1. INTRODUCTIONInterferometers with low-coherence illumination arewidely used for three-dimensional (3-D) reconstruction ofsurface topography,1,2 in surface profilometry,3 and for in-vestigation of subsurface structure along the penetrationdepth of the incident light in optical coherencetomography.4,5 The basic principle of this technique is tolocate the maximum position of low-coherenceinterference-fringe envelopes obtained under controllabledisplacement of the reference mirror in the interferom-eter. Local fringe amplitude can be evaluated with thewell-known phase-shifting technique1,3 or with fringe am-plitude demodulation by fringe signal squaring.2,6 Bothof these approaches are based on nonlinear transforma-tions applied to determinate interferometric data andtherefore methods are generally nonoptimal, especiallywhen interference fringes are distorted by noise.
A new approach to optimal nonlinear fringe processingwas recently described in detail7 and applied successfullyto 3-D surface-relief reconstruction,8,9 random-tissueevaluation,10 and fringe analysis in spectralinterferometry.11 Our approach is based on the definitionof the variation of interference-fringe intensity as a non-linear transform of the assumed dynamic evolution of thesystem, where system parameters such as fringe enve-lope, frequency, and phase are described by discrete sto-chastic differential equations. The approach allows real-ization the optimal recurrence prediction-correctionalgorithms for dynamic fringe processing.
In analyzing low-coherence interference fringes, thefringe envelope is what must mainly be evaluated. Itwill be shown that the general methodology7 can be sim-plified in this case and reduced to the well-known Kalmanfiltering method.12 In this paper, the discrete Kalman fil-tering method is investigated with application to low-coherence interference-fringe demodulation. It has beenrealized in a vectorial implementation for 3-D rough-surface-relief reconstruction.
2. THEORETICAL BACKGROUNDWe consider the simplified scheme of the low-coherenceinterferometer shown in Fig. 1. In this scheme, the low-coherence light source with mean frequency n0 and spec-tral band Dn ! n0 is used. The emitted light is split by ahalf-mirror and illuminates the reference mirror and theobject surface. The reference mirror changes only thephase of the incident light wave by a factor f1 . Themeasured surface with a reflectance r is placed at the po-sition z 5 D/2, where D is the optical path difference(OPD) in the interferometer. In the case of low-coherence illumination one can express the values of theelectric field of the reference mirror and the measuring in-terfering waves at the fixed observation point (x, y), re-spectively, in the following way,
E1i~t ! 5 ai~t !exp~ j2pn it 1 f1!, (1)
E2i~t ! 5 rai~t 1 t!exp@ j2pn i~t 1 t!#, (2)
where the index i denotes various wave trains with enve-lopes ai(t) and frequencies n i inside the narrow spectralband Dn, f1 is the initial phase that can be omitted with-out loss of generality, t 5 D/c 5 2z/c is the measuring-wave delay time due to OPD, and c is the velocity of light.The wave trains ai(t) are assumed to be incoherent witheach other, and the light intensity at the interferometeroutput is determined by averaging over the wave-trainsensemble as
where I0 is the background component independent of t,V(t) 5 2rg (t) is the fringe visibility function, and
g ~t! 5 ~1/I0!^ai~t !ai~t 1 t!& i (4)
is the normalized coherence function.
2004 Optical Society of America
Gurov et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 243
The expression for the normalized interferometric sig-nal s, which is proportional to the intensity, s 5 mI, canbe conveniently described as a function of the reflectingsurface-relief height z in the form
s~z ! 5 1 1 A~z !cos F~z !, (5)
where A(z) is the wave-train envelope’s normalized auto-correlation function, F(z) 5 4pz/^l& is the fringe phase,and ^l& 5 c/n0 is the averaged value of the light wave-length. In the case of a Gaussian power spectrum of thelight-source radiation, the autocorrelation function A(z)is of Gaussian form as well, being the inverse Fouriertransformation of the power spectrum following the well-known Wiener–Khinchin theorem. The signal, Eq. (5), isillustrated in Fig. 2.
If the measuring wave, Eq. (2), is randomly scattered,then the disturbance of the useful component of Eq. (5)should be taken into account, and this component must bepresented in the form
s~z ! 5 A~z !cos f~z ! 1 n~z !, (6)
where f(z) 5 F(z) 1 c (z), A(z) and c (z) are randomlyvaried functions, and n(z) is the observation noise includ-ing stochastic background variations. Note that c is arandomly variable value over the ensemble of speckles,i.e., over the (x, y) domain, being fixed for each separatespeckle. The dependence c (z) is introduced to take intoaccount possible random phase variations, e.g., intro-duced by an inaccurate positioning of the reference mirrorthat is assumed unified for all (x, y) points. The maxi-mum value of the signal, Eq. (6), is achieved at the pointof zero OPD, D 5 0, provided by movement of the refer-
Fig. 1. Schematic of a low-coherence interferometer.
Fig. 2. Normalized low-coherence fringe signal.
ence mirror. The location of such zero points for all ob-servation points (x, y) allows us to find the values z5 z(x, y) that describe the surface relief being evalu-ated.
