procedure. While all of thesemethods are remarkably similar and the computa-tional effort is less than 1 min on a personal compu-ter, the semianalytical technique using piecewiseapproximation by a linear profile is found to besuperior to other techniques in terms of both accu-racy and propagation of measurement error.

2. Inversion Algorithms

The algorithms addressed in this study are

1. numerical integration using Simpsons’1=3rd rule,

2. a semianalytical technique with piecewise ap-proximation in a segment by a constant profile (one-point formula),

3. a semianalytical technique with piecewise ap-proximation in a segment by a linear profile (two-point formula), and

4. a least-squares approximation of a datasetusing the Guass–Hermite product polynomial(GHPP).

These techniques to evaluate the integral in Eq. (4)are compared assuming that the line-of-sight deflec-tion angle data are available at equally spaced inter-vals (N segments). Following Dasch [30], all of thesetechniques can be recast in the following form:

δðriÞ ¼XNþ1

j¼i

Dij ·Θj: ð6Þ

10 July 2009 / Vol. 48, No. 20 / APPLIED OPTICS 3895

Here ΘðrjÞ represents the deflection angle data atspacing of Δr, δðrjÞ is the discrete refractive-indexdifference field, and rj ¼ ðj − 1Þ:Δr is the radial dis-tance. Here i and j are indices corresponding to datapoints varying from 1 to N þ 1. The linear operatorcoefficients Dij are independent of the data spacingΔr except for in the GHPP method. Nevertheless,Dij coefficients can be determined a priori for eachinversion algorithm. The procedure to recast inver-sion techniques in the form of Eq. (6) is discussedin the following.

A. Simpson’s 1=3rd Rule

The Simpson’s 1=3rd rule applied to discretized formof Eq. (4) results in Eq. (7a):

δðriÞ ¼−1π

ZRri

ΘðrÞdrffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 − r2i

q ¼ −1π

XN−1

j¼i;incr¼2

Zrjþ2

rj

ΘðrjÞdrjffiffiffiffiffiffiffiffiffiffiffiffiffiffir2j − r2i

q

¼ −1π

XN−1

j¼i;incr¼2

�Δr3

·� Θjffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2j − r2iq þ 4Θjþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2jþ1 − r2iq þ Θjþ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2jþ2 − r2iq ��

:

ð7aÞ

Equation (7a) can be recast in the form of Eq. (6),where

Dij ¼−13π ·

�2þ

�1þ ð−1Þðj−iþ1Þ

��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðj − 1Þ2 − ði − 1Þ2

pif j > i and j ≠ N þ 1;

¼ −13π ·

�2þ

�1þ ð−1Þðj−iþ1Þ

��2 ·

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðj − 1Þ2 − ði − 1Þ2

p if j ¼ N þ 1;

¼ 0 if j < i: ð7bÞ

The singularity at j ¼ i is eliminated by extrapolat-ing from remaining Dij coefficients, as shown inEq. (7c):

Dii ¼ Diðiþ1Þ for i ≤ N; ¼ 0 for i ¼ N þ 1:

ð7cÞ

B. One-Point Formula

One-Point formulation is quite similar to trapezoidalintegration with the difference that, instead of thecomplete integrand in Eq. (4), which results in singu-larity at x ¼ r, the deflection angle in each segment isrepresented by a constant profile. The resulting inte-grand for each segment is then solved analytically.Thus,

δðrÞ ¼ −1π ·Z∞r

ΘðxÞ · dxffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 − r2

p

¼ −1π ·XNj¼1

�ZΔr

0

�Θj þΘjþ1

2

�·

dlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrj þ lÞ2 − r2i

q �;

ð8Þ

Here the deflection angle is constant over a segmentand it is computed as the average of the two neigh-boring data points. Note that l is the width of the seg-ment. Substituting rj ¼ ðj − 1Þ ·Δr and ri ¼ ði − 1Þ·Δr, Eq. (8) reduces to

δðriÞ ¼ −1π ·XNj¼1

�Z10

�Θj þΘjþ1

2

�

·dβffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ððj − 1Þ þ βÞ2 − ði − 1Þ2p �

; ð9aÞ

where β ¼ lΔr. After analytical integration, Eq. (9a)

can be recast in the form of Eq. (6), where

Dij ¼−12π · ln

�jþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 − ði − 1Þ2

pðj − 2Þ þ


p �

if j > i and j ≠ 2;

¼ −12π ·

�2þ ln

�jþ


pðj − 1Þ þ


p ��

if j > i and j ¼ 2;

¼ −12π · ln

�jþ


pðj − 1Þ þ


p �

if j ¼ i and i ≠ 1;

¼ 0 if j < i or j ¼ i ¼ 1: ð9cÞ

The analytical integral in Eq. (9a) is undefined in thefirst segment, where the deflection angle is assumedto vary linearly to yield Di¼1j¼1 ¼ 0.

