tion coefficient pm,,=Dzm/(D~iDmm) 1 1 2 between the modeI and in is p2*4= 0. 003 and p14 , 16 = 0. 0002. As a, be-comes larger the correlation between the modes tendsto increase, especially for higher-order modes, e.g.,for a,= 2o, p2, 4 = O. 016, plo, 12 = 0. 04 3, and P12, 14= 0. 863.But we have seen that in this case and for 10% noiselevel only 10 modes should be included. This meansthat the modes which have a significant level are ap-proximately uncorrelated. As a, increases further themodes show a considerable degree of correlation. Forexample, for a,=4a, and 10% noise only 7 modes aresignificant and we get P4,6 = 0. 37 and p6 8a = 0. 95 3 . Inorder to determine the number of significant uncorre-lated modes one has to determine the eigenvalues of thematrix Dim.. Since we do not compute the odd modes,we are unable to perform this.

Objects with rectangular spectrum exhibit similar re-sults. For a, = 0. 5a3 the modes are practically uncor-related. For a, = a, the twelve significant modes havecorrelation coefficients varying from P214 = 0. 0023 top 0,,12 = 0. 14 and for a, = 2a, the six significant modes

have correlation coefficients varying from P2,4 = 0. 014to P4,6 = 0. 13. As a, increases, the number of modesdecreases but their cross-correlation increases, e.g.,for ae=4a 3 p2, 4 =0. 67.

'G. Toraldo di Francia, "Degrees of freedom of an image,"

J. Opt. Soc. Am. 59, 799-804 (1969).2C. K. Rusliorth and it. W. Harris, "Restoration, resolution

and noise, " J. Opt. Soc. Am. 58, 539-545 (1968).3 H. Brown, "Effect of truncation on image enhancement by

prolate spheroidal functions, "s J. Opt. Soc. Am. 59, 228-229(1969).

4B. R. Frieden, "Evaluation, design and extrapolation methodsfor optical signals, based on use of the prolate functions, "in Progress in Optics, IX, edited by E. Wolf (North-Holland,Amsterdam, 1971), pp. 311-407.

5 F. Gori and G. Guttari, "Shannon number and degrees offreedom of an image, " Opt. Commun. 7, 163-165 (1973).

6 F. Gori and G. Guttari, "Degrees of freedom of images frompoint-like-element pupils, " J. Opt. Soc. Am. 64, 453-458(1974).

7M. Bendinelli, A. Consortini, L. Ronchi, and B. R. Frieden,"Degrees of freedom and eigenfunctions, for noisy image,J. Opt. Soc. Am. 64, 1498-1502 (1974).

tC. W. Helstrom, "Modal decomposition of aperture fieldsin detection and estimation of incoherent objects, " J. Opt.Soc. Am. 60, 521-530 (1970).

5C. W. Helstrom, "Resolvability of objects from the stand-point of statistical parameter estimation, " J. Opt. Soc. Am.60, 659-666 (1970).

'0F. Gori, "Integral equations for incoherent imagery, " J.Opt. Soc. Am. 64, 1237-1243 (1974).

HF. Gori and C. Palma, "On the eigenvalues of the sinc2Kernel, " J. Phys. A 8, 1709-1719 (1975).

12A. Fedotowsky, thesis (Laval University, Canada) (1972).

Image restoration subject to surface area or arc length constraints

Eric G. HawmanOptical Sciences Center, University of Arizona, Tucson, Arizona 85721

(Received 21 June 1976)

The restoration of edges is often degraded by spurious oscillations (Gibbs phenomenon). Constraining therestoration surface area or arc length is an effective method of suppressing these oscillations. Results ofiterative restorations using both arc length and positivity constraints show edge gradients that can beenhanced by 5:1 over those of the image, while spurious oscillations can be nearly eliminated.

INTRODUCTION

Image restoration methods that enhance resolution usu-ally produce spurious oscillations when applied to edges.In this paper we attempt to overcome this difficulty byrestricting the amplitude variability of the restoredimage. It is intuitively appealing to do this by con-straining the surface area of the restoration.

