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Vol. 11, No. 10/October 1994/J. Opt. Soc. Am . A 2571

Robust, noniterative, and computationally efficientmodification of van Cittert deconvolution optical figuringC. B. Zarowin*

Research Department, Hughes DanburyOptical Systems, Inc., Danbury, Connecticut06810Received February 17, 1993; revised manuscript received February 22, 1994; accepted February 23, 1994

A modification of van Cittert deconvolution (VCD) is introduced and shown to yield a robust, noniterative (orclosed-form), and numerically efficient method of deconvolution. This modification removes the restrictionslimiting the applicability of conventional VCD only to shapes and relative positions of the convolved functionsfor which it converges, while also avoiding the ill effects of zeros in these functions. Th e resulting method iscomputationally efficient because it is noniterative and uses the fast Fourier transform. In contrast to theconvergences obtained with VCD, those obtained with this modified method are ensured by their expansion interms of an introduced auxiliary function rather than the convolved functions. This permits both the generalremoval of the above limitations and arbitrarily accurate deconvolution of infinitely sampled input data evenin the presence of input data noise. For discretely sampled input data the accuracy of this modification isshown to be limited only by the implicit bandwidth of the input data density. To exemplify its numerical andanalytical advantages, I apply the method to computer control of optical surface figuring. I also demonstratean intrinsic means of optimal frequency filtering of raw input data made available by this modification. Theadvantages of this procedure are also applicable to image restoration.

1. INTRODUCTIONTh e original iterative van Cittert deconvolution1 (VCD)has been shown by Hill and Ioup2 to be restricted to con-volved functions for which this procedure converges. Onthe other hand, as these authors show, many convolvedfunctional shapes and their relative positions interferewith its convergence. While they explore the origins2 ofsuch divergence and for certain cases offer ways of modi-fying the functional forms that cause it, they propose nogeneral means of overcoming these restrictions. In sum-mary, the convolved functions for which VC D diverges andfor which its use is perilous are, among others,2 those thathave a discontinuous edge, e.g., a top hat; those with arbi-trary relative positions; and those for which the maximumordinate is not at the origin, e.g., a functional shape thathas a minimum at the origin between peaks.

Our objective is to obtain a noniterative modification ofVC D that removes these restrictions, first showing thatthis modification is consistent with closed-form or non-iterative deconvolution using the multiplicative algebraicproperties of Fourier transforms (FT's) for convolutions. 3To elucidate the cause of these restrictions, we rederiveVC D from a slightly different viewpoint, showing thatit is equivalent to the series expansion of the FT of thereciprocal of a convolved function from that about zero,where a singularity occurs, to that about unity, which ex-cludes this singularity; that is, it is equivalent to a seriesexpansion about the convolutional identity element, unityin spatial frequency space or the delta function in coordi-nate space. Its equivalence to the description of VC D byJansson 4'5 is noteworthy: "van Cittert recognized that"one of the deconvolved functions "could be considered" ..."a first approximation" in his deconvolution method.From a clarification of the basis of these restrictions wepropose to obtain a numerical deconvolution procedurethat is robust because it avoids the convergence depen-

dence on the convolved functions and the singularity thatoccurs when one of the convolved functions reaches zeroat some spatial frequency. This modification of VCDwill be seen to be computationally efficient because it isnoniterative and uses the fast Fourier transform (FFT).Finally, although the modification introduced here grewspecifically from our exploration of the behavior of theVC D method, it appears that it is a particular imple-mentation of a more general regularization of ill-posedFredholm integral equations,6 7 of which the convolutionequation [Eq. (la)] below is an example.Of the many applications 4 5 that require solving a con-volution equation for one of its convolved functions, ourinterest arose out of the need to determine reliably andaccurately the dwell time map (x, y) that produces agiven material thickness correction map Aho(x, y) for agiven (tool) vertical removal rate map (x, y) over thesurface area S0 for which Aho(x, y) is defined. For ex-ample, we require such a deconvolution in order to obtainthe dwell time map for the mechanical polishing tool ofcomputer-controlled polishing8 or for the removal tool ofplasma-assisted chemical etching,9 giving either the tooldwell time or the velocity that it must travel at each pointof the map to achieve the required correction.Th e relation between these quantities is defined by aconvolution equation:

Aho(x, y)So = ff i(x - x', y - y')T(x', y')dx'dy',(la)

which requires that the tool vertical removal rate mapi(x, y) be convolved (or generically scanned over the run-ning coordinates, x' and y'), weighted by the dwell timer(x, y) that we require for generation of the necessary cor-rection map Aho(x, y) at each point over an area So. A

0740-3232/94/102571-13$06.00 1994 Optical Society of America

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2572 J. Opt. Soc. Am. A/Vol. 11, No. 10/October 1994negative tool removal rate implies an additive rather thana subtractive process. If, of a variety of possibilities, wechoose to implement physically the generic convolutionwith an x-y scan, i.e., with So = lly, scanning along xand stepping along y by Ay, the convolution equation (la)becomes

