UNIQUENESS PROPERTIES OF HIGHER-ORDER AUTOCORRELATION FUNCTIONS

The kth-order autocorrelation function of an image is formed by integrating the product of the image and k independently shifted copies of itself: The case k = 1 is the ordinary autocorrelation; k = 2 is the triple correlation. Bartelt et al. [Appl. Opt. 23, 3121 (1984)] have shown that every image of finite size is uniquely determined up to translation by its triple-correlation function. We point out that this is not true in general for images of infinite size, e.g., frequency-band-limited images. Examples are given of pairs of simple band- limited periodic images and pairs of band-limited aperiodic images that are not translations of each other but that have identical triple correlations. Further examples show that for every k there are distinct band-limited images that have identical kth-order autocorrelation functions. However, certain natural subclasses of infinite images are uniquely determined up to translation by their triple correlations. We develop two general types of criterion for the triple correlation to have an inverse image that is unique up to translation, one based on the zeros of the image spectrum and the other based on image moments. Examples of images satisfying such criteria include diffraction-limited optical images of finite objects and finite images blurred by Gaussian point spreads.


INTRODUCTION
The autocorrelation of a real-valued function f is another real function af created by integrating the product of f and a shifted copy of itself: a function of the general form af(s) = f f(x)f(x + s)dx. It is a basic and sometimes frustrating fact of Fourier analysis that the autocorrelation function (ACF) a completely identifies the amplitude spectrum of f but provides no information about its phase spectrum. In recent years there has been growing interest in the possibility of recovering phase information from higher-order ACF's created by integrating the product of a function and multiple shifted copies of itself.' 9 Generalizing the concept of autocorrelation, one can construct a sequence {ak, f: k = 1, 2 ... } of ACF's of a real function f where the kth-order ACF a,f(S1,. .-Sk) is created by an integral of the form f f(x)f(x + l)... f(x + s)dx. In this sequence a is the ordinary ACF and a 2 ,f is the triple-correlation function, which has been widely applied in optics' and is beginning to find uses in vision research. 4 ' 1 0 (The Fourier transform of the triple correlation is commonly known as the bispectrum, so our numbering agrees with standard terminology in the spectral domain.) In the optics literature, application of the triplecorrelation function to phase-recovery problems is usually justified by reference to a uniqueness theorem that is due to Bartelt et al. showing that, if a real function f has bounded support, a 2 , determines f up to a translation [i.e., up to the form f(x + c), where c is an unidentifiable centering parameter]. In other words, the triple correlation of any image of finite size contains sufficient information to identify both the amplitude spectrum and (except for a centering term) the phase spectrum of that image. A natural question is whether this is also true for images of infinite size: frequency-band-limited images, for example, or finite images blurred by Gaussian pointspread functions. This question does not seem to have been directly addressed in the recent literature, but algorithms for recovering infinite-duration temporal signals from their triple correlations have appeared,1 2 and one might be led to think that images with infinite support, like finite images, are always uniquely determined up to translation by their triple correlations.
However, simple counterexamples show that this is not the case. Figure 1 illustrates a pair of nonnegative band-limited integrable functions that have the same triple-correlation function (as is shown below in Subsection 2.D.3) but that are not translations of each other: The functions are sinc 2 (x) (1 + cos 6x) and sinc 2 (x) (1 + sin 6rx), where sinc(x) = sin 7rx/7rx. Figure 2 illustrates a pair of nonnegative periodic functions with the same property: The functions are 2 + cos 27rx + cos 6x and 2 + cos 27rxcos 67rx. (This example is due to Klein and Tyler. 4 ) More generally, it can be shown that for every k there are pairs of nonnegative band-limited integrable functions and also pairs of nonnegative band-limited periodic functions that have identical kth-order ACF's but are not translations of each other. (Examples are given in Subsections 2.D and 2.E.) Thus without the assumption of bounded support, neither the triple-correlation function nor any other finiteorder ACF uniquely determines every image up to a translation: The best one can do is identify useful special classes of infinite images that are so determined.
The identification of such classes is the main purpose of this paper. We examine the uniqueness properties of higher-order ACF's of functions that represent monochromatic images: nonnegative real functions defined on the line R or the plane R 2 . Two general classes of image function are considered: integrable functions (i.e.,  our chief concern) and infinitely extended periodic functions. For integrable images with bounded support we re-prove the triple-correlation-uniqueness theorem of Bartelt et al." by using a different approach, one that can be extended to certain classes of images with infinite support. The proof of Bartelt et al. relied on the fact that the Fourier transform of any function with bounded support is determined, up to a translation factor, by the zeros of its analytic continuation, which can be identified from the zeros of the analytic continuation of its bispectrum.