We formulate the equations for the digital restorationof two-dimensional image subject to a surface area con-straint. The resulting two-dimensional equations ap-pear to be very demanding of computer resources.There may be more efficient algorithms for their solu-tion than we have considered. However, our primarypurpose is to verify the possible usefulness of the sur-face area constraint in image restoration. Hence wehave specialized two-dimensional image restorationequations to the one-dimensional problem with con-strained arc length. One-dimensional computer simu-lations are shown illustrating the usefulness of the arc

76 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977

length constraint for the enhancement of edges. Also,the one-dimensional restoration is of optical interest in

its own right for application to problems where thespread function is separable and the object is one-di-mensional, such as an absorption spectrum.

RESTORATION PROBLEM

Incoherent image formation can be described in termsof a superposition relationship. Let o(x) be the objectdistribution and i(t) the image distribution, where xand t denote (x, y) and (Q,71). These are related by theequation

i(V) = fs Q, x)o(x)dx+nl7 ), (1)

where s(, x) denotes the optical spread function, and11(y) denotes the detection noise. In the restorationproblem we treat here, we seek to estimate the objectfrom image data and a known spread function.

Copyright (D 1977 by the Optical Society of America 76

Equation (1) is a linear integral equation of the first

kind. It is well known that a direct solution of Eq. (1)

is improperly posed. 1-4 The kernel s(Q, x) may nothave an inverse, and if it does, the inverse may have

very large values. The presence of small amounts ofimage noise then causes large spurious oscillations inthe estimate of the object.2

One method of overcoming this problem of instabilityis the method of selection. 5 Here the object is re-stricted to a physically permitted class of functions.

Continuous variations are made on objects within theclass until the image matches the measured image datawith sufficient accuracy. Based on this notion of class

restriction, Lavrentiev has given a procedure for solv-

ing Fredholm equations of the first kind using approxi-mate data.

Somewhat easier to apply are methods based on regu-larization, penalty functions, and constrains which use

additional information about the object in order to sta-bilize the solution. This approach was used by Phillipsin developing a practical algorithm for the solution of

the one-dimensional form of Eq. (1) when there is a

substantial amount of noise in the data. 2 In the two-dimensional generalization of Phillips' approach, onewould minimize a linear combination of the square of

the image residuals and the square of the Laplacian of

the obj ect, i. e.,

J [Ss( ,x)o(x) dx-i iQ dt + aJ[V2o(x)]2 dx= min.

(2)In one-dimensional form, this approach works quite

well and spurious oscillations are greatly reduced.The parameter a controls the degree of smoothing.For a = 0, the problem reduces to direct inversion andthe solution is unstable. The estimate of o(X) obtainedfrom Eq. (2) is a linear function of the image data;consequently, the bandwidth of the object solution doesnot exceed that of the image; also, the object does notnecessarily obey physical constraints such as positivity.

Phillips' method can be generalized as follows:

X [f s s(, x)o (x) dx - i t)] dt, + ao (x) = min . (3)

Here, ¢(x) is a suitably chosen function that represents

auxiliary information about the object.7 '8 This generalapproach is known as Tikhonov regularization. Twomeyconsidered as an example (for the one-dimensionalproblem) the case where 4p(x) = f[o(x) -p(x)]2 dx and p(x)is a known profile or trial solution.9 The regularizingfunctional O(x) =af[o'(x)l2 dx+01f[o(x)]2 dx was first pro-

posed by Tikhonov, and was applied to image restorationby Barakat and Blackman.'

All of the aforementioned regularizing techniques use

a function P(x} which is quadratic in the object; conse-quently the resulting object solutions obtained from Eq.

(3) are linear functions of the image data. It thereforefollows that the solutions are not necessarily positiveand do not have greater bandwidths than the image.