Y. Aho(x, y)i X [ix' fYlxlx'}y.'x [r(x', yi')Ilxlyldx' Ay', (lb)

with step indices i = 1, 2, ... , m. Defining the tool scanvelocity along x asv(x, yi) = 1x/r(x, yi) (lc)

yields the convolution equation in terms of the tool scanvelocity v(x, yi) rather than the dwell time T(x, yi):> Aho(x, yi) = Vly Y- { f(x - x', yi)X [1/v(x', Yi')]dX'}Y'. (Id)

Given maps of the departure of a surface from its de-sired shape (the required correction) and of the removalrate of a tool, we require that unique map of the dwelltime or the scan velocity of the tool that yields the desiredfigure correction. Deducing the dwell time requires solv-ing the above convolution equation, for which, in 1931,van Cittert' described a method (VCD) that iterativelyextracts a successively more accurate approximation of aone-dimensional equivalent of the dwell time map r(x, y)in our example. Thus Eqs. (1) are an exact (volume) cor-rection over the area So, which has been shown' to besolvable for r(x, y) to any approximation in a finite num-ber of iterations, in either a coordinate or a Fourier im-plementation of VCD, so long as the procedure convergesand the consequences of zeros in f(x, y) are avoided.2. SERIES FORM OFVAN CITTERT ITERATIONWith the objective of finding a modification that over-comes the convergence restriction stemming from the na-ture of the convolved functions, we rederive the seriesequivalent to VC D from a slightly different perspective;without such a modification, we will see that this re-sults only in a method identical to the frequency space FTiterative form of VC D outlined in Appendix A and to thecoordinate space form in Appendix B. As we will alsosee below, the advantage of the frequency space FT isthat the convolved integrands in Eqs. (1) become algebraicas a consequence of the Fourier convolution theorem,3permitting a noniterative (or closed-form) asymptotic ap -proximation of the deconvolution to any accuracy for in-finitely sampled input data. Th e following (re)derivationwill be seen to miuggeit an artifice for avoiding the, re-strictions on the shapes and the relative positions of theconvolved functions to ensure that VC D converges while,at the same time, avoiding the ill effects of zeros in theconvolved functions.

By defining the tool's (static) volume removal rate V =ff o (x, y)dxdy, we may normalize the tool removal ratei(x, y) (positive for a subtractive process and negative foran additive process), so that the integral of the resultingtool shape factor f (x, y) = i(x, y)/V over the area So is

II f (x, y)dxdy = 1.so

(2)The spatial frequency characteristic F(kX, ky) of the toolshape factor f(x, y) is given by its Fourier transform:

F(k., y) = "SO f(x, y)exp[-i(kxx + kyy)]dxdy, (3)where k y are the spatial frequencies along x and y,so that Eqs. (2) and (3) can be shown 3 to satisfy 0 cJIF(k, ky)l s IF(0, 0)1 = 1.From the Fourier convolution theorem3 we can writethe convolution equation [Eqs. (1)] describing the surfaceshaping in terms of FT's similar in form to that of Eq . (3):AHo(kx, ky) = FT[Aho(x, y)], T(kx, ky) = T[T(x, y)],F(kX, ky) = FT[f(x, y)], and

AHo(k,, k,)S0 = F(k., k,)T(k., k), (4a)effectively showing that IF(k,, k,)l is the tool efficiencyfo r removing spatial frequencies (ks, ky)and that, at thosespatial frequencies, the product of the tool shape factor FTand the dwell time FT is set by AHo(k., k,)So/V. For ex-ample, when the tool efficiency is zero at spatial frequen-cies (kr, ky), or IF(k., k,)l => 0, the required dwell timefor a finite required correction FT , AHo(k, ky), becomesinfinite, or T(kX, k,) => A, whereas, when IF(kX, k,)I X 1,the tool reaches its maximum efficiency in removing thesespatial frequencies, or the dwell time required is a mini-mum. Further, since AHo or Aho(x, y) and F or f(x, y)are given, we solve for T or (x, y):

T(k,, ky) = AHo(kx, ky)So/VF(k, ky), (4b)manifesting a singularity at Fl = 0. Since Eq. (4b) isimplicitly an expansion of 1/F about (and including) thissingularity, one desires a way of obtaining a series expan-sion that excludes it. Remembering that a delta functiontool shape factor f(x, y) has a FT[ff(x, y)] = F(kx, ky) = 1guarantees a valid (but trivial!) deconvolution; i.e., thedelta function is a convolutional identity operator whenF = F8 = 1, or (V/So)F8 T8 = (V/So)T, = AHo(k., ky). Aswas suggested above, this is equivalent to van Cittert'sstarting approximation, which one makes by taking thefirst term in a series or an iterative expansion of the dwelltime deconvolution as the required correction AHo(k, ky).Taking advantage of this property, we shift the expansionof 1/F from that about zero to that about 1/F8 = 1,