The stochastic interferometric signal, Eq. (6), can bedefined by a random vector of useful parameters u5 (A, f )T, and the evolution of its components may bedescribed by means of stochastic differential equations.7
One of the possible descriptions of the dynamics of thestochastic system is based on the Langevin equation inthe vectorial form
du/dz 5 f ~u ! 1 w~z !, (7)
where f (u ) is a determinate term; w(z) is a stochasticvector with uncorrelated components, so that ^wi(z)&5 0, ^wi(z)wj(z8)& 5 (Gi/2)d ijd (z 2 z8), where Gi is the
spectral density of ith noise component; d ij is the Kro-neker symbol, and d (z) is the delta function. The scalarLangevin equation is used, as is well known, to describethe rate of a Brownian particle motion under the condi-tion f(u) 5 2au, where a is a constant that defines theattenuation of the particle’s rate of motion.
As applied to the description of the low-coherenceinterference-fringe parameters, Eq. (6) defines the varia-tions subject to a random disturbance of the componentsof the vector of the parameters, u 5 (A, f )T, namely, thefringe envelope A(z) and fringe phase f(z), as follows:
dA/dz 5 22s~z 2 z0!A~z ! 1 wA~z !, (8)
df/dz 5 4pu0 1 wf~z !, (9)
so that in Eq. (7), f (z, u ) 5 @22s(z 2 z0)A, 4pu0#T. Itis evident that the solutions of undisturbed Eqs. (8) and(9) under the condition w(z) 5 0 correspond, respectively,to the Gaussian form of the fringe envelope, namely,A(z) 5 exp@2s(z 2 z0)
2#, and the linear fringe phasefunction F(z) 5 4pu0z. In Eqs. (8) and (9), z0 is the en-velope’s maximum position, s [ 2/Lc
2, Lc is the coherencelength of the light, and u0 is the fringe frequency. Thefringe frequency is a priori known, being defined by theaveraged wavelength ^l&. The stochastic components inEqs. (8) and (9) can be defined by the Langevin equation,
dw/dz 5 2aw~z ! 1 w0~z !, (10)
where w0(z) is the forming uncorrelated (white) noiseand w(z) is the stochastic data realization with a limitedspectrum7 defined by the parameter a. The smaller theparameter a, the narrower the spectral band. Equations(8)–(10) define the dynamic evolution of the system oflow-coherence interference fringes.
A few selected solutions of Eq. (8) are presented in Fig.3(a). This equation describes the stochastic process thatgenerates all possible envelope realizations under somefixed values of s, z0 , and forming-noise parameters. Thedispersion of the envelope values within the stochasticprocess is proportional to the dispersion sw
2 of the formingnoise w0(z) in Eq. (10) under the zero initial condition forEq. (8). It was found that the lower the dispersion sw
2 ,the lower the envelope’s maximal value averaged over theensemble of the envelope realizations. Note that solu-tions of Eq. (8) exist that correspond to a negative enve-lope sign.
244 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Gurov et al.
Figure 3(b) shows three possible solutions of Eq. (9)with different values of fringe frequency u0 . In these so-lutions, the phase’s forming noise wf(z) corresponds todifferent values of the coefficient a and of the dispersionof the initial forming noise w0(z) in Eq. (10). The follow-ing relations among the values mentioned above corre-sponding to the upper-to-lower curves in Fig. 3(b) were ac-cepted: u01 . u02 . u03 ; a1 , a2 ; a2 5 a3 ; sw1
2 , sw32
, sw22 . It is evident that the signal modeled under the
presented realizations of the fringe parameters is close tothe form of the typical low-coherence fringe signal (seeFig. 2) modified by the influence of noise.
The main problem is to estimate the value A(z) in Eq.(6) by processing the observed stochastic signal s(z) andusing the system description provided by Eqs. (8)–(10).In practice, this problem is solved by discrete signal-processing methods, in particular, by the discrete Kalmanfiltering method, which is generally based on a descrip-tion of the stochastic system by differential equations7
[see also Eq. (7)] in discrete form when the interferomet-ric signal is presented by a discrete-sample series at thepoints zk 5 kDz, k 5 0, 1,..., K, and Dz is the discretiza-tion step.
3. DISCRETE KALMAN FILTERING OFLOW-COHERENCE INTERFERENCE FRINGESThe discrete Kalman filtering procedure is based on pre-diction of the signal value to the next discretization stepk, taking into account all the statistical information avail-able before this step. The difference between the ob-served and the predicted values is used for dynamic cor-rection of the signal parameters. Discrete linear Kalmanfiltering is usually defined (see, e.g., Ref. 13) by the vec-torial observation equation
s~k ! 5 H~k !u ~k ! 1 n~k ! (11)
and the system equation in the state space
u ~k ! 5 F~k 2 1 !u ~k 2 1! 1 w~k !. (12)
Fig. 3. (a) Fringe envelope realizations and (b) unwrappedphase realizations modeled by Eq. (7).