That approach fails for functions with infinite support, whose complex transforms may be nonvanishing (e.g., Gaussians). Our proof is based on a functional-equation argument combined with the fact that all integrable functions with bounded support are uniquely determined by the values of their Fourier transforms in a neighborhood of the origin (that is, by the derivatives of the transform at zero, which determine the entire transform by means of a Taylor series). From a functional-equation standpoint the key fact about two images f and g with the same triple correlation is that their Fourier transforms F and G must satisfy a relationship of the form for all arguments u,u Thus the uniqueness properties of the triple correlations of images are intimately related to the solutions of Eq. (1) with F and G complex functions on R or R 2 . If f and g belong to a class of functions for which all solutions of Eq. (1) take the form G(u) = exp(i27rc-u)F(u) (2) for some constant c E R or R 2 , then f and g have the same triple correlation if and only if g(x) = f(x + c). This is true for all integrable image functions with bounded support, because it can be shown that Eq. (1) implies that Eq. (2) holds for all u in some neighborhood of the origin and the transforms of finite images are determined everywhere by their values in such a neighborhood. For infinite images it remains true that Eq. (1) implies Eq. (2) in a neighborhood of the origin, but in general it is no longer the case that the image transform is completely determined by its values in such a neighborhood. Thus for infinite images one needs to impose additional constraints to guarantee that Eq. (2) is the only solution to Eq. (1). Two kinds of constraint suggest themselves. One involves restricting the zeros of the image transform in such a way that solution (2) can be recursively extended from a neighborhood of the origin to all values of u. From an image-reconstruction standpoint, this approach corresponds to the recursive algorithms proposed by several authors for recovering the phase spectrum of an image from that of its bispectrum.""-' 5 Such algorithms implicitly rely on the bispectrum's having adequate support, which is always guaranteed for finite images but not for infinite ones unless the zeros of the image spectrum are constrained.
The other approach is to use the fact that, when Eq. (2) holds in a neighborhood of the origin, the moments of g(x) are identical to those of f(x + c) for some c, which may be enough to guarantee that g(x) = f(x + c). This approach to triple-correlation uniqueness corresponds to a second type of image-reconstruction algorithm, in which the moments of the image are recovered from moments of its triple correlation and the image is reconstructed from its moments. Such a reconstruction is always possible in principle for finite images but not for infinite ones unless the image moments are appropriately constrained.
Using these ideas, we show that the triple-correlation function uniquely determines, up to translation, the following types of one-dimensional (1-D) and twodimensional (2-D) integrable image: (1) Images whose Fourier transforms are nonvanishing everywhere (e.g., Gaussians, Gabor functions, exponential and gamma densities, Cauchy densities, and any convolution of such images).
(2) Band-limited images whose transforms have no zeros [e.g., sinc'(x), J,'(r)/r 2 ], or at most a finite number of zeros [e.g., sinc 2 (x) or Jl'(r)/r' convolved with any image of finite size], below the frequency cutoff' 6 In two dimensions the condition refers to zeros along the axes corresponding to each spatial-frequency orientation. This result implies that the triple-correlation function determines all the diffraction-limited incoherent optical images of finite objects formed with (for example) square or circular exit pupils.
(3) Images whose transforms have at most a finite number of zeros in every finite interval. (In two dimensions, this refers to intervals along the axes corresponding to each spatial-frequency orientation.) Examples include any of the functions cited in condition (1) convolved with any image of finite size, in particular, any finite image blurred by any Gaussian point-spread function.
(4) Images with the property that every point in frequency space at which the transform is not zero can be finitely linked to a neighborhood of the origin in which the transform is nonvanishing. This is a technical condition that is not easy to state concisely but that is useful for establishing triple-correlation uniqueness in cases in which the transform of a band-limited image vanishes over an interval below the frequency cutoff. For example, it shows that the triple correlation determines the functions sinc 2 (x) (1 + cos 57rx) and sinc 2 (x) (1 + sin 5irx), which are similar to the counterexamples of Fig. 1  If the moments of an image satisfy that condition, its complex Fourier transform is analytic in a strip containing the real axis and consequently is uniquely determined for all real arguments by its derivatives at the origin, which can be obtained from certain derivatives of the bispectrum. [In that case the transform cannot have infinitely many zeros in any finite interval of the real axis, so the moments condition becomes redundant with condition (3) above, although perhaps easier to verify in some contexts.] However, there are also images that are determined by their moments and thus by their triple correlation but that do not satisfy the limit condition: f(x) = exp(-x"l') (x 0) is a 1-D example.
Thus certain natural subclasses of integrable images with infinite support are uniquely determined up to translation by their triple correlations. But this is not true of the entire class, and the imposition of limits on the image bandwidth does not improve the situation, because there are images with arbitrarily small bandwidths (e.g., rescalings of the images in Fig. 1) that are not determined by their triple correlations. One's next hope might be that the entire class of integrable images would be determined by the kth-order ACF for some fixed k > 2. But that is not the case: as was noted above, for every k one can find pairs of integrable band-limited images that are not translations of each other but whose ACF's agree for all orders 1 through k. However, one can show that, if all the ACF's of two integrable images f and g are identical (i.e., a,g = ak,f for all k), then f andg must be identical except for a translation.