There has been much progress in the last eight yearson the development of restoration algorithms which are

77 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977

super-resolving, i. e., the bandwidth of the object re-

storations exceeds that of the image data. Frieden hasgiven a useful survey of some of these new algorithms.'Most of the new algorithms achieve super-resolution byforcing the object estimate to be both positive and con-sistent with the image data. Generally, results arevery good for impulse-type objects, but for extendedobjects they do not perform much better than by linearmethods (an exception is the method of Friedenl0 ; seebelow). The restoration of edges and steep edge gradi-ents are generally accompanied by a large amount ofspurious oscillation or ringing-the steeper the edgegradient of the restoration, the greater the ringing.Hence there is a need for restoration algorithms whichpermit the restoration of the steep edge gradients andwhich do not permit excessive ringing, and which areconsistent with physical constraints such as possitivity.One algorithm addressing this need has recently beenpresented. 10 Here we propose a second approach,based on the use of surface area for regularization.

SURFACE AREA PENALTY

One way to suppress ringing in a restoration is topenalize surface area (or arc length in a one-dimen-sional problem). The usefulness of a profile lengthconstraint has been indicated by Lucy. 11 If we case therestoration problem in the form of Eq. (3), then thesurface area penalty function is

(x)= f[1+2I Vo(X)w2 1'/2dx, (4)

where i3 is a scale factor necessarily required by thedifferent dimensions of o and x. There do not appearto be any prior values of 13 which have special signifi-cance. Therefore we treat 13 as a free parameter andselect those values which give the desired control ofspurious oscillation and resolution. Note that ¢(x) inEq. (3) can be thought of in two different ways: as aregularizing function, or as a penalty function. "Regu-larization" derives from Tikhonov's approach to solvingFredholm equations of the first kind, and "penalty" de-rives from optimization theory. 12 In the sense of opti-mization theory, Eq. (3) can be described as a least-squares problem with a nonlinear penalty.

Since we use numerical methods to obtain the solutionof Eq. (3), we now write out the functional to be mini-mized in discrete form:

F[o] = E(2 Sm,n~,i mmn ii niJi~ -n

(5)+ a E {1 + p32[(AXoi, j)2+Q (AXO, "j)21}h /2 = min.,

ii

where oij =o(xi, yi), Sm,n;i, =s( m,711n;Xi) Yj), m,n

=ij(m97n)9 A/xi,j=0 i+1,J 0i,- A \yi,,=°iJ1 -°, and°= (o,,,, 01,2 .*.., ON,N). The object samples and imagesamples are N by N and M by M arrays, respectively.

By taking the partial derivative of F with respect toeach object variate o.,, and setting it to zero, we ob-tain the following system of nonlinear equations for the

object:

Eric G. Hawman 77

.U"(0)=' aO =1:(1Sm,n,i,J0i,J -mn)SM )2 8o0,,vrnk i inu 2 'd[1+ / 2(A"o,,.v)2 + ( 2(AO .)2]i /2

Axiu+( ) v[ 1 + 2(AX0U_-l V)2 + f A ,0_1' V)211/2

[ v-l \

[l +A(Ao 0 )+2A°Uvl

for it,v=1, ... ,N.

In principle, this system of N 2 simultaneous nonlin-ear equation in N2 unknowns can be solved by the multi-variate Newton-Raphson method. 13 This is an iterativeprocess whereby trial solutions are corrected at eachstep; the corrections are determined by solving a lin-ear system.

The arrays {fuj and {oi j} can each be indexed by asingle subscript, i.e., {furj - {t%} and {o,,} -{or},where q=(C-1)N Vand r=(i-l)N+j. Let oP={°o} bethe pth trial solution of Eq. (6) and Etq be the residual ofthe qth equation at the pth step, i. e.,

f,(°t)=EaP, q=l,...,N2 .7

By solving the following linear system

rl 9°r |p 6o° =EP, q =1,...,N2 (8)r= 80 , q

for the 6°or, we obtain corrections to the pth trial solu-tion. Hence the next trial solution is ot' =-oP+ 6oP forr=1, .. .,N

2 . Iteration of this procedure is continueduntil the residuals are negligible.