In Eqs. (11) and (12), H(k) is the observation matrix, u(k)is the vector of the parameters, n(k) is the observationnoise, F(k) is the transition matrix, and w(k) is the sys-tem’s forming noise.
Taking into account Eqs. (6) and (8)–(10), one can de-fine the observation and transition matrices in Eqs. (11)and (12) for a discrete scalar signal s(k) 5 s(zk)5 s(kDz) as follows:
H~k ! 5 @cos f~k ! 0#, (13)
F~k ! 5 F1 2 2s~k 2 k0!Dz 2 aA 0
0 1 2 afG ,
(14)
where the coefficients aA and af define the power spec-trum cutoff frequency of the stochastic components inEqs. (8) and (9), taking into account Eq. (10).
It can be shown12 that the a posteriori optimal estimateof the vector of the parameters, u(k), provided by discreteKalman filtering is expressed as
u ~k ! 5 F~k 2 1 !u ~k 2 1 !
1 P~k !@s~k ! 2 H~k !F~k 2 1 !u ~k 2 1 !#,
(15)
where the first term is considered to be the predictedvalue, the second one is the a posteriori correction value,and the Kalman-filter amplification factor P(k) is calcu-lated from Eq. (A4) (see Appendix A). So the local fringeenvelope A(k) as the first scalar component of the vectoru(k) can be calculated from Eqs. (12)–(15) as follows:
A~k ! 5 A~k/k 2 1 !
1 PA~k !@s~k ! 2 A~k/k 2 1 !cos f~k 2 1 !#,
(16)
where A(k/k 2 1) is the predicted envelope value fromthe (k 2 1)th step to the kth step,
It is important to emphasize that the known determinatephase component F(k) 5 4pu0kDz is omitted for sim-plicity in Eqs. (16) and (18) and later.
The amplification factor PA(k) in Eq. (16) can be foundfrom modification of Eq. (A4) to the case of the scalarfringe signal s(k) as follows:
PA~k ! 5 sA2 cos f~k 2 1 !@sA
2 cos2 f~k 2 1 ! 1 sn2 #21
5 ~ sA2 /sn
2 !cos f~k 2 1 !, (19)
where sA2 is the dispersion of fringe-envelope prediction
error, sn2 is the dispersion of the observation noise, and sA
2
is the dispersion of the a posteriori fringe-envelope esti-mate. The ratio sA
2 /sn2 can be regarded in some cases as
a constant value sA2 /sn
2 5 P0 that is determined a priori,taking into account the assumed useful local variations of
Gurov et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 245
the fringe envelope with respect to observation noise. Fi-nally, local fringe amplitude is calculated as
A~k ! 5 A~k/k 2 1 ! 1 P0 cos f~k 2 1 !
3 $s~k ! 2 A~k/k 2 1 !cos f~k 2 1 !%. (20)
Kalman filtering allows estimation of the local fringephase by using Eq. (18) as well. The amplification factorPf(k) in Eq. (18) cannot be calculated directly by usingEq. (A4) because of the nonlinear dependence of the sig-nal on its phase. To overcome this difficulty, one can usethe linearization method; that is, in Eq. (11) one can as-sume that
H~k ! 5 @cos f~k 2 1 !,2A~k !sin f~k 2 1 !#,
u 5 ~A, df !T, f~k ! 5 f~k 2 1 ! 1 df~k !.
After such linearization, one can use Eq. (A4). As a re-sult, one can obtain
where sf2 is the dispersion of the fringe-phase prediction
error.
4. INTERFEROMETRIC SYSTEM ANDSIMULATION RESULTSThe Kalman filtering method described above was appliedto 3-D surface-relief reconstruction to verify the stabilityand accuracy of the method. Interferometric data werecalculated (see Appendix B) for the optical scheme of thelow-coherence interferometer shown in Fig. 4. In thisscheme, light radiation from the low-coherence light
Pf~k ! 5 2sf
2 A~k/k 2 1 !sin f~k 2 1 !
sA2 cos2 f~k 2 1 ! 1 sf
2 A2~k/k 2 1 !sin2 f~k 2 1
source, 1, after the collimator, 2, illuminates the interfer-ometer. The specimen, 3, is placed in one interferometerarm; the reference mirror, 4, is mounted in another one.The output objective includes the lenses L1 and L2 , whichimage the specimen surface with a low-coherence back-ground to the observation plane, 5, of the two-dimensional (2-D) array of sensitive elements.
The input data about the rough-surface relief of thespecimen, 3, were obtained from recently published re-search results.14 In that research, the metal specimenhad been used with the surface deformed by a stamp inthe form of the digit ‘‘4.’’ The initial surface relief is pre-sented in Fig. 5(a) as a gray-level map and in Fig. 5(b) asa 3-D plot, which was recovered by using the fringe-projection technique with application of a few additionalprocessing procedures14 to enhance initial relief data. Inother words, data were used in the form of the knownrelief-height function z 5 z(x, y) as standard input data
for the Kalman filtering algorithm. The lateral dimen-sions of the useful surface area are equal to 33 2 mm, and the surface-relief-deviation range is equalto 180 mm. The surface-relief deviations are presented inFig. 5 and later for a convex relief form that is more vi-sual in illustration.