We also sketch the uniqueness properties of the higherorder ACF's of infinitely extended periodic images, drawing on the work of Klein and Tyler. 4 Here again two images f and g have the same triple-correlation function if and only if their transforms satisfy Eq. (1), but the discreteness of the spectrum in this case weakens the force of that constraint, making the uniqueness problem more difficult. For example, it is easy to prove that every integrable image is determined up to translation by its entire set of ACF's of all orders, but it is not so obvious whether this is also true of periodic images. As in the integrable case, for every k one can find pairs of distinct band-limited periodic images that have the same kthorder ACF, and one can quickly show that the set of ACF's of all orders determines any periodic image whose spectrum is nonvanishing at the fundamental frequency. But we do not know whether this is true for arbitrary periodic images, and we leave that as an open problem.' 9 On the positive side, we show that the triple correlation uniquely determines all periodic images that satisfy conditions analogous to condition (1) or (2) above.
The uniqueness properties of higher-order ACF's are essentially the same for 1-D and 2-D images. We deal first (in Section 2) with the 1-D case in some detail, and then (in Section 3) we show that the same reasoning can be readily extended to 2-D images. It will be seen that the proofs in Section 3 are actually independent of dimensionality, so with natural rephrasing the theorems hold for nonnegative real functions on RW for any n.
From a reconstruction standpoint, the fact that some infinite images are not uniquely determined up to translation by their triple correlation raises two immediate questions, which we address in Subsection 2.F In such cases a given triple-correlation function could have been generated by at least two disjoint families of images, say {f(x + c)} and {g(x + c)}. What happens if one tries to invert such a triple-correlation function? Is one of the possible inverse-image families arbitrarily singled out, or does the inversion procedure simply fail to give any result? (For the two inversion procedures considered here the answer is the latter.) The other question concerns the fact that in principle any finite portion of an infinite image can be uniquely reconstructed from its triple correlation, even though the triple correlation of the entire image may not have a unique inverse. What effect does the nonuniqueness of the triple correlation of the entire image have on one's ability to reconstruct a finite portion of it? (Basically, the answer here is that reconstruction becomes increasingly unreliable as the size of the finite portion grows, because one is forced to assign definite phase values to the bispectrum at points where its absolute value is suspiciously close to zero. In a sense, the more we know about the bispectrum of such an image, the less certain we become about the image itself.) To avoid possible misunderstanding, it is noted explicitly that, while the following analysis often relies on probabilistic arguments (exploiting the formal similarity between images and probability distributions), the images that concern us are always deterministic: the paper does not deal with higher-order ACF's of stochastic processes.

HIGHER-ORDER AUTOCORRELATIONS OF 1-D IMAGES A. Definitions
We think of a function f :R -R as representing a 1-D image in the sense that the total amount of light in any interval I is f f(x)dx. We will say that f is an image function (or, simply, an image) if f is nonnegative and integrable over every finite interval. 2 0 If two functions f and g have the same integral for every I so that f = g almost everywhere, then f and g represent physically indistinguishable images, and we write simply f = g. In particular, if f .,f(x)dx = 0, then (since f is nonnegative) f = 0. We consider the ACF's of two classes of image: integrable and periodic. If f f(x)dx is finite, f is an integrable image and its kth-order ACF is denoted ak,f and defined as ak, f(S1, *Sk) = f(X)f(X + S) . f(X + Sk)dx. (3) A probabilistic argument shows that ak,f is integrable over Rk for all k. Let a = f Xf(x)dx.
By definition a is finite, and it is positive unless f = 0, in which case ak,f = 0 and the claim is trivially true. Assuming that a > 0, let pf (X) = f(x)/a. Then pf is a probability-density function.
Let x0, x1,.. , Xk be k + 1 independent random variables, each having the density function p . Then the probability-density function of the k-dimensional random vector (x1 -xO,x2 -xO,...,Xk -xo) at the point Since the density of the random vector must be integrable over Rk, so is ak, f.
which is equivalent to the definition akf(sl,...,Sk) = (lip) The function akf is periodic in Rk. Its integral over every k-dimensional cube with sides of length p is (/p)pk+l, where 6 is the integral of f over a single period.
[In the first paper applying higher-order ACF's to visual perception, Klein and Tyler 4 analyzed what they called the "generalized autocorrelations" of periodic images, which are the same as the kth-order ACF's defined by Eq. (4) except for numbering: the "kth-order generalized autocorrelation" in their terminology is our (k -1)storder ACF ak-1,f]

B. Fourier Transforms
The Fourier transform F of an integrable image f is defined as The Fourier transform of its kth-order ACF ak, f is denoted Uk) It is related to F by the following calculation: The transforms Ak, f are called the higher-order spectra or polyspectra of f A2,f being the bispectrum. Equation (5) implies that integrable images f and g have the same triple correlation if and only if their transforms F and G satisfy Eq. (1).
If f is a periodic image with period p, its Fourier transform (in the sense of generalized function theory) is where () is the Dirac delta function and Fp(u) is the transform of the single-period segment fp: The (iii) If images f and g have the same kth-order ACF and both are convolved with any third function h, the convolutions h * g and h * f have the same kth-order ACE (iv) If images f and g have the same kth-order ACF, so do the convolutions f * f and g * g.
(v) If images f(x) and g(x) have the same kth-order ACF, so do f(ax) and g(ax) for any constant a • 0.
Properties (i)-(v) follow immediately from Eq. (5) for integrable images or from Eq. (6) for periodic images: For k, f the same argument holds when Eq. (6) is used.