C. Two-Point Formula

The two-point formula follows the same procedure asthe one-point formula except that the deflectionangle is approximated by a linear profile in eachsegment. Thus, Eq. (4) results in the followingapproximation:


δðrÞ ¼ −1π ·Z∞r

ΘðxÞ · dxffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 − r2

p

¼ −1π ·XNj¼1

�ZΔr

0

�Θj þ l ·

ðΘjþ1 −ΘjÞΔr

�

·dlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðrj þ lÞ2 − r2i

q �:

ð10aÞ

After analytical integration, Eq. (10a) can be recastin the form of Eq. (6), where

Dij ¼1π · ðAi;j − Ai;ðj−1Þ − j · Bi;j þ ðj − 2Þ · Bi;ðj−1ÞÞif j > i and j ≠ 2;

¼ 1π · ðAi;j − j · Bi;j − 1Þ if j > i and j ¼ 2;

¼ 1π · ðAi;j − j · Bi;jÞ if j ¼ i and i ≠ 1;

¼ 0 if j ¼ i ¼ 1 or j < i ð10bÞ

Ai;j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 − ði − 1Þ2

q−


q;

Bi;j ¼ ln�

jþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 − ði − 1Þ2

pðj − 1Þ þ


p �: ð10cÞ

D. Gauss–Hermite Product Polynomial Method

In this method, wherein the deflection angle data arerepresented by the product of Gaussian and Hermitepolynomials with the first seven odd polynomials,i.e., the 13th-order polynomial is the highest-orderpolynomial [34],

ΘðxÞ ¼�X6

p¼0

að2pþ1Þ ·X6q¼0

Hð2pþ1Þ;ð2qþ1Þ · yð2qþ1Þ�· e−x2 :

ð11Þ

Substituting Eq. (11) into the discrete form of Eq. (4),we obtain

δðriÞ ¼−1π ·

Z∞ri

ΘðxÞ · dxffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 − r2i

q

¼ −1π ·

Z∞ri

�X6p¼0

að2pþ1Þ ·X6q¼0

Hð2pþ1Þ;ð2qþ1Þ · xð2qþ1Þ�

· e−x2 ·dxffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 − r2i

q ⇒ δðriÞ

¼ −1π ·

�X6p¼0

að2pþ1Þ ·X6q¼0

Hð2pþ1Þ;ð2qþ1Þ · Iq;i

�; ð12Þ

where Iq;i ¼R∞rixð2qþ1Þ · expð−x2Þ · ðx2 − r2i Þ1=2 · dx has

an exact analytical solution as given by Rubinsteinand Greenberg [34]. After algebraic manipulation,Eq. (12) can be recast in the form of Eq. (6), where

Dij ¼−1

π3=2·X6p¼0

0BB@

1ð2pþ1Þ!·2ð2pþ1Þ ·

P6q¼0

�Hð2pþ1Þ;ð2qþ1Þ · ðj − 1Þð2qþ1Þ

·ðΔrÞð2qþ2Þ

�

·P

6q¼0 Hð2pþ1Þ;ð2qþ1Þ · Iq;i

1CCA if j ¼ 1 or N þ 1;

¼ −1

π3=2·X6p¼0

0BB@

1ð2pþ1Þ!·2ð2pþ1Þ ·

P6q¼0

Hð2pþ1Þ;ð2qþ1Þ ·

�2 · ðj − 1Þð2qþ1Þ

�·ðΔrÞð2qþ2Þ

!

·P

6q¼0 Hð2pþ1Þ;ð2qþ1Þ · Iq;i

1CCA if 1 < j < N þ 1: ð13aÞ

The above analysis shows that all techniques canbe recast in the form of Eq. (6), which can be repre-sented by the matrix equation δ ¼ D ·Θ where δ andΘ are column vectors of length N þ 1 and D is theinversion coefficient matrix of size ðN þ 1Þ × ðN þ 1Þ,known a priori depending upon the inversiontechnique.