Restoration with constrained arc length is not neces-sarily non-negative. However, positivity can be en-forced by a transformation of variables in the mini-mization problem. 14 That is, we let o° =o(z;), which isa function such that o; '0 for all real values of z;. Wethen find the set of zj for which F[oi(zj)] is a minimum.For example, the transformations Oc =zj2 or oj=cezguarantee positive object values.

Solving the system of Eq. (8) requires computation ofthe Hessian matrix:

H = [Ha,] [2 (9)

This matrix is symmetric and generally, for small val-ues of a, it is positive definite. When positivity is en-forced by means of the transformation oj = z2, then thenew Hessian matrix becomes

H'- Hqr -] LaF6'+ 2ze~zr 8 2 (1 0)

[H'] [8F" r rJ .

This matrix is generally not positive definite. In thenumerical solution of system (6) when the transforma-tion oi =z2 is used, it is found useful to use the modi-fied Newton-Raphson method of Goldfeld. 15 At eachiteration, a sufficiently large constant is added to eachelement of the Hessian matrix diagonal such that theresulting matrix is positive definite. This aids in in-suring that at each iteration step, the value of F(o) isdecreased. The added constant is chosen to be a smallmultiple of the magnitude of the most negative elementon the diagonal of H'.

78 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977

For N 2 object elements, the Hessian matrix has N4elements. For objects of practical size, say N ' 100,the solution of Eq. (6) by use of the Newton-Raphsonmethod and Eq. (8) would be formidable with present-day computers. It may be possible to solve Eq. (6) byrelaxation methods (e. g., nonlinear Gauss-Seidel meth-od), which do not require a Hessian matrix.

We can specialize Eq. (6) to the one-dimensionalcase, that of constrained arc length. It is important toshow that the method is useful in one dimension; if it isriot, further work on the two-dimensional problem wouldbe contraindicated. Also, the one-dimensional formu-lation is important in its own right. For example, ifthe spread function is separable, i. e., s( , '; x, y)-=s(1, X)S 2(q, y), then restoration on a line basis is jus-tifiable.

The one-dimensional form of Eq. (6) is obtained bysetting all the A o factors to zero and dropping secondindices. This gives

fu(o) = E Smii im)Smu

ap Aou AOu- =12~~~7 F~+2tu2]12[1+:2(Ao -)2]1/2 ~ (11)

for u = 1, .. ., N. This one-dimensional form is used forthe illustrations below.

ILLUSTRATIONS OF THE METHOD

We consider the case of the diffraction-limited pupilwhere the spread function is

s(x, y) = (1/r) sinc2 [(x - y)/rl

r is the Rayleigh distance and sinc(x) = (sin7rx)/Arx). Asthe first test case we used a rectangular object with awidth 3 times the Rayleigh distance r. Uniformly dis-tributed noise with a maximum amplitude 1% of theideal image was added to the image prior to restoration.The object was specified by 53 sample points and theimage was sampled at the Nyquist interval, every thirdgrid point in this case. Figure 1(a) shows the object,the noisy diffraction-limited image, and the restorationobtained by inverse filtering.

The same image data was restored using an arclength penalty function and enforcing positivity with thetransformation oj =AZ2. For the penalty function weightce = 6.67x 10-5 and the scale factor 3 = 16. 7, the resultsshown in Fig. 1(b) were obtained. Comparing the tworestorations, we find the one shown in Fig. 1(b) hasedge gradients at least 5 times greater than the imageor 22 times the inverse-filtered restoration. Also, thespurious oscillations have been greatly reduced. If wedefine the modulation of these spurious oscillations bythe factor (max - min)/(max + min), then for the case

Eric G. Hawman 78

(6)

magnitude. However, note that the enforcement of

v \positivity alone effectively gives total suppression ofoscillations about the zero level.

(a) | \The application of the arc length constraint without

enforcement of positivity leads to results such as shownin Fig. 1(d). Here, edge gradients of the restorationare about 31 times those of the image, which is not

- /quite so good as when positivity is also enforced [seeFig. 1(b)]. However, it is still better than the inverse-filtered result. There is residual oscillation about boththe upper level and the zero level. The magnitude ofthe upper level oscillation is at least 5 and 10 times

(W smaller than those of the inverse-filtered restorationand the positive restoration, respectively.