The following symmetry properties of the bispectrum are also immediate consequences of Eqs. (5) and (6): Proof: "If" follows from relation (ii) in Subsection 2.C.
To prove "only if, " we start with the fact that if a 2 , = a 2 , f then A 2 , = A 2 , f, so, from Eq. (5), for all u and u. We , and in that case g = 0 as well. So we assume F(0) and G(0) are not zero, and without loss of generality we can assume that their common value is 1. [If it is not, we can divide a 2 , . by F(0)' and a 2 ,g by G(0)' and show that with bounded support is uniquely determined by the values of its characteristic function in a neighborhood of the origin. Since F(0) = G(0) = 1, property (i) implies that there is an interval around the origin, say (-, 13), in which both F and G are nonvanishing. Now write F and G in exponential form: Since F(u) and G(u) are nonvanishing and continuous for u E (-, 13), PhaG(u) and PhaF(u) are defined and continuous in that interval, and for all values of u, v and u + v for which F and G are not Equation (10) is the classic Cauchy functional equation. Aczel 22 shows that, if D is continuous and Eq. (10) holds for all u, v in any interval containing the origin, then over that interval D(u) = bu for some constant b. Setting b = 27rc, we have PhaG(u) = PhaF(u) + 2rcu, so G(u) = exp(i27rcu)F(u) in a neighborhood of the origin. Consequently, in that neighborhood the characteristic function of the probability-density function g(x) agrees with that of some density f(x + c). Since f(x) has bounded support, f(x + c) does also, and thus its characteristic function is completely determined by its values in a neighborhood of the origin. Thus for all u, so g(x) = f(x + c), and Theorem 1 is proved.

Reconstruction Algorithmsfor Finite Images
Theorem 1 guarantees that every finite-sized 1-D image f is uniquely determined up to the form f(x + c) by its triple correlation but does not show explicitly how the family {f(x + c)} can be recovered from a2,f. We are aware of two basic approaches to this problem. One involves a recursive reconstruction of the Fourier transform F from the bispectrum A,f. The other uses the derivatives of the bispectrum (or, equivalently, certain moments of the triple correlation) to recover the moments of f which determine the power-series expression for its trans-form. We discuss first the recursive approach and then the one based on moments.
Equation (5) shows that the amplitude spectrum F(u)l can be obtained immediately from A 2 ,f(u,V) through the relationship However, there is no analogous direct expression relating the phase spectrum PhaF(u) to A 2 ,f(u,v). To recover PhaF(u), several authors" 113-5 have proposed recursive algorithms based on the fact that Eq. (5) implies that the phase of F is related to the phase of A 2 , f by (12) for all u, v, u + v at which F does not vanish. Since A 2 , f determines F only up to the form exp(i2rrcu)F(u), we can assign one frequency an arbitrary value, say, PhaF(u ) = E) (0 = 0 being the natural choice). Then to recover PhaF at frequencies that are multiples of ul one could try to use the simple recursion For example, if f is assumed to vanish outside a finite interval (-w/2,w/2), one might take ul to be 1w, since in that case the sampling theorem implies that F is determined by its values on the set {n/w: n = 0, 1, ±2, .. .
If F is nonvanishing on all multiples of ul, recursion (13) will automatically deliver all the phases PhaF(nul).
However, if F vanishes on the sequence ul,2ul,.... recursion (13) cannot be used beyond the first n for which F(nul) = 0: at that point PhaF(nul) is undetermined, and PhaF[(n + 1)ul] cannot be calculated from it. In this case Pha[(n + 1)ul] may still be recoverable by means of Eq. (12) by using some combination Pha(kul) and Pha[(n + 1 -k)ul] with k < n: the bispectrum must be examined to determine whether this is possible. Failing that, a finer sampling lattice can be tried. In general one cannot specify in advance a frequency ul for which recursion (13) is guaranteed to succeed: one needs first to identify the zeros of F and then to find a ul for which F(nul) is never zero and 1/ul is an adequate sampling rate for F Bartelt et al." note that, in principle, a value of ul that will work for all frequencies nu, up to any desired limit can always be found, because the Fourier transform of a function with bounded support can have only a finite number of zeros in any finite interval. However, in practice, phase recovery for finite portions of images whose infinite versions are not uniquely determined by their triple correlations (e.g., those in Fig. 1) will become increasingly difficult as the size of the observation window grows and the bispectrum of the windowed portion converges to that of the infinite version. (This point is discussed below in Subsection 2.F.) An alternative approach to image reconstruction can be based on the fact that the derivatives of the bispectrum A 2 , f(u,v) along the line v = u determine the moments of f [more precisely, the moments of a certain member of the family {f(x + c)}], which in turn can be used to reconstruct the transform F(u) [up to a factor exp(i27rcu)] and thus f itself. This approach may be too fragile for practical use, but the argument has mathematical interest because it provides a constructive proof of Theorem 1.
