efficient nonlinear algorithm for envelope detection in...

832 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 Kieran G. Larkin

Efficient nonlinear algorithm for envelopedetection in white light interferometry

Kieran G. Larkin

Department of Physical Optics, The University of Sydney, NSW 2006, Australia

Received October 11, 1994; accepted September 19, 1995; revised manuscript received October 25, 1995

A compact and efficient algorithm for digital envelope detection in white light interferograms is derived froma well-known phase-shifting algorithm. The performance of the new algorithm is compared with that ofother schemes currently used. Principal criteria considered are computational efficiency and accuracy inthe presence of miscalibration. The new algorithm is shown to be near optimal in terms of computationalefficiency and can be represented as a second-order nonlinear filter. In combination with a carefully designedpeak detection method the algorithm exhibits exceptionally good performance on simulated interferograms.

Key words: interferometry, white light interferometry, interference microscopy, phase-shifting algorithm,low coherence, nonlinear filter, envelope detection, demodulation. 1996 Optical Society of America

1. INTRODUCTIONRecently there has been much interest shown in the areavariously known as white light interferometry (WLI),1

coherence radar,2 coherence probeyscanning,3,4 correla-tion microscopy,5 – 7 interference microscopy,3,8,9 and low-coherence interferometry.10 The main reason for suchinterest is that the ambiguity present in conventionalmonochromatic interferometers is not present in WLI.White light interferometers have a virtually unlimitedunambiguous range, whereas their monochromatic coun-terparts are usually limited to not more than half a wave-length (slightly more for systems using high-aperturemicroscope objectives11 – 13). The close parallel betweenWLI and confocal (as well as conventional) microscopywas noted in the early literature3 but has been largelyignored since. Like confocal microscopy, WLI allows sur-face profiling with high accuracy over a large range, butunlike confocal microscopy WLI allows the entire imagefield to be captured in one instant without the need forscanning apertures.

Although the objective of WLI can be simplystated—to find the location of peak correlation (or fringevisibility)—a problem arises because of the largethree-dimensional sample data sets and the associatedcomputational burden. A typical system8 collects imagescontaining 256 3 256 pixels over a series of 64 equispacedsections. If one processed these data using the exactFourier method5 to ascertain the peak correlation depthat each pixel, then at least 128 3 6 multiplications mustbe evaluated at each pixel, resulting in approximately5.6 3 107 multiplications. A new algorithm that is manytimes faster (approximately eight times faster on theabove data set) than the Fourier method is developedfrom a generalized form of the well-known five-stepphase-shifting algorithm.14,15 The application of a spa-tial carrier phase-shifting algorithm to WLI is, I believe,novel, although a temporal phase-shifting algorithmusing three full data sets has been used previously.2

The original aim of this paper was to investigate thesuitability of phase-shifting algorithms for white light

0740-3232/96/040832-12$06.00

fringe analysis, but as the analysis and the simulationsprogressed, it became clear that one particular algorithmneatly16 combined the properties of simplicity and robustefficiency. The intent of this paper is to present therather circuitous development of this new algorithm andto demonstrate its suitability for efficient white lightinterferogram analysis.

This paper is arranged as follows:Section 2 outlines the essential details of the structure

of white light interferograms.Section 3 considers the elements of ideal envelope and

phase detection.Section 4 develops an approximation to the Hilbert

transform method outlined in Section 3 that correspondsto the phase-shifting algorithm of conventional digitalinterferometry. Some observations regarding samplingare made.

Section 5 introduces a unique nonlinear algorithm thatcombines filtering and demodulation in an elegant way.

Section 6 compares envelope detection using the non-linear algorithm with that of two common phase-shiftingalgorithms. Performance in the presence of extreme mis-calibration errors is evaluated.

Section 7 looks at the interpretation of the calculatedenvelope and phase. The salient features of peak detec-tion schemes are compared.

Section 8 concludes the paper.A revealing comparison of several well-known al-

gorithms with the new nonlinear algorithm operatingupon simulated, noisy interferograms is included as anappendix.

2. STRUCTURE OF A WHITELIGHT INTERFEROGRAMThe light intensity, g, measured in a white light interfero-meter that is spatially incoherent has the following form(see Chim and Kino,6 for example):

gsx, y, zd asx, yd 1 bsx, ydcfz 2 2hsx, ydg

3 cosf2pw0z 2 asx, ydg . (1)

1996 Optical Society of America

Kieran G. Larkin Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 833

Coordinates x and y correspond to the conventional trans-verse object or image coordinates, and the coordinate zindicates the axial location or defocus of the object. Thequantity asx, yd is an offset related to the reference andobject beam intensities. It is the reflected beam inten-sity that determines bsx, yd. The interferogram envelopefunction c is related to the spectral profile of the whitelight, and the spatial frequency of interference fringes inthe z direction, w0, is related to the mean wavelength ofthe light. A phase change of reflection that is due to thecomplex reflectance of the surface determines the parame-ter a. Many papers have assumed that a 0, althoughthis is generally not the case. Arbitrary control of thefringe phase, a, using the geometric phase is now pos-sible and has been recently demonstrated.17 I shall usethe term correlogram2 to describe the function gszd whenthe emphasis is upon the z variation and its character-istic form of fringes within an envelope determined byspectral correlation.

The exact form of the envelope c cszd is not criti-cal, although it is usually approximated by a Gaussianfunction for simplification of the calculations, especiallythose in the Fourier domain. In interferometer systemswithout suitably matched reference and object paths cszdmay not be symmetrical because of dispersion. Disper-sion has been ignored in the following analysis.

A spatially incoherent source ensures that correlogramscan be considered independent of sx, yd location. In apractical system intensity measurements are performedover a uniform array of x, y, and z values. The x andy array values are determined by the pixels of a CCDarray. Generally CCD arrays of 256 3 256 pixels or512 3 512 pixels are common, with even larger formatsbecoming popular. The sequence of values of z at whichthe intensities are sampled is determined by the sequenceof positions of a piezoelectric transducer. The samplingin the x, y, and z directions must satisfy certain con-straints. In the transverse directions these constraintscan be summarized in terms of the conventional sx, ydimage bandwidth. The sampling requirement in thez direction will be covered in more detail in Section 4.