1/F = 1/[1 - (1 - F)] = Y (1 - F)P,p=l (5a)and obtain an infinite series that converges if 11 - Fl

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Vol. 11 , No. 10/October 1994/J. Opt. Soc. Am. A 2573(ks, ky)So/VF(k, ky) emphasizes that this is an exact(or infinitely iterated) dwell time FT. Th e infinite sumin Eq. (5b) can be further partitioned into a finite sumand a residue:

n-1E(1 -F)P = E (1 -F)P + E(1 -F)P,p=1 p=1 p=n

yieldingx x~~~~~~~nZ (1-F)P= [1 -(1 -F)]/F+ Yj (1-F)P, (5c)p= 1 p=n

since the first (finite) sum is a geometric series. Thus theexact (infinitely iterated) dwell time FT from Eq. (4b) isTc(kx, ky) = [AHo(k., ky)S 0 /VF]

x [1-(1- F)n] + F j (1-F)P (6a)p=n

from which we identify the nth virtual (closed-form) itera-tion (or approximation) of the dwell time FT asTn(k,, ky) = T.(kx, ky){1 - [1 - F(k., ky))]}, (6b)

where AHo(k., ky)So[F(kx, ky)V] = Ta(kx, ky). Th e nthvirtual iteration of the coordinate space dwell time maprn(x, y) can be obtained from its inverse FT :

given by the difference between the required correctionAHo(k., k,) and the nth approximate correction residueAHnR(kx, ky)H

Rn(k.,, ky) AHo(k., ky) -AHn(k., y)= AHo(kx, ky)[1 - (1 - fln], (6g)

whose inverse FT then yields the coordinate space re-moval equivalent r(x, y); note that, as n X , Eq. (6g)yields R(kx, ky) = AHo(k., ky), i.e., the exact (or infi-nitely iterated) removal necessarily becomes the requiredcorrection.For a given input data pair the FT of the requiredcorrection map Ho(kx, ky) and the tool shape factorF(k., ky), the number of virtual (or noniterative) itera-

tions n that we require for a given residue to be achievedat each spatial frequency k = (kx + k 2)- 2 AHn(k) isuniquely determined by Eq. (6f). Among other possibili-ties, one might choose to identify the number of these vir-tual iterations n that we require in order to achieve therms residue over al l spatial frequencies

1S12Lmsn(1 'SO j~nX'xy(a

since, by Parseval's theorem, 3

T(kx, ky)exp[i(kxx + kyy)]x dkxdky/(2(6c)2)

rms(n) = (1/2r)[ (/So) ffX dkxdkx (

and the dwell time residue FT isATn(kx, ky) = T.(kx, ky)F Z.1 - F)P.p=n

As required, the exact dwell time FT is the sum of the nthvirtual iteration and its residue:

T.(kx, ky) = T(k., ky) + ATn(k., ky), (6d)permitting us to express the dwell time residue FT,ATn(k., ky), in closed form:

ATn(k,, ky) = T(kx, ky) - Tn(k., ky)= T,(kx, ky){1 -[1 - (1 - F)n]l= T.(k., ky)(1 -F)n. (6e)

As n approaches infinity, the dwell time FT residue ap-proaches zero as long as 11 - Fl < 1. Recognizing thatthe nth iterative correction residue FT , AHn(kx, ky) =VF(k,, ky)ATn(kx, ky)/S 0, we can obtain this quantity ex-plicitly:

AHn(kx, y) = AHo(k., ky)(1 - F)n, (6f)as well as the equivalent coordinate space nth itera-tive correction residue from its inverse FT, Ahn(x, y) =FT[AHo(kx, ky)]. One may also recognize the nth ap-proximation of the FT of the removal, Rn(kX, ky), to be

and, with Eq. (6f),

rms(n) = (1/2 w) (/So) fO f1/2

X 11 -F(kx, k ) 2n dkxdkx > (7c)showing that the rms correction residue depends only onAHo(kx, ky), F(kx, ky), and n for infinitely sampled in-put data; i.e., the integration is over al l k in Eq . (7c)and always approaches zero as n => . In contrast, forfinitely sampled input data the rms correction residuedoes not (necessarily) approach zero because the data den-sity of the input data fo r Aho(xi, yi) and f(xi, yi), i =1, 2, ... , m, effectively band limits their spatial frequencycontent, as implied by Eq . (7c). This stems from theWittaker-Shannon sampling theorem,3 requiring that aband-limited function AHo(kx, ky) or F(kx, ky) be zeroabove kmaX = x = m. irux and kmaX = 7r/Ay = myrly.The surface is uniquely representable only when it is at2kmax and 2kymax (the Nyquist frequencies) with data in-put maps of (2mx)(2my). Conversely, if we sample with(2mx)(2my) points, correction beyond spatial frequencieskmax and kmaX is empty. A plot of rms(n) calculated for* = my = 33, 49, and 65 data matrices for Aho(xi, y) andf(xi, yi) is given in Fig. 8 below and shows the asymptoticconvergence of these residues to a smaller rms as the datadensity mx = my increases.