The effects of increased noise levels on the restora-tions were examined. Figure 2(a) again shows the

image and inverse-filtered restoration of the rectangu-lar object when the image data now has 8% relativenoise. The distortion of the image is clearly evident.The amplitude of spurious oscillations in the restora-tion is roughly twice that shown in Fig. 1(a). Figure2(b) shows a restoration of the same image data ob-tained using the arc length constraint and enforcing

(c) positivity. For the case shown, the penalty functionweight x= 2. 00x 10-4 and scale factor 3 = 16. 7. Thisrestoration shows increased spurious oscillation, butthe magnitude is approximately 3 times smaller thanthose of the inverse-filtered result. There is alsosome degradation in the edge gradients. The left andright edge gradients are approximately 1. 5 and 2 timesgreater than those of the inverse-filtered restoration.

As a second test object, we used two closely spacedimpulses separated by one-half the Rayleigh distance.

(d) Figure 3(a) shows the diffraction-limited image and theinverse-filtered restoration. The impulses are clearly

FIG. 1. (a) Inverse-filtered restoration. The object is the (a)rectangular pulse. The diffraction-limited image (dashed (a)curve) contains 1% relative noise. The solid curve is the in-verse-filtered restoration (b) Restoration using an arc length

penalty and enforcing positivity. The image data is the sameas in (a). The values of penalty function weight and the scalefactor are a = 6. 67 x 10-5 and /3=16. 7. (c) A pure positive res- ---toration (a = 0) . The image data is the same as in (a). (d) Anarc length constrained restoration without enforced positivity.The image data is the same as in (a): a = 10-4 and /3 = 10. 0

shown, these oscillations have been reduced 11-fold. (b)

It is of interest to know how much of the improvementof the restoration shown in Fig. 1(b) can be attributedto the arc length constraint and how much to the enforce-ment of positivity. A pure positive restoration can be

obtained by setting the penalty weight to zero while still.. .. .FIG. 2. (a) Inverse-filtered restoration. In this case, the

enforcing positivity. Figure 1(c) shows such a positive i ( cimage (dashed curve) contains 8% relative noise. Compare

restoration without an arc length constraint. The edge with Fig. 1(a). (b) An arc length penalty restoration with en-

gradients are approximately 4 times those of the image, forced positivity. Image data was the same as in (a). a =2. 00

and the spurious oscillations have nearly doubled in K 10-4 and / = 16. 7.

79 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977 Eric G. Hawman 79

(a)

(b)

FIG. 3. (a) Inverse-filtered restoration of a pair of impulses.Image data (dashed curve) contains 1% relative noise. The im-pulses are separated by the Nyquist interval, one-half the Ray-leigh distance. (b) Two arc length penalty restorations with en-forced positivity. The image data is the same as (a). Thesolid and dashed restorations are for scale factors of 3 = 1. 0and P = 10. 0, respectively. In both cases, a = 1. o x lo-8 .

not resolved by inverse filtering. The image contains1% relative noise. Figure 3(b) shows two restorationsobtained by using an arc length penalty regularizingfunction and enforced positivity. As the scale factor 13is increased, the impulses are less clearly resolved inthe restoration.

Restoration of this second test object was also triedusing the arc length constraint alone without enforcedpositivity. In this case, the impulses could not be re-solved.

Generally, the restorations shown in the figures withconstrained arc length and enforced positivity requiredabout 10 iterations. Each iteration required about 1 son the CDC-6400 computer.

SELECTION OF PARAMETERS at AND 13

In practice, the choice of the penalty function weighta and the scale factor 13 is mostly a matter of trial anderror. There are several things which may help intheir selection. For instance, if some features in animage are known, such as an edge, then a, 13 could beselected to give good restoration of this feature. Hope-fully, then other unknown features would be simulta-neously enhanced. Also, our numerical experimentsshow that spurious oscillations are more sensitive intheir position and amplitude to variations of the param-eters than are real features. The parameters may thenbe selected to eliminate to the greatest extent the spuri-ous features in a family of restorations.