Figure 1 compares the inversion coefficient (Di¼10;j)for different algorithms. Results show that theDij forvarious inversion techniques are similar and followan asymptotic trend with increasing radius exceptfor the GHPP method. The inversion coefficients forthe GHPP method depend on the data spacing and,hence, the accuracy of inversion is compromisedas discussed in Section 3. The asymptotic trend


signifies diagonal dominance in the inversion coeffi-cient matrix, which means that the regions localizedto the point under consideration (i ≈ j) have greaterweight than regions farther away (j > i) from it. TheGHPPmethod, however, results in a significantly dif-ferent inversion coefficient profile. All techniques ex-cept the GHPP method result in an upper triangularform of the inversion coefficient matrix, indicatingthat the interior points (j < i) have no effect on theinversion. In contrast, the GHPP method employsa single curve to fit the entire dataset; thus, the in-version coefficient matrix is a full matrix and data onboth sides of the point under consideration affect theinversion.

3. Inversion Accuracy

In this study, a test profile of refractive-index differ-ence taken as the sum of Gaussian distributions gi-ven by Eq. (14) is chosen:

δðrÞ ¼�e−ðr−1Þ2 þ e−ðrþ1Þ2

�: ð14Þ

The corresponding line-of-sight deflection angle dataare obtained using a semianalytical numerical inte-gration procedure outlined by Shenoy et al. [38]. In-tegration was carried out with 1000 segments toobtain deflection angle data with high accuracy.However, only 51 equispaced deflection angle datapoints were used as input to verify the accuracy ofthe proposed inversion algorithms. Figure 2 showsthe refractive-index difference profile and the corre-sponding deflection angle profile used for algorithmverification.Figure 3(a) compares refractive-index difference

profiles reconstructed using different algorithms dis-cussed previously. Results show that all of the inver-sion algorithms produce accurate results, except forthe GHPP method, which incurs large errors in thecenter region (r < 1:0). The inversion error defined as

the difference between the original [Fig. 2(a)] and re-constructed profiles is shown in Fig. 3(b) for all butthe GHPP method. Results show the highest inver-sion error for the Simpson’s 1=3rd formula and theleast inversion error for the two-point formula. Simp-son’s 1=3rd rule accuracy was dependent on thetreatment of the singularity of the improper integral.The authors investigated open and semiopen ex-tended formulation of the trapezoidal and Simpson’s1=3rd rule [39] to avoid the singularity; however, theinaccuracy in the reconstructed field rendered thesetreatments unusable.

The GHPPmethod employs a single curve to fit theentire dataset. Thus, the curve-fit accuracy affectsthe inversion accuracy. The curve fit is generatedfrom a linear combination of Gauss–Hermite polyno-mial basis functions varying asymptotically with ra-dius and, hence, they provide accurate curve fit onlyif data show asymptotic behavior with radius (at

Fig. 1. Comparison of inversion coefficient (D10j) for different al-gorithms (N ¼ 25).

Fig. 2. Test profiles of (a) refractive-index difference (to be recon-structed) and (b) transverse deflection angle (input data).


r ≈ π). This requirement can be imposed by adding abuffer region (dummy data points) to the profile inFig. 2(b), such that Θ ¼ 0 for 3:5 < r < 7:0. Figure 4shows that the GHPP method with buffer nearlyeliminates the curve-fit error in the deflection angleprofile. An increase in the number of input datapoints will, however, increase the computational ef-fort for the GHPP method with buffer.Figure 5 shows a vast improvement in the inver-

sion accuracy of the GHPP method with buffer suchthat the inversion error for the GHPP method is si-milar to that of the two-point formula. Evidently, thereconstruction accuracy of least-squares approxima-tion methods depends largely on the shape of the in-put profile. The inversion accuracy is higher if theinput profile is asymptotic over a sufficiently largeregion. If experimental data are not asymptotic,dummy data must be appended to improve the inver-sion accuracy. Methods employing Fourier series ororthogonal polynomials (namely, Laguerre, Hermite,

Legendre, Chebyshev, Jacobi, Gegenbauer) to ap-proximate data by least-squares curve fit suffer froma similar drawback and they are accurate only forcertain types of input data profiles. GHPP methodalso has severe drawback if the maximum radiusis very small or large. The radial coordinate must benormalized such that the maximum value is around‘π’. The above quantitative comparison supplemen-ted with results for several other input data profilesnot shown here, suggests that the two-point formulais by far the best method. GHPP method provides ac-curacy comparable to two-point method only if inputdata are buffered and/or the radial coordinate is nor-malized to maximum value around ‘π’.