Three-dimensional data sets need significant memorystorage capacity. For example, a 256 3 256 3 64 datasets at 1-byte resolution requires 4 Mbytes of memory.Memory cost is rapidly becoming much less significant indigital instrumentation; in fact, the renewed interest inWLI has been partly stimulated by these lower costs.

3. IDEAL ENVELOPE ANDPHASE DETECTIONThe usual purpose of WLI is to determine the profile[characterized by hsx, yd] of a surface too steep formonochromatic interferometry. Also of importance isthe phase change on reflection [related to asx, yd], whichis determined by the dielectric properties of the surface.Although WLI is often proposed for profiling of rough sur-faces, it is important to note that surfaces must be smoothat the scale of the system resolution for both hsx, yd andasx, yd to be meaningful.

Figure 1 shows a typical correlogram intensity distri-bution as a function of z. In this particular instance theenvelope is Gaussian. Generally it is possible to derive

hsx, yd and asx, yd from a sequence of intensity samples[given by Eq. (1)] over a range of z values. Some in-sight can be gained if one first considers the (continuous)Fourier transform of Eq. (1) with respect to the z coordi-nate:

Gsx, y, wd Z `

2`

gsx, y, zdexps22piwzddz . (2)

Hence

Gsx, y, wd asx, yddswd

1bsx, yd

2Cswdexpf24piwhsx, ydg

p fexpsiaddsw 2 w0d 1 exps2iaddsw 1 w0dg . (3)

The symbol p indicates one-dimensional convolution.18

Spatial frequency in the z direction is denoted by w, anddswd is the Dirac delta function. The Fourier transformof cszd is Cswd. Explicit sx, yd variation can be ignored inthe following analysis as long as it is remembered that thecalculations are performed over an array of points sx, ydin the sampled data.

Equation (3) can be rewritten as

Gswd adswd 1b2

expf1isa 1 4pw0hdg

3 Csw 2 w0dexps24piwhd

1b2

expf2isa 1 4pw0hdg

3 Csw 1 w0dexps24piwhd . (4)

In terms of phase and modulus

Gswd jGswdjexpfifswdg , (5)

with

jGswdj > adswd 1b2

jCsw 2 w0dj 1b2

jCsw 1 w0dj , (6)

fswd >

(2a 2 4psw 2 w0dh w . 01a 2 4psw 1 w0dh w , 0

. (7)

Thus relation (6) shows that Gswd has an impulse at theorigin and sidelobes centered at frequencies w 6w0. Atypical plot of the modulus, jGswdj, is shown in Fig. 2.The approximate equality in the previous equations isachieved when there is minimal overlap of the sidelobes.

Fig. 1. Typical white light correlogram, gszd. Note the fringephase offset a py4.


Fig. 2. Modulus of the Fourier-transformed correlogram,jGswdj. The normally large dc term has been reduced inmagnitude to fit within the chosen scale. Also shown forcomparison is the FT F1swd of the finite-difference filter f1szdwith separation D 1y4w0.

Generally the impulse is somewhat spread out by noiseand a variation of a with z. The two lobes in Fig. 2 arewell separated; that is to say, the separation is typicallygreater than the bandwidth of the lobe. The bandwidthis inversely related to the envelope width in the spatialdomain. Careful analysis of the phase function f revealsthat the two parameters of interest, h and a, can be sim-ply extracted from the slope and the intercept of the linearportion of the curve. Nevertheless one must take carein performing a linear regression on the phase. Onlyin the regions where jGswdj has significant, nonzero val-ues will the phase have meaningful values. A weightedleast-squares fit (LSF) to the phase using jGswdj2 as theweight automatically gives good estimates of h and a.In this context h actually corresponds to the position ofthe centroid of the envelope squared, as can be shownby a Fourier correspondence theorem.19 Unfortunately,calculation of h and a using the above method is compu-tationally intensive. The fast Fourier transform (FFT) ofa real array alone requires (of order) N log2 N floating-point multiplications followed by N multiplications foran optimized LSF over the positive frequency region. Aweighted fit requires an additional 2N multiplications.Here N is the total number of samples in the z direction(typically N 64). The calculation is then repeated foreach element in the x–y array (typically 256 3 256).

Conceptually, perhaps the easiest way to obtain thephase and the envelope is by use of the transformtechnique that has been outlined in several papers.5 – 7

Briefly, the method entails fast Fourier transformationof the raw data (in the z direction) and then removal ofthe negative- and zero-frequency components. Finallythe transform data are recentered at the midpoint of thesidelobe and then inverse transformed. In fact, the verysame technique is better known as the FT method offringe analysis in interferometry. The signal sszd thatresults has the form

sszd bcsz 2 hdexpfisa 2 4pwohdg . (8)

The modulus of sszd is the envelope that is required, andthe argument of sszd contains the phase offset a. Themethod outlined above is even more computationally in-tensive than the LSF method because forward and inverseFFT’s are required in addition to the squaring operationsrequired to determine the modulus.

More recently a much improved, computationally ef-ficient method of determining the envelope has beendeveloped.8 The method relies on a real-space imple-mentation of the previous double FFT method. The cru-cial point is the introduction of the Hilbert transform (HT)kernel as a digital finite impulse response filter.20 Com-putational efficiency is gained by the realization that anapproximate HT can give near-perfect results for typicalinterferograms. A real-space implementation of the HThas also been proposed for two-dimensional interferogramanalysis by Zweig and Hufnagel.21 As I shall show in thefollowing sections of this paper, the real-space implemen-tation can be greatly simplified, so much so that the finalalgorithm requires only two multiplications per point toproduce the envelope squared at each point. In terms ofmultiplications alone this represents a lower limit uponnumerical envelope detection using quadrature functions.