To simplify matters, assume that f 'f(x) if not, we can begin by dividing A 2 ,f by A 2 ,f(0,0) and recover f(x + c)/F(0)]. Let fo(x) be the unique member of the family {f(x + c)} for which f xf(x + c)dx = 0. Since fo has finite support, its transform Fo can be expressed as a Taylor series about the origin, which is valid for all u: where Ai is the nth moment of fo: Attn = fxfo(x)dx If the moments nU can be determined, then Fo can be constructed from Eq. (14) and fo can be recovered by Fourier inversion. We know that ,o = 1, and l,, = 0 by construction. To obtain the other moments, let Q(u) = A 2 , f(u, u).
where xj, j = 0, 1, 2, are independent random variables with common density fo; i.e., Q(u) = E{exp[-i2vru(xl + X2over, Q is the characteristic function of a random variable that has finite absolute moments of all orders (since the density of y has bounded support), so Q is infinitely differentiable at the origin, and thus log Q is also. Now let LQ(u) = log Q(u/-27r) and LF(U) = log Fo(-27ru). Then and differentiating this equation n times yields LQ (n)(u) = 2LF(n)(u) + (-2)nLF(n)(-2u).
for n = 2,3,.... The quantities Kn = i-nLF(n)(0) on the left-hand side of Eq. (15) are the cumulants of the unknown density function fo whose moments we seek, and Eq. (15) shows that, for n 2 2, Kn can be obtained from the known derivative LQ(n)(0). It is a fact of probability theory that the moments An of a density function can be calculated directly from its cumulants: Al = Kl,, 1

(Lukacs 2 3 gives a general
formula for calculating the moments of a density from its cumulants.) Here , 1 is 0 by construction, and Eq. (15) shows that all the remaining moments of fo can be obtained from the successive derivatives of log A 2 , f (-u/27T, -u/27r). Consequently, Fo and thus fo can be reconstructed from A 2 f, as is claimed. The argument provides a direct proof of Theorem 1 and shows in addition that, when f has finite support, all the information in its triple-correlation function is carried by a countable set of values: the derivatives of A 2 , f(u, u) at u = 0 or, equivalently, the triple-correlation moments that correspond to those derivatives: f J -i2X.wrsi + s 2 ) a 2 ,j(sl,s 2 )dslds 2 = A2,

Higher-OrderAutocorrelations of Infinite Images: Examples of Nonuniqueness
Theorem 1 shows that, for integrable images of finite size, the information available from the entire set of higherorder ACF's of an image is already fully contained in its triple correlation: a 2 ,j determines f up to the form f(x + c), and, in view of relation (ii) in Subsection 2.C, the unidentified constant c cannot be obtained from any higher-order ACE For images with infinite support the uniqueness situation is not so straightforward. While Theorem 1 can be extended to certain useful classes of infinite images, as is shown below by Theorems 3-5, in general it is not the case that all integrable images with infinite support are uniquely determined up to translation by their triple-correlation functions. In fact, no ACE of any fixed order uniquely determines all the integrable images up to translation: for every k there are images f and g for which ak, = a,f but g(x) f(x + c) for any c. This point is demonstrated by the following example: For any integer k 2 the integrable images have the same kth-order ACF and clearly are not translations of each other: f is symmetric; g is not. (Figure 1 illustrates f and g for k = 2.) The fact that ak,f = a g for Eqs. (16) and (17) follows from the fact that the transforms of f and g satisfy the relationship for all arguments (ul,..., u), and thus, from Eq. (5),
[The same argument can be used to show that Eqs. (21) and (22) have the same triple correlation when sin 27r3x in Eq. (22) is replaced by -sin 27r3x or by -cos 27r3x. A geometrical understanding of the lack of uniqueness for these images can be gained from Fig. 5 in Subsection 2.F, which illustrates the common support of their bispectra.] The proof that Eqs. (16) and (17) satisfy Eq. (18) for an arbitrary k 2 involves the same sort of argument checking on a more elaborate scale. 7 When f and g have the same kth-order AC, the same is also true of h * f and h * g for any third function h [property (iii) in Subsection 2.C] and of f * f and g * g [property (iv)], so functions (16) and (17) can be used to construct an infinite variety of pairs of distinct bandlimited images f and g with identical kth-order ACF's. And since resealing such images will not alter the identity between their kth-order ACF's [property (v)], we can construct examples of distinct band-limited functions with an arbitrarily small bandwidth that have identical kth-order ACF's. In view of such examples, the only completely general uniqueness theorem that one can prove for integrable images is the following: Theorem 2. If f and g are integrable images and ak, = ak,f for all k, then g(x) = f(x + c).
Proof: The proof of Theorem 1 shows that if a 2 , g = a 2 , f there is a neighborhood A of the origin in which the transforms G and F are nonvanishing and G(u) = exp(i27rcu) F(u). Since ak,g = ak,f for all k, Eq. (5) implies that F(-ku)F(u) = G(-ku)G(u)k for all u and k. For any v E R -A there is a u E A such that v = -ku for some k, and since G(u) = exp(i27rcu)F(u) 0 for all u A, we have Dividing both sides of the second equality by [exp(i27rcu) x Since this holds for all v, g(x) = f(x + c).