In our numerical experiments, the restorations werefound to be more sensitive to selection of the scale fac-tor 13 than to the selection of a. Also, it was desirableto make 13 as large as possible without making the sys-tem of equations numerically unstable. From Eq. (5)

80 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977

we see that an optimal value of a will be proportional tothe ratio of noise variance and object arc length. If ais too large, the restorations are overly smooth andresolution is degraded. If a is too small, the resolu-tion may be good, but the spurious oscillation is ex-cessive. Fortunately, there is considerable latitude inthe choice of a suitable a. For example, with 13 = 16. 7,the restorations of rectangular objects [see Fig. 1(b)]were nearly as good for any a in the range lo-3 -10- 7 .

CONCLUSIONS

As the restoration of the two impulses demonstrates,the restoring algorithm with penalized arc length andenforced positivity is super-resolving. The ability toresolve closely spaced impulses is due here to the en-forcement of positivity. However, both the arc lengthconstraint and positivity enforcement contribute to theedge enhancement of extended objects. More impor-tant, the arc length penalty function offers a very ef-fective method of control over spurious oscillations.Finally, this suppression of spurious oscillations can beachieved without destroying the sharpness of restorededges.

The success of the arc-length constraint in the one-dimensional problem justifies further work for develop-ing efficient procedures of solving the equations for sur-face area constrained restorations.

ACKNOWLEDGMENTS

I am grateful to Professor B. Roy Frieden, Dr.Ronald S. Hershel, and Dr. James J. Burke for theircomments and discussions of this work.

'B. R. Frieden, "Image enhancement and restoration," in Pic-ture Processing and Digital Filtering, edited by T. S. Huang(Springer-Verlag, New York, 1975), pp. 177-248.

2 D. L. Phillips, "A technique for the numerical solution ofcertain integral equations of the first kind," J. Assoc. Compt.Mach. 9, 84-97 (1962).

3 R. Barakat and E. Blackman, "Application of the Tichonovregularization algorithm to object restoration," Opt. Commun.9, 252-256 (1973).

4 M. M. Lavrentiev, Some Improperly Posed Problems ofMathematical Physics (Springer-Verlag, New York, 1967).

5A. DI. Tikhonov, "On the stability of inverse problems," Dokl.Akad. Nauk SSSR 39, 176-180 (1943).

6 M. M. Lavrentiev, "On integral equations of the first kind,"Dokl. Akad. Nauk SSSR 127, 31-33 (1959).

7R. Bellman, Introduction to Matrix Analysis, 2nd Ed. (Mc-Graw-Hill, New York, 1970), p. 362.

8A. N. Tikhonov, "Solution of incorrectly formulated problemsand the regulation method, " Sov. Math. -Dokl. 4, 1035 -1039,(1963).

9S. Twomey, "On the numerical solution of Fredholm integralequations of the first kind by the inversion of linear systemsproduced by quadrature," J. Assoc. Compt. Mach. 10, 97-101 (1963).

10B. R. Frieden, "A new restoring algorithm for the preferen-tial enhancement of edge gradients," J. Opt. Soc. Am. 66,280-283, (1976).

11L. B. Lucy, "An iterative technique for the rectification ofobserved distributions," Astron. J. 79, 745-754 (1974).

Eric G. Hawman 80

12L. C. W. Dixon, Nonlinear Optimization (Crane, Russak &

Co., Inc., New York, 1972), Chap. 6.13G. Dahlquist, A. Bjorck, and N. Anderson, Numerical Meth-

ods (Prentice-Hall, Englewood Cliffs, N. J., 1974), p. 249.

14S. L. S. Jacoby, J. S. Kowalik, and J. T. Pizzo, IterativeMethods for Nonlinear Optimization Problems (Prentice-Hall,

Englewood Cliffs, N.J., 1972), p. 31.15 Reference 14, p. 121.