4. APPROXIMATIONS TO HILBERTTRANSFORM ENVELOPE DETECTIONThe perfect HT can be considered equivalent to a wide-band 90± phase-shift operation,21 but, as we have alreadyseen, a typical white light interferogram is a bandlimi-ted signal. That is to say, the spatial frequency con-tent is limited to a region centered around a carrierfrequency. The wideband property of the perfect HTis not required for such signals. Thus the constraintsupon an approximation to the HT can be relaxed—it onlymust have a 90± phase shift over a limited frequencyband. Outside this band the transform can have anyphase shift, but the modulus of the response should below, thus suppressing noise present outside the pass-band. In the case of discrete sampled data with whitenoise it is desirable for the transformer to have zero ornear-zero response at frequencies below and above thepassband. An important practical requirement for a nu-merical discrete envelope detector is computational effi-ciency. A discrete implementation standard of efficiencywith which to compare any method is the method ofChim and Kino,8 in which the main computational bur-den for N samples is due to the 6N multiplications andthe N square roots required to obtain the envelope. Allthe above requirements can be met by a pair of quadra-ture filter functions well known to researchers workingin the area known as phase-shifting interferometry. Thecrucial Fourier properties of these functions have beenderived22 and extended23 but have not previously beenapplied to the analysis of white light interferograms.Freischlad and Koliopoulos22 introduced the two filterfunctions f1std and f2std, which are correlated with a gen-eralized interference pattern gsx, y, td to produce twoquadrature functions, the ratio of which gives the tangentof the phase sought by the technique while the envelope(or the modulation) is given by the root sum of squares.A well-known algorithm that uses five samples14,15 hasthe discrete filter functions

f1std 2fdst 2 Dd 2 dst 1 Ddg , (9)

f2std 2dst 2 2Dd 1 2dstd 2 dst 1 2Dd . (10)

Here D is the step between samples. Those familiarwith signal processing or digital filtering may recognize


Fig. 3. FT’s F1swd and F2swd of the five-step phase-shiftingfilters f1szd and f2szd. Note particularly the stationary pointsat w 61y4D.

these as simple finite impulse response filters. In thecase of white light interferograms the temporal parame-ter t is replaced by a spatial coordinate parameter z. Infact, f1 and f2 are symmetrical and antisymmetrical lin-ear phase digital filters, respectively.24 The function f1

has the property that its FT is an imaginary odd func-tion, whereas that of f2 is a real even function. Figure 3shows F1 and F2, the transforms of f1 and f2. The choiceof the functions f1 and f2 is important. There are a mul-titude of possible choices, with three samples being theminimum number required for such an algorithm. Thefive-sample algorithm, however, has several importantproperties not possessed by any other algorithms. Themost important of these23 is related to the matched gradi-ents of the FT’s F1 and F2 at the fundamental frequencyw 1y4D. In Fig. 3 the gradients at the fundamentalfrequency can be seen to be near zero and slowly varyingin this region. A consequence of the zero gradient is thatboth phase and envelope calculations for the five-step al-gorithm are somewhat insensitive to sample spacing (orstep size) errors; essentially the filter responses changelittle in this region. Conversely, such an algorithm is in-sensitive to fringe spacing variation when the samplingis fixed.

At this point it is worth considering the sampling re-quirements for WLI. Most sampling schemes greatlyoversample the correlogram data. A typical example,Chim and Kino8 use approximately 8–10 samples perperiod. This exceeds the Nyquist criterion by at leasta factor of 4. For a typical correlogram this samplingrate gives an unbalanced distribution of information inthe Fourier frequency domain. The transform data arecentered on the value 1y8D, which is one-quarter of theNyquist frequency. A balanced distribution of frequencycomponents requires approximately four samples perfringe and keeps the peak frequency midway between dcand the maximum frequency 1y2D.

More detailed analyses of the optimum sampling re-quirements for bandpass signals have been consideredelsewhere.25,26 It is generally agreed that it is thebandwidth of a narrow-band signal that determines thesampling frequency, not the carrier plus bandwidth asrecently suggested by Caber.27 However, the proposi-tion that 4fc 1 2B is the minimum sampling frequencynecessary to avoid aliasing when the signal is squared(here fc is the fringe frequency, and B is the envelopebandwidth) fails to account for the fact that the alias-

ing that results when one is sampling at 4fc occurs atfrequency 4fc 2 2B and consequently has no effect afterlow-pass filtering (with a cutoff below 2fc).

In spite of this reduced bandpass sampling require-ment, a criterion of just four samples per fringe is consid-ered an adequate compromise here because it correspondsto the optimum sampling for a 90± step phase-shiftingalgorithm. A common misconception is that increasedsampling frequency gives a proportionate increase in ac-curacy, but this is simply not the case. All the followinganalysis is based on a nominal sampling frequency of foursamples per fringe (practically, this will cover the rangefrom approximately three to eight samples per fringe) butis easily modified for other values.

Recently a method has been proposed to sample correlo-grams at a frequency determined by the bandwidth,28,29

although such undersampling was previously used in1991.10 The technique has been called sub-Nyquist sam-pling. Instead of sampling the fringe pattern at foursamples per carrier fringe the undersampling is by an oddinteger factor, so there are 4ys2L 1 1d samples per fringe,where L is typically 1 or 2. Undersampling of this kindavoids common aliasing problems so long as the envelopebandwidth alone is adequately sampled. An inevitableconsequence of undersampling is that the allowable er-ror in the initial prediction of the mean wavelength isinversely proportional to the undersampling factor. Un-dersampling also increases the relative bandwidth of thebandpass signal and can be expected to degrade the per-formance of demodulators in the presence of noise. Thereare some issues regarding the effects of sampling upon theenvelope peak detection process that shall be discussedin Section 7.

Another issue worth mentioning is optimality with re-gard to computational efficiency. It is generally agreed(see Chim and Kino,8 for example) that mathemati-cal operations such as multiplication, division, and theevaluation of transcendental functions and square rootsare much more significant to calculation time than theadd and subtract operations. So, for the estimation ofthe computational burden of a numerical procedure, itis convenient to ignore the add and subtract operationsand just count the other operations. Envelope detectionalgorithms can be compared in efficiency with an ideal-ized numerical scheme that cannot be realized in practicebut gives an idea of the limits to efficiency. A perfectenvelope detection scheme could consist of the followingsteps:

1. Read in function values gn and the perfect quadra-ture function values gn for n 1 ! N . (The sequence gn

is assumed to be zero mean.)2. Calculate envelope values en

pgn

2 1 gn2.