Uniqueness Theorems for Special Classes of Infinite Images
The examples given in Subsection 2.D.3 show that, in general, the triple-correlation function of an integrable image with infinite support does not uniquely determine that image up to translation. However, it is possible to extend Theorem 1 to certain natural classes of infinite images. One approach is to show that the relationship PhaG(u) = PhaF(u) + 27rcu, which holds in some neighborhood of the origin for any images f and g that satisfy a 2 , 5 = a2,f, can be extended by using Eq. (9) to any point u at which F and G are nonvanishing, provided that F has at most a finite number of zeros below u. The following theorem summarizes the results of this approach. Theorem 3. If f is an integrable image and a 2 ,g = a 2 , for another image g, then g(x) = f(x + c) for some constant c if the Fourier transform of f satisfies any one of the following conditions:   Examples of the various cases were mentioned in Section 1. We note that, if a nonzero image has bounded support, its transform can have only a finite number of zeros in any finite interval, since in this case its complex transform is an entire analytic function (e.g., see Lukacs, 2 3 Theorem 7.2.3) and cannot vanish infinitely often on any finite interval of the real axis without vanishing everywhere. Thus condition (c) implies that when, e.g., sinc 2 (x) is convolved with any finite image, the resulting image is determined up to translation by its triple correlation. Part (d) shows that the same is true of finite images convolved with impulse responses whose transforms are nonvanishing, e.g., Gaussians.
Theorem 3 excludes images whose spectra contain nonzero regions separated by an interval of zeros.
Images (21) and (22), which are not determined by their triple correlations, have transforms of that sort, but the fact that the spectrum of an image contains intervals of zeros does not necessarily mean that the image is undetermined by its triple correlation. The critical factor is the size of those intervals-in particular, their size relative to the width of the neighborhood of the origin in which the transform is nonvanishing. Let Af be that neighborhood for an image f We say that a point u, at which the transform F(u) • 0, can be finitely linked to Af if there is a point p0 in Af and a sequence of numbers a,,... *a X°n, < ai 1 such that U ( + aiPo (24) and at each pointp = (1 + a 1 + + ai)po, F(pi) 0.
(The points pi, 0 < i < n serve as stepping stones linking u to po.) Theorem 4 If f is an image for which every point u at which the transform F(u) # 0 can be finitely linked to Af and if g is another image with a 2 ,g = a2jf, then g(x) = f(x + c) for some constant c.
As an example, f(x) = sinc 2 (x) ( ] Thus whenever f is uniquely determined by its moments, it is also uniquely determined up to translation by its triple correlation. The problem of characterizing classes of functions that are uniquely determined by their moments has a large literature that we will not attempt to summarize; Shohat and Tamarkin' 7 and Akhiezer' 8 provide reviews. Instead we describe a single condition involving moments that is sufficient to guarantee that an image is determined by its triple correlation. Characteristic-function theory 2 3 shows that, if all the moments of an image f are finite and satisfy the condition lim sup(/.tnJ/n!)"" = A < a, (25) then F(u) has a unique extension to a function F(z), with z complex, which is analytic in an open disk Jz < 1/27A. In this case F(z) is analytic for all z in an open horizontal strip containing the real axis and is determined by analytic continuation for all real z by the derivatives F')(O).
Since exp(i27rcz) is an entire function for any constant c, exp(i27rcz)F(z) will be analytic in the same strip and thus be determined for all real z by its derivatives at zero. So we have the following: Theorem 5. If f is an integrable image that is uniquely determined by its moments [for example, if its moments satisfy Eq. (25)] and g is another image for which a 2 ,g = a 2 ,, then g(x) = f(x + c).
[We note that an image may be uniquely determined by its moments without satisfying Eq. If f and g are two images whose individual transforms F and G satisfy Eq. (25) and thus are both analytic in some neighborhood of the real axis, the product FG is also analytic in such a neighborhood, hence determined by its derivatives at 0, so f*g is determined up to translation by its triple correlation. The Gaussian f(x) = exp(-ax 2 ) satisfies condition (25), so we have another proof that any finite image convolved with a Gaussian impulse response is determined by its triple correlation.
It was shown in Subsection 2.D.2 that all the moments of a finite image f can be recovered from the derivatives of A 2 f (U, u) at u = 0. The same procedure will work for any image whose moments satisfy Eq. (25), so for all such images it is the case that all the information in the bispectrum 1/2v are still determined in principle by its derivatives at u = 0 (i.e., by the moments n), but their actual calculation by analytic continuation would not be straightforward. Thus Theorem 5 guarantees that images whose moments satisfy Eq. (25) are uniquely determined up to translation by their triple correlations but does not provide a practical way of reconstructing all such images from those moments.

E. Uniqueness of Higher-Order Autocorrelations of Periodic 1-D Images
By comparison with the rich subset of integrable images that are determined up to translation by their triple correlations, the uniqueness properties of the higher-order ACF's of infinitely extended periodic images seem rather bleak. In general, for every integer n, there are pairs of band-limited periodic images that are not translations of each other but whose ACF's agree for all orders up through n, and there do not seem to be many interesting subclasses that escape this ambiguity. Klein 26) and (27) shows that the same is true for any images of the forms f(x) = L(2 + C 1 cos 2kx + C 2 cos 2Tkox), g(x) = L(2 + C 1 cos 2ox -C 2 cos 2kox), with k 2, L > and k > 0, and 0 < C 1 ,C 2 1.