These initial data have been used to recalculate thesurface relief dynamically by the Kalman filtering methodand to compare the initial data with the data recovered bythis method. The video frame set was modeled for differ-ent positions of the reference mirror, 4, in the opticalscheme in Fig. 4 by the method described in Appendix B.Mathematical modeling of the low-coherence fringe-formation process has been used to verify the Kalman fil-tering algorithm properties without the influence of pos-sible unregulated errors introduced by the optical system.To decrease the amount of data that have to be processed,the surface-relief-height scale was twice reduced and arelief-height range equal to 90 mm was accepted.
The low-coherence light-source spectrum was assumedto be of Gaussian form. The wavelength step Dl usedwas equal to 2.5 nm within the spectral band B5 20 nm. It can be easily shown that continuous spec-trum replacement with the wavelength set of nDl is cor-rect under the condition Dl , l2/D. In the modeling ex-ample, the OPD D must be less than 200 mm, whichcorresponds to a surface-relief height up to 100 mm andsatisfies the reduced surface-relief-height range from 50to 90 mm mentioned above. The examples of low-coherence patterns obtained for the surface-relief heightsof 45, 60, 75, and 90 mm are shown in Figs. 6(a), 6(b), 6(c),and 6(d), respectively.
The nonlinear Kalman filtering algorithm based on themodel Eq. (5) and formulas Eqs. (16)–(18) was used to cal-culate the fringe envelope dynamically and to find the
sn2 , (21)
! 1
Fig. 4. Schematic of the interferometric setup.
246 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Gurov et al.
fringe envelope’s maximum positions by recurrence step-by-step processing the pixel evolution signals obtained ateach observation point of 2-D sensitive array from frameto frame while the reference mirror position changed.
With change of the OPD in the interferometer by move-ment of the reference mirror step by step in the range ofcoherence length, the signal registered at each ith pixel ofvideo frames is changed on the k video frame number, asillustrated in Fig. 7 for two selected pixels of the sensitivearray. Note that the interference-fringe spacing is equalto ^l&/2; using a superluminescent diode with ^l&5 820 nm and coherence length 15 mm, we obtain ap-proximately 37 fringes inside the fringe-envelope FWHM.The deformed signal-envelope form in Fig. 7(b) is ex-plained by the small nonzero surface area observed by thesingle sensitive element, which shows that two parts ofthis small area with different heights give rise to one sig-nal that results in the overlapped signal Fig. 7(b) with ex-tended envelope width.
Calculations were performed for the frame format of1000 3 1000 pixels in the observation plane, i.e., the Kal-man filtering method was applied to 106 parallel channelsof pixel evolution. The full number of reference-mirrorsteps (number of frames or discrete samples for eachchannel) was equal to K 5 801 in the relief-height rangefrom 50 to 90 mm.
Fig. 5. Initial 3-D rough-surface relief presented as (a) a gray-level map and (b) a 3-D relief reconstruction shown as a convexform.
The fringe-visibility maxima were observed at thepoints of the sensitive-array plane (j, h) at different posi-tions of the reference mirror in the schematic of Fig. 4.The reference-mirror position under the fringe-envelopemaximum location determines the relief height at the cor-responding point (x0 , y0) of the surface to be tested.
The result of the surface-relief reconstruction is pre-sented in Fig. 8(a). The stability of the low-coherencefringe-envelope calculations by the Kalman filteringmethod was confirmed. No calculation instability wasfound in any channel. The difference between initial andrecovered surface relief is illustrated in Fig. 8(b). Thegray-level map scale was enlarged in Fig. 8(b) to presentclearly the spatial error distribution. In this gray-levelmap, the lower the brightness is, the lower the relief re-construction error is. Bright contours around the surfacedeformation area, i.e., the increased relief reconstructionerrors, are explained by the limited height range from 50to 90 mm processed; the limited height range is why onlya part of the extent of the fringe envelope could be ob-tained near the lower height range border of 50 mm. Themaximal deviation from the initial relief (see Fig. 5), i.e.,the PV error module, did not exceed 5 mm over the fulluseful surface-relief area, which corresponds to a relief re-construction rms error estimate close to 1 mm.
Figure 9 shows the relief reconstruction error histo-gram, i.e., the estimate of the error probability densityfunction, obtained inside the useful relief area after re-moval of the increased reconstruction errors pointed outabove and indicated by the bright contours in Fig. 8(b).The histogram of Fig. 9 was calculated over an ensembleof 22,801 useful relief points with error value resolution of0.25 mm. The histogram curve is non-Gaussian, which isinherent in statistical properties of random-coherencelight fields.15
When one is obtaining low-coherence fringes, the fringefrequency may vary because of the nonuniformity of theaxial translation stage scan and the influence of externalvibrations. We have processed such a distorted low-coherence fringe signal to verify in addition the stabilityof the Kalman filtering calculation procedure describedabove.