In this idealized scheme evaluation of the envelope func-tion requires 2N multiplications and N square-root opera-tions. The sequences gn and gn are considered known.In practice, gn must be derived from the sequencegn (which itself must be made zero mean), and morecomputational steps must be involved. As was men-tioned above, the recent computational scheme of Chimand Kino8 requires 6N multiplications and N squareroots to yield the envelope, almost a factor of 3 belowoptimality.


The difficulty of implementing a perfect HT with theuse of a real-space filter function is closely related to thewell-known problem in communication theory of imple-menting a wideband 90± phase shifter for single sidebandmodulation.30 One way around this is to start with theoriginal function gszd and produce two new functions gaszdand gbszd. The two new functions are in quadrature toeach other, but the phase relationship with gszd is notconstrained. This gives an extra degree of freedom thatallows a practical implementation using realistic avail-able electronic components. For example, instead of re-quiring a 90± phase shifter, one would consider a 145±

and a 245± shifter. In terms of numerical filter func-tions it is in principle trivial to derive a 245± shifter oncethe 145± shifter is known. Consider a real filter func-tion f szd and its FT F swd. This can be conventionallyrepresented as

f1szd f szd , F swd jF sudjexpfixswdg . (11)

The symbol , represents Fourier transformation, andxswd is the FT phase. A filter function with the oppositephase shift is simply

f2szd f s2zd , Fpswd jF sudjexpf2ixswdg . (12)

In both cases the modulus of the frequency response isthe same, jF j. The two functions above, f1 and f2, arethe bases for a whole series of quadrature functions thatare linear combinations of f1 and f2. For example, a zerophase function f0 can be defined as

f0szd f szd 1 f s2zd , 2jF jcos x . (13)

In a similar way a 90± phase function can be defined as

f90szd f szd 2 f s2zd , 2ijF jsin x . (14)

In this case, although f0 and f90 are exactly 90± in phase,their moduli are no longer necessarily equal at all fre-quencies. Compare these with f1 and f2, which haveequal moduli but the phase difference is 2x, which doesnot necessarily equal 90± at all frequencies. The functionpair f1 and f2 or f0 and f90 can be used to generate ap-proximate quadrature pairs of functions from the correlo-gram gszd simply by correlation (or by convolution withz-reversed functions):

g1szd f1szd ≠ gszd , g2szd f2szd ≠ gszd . (15)

The symbol ≠ indicates the correlation operation.18

5. CALCULATION OF PHASEAND MODULATION USINGPHASE-SHIFTING ALGORITHMSThe following analysis will consider continuous functionsand continuous FT’s. However, the techniques outlinedare applicable to discrete sampled data and discrete FT’s.Important differences between the continuous and thesampled case will be noted as necessary.

The idea of using spatial carrier phase-shifting algo-rithms for white light envelope detection has not, to myknowledge, appeared in the literature. Typically such al-

gorithms are used in interferometry to detect the phase ofa wave front over a two-dimensional array of points whenonly one interferogram is available. Initially it is simplerto consider temporal phase shifting for the derivation ofthe new algorithm. In the case of the five-step algorithmfive interferograms are captured by a digital imaging sys-tem. At each pixel location in the image one combines allfive intensity values to extract the phase fsx, yd at eachpixel. The algorithm can be simply defined as

fsx, yd tan21

(2fI2sx, yd 2 I4sx, ydg

2I1sx, yd 1 2I3sx, yd 2 I5sx, yd

), (16)

where I1 to I5 are the interferogram intensities. Themodulation at each point, Msx, yd, can also be calculatedeasily from

Msx, yd 1y4f4sI2 2 I4d2 1 s2I1 1 2I3 2 I5d2g1/2 . (17)

Both these formulas are exact when the phase shift be-tween interferograms is 90±. For phase shifts in theregion near 90± the errors in both f and M are smallbecause the first-order error terms cancel.15 This error-compensating property of the five-step algorithm hasmade it popular in many digital interferometer sys-tems. Other error-compensating algorithms exist, themost well known being that of Carre.31 The interestingpoint about the Carre algorithm is that it compensatesexactly for step errors and requires only four interfero-grams, whereas the five-step algorithm compensates onlypartially (second-order residuals are present). The prob-lem with the Carre algorithm is that it is more complexand hence more time consuming to compute. A littleknown fact is that an exact compensating form of thefive-step algorithm exists and is somewhat less complexthan the Carre algorithm. The essential background tothis exact compensating five-step algorithm is present inthe paper of Hariharan et al.,15 but it is not shown ex-plicitly. It can be easily shown that the following phaseand modulation expressions are exact:

f tan21

(f4sI2 2 I4d2 2 sI1 2 I5d2g1/2

2I1 1 2I3 2 I5

), (18)

M sI2 2 I4d2hf4sI2 2 I4d2 2 sI1 2 I5d2g 1 s2I1 1 2I3 2 I5d2j1/2

4sI2 2 I4d2 2 sI1 2 I5d2.

(19)

The phase step between interferograms here is c, whichcan have any value not equal to an integral multiple ofp (this condition also applies to most phase-shifting algo-rithms including the Carre algorithm). The denomina-tor of Eq. (19) can also be expressed in a form that avoidsproblematic zero-by-zero divisions whenever sin f 0.

A serendipitous simplification of Eq. (19) leads to

4M2 sin4 c sI2 2 I4d2 2 sI1 2 I3dsI3 2 I5d , (20)

M2 / sI2 2 I4d2 2 sI1 2 I3dsI3 2 I5d . (20a)

For values of c near 90± (and odd multiples thereof) thesine factor is near unity, and so M can be calculated withjust two multiplications and one square-root operation;precisely the number of operators required for the ideal


detection scheme. The division operation [Eq. (19)] andits associated computational problems are neatly avoided.The right-hand side of relation (20a) defines what is nowreferred to as the five-sample-adaptive (FSA) nonlin-ear algorithm. Both the Carre algorithm and the con-ventional five-sample algorithm require an additionalmultiplication to calculate M. Remarkably the optimumsampling (Nyquist and sub-Nyquist) is predetermined bythe maxima of sin4 c, which is entirely in accord with thebandpass signal sampling discussed earlier in this section.