Property (iv) in Subsection 2.C shows that the pair f*f and g*g constructed from any of these functions will also have identical ACF's of order k -1.
In view of these examples, the only completely general uniqueness theorem that one can hope to prove here is that every periodic image is uniquely determined up to translation if all its ACF's are known. We showed earlier (Theorem 2) by a simple argument that this is true of all integrable images, but the periodic case is more difficult to decide one way or the other. We leave it as an open problem 9 and prove instead an easy weaker result: Every periodic image with period p is determined up to translation by its ACF's of all orders if its spectrum is nonvanishing at the fundamental frequency lp: Theorem 6. If f is a periodic image with period p and F(l/p) • 0, then ai, 5 = k,f for all k for another image g if and only if g(x) = f(x + c).
Proof "If" is property (ii) in Subsection 2.C. To show "only if," we start with the fact that 2,g = a 2 , implies, from Eq. (6), that IG(nlp)l = IF(nlp)l for all n.

G(u) = exp(i27rcu)F(u) for all u, and g(x) = f(x + c).
The next result is a periodic analog to conditions (a) and  Eq. (11) and the phase spectrum recursively by using Eq. (12) and (ii) determine the cumulants, and from them the moments of the image, from the derivatives of the bispectrum at zero (or, equivalently, from moments of the triple correlation), and reconstruct the image transform from the moments. Method (ii) cannot be applied to infinitely extended periodic images (which always have infinite moments) or to the integrable images (21) and (22) whose triple correlations were shown in Subsection 2.D.3 to be nonunique (since their even moments are all infinite). But there is no immediate reason why method (i) cannot be successfully applied to periodic images (for example, those that satisfy either condition of Theorem 7) or to some integrable images with infinite support even if they happen to have infinite moments (e.g., those covered by Theorem 3 or 4). However, it must fail somehow for images whose triple correlations do not have an inverse that is unique up to translation. This section explains the relationship between image reconstruction by method (i) and the infinite images whose triple correlations were shown in Subsections 2.D.3 and 2.E to be nonunique. We show first how the method fails for such images in cases in which the bispectrum of the entire image is assumed to be available and then discuss what happens when it is applied to the bispectrum of a finite portion of an image whose infinite version has a nonunique bispectrum. In the latter case Theorem 1 implies that the bispectrum of the finite portion always has an inverse that is unique up to translation, and in principle that inverse should be recoverable by method (i) no matter what the size of the observation window is. However, since this is not true in the limit, one expects recovery to become more difficult in some sense as the size of the window grows. We show by example the form that this difficulty takes.  then, is to recover PhaF(u) at those frequencies. TI phase at u = 1 can be set arbitrarily, and PhaF(u) PhaF(1) for u = 2,3, 5 can be reconstructed recursive by using Eq. (12), which corresponds to the zigzag dotty path connecting (1,0) to (5,0). For any given frequen u', a necessary condition for PhaF(u') to be determinE by the phase at some lower-frequency p by means recursion (12), i.e., by 2 + cos 2 7rx -cos 67rx. [The lower graph in Fig. 4 shows the normalized amplitude spectrum of the image, i.e., IF(u)IiF(0).] It can be seen that the frequency u = 3 fails the test just described: PhaF(3) cannot be reconstructed from PhaF(1) because the line from (3,0) to the u = u diagonal is empty: A 2 ,f(p, 3 -p) is zero for allp. Thus the bispectrum can tell us nothing about PhaF(3) -PhaF (1). Figure 5 (heavily striped areas) shows the support of the bispectrum of the integrable image sinc 2 (x)(1 + cos 6x), whose triple correlation is the same as that of sinc 2 (x) (1 + sin 67rx). Here the amplitude spectrum IF(u)l (shown in the bottom part of the figure) is nonzero on the intervals (0,1) and (2,4), but there is no frequency u' in the interval (2,4) whose phase can be reconstructed onfrom that of any frequencyp < 2, because, if u' > 2, either in

PhaF(u') = PhaF(p) + PhaF(u' -p)
-PhaA 2 ,f(pu'-P), is that the bispectrum A 2 ,f(u,v) be nonzero at the point (p,u' -p), so that PhaA 2 ,f(p,u' -p) is defined, and at (p, 0) and (u'p, 0), so that PhaF(p) and PhaF(u' -p) If there is no frequency p with u'/2 p < u' for which all three conditions are satisfied, then the phase at u' cannot be recursively reconstructed from any lower frequencies. For a given u' one can quickly check whether there is anyp that satisfies the first two conditions by examining the bispectrum along the line from (u', 0) to the v = u diagonal and at the point on the u axis directly below any point (p, u'p) at which A 2 ,f(p,u'p) 0. Figures 4 and 5 illustrate how recursive-phase reconstruction fails for images whose triple correlations do not have inverses that are unique up to translation. The upper part of Fig. 4 shows the support (for uv 0) of the bispectrum of the infinitely extended periodic image f(x) = 2 + cos 2rx + cos 6-x, whose triple correlation was shown in Subsection 2.E to be the same as that of  can be reconstructed from its bispectrum. However, as the window