The signal example with the fringe frequency changedessentially inside the envelope is shown in Fig. 10(a). Inspite of such essential signal-frequency deviations, theKalman filtering procedure based on use of Eqs. (18) and(21) allows us to obtain a dynamic estimate of the fringephase. The fringe’s unwrapped phase recovered by theKalman filter is shown in Fig. 10(b). This curve containsinformation about the nonlinearity of the fringe phase,which can be taken into account, in particular, to estimatethe fringe envelope accurately by rescaling the horizontalcoordinate axis as provided by the linear unwrapped-phase-change law.
If the surface to be tested is reflecting without scatter-ing and the signal-to-noise ratio is high enough to identifythe central fringe inside the fringe envelope, it is possibleto use the information about the low-coherence fringephase, and the axial resolution can be essentially in-creased by locating the central fringe and calculating thelocal fringe phase. In this case, the axial resolution is es-timated as
Gurov et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 247
Fig. 6. Low-coherence images obtained for different reference-mirror positions in the schematic of Fig. 4.
where d F is the unwrapped fringe phase error. It isclearly seen from Eq. (22) that the information aboutfringe phases provides increased axial resolution typicallyup to two orders of magnitude. It corresponds to smallerrelative error of the surface-relief evaluation with respectto conventional low-coherence interferometers with evalu-ation of only the fringe envelope.
5. DISCUSSIONThe discrete Kalman filtering method described abovehas been used to obtain the a posteriori estimate of thevector u(z) by using the Langevin equation [Eq. (7)]which is applied to the description of the evolution of thelow-coherence fringe envelope and phase.
It is known that all available information about thevector of the parameters, u(z), is contained in the a pos-teriori probability density function ps(z, u), which can beobtained from the well-known equation first achieved byStratonovich,7
]ps~z, u!
]z5 T$ ps~z, u!% 1 @F~z, u! 2 ^F~z, u!&#ps~z, u!,
(23)
where T$ p(z, u)% denotes the Fokker–Planck–Kolmogorov operator,
F~z, u! 5 ~1/Gn!@sobs~z ! 2 s~z, u!#2
248 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Gurov et al.
is a derivative by the variable z from the likelihood-function logarithm, Gn is the (uniform) power spectraldensity of the observation noise n(z), sobs(z) is the ob-served fringe signal, ^F(z, u)& is the mathematicalexpectation value, and ^F(z, u)& 5 *RnF(z, u)p(z, u)du,calculated over n-dimensional Euclidian space Rn.
Solution of Eq. (23) provides the optimal a posteriori es-timate of the stochastic vector u(z), including the cases ofnonlinear signal models s(z, u).7 In the general case,the Eq. (23) solution is very complex. If the probabilitydensity functions of the observation noise and the predic-tion error can be assumed to be of Gaussian kind, one canobtain a dynamic estimate and covariation matrix of vec-tor u(k) by the discrete Kalman filtering procedure.Thus the discrete Kalman filtering method used and con-sidered in this paper belongs to the simplified Stratonov-ich methodology for the case of Gaussian probability den-sity functions and the linear kind of Eq. (7).
It is interesting to compare the Kalman filteringmethod with other known methods, first of all withWiener filtering. Such consideration is given in Kal-man’s original paper12 and discussed in detail in otherpublications (see, e.g., Ref. 16). The main points of thisconsideration are the following.
As is known, conventional Wiener filtering theory is ap-plicable for stationary signals, in other words, for signalswith invariable statistical properties over the ensemble of
Fig. 8. (a) Surface-relief reconstruction and (b) relief recon-struction error presented by gray-level maps.
realizations and within a single realization (ergodic prop-erty). The low-coherence fringe-formation process ismore complex and cannot generally be classified as a sta-tionary ergodic process. The Kalman filtering methodovercomes this difficulty as a result of its locally variableparameters such as the filter amplification factor P(k) inEq. (15), which generally depends on the local statisticalproperties of the signal and noise. In the stationary case,the Kalman filter with fixed parameters provides thesame characteristics as the Wiener filter.
Wiener filtering theory has been developed mainly forlinear systems; this is why the fringe phase can not be de-modulated directly by the Wiener filter. For this pur-pose, one should use an additional nonlinear transformsuch as arctangent transformation, which is generallynonoptimal with respect to rms error minimization. Inaddition, the recovered phase is wrapped, and calcula-
Fig. 9. Experimental estimate of the probability density func-tion of the surface-relief reconstruction error.
Fig. 10. (a) Low-coherence fringes with variable frequency and(b) unwrapped fringe phase recovered dynamically by the dis-crete nonlinear Kalman filtering method.
Gurov et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 249
tions are nonstable under the influence of noise. Thenonlinear Kalman filter overcomes these problems by alinearization that is easy to provide at local discretizationpoints to demodulate the phase directly and by recurrencecalculations with an accumulation of local phase incre-ments to recover dynamically the unwrapped fringephase.