The temporal phase-shifting analysis immediatelypreceding, strictly speaking, applies only where themodulation and the offset remain constant between inter-ferograms. A technique known as spatial carrier phasedetection32,33 applies the same algorithms to a single in-terferogram. The intensity values I1 to I5 now representadjacent pixels instead of separate interferograms. Ifthe modulation and the offset are assumed to vary slowlyacross the interferogram, then the algorithms are approxi-mately correct. Generally, it is necessary to introduce alarge number of tilt (or carrier) fringes into the interfero-gram. This is because the maximum phase variationdetectable is proportional to the mean phase variation,which in turn is related to the total number of fringes.The application of spatial carrier phase detection algo-rithms to white light correlograms initially appears coun-terintuitive because spatial phase techniques normallyassume slowly varying offset and modulation, whereaswhite light correlograms, generally, have rapidly varyingmodulation along the z direction.

6. APPLICATION OF (SPATIALCARRIER) PHASE-SHIFT ALGORITHMSTO ENVELOPE DETECTIONPerhaps the easiest way to investigate envelope detec-tion with the use of phase-shift algorithms is by com-puter simulation.34 Three representative algorithms arecompared in this section. It should be emphasized thatfrom here on the values I1 ! I5 represent adjacent samplevalues in the z direction. The Carre-derived envelope al-gorithm has been omitted because it requires three mul-tiplications to be evaluated (also, preliminary simulationsindicate poor performance). Two of the (five-step) en-velope algorithms have been mentioned in the previoussection. The third is based on the simplest three-stepalgorithm utilizing 90± phase steps.35 The modulationfactor in this case is

Msx, yd 1y2fsI3 2 I2d2 1 sI2 2 I1d2g1/2 (21)

and is correct only for exact 90± steps. The above al-gorithm has been selected because it requires only twosquaring operations and one square root (both opera-tions are possible with the use of fast lookup tablecomputation).

A number of other three-, four-, or five-step algo-rithms36 could have been chosen with and without error-compensating properties. Initial testing has shown thatconventional phase-shifting algorithms have inferior per-formance to the algorithm defined in relation (20a). Thethree algorithms chosen here illustrate the performanceof no error compensation [Eq. (21)], partial (first-order)

error compensation [Eq. (17)], and exact error compen-sation [relation (20a)]. For convenience the algorithmsshall be denoted (1), (2), and (3), respectively. Each algo-rithm has been applied in turn to a simulated white lightcorrelogram without noise. The correlogram is shownin Fig. 1. The calculated envelopes are shown in Fig. 4for the case in which the carrier, or rather the meanfringe spacing, is known precisely and the algorithm stepsize and sampling step are set at 90± or, equivalently,one quarter of the mean period. Figures 1 and 4 showcontinuous functions, but it can be easily shown that thediscretely sampled case involves samples that occur atpoints located on the continuous curves.

Often the exact value of the carrier frequency is un-known before a measurement is made. The approxi-mate value can be readily calculated from a knowledgeof the spectral profile of the illumination and the nu-merical aperture of the imaging system (typically a mi-croscope objective). The exact value depends on othersystem parameters and the spectral reflectance of thesample being measured. Therefore it cannot be knownexactly before a measurement is made. Calibration of

Fig. 4. Envelope detection for all three algorithms with use ofa 90± step size. In this particular case the curves produced byalgorithms (2) and (3) are the same. Algorithm (1) has notice-able fringe structure.


Fig. 5. Envelope detection for all three algorithms with use of a45± step size. Both algorithms (1) and (2) have significant fringestructure visible. Algorithm (3) performs exceptionally well andhas a 50% reduction as predicted.

the system for each sample can be a rather tedious addi-tion to a measurement procedure. A preferable methodrequires a self-calibrating (or error-compensating) al-gorithm that works effectively over a range of carrierfrequencies. To show the effects of carrier frequencyvariation, I have calculated the envelopes for two ex-treme values of the frequency. In Fig. 5 the frequencyis 0.5 times its nominal value, and hence the samplesare now 45± apart. Figure 6 shows the envelopes calcu-lated when the frequency is 1.5 times its nominal valueand the samples are thus 135± apart. In a system withwhite light in the range 400 nm to 700 nm the extremefrequency variation that is due to spectral effects alone isin the range 0.72–1.28 times nominal. Such extremescan be achieved only if the reflected light is narrow bandat either 400 nm or 700 nm, in which case the envelopebecomes broad and the FSA algorithm can again be ex-pected to perform well. The dotted curves in Figs. 4–6represent the ideal envelope.

A detailed analysis of the effect of noise in the cor-relogram will be the subject a subsequent paper. The

performance of the FSA algorithm versus that of threeother well-known white light profiling algorithms is cov-ered in Appendix A; nevertheless some general observa-tions can be made. The linear filtering properties of allthree algorithms22,37 as well as the perfect HT method canbe determined. To summarize: (1) is a high-pass filter,(2) and (3) are bandpass filters centered on the nominalcarrier frequency, and the HT is a wideband (all-pass) fil-ter. In the presence of zero-mean Gaussian white noise,algorithms (2) and (3) suppress spectral components ofnoise outside the signal bandwidth and can therefore beexpected to perform well compared with both (1), whichboosts high-frequency noise, and the HT method, whichneither suppresses nor boosts noise.