4. Error Propagation

So far, we have considered only a smooth input dataset. In practice, however, experimental data would becorrupted by unavoidable random noise inthe measurements. Therefore, knowledge of noisedamping or magnification in the inversion processis important to assess the reliability of the algorithm.In this study, the well-established uncertainty analy-sis principle [40] is used to compute the error prop-agation by the inversion algorithm. The elementalerror source is the measurement error at discretedata points. This error source is propagated by thenumerical procedure employed to evaluate theAbel integral using the data reduction Eq. (6). There-fore, the total uncertainty in the inverted function isgiven by

UδðiÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNþ1

j¼i

��∂

∂Θj

�XNþ1

j¼i

Dij ·Θj

��2·U2

Θj

�vuut

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNþ1

j¼i

ðDij ·UΘjÞ2

vuut ; ð15Þ

Fig. 3. Inversion accuracy of different algorithms: (a) invertedprofile and (b) error in inversion.

Fig. 4. Effect of buffer zone on curve-fit error using GHPP.


Here, UδðiÞ and UΘjare. respectively. the standard

errors in the inverted function (refractive-indexdifference) and deflectometric measurements. Equa-tion (15) shows that the propagated error is indepen-dent of the magnitude of the experimental data andonly depends on the standard error profile. Thus, er-ror propagation analysis for various standard errorprofiles can be performed without the knowledgeof the actual deflection angle profile although, inpractice, the error profile would depend upon themeasured deflection angle profile.For uniform standard error in input data, Uθi ¼

1:0, Figure 6 shows the propagated error, which ishighest at the center (r ¼ 0) for all of the algorithms.The error propagated for the GHPP method (withbuffer) is much larger compared to the other algo-rithms, of which the two-point formula is the best.

For the GHPP method, although the buffer zone im-proves the inversion accuracy (Fig. 5), it has no effecton the propagation of the measurement error.Figure 7(a) shows a random distribution error pro-file. The corresponding propagated error profiles fordifferent algorithms are shown in Fig. 7(b). Algo-rithms other than the GHPP method show a similartrend in error propagation with no clear advantage ofone over the other; however, the error amplificationof the GHPP method is much larger than all theother algorithms. The maximum propagated erroroccurs at the center (r ¼ 0) for all the algorithms.Figure 8 shows a Gaussian distribution standard er-ror profile and the corresponding error propagationprofiles for different algorithms. Again, the erroramplification for the GHPP method is much highercompared to the other algorithms. The maximumpropagated error location for the GHPP method isat the center whereas, for the other algorithms,Fig. 5. Inversion error with and without a buffer zone in the

GHPP method.

Fig. 6. Error propagation for different algorithms using a uni-form error in input data.

Fig. 7. Error propagation for different algorithms: (a) random er-ror in input data and (b) propagated error profiles.


the peak propagated error location coincides with thepeak in the standard error profile.

5. Concluding Remarks

Four Abel inversion techniques using deflectometricmeasurements are compared for accuracy and propa-gation of measurement error. These techniques coverexisting approaches for Abel inversion and include(1) numerical integration using Simpson’s 1=3rdrule, (2) the semianalytical method with piecewiseapproximation by a constant or linear profile, and(3) least-squares approximation by GHPP. All thesemethods can be recast in a linear operator form thatconverts line-of-sight deflection angle data into a lo-cal refractive-index difference field. Results showthat numerical and semianalytical techniques yieldgood inversion accuracy, with the two-point formulaproviding the best accuracy. For the GHPP method,the function value in a segment uniquely determinesits values over the entire interval. Thus, this method

does not perform well, and it requires a buffer regionof dummy data to impose an asymptotic nature in theinput data. Error propagation analysis reveals thatSimpson’s 1=3rd rule, the one-point formula, andthe two-point formula perform similarly, althoughthe two-point method is superior. The maximum pro-pagated error location depends on the nature of thestandard error input profile for these three methods.The propagated error for the GHPP method is muchlarger and is unaffected by the buffer zone. Thus,Abel inversion of deflectometric data should be car-ried out using the two-point formula to achieve highaccuracy with minimum propagation of measure-ment error.

This work was supported in part by the PhysicalSciences Division of NASA’s Office of Biologicaland Physical Research under grant NNC05AA36A.

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abel inversion of deflectometric data: comparison of accuracy and noise propagation of existing...

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