In our research, possible simultaneous random varia-tions of the fringe envelope, frequency, and phase weretaken into account; i.e., a generally nonstationary, nonlin-ear case is considered. Fringe-frequency variations (seeFig. 10) may be caused, e.g., by a nonuniform movementof the reference mirror in the interferometer. If one usesthe conventional single-sideband filtering method, fringefrequency should be strongly stable and known a priori.Apart from it, the filter spectral band should be set wideenough to transmit a signal with the extended spectralband caused by the signal-envelope width and by possiblefrequency and phase variations. It may decrease thesignal-to-noise ratio under the influence of an observationnoise of uniform spectral density. The Kalman filter istuned automatically to local fringe frequency and phasewithout any additional means such as a tracking single-sideband filter. Kalman filtering theory (in general, non-linear recurrence filtering theory7) proves that this filteris optimal with respect to rms error minimization underthe usually satisfied assumptions, including nonstation-ary nonlinear cases.
Finally, consideration of signals in the frequency do-main (within Fourier methods) involves the transition ofa signal with limited extent in the frequency domain.This is equivalent to convolution of delta functions (rep-resenting the spectrum of the unlimited cosine function)with a Fourier transform of the signal envelope. In thiscase, spectral resolution is limited by the envelope width.The Kalman filtering method is based on the mathemati-cal formalism of stochastic differential equations, and anysignal realization with a complicated locally varied spec-trum corresponding to the initial signal model (i.e., anysignal realization that is a solution of this differentialequation and is dynamically recognized by the Kalmanfilter as the most probable within the ensemble of possiblerealizations) is passed through the Kalman filter withoutany decrease of spectral resolution. In addition, in somecases conventional optimal filters such as matched filterscannot be realizable under the causality principle. TheKalman filter, being a feedback filter, is realizable in anycase.
6. CONCLUSIONSThe possibility of describing low-coherence interferencefringes by stochastic difference equations has been dem-onstrated. The nonlinear Kalman filtering method basedon stochastic signal description, being the simplified ver-sion of general nonlinear filtering methodology,7 is an al-ternative approach to low-coherence fringe processingwith respect to well-known conventional fringe-demodulation methods that are applicable primarily todeterminate signals such as the phase-shifting technique
or fringe-amplitude demodulation by fringe-signal squar-ing or to stationary signals assumed in the Wiener filter-ing method.
The Kalman filtering method belongs to the category ofparametric stochastic processing methods, in which thedynamic evolution of random signal parameters is evalu-ated in an independent-variable domain. The main ad-vantage of the Kalman filtering method consists in dy-namic optimal signal processing based on stochastic-difference-equation formalism as opposed to the well-known Fourier transformation method, which is based onthe integral transform. Using stochastic differentialequations, one can generally synthesize the optimal dy-namic processing algorithms for evaluating, in parallel,fringe background, envelope, frequency, and unwrappedphase7 with a high immunity to noise.
The advantages of nonlinear recurrence filtering are asfollows: optimal estimation of useful parameters that de-fine the fringe intensity nonlinearly; solving the phase-unwrapping problem automatically by recurrence calcula-tions; possibility of using the algorithm with changingsignal parameters and phase-shifting parameters in awide range because of the adaptive character of the algo-rithm; and high-speed processing under dynamic observa-tions.
The Kalman filtering method can be applied not onlyfor evaluating rough-surface relief but also for investigat-ing the internal structure of random tissues in optical co-herence tomography.
APPENDIX A: OPTIMAL DISCRETE LINEARKALMAN FILTERINGAccording to Eq. (12), the predicted estimate of the vectorof the parameters u (k) can be assumed in the simplestcase as F(k 2 1)u (k 2 1). The difference between ob-served and predicted signal values is v(k) 5 s(k)2 H(k)F(k 2 1)u (k 2 1). The estimate of the vectorof the parameters can be expressed as
u ~k ! 5 F~k 2 1 !u ~k 2 1 ! 1 P~k !v~k !, (A1)
where P(k) is the Kalman filter amplification factor.This factor must be optimized to minimize the influence ofthe observation noise n(k) in Eq. (11), taking into accountthe system variations w(k) in Eq. (12). Such optimiza-tion can be achieved by consideration of the covariationmatrix R(k) of the a posteriori estimate error of the vectorof the parameters. The matrix is expressed as12
R~k ! 5 @I 2 P~k !H~k !#Rpr ~k !@I 2 P~k !H~k !#T
1 P~k !RnP~k !T, (A2)
where Rpr(k) is the covariation matrix of the predictionerror of the vector u, Rn is the observation-noise covaria-tion matrix, and I is the unit matrix. The diagonal ele-ments of the R(k) matrix are the dispersions of the vectoru components; therefore these elements should be mini-mized by appropriate choice of the amplification factorP(k).
By differentiating the R(k) diagonal elements on theP(k) components in Eq. (A2), one can obtain the condition
250 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Gurov et al.
According to Eq. (12), Rw is the system noise covariationmatrix. Note that Eqs. (A4) and (A5) are independent ofthe input signal s(k).