7. INTERPRETATION OF CALCULATEDCORRELOGRAM ENVELOPESIn the previous section three algorithms have been usedto estimate the envelope of the white light correlogram.In all cases some fringe structure propagates through into

Fig. 6. Envelope detection for all three algorithms with useof a 135± step size. Again both algorithms (1) and (2) havesignificant fringe structure visible, whereas algorithm (3) showsonly a trace of the second-harmonic fringe structure.


the calculated envelopes. This problem does not occur inthe FT method and occurs only to a minuscule degreein the real-space HT technique. Of the algorithms,(3) has by far the smallest residual of fringe structureover the full range of sampling intervals from 45± to135±. This factor is important in the process of findingthe envelope peak, which is the crucial parameter. InSection 2 the height of the sample surface at any point,hsx, yd, was directly linked to the ideal envelope peakposition zpsx, yd:

zpsx, yd hsx, yd . (22)

Inevitably, the three algorithms tested only approximatethe desired envelope. Applying a simple point-to-pointpeak detection process38 to the calculated envelope cangive significant errors with respect to the ideal peak po-sition and requires significantly more than four samplesper fringe to work correctly. A better way to find thepeak is through use of the overall shape of the envelopearound the approximate peak position. Simple curve fit-ting using three points has been proposed in an alterna-tive approach to the envelope detection process.27,39 Inthe region of the peak the calculated envelope can beexpected to be well approximated by a Gaussian func-tion, exps2bz2d, where b characterizes the ideal enve-lope. A better estimate of the peak position can thusbe obtained from a LSF to this function. For example,the nearest-neighbor and next-nearest-neighbor samplescan be used for a five-point, symmetrical LSF to the func-tion’s exponent, fb0 2 bsz 2 zpd2g, which is a quadratic inz. The use of a symmetrical LSF greatly simplifies thecalculation.40,41 The full process is simply implementedif one takes the logarithm of the calculated envelope val-ues and computes the peak position from an explicit so-lution of the symmetrical LSF.

The peak detection process can just as easily be ap-plied to the envelope squared, as this produces only afactor of 2 in the envelope exponent. Thus the overallcomputation can be reduced by N square-root operations.The increased computational burden of the five-pointLSF is only 19 operations (5 logarithms, 12 multiplica-tions, and 2 divisions). Initial analysis indicates that thedominant error in the calculated envelope occurs at thesecond harmonic of the carrier frequency, which is typi-cal of a second-order nonlinearity. As a result theconventional LSF gives a significant error in the peakprediction. However, it is possible to define a weightedfive-point LSF that is insensitive to second-harmonic er-rors and thus gives much improved peak prediction. Fiveis the minimum number of points required to satisfy bothLSF and harmonic criteria.42 Equation (23) defines thefive-point, frequency-selective, LSF peak predictor, wherethe symbol Ln represents the logarithm of the envelopevalue In and the distance zp is measured from the thirdsample:

zp 0.4D

√L1 1 3L2 1 0L3 2 3L4 2 L5

L1 1 0L2 2 2L3 1 0L4 1 L5

!. (23)

Figure 7 shows the result of applying the aforementionedpeak detection process to the algorithm (3) envelopeshown in Fig. 4. Again a continuous function analysis

has been performed, but the result for a sampled functionis just one point on the curves shown. In this particu-lar instance the error of the proposed procedure is lessthan 1y20 sample for calculations with initial estimatesof peak location within two samples of the actual value.The unweighted LSF is 1 order of magnitude less accu-rate. Careful scrutiny of Fig. 7 reveals a small bias inthe predicted value; a bias related to the nonzero phasechange (in this case a py4).

Once the peak position of the envelope has been es-timated, it is a straightforward task to find the phaseat that position. The fully compensating five-step algo-rithm phase given in Eq. (18) can also be shown to havemuch smaller errors (which are due to miscalibration)than those of the two other algorithms. The phase atthe estimated peak must be interpolated from actual cal-culations of the phase at sample positions on either sideof the peak. The expected form of the phase near thepeak is linear with respect to z. Hence a two-point lin-ear interpolation will give a good estimate of the phase atthe peak, in other words, an estimate of asx, yd. Morepoints could be used for a linear least-squares estimate.However, the main difficulty in estimating the phase isthe occurrence of phase discontinuities that are due tothe modulo 2p restriction of the arctangent function. Aphase discontinuity in the region of the peak renders theinterpolation useless. To avoid such discontinuities, onemust subtract a linear phase component from the calcu-lated phase and reevaluate modulo 2p:

f1 mod2p

3

√√√tan21

(f4sI2 2 I4d2 2 sI1 2 I5d2g1/2

2I1 1 2I3 2 I5

)2 4pw0z

!!!.

(24)

The interpolation scheme can then be applied andthe mean phase term added afterward to give a0

p. Arough estimate of w0 is sufficient because errors cancelcompletely:

Fig. 7. Error in the predicted peak position with the weightedLSF defined by Eq. (23). Note that, for an initial estimate ofpeak position within two samples of the actual value (delimitedby the vertical dotted lines), the peak prediction has only 1y20sample error. The small bias in the predicted peak is relatedto the nonzero phase change on reflection. The range of z isjust half that shown in Figs. 1 and 4–6. Clearly the erroris related to the second harmonic of the original fringe. Asecond application of the peak detection process ensures thatthe estimate is within one-half sample, and errors decreaseproportionately.


a0p f1 1 4pw0z . (25)

This process works well in the region of the envelope peakfor values of a not equal to 6py2 or 6p. The number ofadditional significant computational steps for a two-pointinterpolation is 10. So for a data set with N values of zthe total number of operations required for an estimate ofhsx, yd and asx, yd is 2N 1 29. This method comparesfavorably with the HT method of Chim and Kino,8 whichrequires 6N nontrivial multiplications just to produce theenvelope squared. Typically N 64, which means thatthe phase-shift algorithm method is nearly two and ahalf times faster than the HT kernel method. Essen-tially this speed gain is due to the remarkable adaptiveproperties of the fully compensating five-step algorithm ofrelation (20a) when applied to envelope detection. Also,by limiting the accurate estimates of h and a to small re-gions near the estimated peak of the envelope, one avoidsmuch global calculation.