Long- baseline optical interferometer for astrometryMichael Shao and David H. Staelin

Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139(Received 9 June 1976)

A proposed astrometric interferometer capable of relative stellar position measurements with 10' arcsec

accuracy is described. The instrument is a long-baseline Michelson interferometer modified to track thefringe motion caused by atmospheric turbulence. Simultaneous fringe amplitude and phase measurements at

two wavelengths are used to correct atmospheric distortion when the field of view is much larger than the

isoplanatic patch. Relative positions of stars brighter than - 10 mag in an - 1° field of view can be measured

with an accuracy of -10' arcsec after several hours of observation. Such an instrument should have a

number of interesting astrophysical and geophysical applications, such as a search for planets around nearby

stars, the gravitational deflection of light around the sun, and changes in the earth's axis of rotation.

1. INTRODUCTION

Stellar astrometry has a long history. The tech-nique that is most widely used at present is photographicastrometry, by which stellar positions are determinedfrom precise measurements of photographic images.Relative stellar positions can be measured with an ac-curacy of 0. 05-0. 1 arc sec from a single photographicplate.1 By averaging measurements from severalthousand plates, a formal error of - 10-3 arcsec maybe achieved. Comparison of data from different obser-vatories or from different sets of plates taken at thesame observatory reveal the presence of systematicerrors that limit the ultimate accuracy in most casesto-10-2 arcsec.

2

It has become increasingly obvious that substantialimprovement in astrometric accuracy will provide an-swers to many astrophysical questions. This has stim-ulated the development of several new types of instru-ments.

One approach is to place a laser-monitored measur-ing engine at the focal plane of a mechanically and ther-mally stable telescope. The accuracy of such an in-strument is only limited theoretically by the atmo-sphere. For example, accuracies of 0. 03 arc sec in4 min of observation have been reported for SCLERA,which observes the position of the solar limb.3 Ifsources of systematic errors could be understood forthis instrument, 10-3 arcsec accuracy should beachievable after - 70 h of observation, and 10-4 arc secaccuracy after - 7000 h.

A second, more promising, approach is to measurethe angles between stars with a high spatial resolutioninstrument. If it is possible to construct a simplehigh-resolution instrument for which an accuratemathematical model of systematic errors can be de-veloped, 10-4 arc sec accuracy should be achievableafter a few hours of operation.

The advantage of high-resolution instruments comes

81 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977

from their ability to make extremely accurate mea-surements in a short time. This is important in orderto evaluate instrumental errors before the instrumentparameters change. To achieve 10-3 arcsec accuracyin photographic astrometry, for example, would re-quire that the telescope be stable to 10-3 arc sec for aperiod of several years while the coefficients of a modelfor systematic errors are measured; hence, it is notpossible to correct 10-3 arcsec instrumental errorsexplicitly in photographic astrometry. But the prop-agation of instrumental errors as a result of averagingdata taken over a period of years may be avoided if thedesired accuracy is achieved within several hours.

High-resolution instruments must first overcomethe effects of atmospheric seeing. Since the inventionof speckle interferometry, 4 techniques have been pro-posed to obtain diffraction-limited information throughthe turbulent atmosphere. 5-13 These techniques cangive diffraction-limited astrometric performance fora 2-3 arc sec field of view. This field of view, oftencalled the "isoplanatic patch," is limited by turbulencethroughout the troposphere, which has been studied ex-perimentally and theoretically. 14-16 Thus high-reso-lution astrometry has been limited to close doublestars.

Another instrument for obtaining diffraction-limitedperformance is the astrometric interferometer de-scribed below. The unique feature is the importantability to measure precisely the angular separationsof stars located in totally distinct isoplanatic patches.The purpose of this discussion is description of thenovel features which make the system possible, andconsideration of some of the important ancillary prob-lems so as to make the total system performance es-timates plausible.

II. ASTROMETRIC INTERFEROMETER

As is well known, a Michelson stellar interferometeris capable of extremely high resolution. By modifying

Copyright © 1977 by the Optical Society of America 81