Tn(X, Y) = ffkky

C. B. Zarowin

|iAHn(kx, k) 2

(6c) (7b)

1^AHo(kx k) 2

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2574 J. Opt. Soc. Am. A/Vol. 11 , No. 10/October 1994pansion with an auxiliary function W(k), as follows:

1/F = (1/F)(W/W) = (W/F){1/[1 - (1 - W)]}= (WF) (1 - WYIp=l

20 40 60 80Fig. 1. Plot of F(k), 1 - F(k), AHo, and AHo(1 - F)n.

A plot of the integrand in Eq. (7c), AHO(l - F)n, for anassumed AHo and F shown in Fig. 1 demonstrates thatrms(n) approaches zero more rapidly for low spatial fre-quencies and considerably less rapidly at higher spatialfrequencies. Thus suitable prefiltering of the spatial fre-quency content of AH o offers the potential of efficaciouslyremoving its less accessible higher spatial frequencies, tothe extent that this is advantageous, and may thus yield asmaller dwell time spatial frequency response to achievea given rms residue. This will be demonstrated below fora case in which raw input data contain artifact high spa-tial frequencies that are due to data edges that are inac-cessible to the tool, where the computational efficiency ofthis approach is used to presmooth the raw data. Whilethe resulting correction will then be valid only for thesmoothed initial shape, one can often achieve moderate-fidelity corrections by using this technique without time-consuming data-edge extensions.

A comparison of the appropriate expressions in Appen-dix A shows that such a truncated series is completelyequivalent to the frequency or coordinate space recursivevan Cittert process.

(8a)where the only requirement is that I1 - WI < 1 [note thatthis also implies that W(k) # 0, the other requirementfor validity]. Thus, in terms of an arbitrary but well-behaved auxiliary function W(k), whose definition will beexplored further below, the dwell time FT [Eq.(6b)] canbe rewritten as

To(kX, ky) = [AHo(k., ky)So/FV]W E (1 - W)P.p= 1 (8b)Performing the same simplifications that yielded Eqs. (6)gives

Tn(kx, ky) = T(kX, ky){1 - [1 - W(k)]}AHn(kX, ky) = AHo(k, k)[1 -W(k)]n'

(8c)(8d)

showing that the convergence of Eq. (8c) can be achievedby the choice of an auxiliary function W for which both11 - W(k)I < 1 and nW/F < oo . We explore the formernext, and the latter will then be shown to be achievableeven when IF(k)I => 0 with a suitable choice of W.Th e convergence requirement 11 - W(k)I < 1 can berewritten as11 - W12= (1 - W)(1 - W* ) = 1 +WW* - (W + W*) < 1or

WW* < (W + W*), (8e)where W* is the complex conjugate. With W = Wr + iWi,this becomes

Wr 2 - 2W + 1 + Wi2 = (Wr - 1)2 + Wi2 < 1, (8f)3. REMOVING THE RESTRICTIONSTHAT ARE DUE TO TH E NATURE OFTHE DECONVOLVED FUNCTIONSHaving reconstructed VC D from this slightly differentperspective, we now use the same procedure to addressthe objective of removing the restrictions imposed bythe convergence requirement. F(k) = IF(k)Iexp[-i'k(k)]does not generally satisfy the convergence inequality11 - F < 1 for all k. As shown in Ref. 2, these re-strictions may arise from the shape of the tool shapefactor f(x, y) and its position relative to the dwell timer(x, y) [or, equivalently, F(kx, ky) and T.o(kX, ky)]. Un-surprisingly, although its cause is not so manifest in itscoordinate space implementation, VCD also suffers fromthe same restrictions on account of the shape and therelative position of the convolved functions.Th e procedure leading to Eqs. (5) implies a generalway of avoiding the above limitation, which we now showyields a completely robust and transparent deconvolutionand, in addition, a means of performing the process non-iteratively or in closed form. To free VC D from therestrictions on the shapes of (say) f (x, y) can be accom-plished by the replacement of F(kx, ky) in the series ex-

which is plotted in Fig. 2. This plot of the imaginary partof W (Wi) versus the real part of W (W,) shows that theconvergence of the expansion of Eq. (8a) is guaranteed in-side the circle (W,. - 1)2 + W,2 = 1, so that, for example,when W = W, (or W, = 0), the series converges for 0