The comparisons in speed are valid only if the meth-ods compared have similar accuracy. Certainly the pro-cedure consisting of algorithm (3) followed by a weightedleast-squares peak prediction given by Eq. (23) has an er-ror of less than a small fraction of one sample interval.There appears to be no published work that considers theperformance of any WLI peak prediction schemes in thepresence of noise and other degradations. A preliminaryanalysis of this kind is included as Appendix A to indi-cate the veracity of the general principle discussed here.All methods that estimate the envelope can also utilizesome form of LSF peak prediction and so, presumably,are capable of subsample resolution. In the past not allmethods have used simple curve fitting to such advan-tage and therefore have a crude resolution limited to halfa sample at best.

A recent paper by Caber27 developed a communicationtheory approach to the interferogram envelope detection(see also Liu et al.43). The well-known demodulationprocess of a bandpass filter followed by a square-law non-linearity followed, in turn, by a low-pass filter is imple-mented as a sequence of digital filters. Although detailsare not given, the known computational efficiency of twodigital infinite impulse response high-pass or low-passfilters20in series with a squarer is lower than that of thecompensating five-step algorithm outlined in the preced-ing sections. The availability of digital signal processingboards with special digital filter hardware may counter-balance the lower efficiency in practice. A minimum ofjust eight frames of data need to be stored at any mo-ment in this scheme, compared with ten frames needed forthe FSA algorithm proposed here. An accuracy of 1y25sample spacing is claimed for this method.39 The ac-curacy of this technique has not been tested in the ap-pended simulation because details of the infinite impulseresponse filters used have not been disclosed in the openliterature.

Some final remarks about the potential accuracy ofthe sub-Nyquist sampling method of de Groot13 follow.The method relies upon a best fit to the phase gradientcalculated from the FT of the sampled data. The phasegradient at the carrier frequency is easily shown to be pro-portional to the first moment (or centroid) of the envelopeby a well-known Fourier correspondence theorem. Simi-

larly the weighted LSF to the phase gradient (weighted bythe magnitude squared) is proportional to the centroid ofthe square of the envelope.19 Several authors have stud-ied the effects of sampling upon centroid estimation,44 – 46

essentially concluding that sampling must satisfy thebandpass sampling requirements mentioned in Section 4.The main point, however, is that phase gradient estima-tion (equivalently FT centroid estimation) is quite distinctfrom peak detection. In statistics the centroid is knownto be susceptible to noise, that is to say, it is not a ro-bust estimator. The effect of noise upon the centroidincreases with the interval over which the centroid isevaluated. In contrast the peak prediction schemesoutlined earlier depend only upon values of a distribu-tion near the peak. A balanced assessment of the twotechniques must compare accuracy versus computationalcomplexity. Both the centroid of the envelope and thecentroid of the squared envelope have been calculatedfor simulated data sets, and the results are presented inAppendix A.

A Fourier description of the mechanism defined by rela-tion (20a) shows some similarities to the Caber method.The essential difference is that the Caber method uses twoconventional infinite impulse response filters to removelow frequency and the second-harmonic components pro-duced by the square-law nonlinearity, whereas the FSAnonlinear filter bandpass filters the signal and then shiftsit to dc (i.e., demodulates) in one operation. From thepoint of view of classification the new procedure can beseen as a (nonlinear) second-order polynomial (Volterraseries) digital filter47 followed by least-squares peak pre-diction. The filter can be defined in general terms as afinite-difference operation followed by a nonlinear differ-ence operation. There are similarities to the quadraturereceiver (see, for example, Whalen48), except that the sineand cosine modulation terms are derived from the signalitself instead of an external source. Yet another classi-fication known as the bilinear (quadratic with memory)transformation49 covers such nonlinearities and offers atractable analysis of noise propagation. The nonlinearfilter can be explicitly defined by fb, where

faszd gsz 1 Dd 2 gsz 2 Dd

finite difference , (26)

fbszd fa2szd 2 fasz 2 Ddfasz 1 Dd

nonlinear difference . (27)

Such a definition is amenable to Fourier analysis, and thefollowing relations can be demonstrated:

Faswd 2i sins2pwDdGswd bandpass filter , (28)

Fbswd Faswd p Faswd 2 fexps22piwDdFaswdg

p fexps2piwDdFaswdg . (29)

The last equation represents zero- and second-harmonicgeneration (from autoconvolution), with out-of-phase com-ponents canceling at the second harmonic.

8. CONCLUSIONA simple but highly effective method for envelope detec-tion in white light correlograms has been introduced anddemonstrated. The speed that is due to increased com-


putational efficiency is between two and three times thatof the real-space HT technique. Combined with a newprocedure for peak prediction using a weighted LSF al-gorithm that removes the residual second-harmonic errorin the envelope, the fully compensating five-sample en-velope detection algorithm is remarkably simple yet iseffective over a wide range of carrier frequencies. Thesimulations presented used a fixed (but wide) bandwidthand show that repeatable subsample accuracy can beattained for envelope peak detection. In fact, the FSAalgorithm, when combined with recursive peak detection,gives exceptionally good results compared with a numberof conventional (and relative inefficient) WLI profiling al-gorithms. In terms of multiplication operations the newalgorithm has been shown to be near the ideal limit oftwo multiplications per sample, suggesting that any fur-ther speed improvements from other methods can only bemarginal. In situations in which the bandwidth is smallenough the proposed algorithm can be combined with sub-Nyquist sampling to improve efficacy further.

The method is not limited to WLI and is applicableto any bandpass signals where either the envelope orthe phase, or both, need to be detected. Optical mea-surement techniques such as confocal interferometry andspatial carrier phase-shifting interferometry could bene-fit from such a method.