The algorithm of the recurrence signal processing byoptimal linear Kalman filtering is defined by Eqs. (A4)–(A7):
u~k ! 5 F~k 2 1 !u~k 2 1 !
1 P~k !@s~k ! 2 H~k !F~k 2 1 !u~k 2 1 !#,
(A6)
R~k ! 5 @I 2 P~k !H~k !#Rpr~k !. (A7)
The amplification factor P(k) is determined in Eq. (A4).The a posteriori estimate is calculated according to Eq.(A6), and the covariation matrix of the a posteriori esti-mate error is evaluated by Eq. (A7).
APPENDIX B: COMPUTER MODELING OFTHE LOW-COHERENCEFRINGE-FORMATION PROCESSThe light intensity at the observation-plane point (j, h) isdetermined as
I~j, h, t ! 5 uE~j, h, t !u2, (B1)
where E(j, h, t) is the resulting radiation electrical field.As was mentioned above, the fringe intensity [see Eq. (3)]is defined by the coherence function, i.e., by the radiationautocorrelation function. This function corresponds tothe power spectrum of low-coherence light under theWiener–Khinchin theorem. Therefore the electric fieldE(j, h, t) can be interpreted as the superposition of thespectral components with complex amplitudesA(j, h, l). It can be easily shown that the spectral com-ponents at the wavelengths l i separated by the spectralstep Dl may be considered to be noncoherent with eachother if the intensity is averaged over the small time in-terval T . l2/cDl, where c is the velocity of light. Thisinterval is usually much smaller than the extent of thephotodetector’s temporal impulse response, and thereforethe light intensity can be calculated as the sum of inten-sities at the discrete wavelengths l i in the form
I~j, h! 5 (i51
n
uA~j, h, l i!u2, (B2)
where n 5 B/Dl and B is the low-coherence radiationspectral band.
Complex amplitudes A(j, h, l i) were calculated in Eq.(B2) for the optical schematic of Fig. 4 following the
geometrical-optics approach. The complex amplitudeA(j, h, l) at the observation-plane point (j, h) can be cal-culated as
A~j, h, l! 5 Ar~l! 1 (~x0 , y0!
A~j, h, x0 , y0 , l!,
(B3)
where A(j, h, x0 , y0 , l) is the complex amplitude of thewave l propagating from the investigated surface point(x0 , y0) and Ar(l) is the reference-wave’s complex ampli-tude independent of the coordinates (j, h) and determinedfor the schematic of Fig. 4 as
Ar~l! 5 C~l!exp@ j~4p/l!~ f1 1 f2 1 Dh !#, (B4)
where Dh is the reference-mirror shift and C(l) is thedistribution of the light source spectrum.
A 2-D discrete-point set was used for computer simula-tions in the spatial domain. For the rectangular elemen-tary cell in the observation plane, the corresponding rect-angles in the principal planes of the lens can be obtainedby means of the light that comes from the surface point(x0 , y0) to this cell. Therefore the complex amplitude ofthe wavelength l arriving from the investigated surfacepoint (x0 , y0) at the observation-plane point (j, h) can becalculated as
A~j, h, x0 , y0 , l!
5 A0~x0 , y0 , l!r~x0 , y0!V~j, h, x0 , y0!
3 exp@ j~2p/l!d~j, h, x0 , y0!#,(B5)
where r(x0 , y0) is the reflection coefficient at the point(x0 , y0), A(x0 , y0 , l) is the complex amplitude at the in-vestigated surface, which is obtained as
h(x0 , y0) is the surface relief deviation from the focusingplane of the lens L1 , and V(j, h, x0 , y0) is the part ofthe wave propagating from the investigated surface point(x0 , y0), which is collected by the optical system into theobservation-plane cell (j, h). The optical path lengthd(j, h, x0 , y0) from the investigated surface point(x0 , y0) to the observation-plane point (j, h) is calculatedas
d~j, h, x0 , y0! 5 d0 1 D1 1 d1 1 D2 1 d2 , (B7)
where d0 5 @( f1 1 h)2 1 (x1 2 x0)2 1 ( y1 2 y0)2#1/2 isthe distance from the surface point to the lens L1 point;d1 5 @( f1 1 f2)2 1 (x1 2 x2)2 1 ( y1 2 y2)2#1/2 is the dis-tance from the lens L1 point to the lens L2 correspondingpoint; phase components D1 5 2(x1
2 1 y12)/f1 and D2
5 2(x22 1 y2
2)/f2 are introduced by lenses L1 and L2 , re-spectively; d2 5 @ f2
2 1 (j 2 x2)2 1 (h 2 y2)2#1/2 is thedistance from the lens L2 point to the observation-planepoint; and (x1 , y1) and (x2 , y2) are the principal-planepoints of lenses L1 and L2 , respectively, through whatlight beam comes from the point (x0 , y0) to the point(j, h).
If the wave scattered at the surface point (x0 , y0) is notspherical, the wave amplitude depends on the diffusion
Gurov et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 251
indicatrix at the surface point (x0 , y0). It can be takeninto account in calculating the value V(j, h, x0 , y0) inEq. (B5).