Note added in final revision: A number of algorithmsclosely related to the FSA nonlinear algorithm in thispaper have appeared recently in a number of papers re-lated to amplitude and frequency demodulation of speechsignals. The derivation of these algorithms uses a con-struct known as the energy operator,50 which is definedfor continuous, oscillating signals. Discrete approxima-tion of the energy operator leads to a number of algo-rithms called discrete energy separation algorithms,51 – 53

one of which is very much like Eq. (27). The derivationof the FSA nonlinear algorithm that I have presented here(from phase-shifting algorithms) is more general in thatthe separation of samples is initially assumed arbitrary[Eqs. (18) and (19)] rather than infinitesimal, as it is forthe energy operator. The energy operator approach alsoassumes that signals have the dc, or background, com-ponent removed in a preprocessing operation. Notablythe adaptive properties of the algorithm are particularlystriking when applied to speech demodulation. Interest-ingly the idea of using these algorithms with undersam-pled (or, more correctly, bandpass-sampled) signals hasbeen overlooked until recently.54

APPENDIX A: NUMERICAL SIMULATIONOF ALGORITHMS APPLIED TO NOISYWHITE LIGHT INTERFEROGRAMSThis appendix contains a summary of results obtained byapplication of a number of well-known algorithms to simu-lated white light correlograms. The simulated data areavailable from the author as tagged image format filescontaining 512 3 64 pixels with 1-byte resolution. Suf-ficient information is provided for interested researchersto recreate simulated data with similar statistical prop-erties. A full analysis of error propagation in WLI is inpreparation.

1. Simulated DataThe fundamental parameters of the simulated dataclosely resemble the experimental data shown in a num-ber of papers by Chim and Kino.5 – 8 Essentially thecorrelogram is sampled at 64 locations in depth (z). Thesampling occurs at a sample spacing of one-eighth themean wavelength and three-eights the mean wavelengthin the undersampling case, corresponding to step sizes of90± and 270±, respectively. The envelope chosen corre-sponds approximately to a spectral range from 400 nm to700 nm. In order that sufficient data exist for useful sta-tistical inferences to be made, there are 512 independentmeasurements of the correlogram. To eliminate somesystematic errors, the correlogram shifts z position pro-gressively over the full 512 range. The total shift is onesample period over the full 512 range. Mathematicallythe image files used can be defined as

gsx, zd INT f128 1 100 exps2zs2ys2dcoss4pzsylmd

1 nsx, zdg . (A1)

The INTs?d function outputs the nearest integer to theargument input. The z sample locations are defined byzs z 2 32D 2 xy512. The sample spacing is defined byD lmy8 or D 3lmy8 in the undersampling case con-sidered. The noise added to the interferogram is nsx, yd.The coordinates x and z take on only the following inte-ger values: x lD and z mD, where 0 # l , 512 and0 # m , 64. For the selected spectrum s 3.85D.

The noise nsx, yd is zero-mean Gaussian-distributedrandom noise with a standard deviation (or rms) valuespecified in the range 0%–8% of the modulation value.The modulation is chosen to be 100 in this case. Notethat even in the case of zero noise the quantization intro-duces some systematic (or correlated) noise. The actualnoise characteristics for WLI are rather complex, beinga combination of such factors as vibration, photon noise,and quantization, to name just a few. A full analysisrequires a multidimensional statistical procedure. Zero-mean Gaussian noise has been chosen as a simple andwell-defined starting point.

All the algorithms tested were set up to predict thez peak position at all 512 values of x. The ideal results lieupon a straight line in the x–z plane. The distributionof actual values around the best-fit (least-squares) line iscomputed in each case, and the standard deviation of theerror is tabulated in Tables 1 and 2.

Four algorithms were tested:(1) The FSA nonlinear envelope demodulator [rela-

tion (20a)] in conjunction with the specialized five-pointpeak detector of Eq. (23). The peak detector is appliedtwice if the first estimate is more than half a samplefrom the raw data peak. This iterative technique re-moves some systematic errors shown in Fig. 7.

(2) The Fourier–Hilbert transform method5 – 7 is usedto generate the envelope, and a simple three-point peakdetector is used.

(3) The envelope is predicted by the exact Fouriermethod as for algorithm (2). The centroid of the enve-lope is then calculated. This is numerically identical tothe instantaneous phase derivative method of de Groot.

(4) The square of the envelope is predicted by the exactFourier method as for algorithm (2). The centroid of the


Table 1. rms Peak Location Error for VariousAlgorithms with Four Samples per Period

Position Error in Sample Spacings

Fourier– Square ofFSA Hilbert Envelope Envelope

rms Noise (%)a Algorithm Algorithm Centroid Centroid

0 0.010 0.003 0.000 0.0001 0.034 0.056 0.175 0.0272 0.064 0.112 0.322 0.0614 0.126 0.233 0.553 0.1608 0.248 0.479 0.856 0.476

aExpressed as a percentage of the modulation.

Table 2. rms Peak Location Error forVarious Algorithms with 4y34y34y3 Samplesper Period (3 Times Undersampling)

Position Error in Sample SpacingsFourier– Square of

FSA Hilbert Envelope Enveloperms Noise (%)a Algorithm Algorithm Centroid Centroid

0 0.055 0.166 0.305 0.1701 0.058 0.166 0.430 0.1752 0.065 0.168 0.625 0.2044 0.094 0.176 0.909 0.3968 0.172 0.206 1.190 1.046

aExpressed as a percentage of the modulation.

square of the envelope is then calculated. This is nu-merically identical to the weighted least-squares phase-derivative method of de Groot.

2. ResultsThe results for ideal sampling and 3 times undersamplingare shown in Tables 1 and 2, respectively. It is interest-ing to note that the FSA algorithm gives the best resultsin the undersampling case and that for full sampling theFSA algorithm is superior for noise levels above 2%.

Note that, unfortunately, the method of Caber could notbe simulated because the digital filter parameters neededto define the algorithm have not been disclosed.

ACKNOWLEDGMENTSThe development of this new algorithm was inspired byan impromptu remark by James C. Wyant about the needfor a simple algorithm to analyze white light interfero-grams (February 1993, Eighth Australian Optical Soci-ety Conference, Sydney). I thank Colin Sheppard for themany enlightening discussions on short coherence lengthinterferometry that stimulated this work. Gordon Kinokindly provided advice on the significance of computa-tional (and general) aspects of WLI, and this furtherstimulated my interest. John Quartel kindly provided asuite of C programs utilized in the production and theanalysis of simulated white light interferograms.

The author’s e-mail address is [email protected].

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efficient nonlinear algorithm for envelope detection in...

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