The coordinates (x1 , y1) and (x2 , y2) mentioned abovecan be found from geometrical-optics equations as
x1 5 2S x0f2
f1 1 h1 j D ~ f1 1 h !f1
f2h
5 2x0
f1
h2 j
~ f1 1 h !f1
f2h, (B8)
y1 5 2S y0f2
f1 1 h1 h D ~ f1 1 h !f1
f2h
5 2y0
f1
h2 h
~ f1 1 h !f1
f2h, (B9)
x2 5 2x0
f1
h2 j
f12 2 f2h
f2h, (B10)
y2 5 2y0
f1
h2 h
f12 2 f2h
f2h. (B11)
ACKNOWLEDGMENTSThe authors are grateful to the Optoelectronics and Mea-surement Techniques Laboratory of University of Oulu,Finland, for the experimental interferometric data pre-sented in Fig. 10(a).
Igor Gurov, the corresponding author, can be reachedby e-mail at [email protected].
REFERENCES1. T. Dresel, G. Hausler, and H. Ventzke, ‘‘Three-dimensional
sensing of rough surfaces by coherence radar,’’ Appl. Opt.31, 919–925 (1992).
2. T. Yoshimura, K. Kida, and N. Masazumi, ‘‘Development ofan image-processing system for a low coherence interferom-eter,’’ Opt. Commun. 117, 207–212 (1995).
3. L. Deck and P. de Groot, ‘‘High-speed noncontact profilerbased on scanning white-light interferometry,’’ Appl. Opt.33, 7334–7338 (1994).
4. J. G. Fujimoto, C. Pitris, S. A. Boppart, and M. E. Bresinski,‘‘Optical coherence tomography: an emerging technologyfor biomedical imaging and optical biopsy,’’ Neoplasia 2,9–25 (2000).
5. A. F. Fercher, ‘‘Optical coherence tomography,’’ J. Biomed.Opt. 1, 157–173 (1996).
6. Q. Wang, Y. N. Ning, A. W. Palmer, and K. T. V. Grattan,‘‘Central fringe identification in a white light interferom-eter using a multi-stage-squaring signal processingscheme,’’ Opt. Commun. 117, 241–244 (1995).
7. I. Gurov and D. Sheynihovich, ‘‘Interferometric data analy-sis based on Markov nonlinear filtering methodology,’’ J.Opt. Soc. Am. A 17, 21–27 (2000).
8. I. Gurov and A. Djabiev, ‘‘3D surface reconstruction by vec-tor nonlinear Markov filtering of white-light interferencefringe visibility,’’ in Optical Biopsies and Microscopic Tech-niques III, I. J. Bigio, H. Schneckenburger, J. Slavik, K.Svanberg, and P. M. Viallet, eds., Proc. SPIE 3568, 178–184(1999).
9. A. Zakharov, M. Volkov, I. Gurov, V. Temnov, K. Sokolovski-Tinten, and D. von der Linde, ‘‘Interferometric diagnosticsof ablation craters created by the femtosecond laser pulses,’’J. Opt. Technol. (English-language translation of Op-ticheskii Zhurnal) 69, 478–482 (2002).
10. E. Alarousu, I. Gurov, J. Hast, R. Myllyla, T. Prykari, and A.Zakharov, ‘‘Diagnostics of internal random structure ofwood fiber tissue by the dynamic stochastic fringe process-ing method,’’ in Proceedings of the International Workshopon Low-Coherence Interferometry, Spectroscopy and OpticalCoherence Tomography [Saint Petersburg Institute of FineMechanics and Optics (Technical University), Saint Peters-burg, Russia, 2002], pp. 16–25.
11. I. Gurov and P. Hlubina, ‘‘Analysis of spectral modulated in-terferograms by recurrence nonlinear filtering method,’’ inLaser Optics 2000: Control of Laser Beam Characteristicsand Nonlinear Methods for Wavefront Control, L. N. Somsand V. Sherstobitov, eds., Proc. SPIE 4353, 281–286 (2001).
12. R. E. Kalman, ‘‘A new approach to linear filtering and pre-diction problems,’’ Trans. ASME 82, 35–45 (1960).
13. A. Lloyd and W. Ledermann, eds. Handbook of Appli-cable Mathematics, Vol. 6 (Statistics), (Wiley-Interscience,Chichester, UK, 1984), Chap. 20.
14. I. Gurov and J. Vozniuk, ‘‘Rough surface shape retrieval ina fringe projection technique by the image enhancementand fringe tracing method,’’ in Proceedings of QCAV’2001,International Conference on Quality Control by Artificial Vi-sion (Cepadues-Editions, Toulouse, France, 2001), Vol. 1,pp. 79–84.
15. J. W. Goodman, ‘‘Statistical properties of laser speckle pat-terns’’ in Laser Speckle and Related Phenomena, J. C.Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–77.
16. H. L. Van Trees, Detection, Estimation, and ModulationTheory, Part I (Wiley, New York, 1968).