widens the situation deteriorates. Figure 7 shows the support of the bispectrum of fw(x) = rect(x/9)f(x), i.e., f(x) seen through a window of width 9. Again, the large dots indicate the points where the bispectrum of fw is 0.001. (The sampling rate has been increased to 10, still greater than the Nyquist rate.) We see from the u axis that the normalized power spectrum exceeds 0.001 at u = 3.0, 1.0, and numerous points below 1.0, but the bispectrum contains no points with absolute values 0.001 along the line from (3,0) to the u = u diagonal. Consequently, PhaFW(3) cannot be determined recursively from any lower frequencies, that is, from any lower frequencies whose normalized spectral powers are at least 0.001. In particular, there is no recursive path from u = 1 to u = 3 connecting the two frequencies that have sizable spectral power. Of course, since fw has bounded support, we know that in principle PhaFW(3) must be recursively determinable by some sequence of frequencies beginning at u = 1. However, it must be a path along which the absolute values of the bispectrum are all near zero. In a real-world problem such values might be zero in fact, spuriously inflated by noise or round-off error. The problem is that, if we use the phases of such suspect points to reconstruct the phase recursively at a frequency that does have significant spectral power, such as u = 3 in this example, their small absolute values are irrelevant: the recursion treats the phases of all frequencies with equal respect, regardless of their spectral amplitudes. But the reliability of the phase that it assigns to any frequency depends on the reliability of the phases assigned to earlier frequencies. If all the possible recursive paths from one significant frequency to another involve intermediate frequencies whose spectral amplitudes are very small and thus likely to be actually zero except for measurement error, the reliability of subsequent phase assignments must suffer accordingly.
From a practical standpoint, then, the fundamental problem posed by the nonuniqueness of the triple correlations of infinite images is essentially statistical. In principle, with perfect measurement and computation, any finite portion of any infinite image can be unambiguously reconstructed from its bispectrum, even if the full image does not have a unique inverse. But in the latter case it is precisely the gaps in the bispectrum of the full image that make its triple correlation noninvertable, and the bispectrum of a windowed version of such an image must converge to zero in those gap regions as the size of the window increases. This forces the recursion linking frequencies across a gap to depend on intermediate frequencies whose true spectral power is becoming vanishingly small and, in the presence of noise, increasingly difficult to distinguish reliably from zero. Consequently, phase reconstruction for such an image becomes increasingly unreliable as the observation window widens: in effect, the more we see of its bispectrum, the less confidence we have in our reconstruction of the image.

OF 2-D IMAGES A. Definitions and Basic Properties
We call a function f :R 2 -> R an image function (or, simply, an image) if f is nonnegative and integrable over every finite rectangle. 2 4 We write (x 1 , x 2 ) as X and denote the integral of f(X) over the rectangle R by fRf(X)dX If fRf(X)dX = fRg(X)dX for every R, we say f = g. If fR2f(X)dX = 0, then f = 0. An image f is integrable if fR2f(X)dX is finite. In that case the kth-order ACF of f is defined to be ak,f(Sl, ,Sk) = f f(X)f(X + S 1 ) . .f(X + Sk)dX, (32) where Sj = (Sj1, sj,2) forj = 1,...,k. [For k = 2, Eq. (32) defines the triple correlation of an integrable 2-D im- Then f and g are probability densities of random vectors Xf = (Xf, 1 , Xf 2 ), Xg = (X, 1 , XI, 2 ), and F and G are the characteristic functions of these densities. We rely again on two general properties of characteristic functions: Every characteristic function is continuous and equals 1.0 at the origin (so F and G are both nonvanishing in some neighborhood of the origin); and, if F is the characteristic function of a density with bounded support in the plane, F is completely determined by its values in a neighborhood of the origin. {For every fixed value of U. U Xf is a 1-D random variable whose density has bounded support on the line, so the characteristic function +(t) = E[exp(-i27rtU Xf)] = F(tU) is determined for all t by its derivatives at t = 0, i.e., by the values of F(U) in a neighborhood of the origin. Thus (1) = F(U) is so determined, and this is true for all U.} Consequently, if G(U) = exp(i27rC U)F(U) in a neighborhood of the origin for some constant C, then the density g(X) must equal f(X + C). Since F(U) and G(U) are nonvanishing in some neighborhood U < 13, Eq. (37) implies that, for all JUJ, lVI < 3/2,

Infinite Images
The 1-D counterexamples (16) and (17)  Proof: When the vectors U and V are substituted for arguments u and v, the proof is the same as that of Theorem 2.
The following triple-correlation-uniqueness theorems for special classes of infinite images are 2-D versions of Theorems 3-5: (That is, f is band limited, and for every spatial-frequency orientation its spectrum is nonvanishing below the cutoff for that orientation.) (c) For every line Lo through the origin of the spatialfrequency plane there is a cutoff bo such that F(r cos 0, r sin 0) _ for Irl 2 bo, and F(r cos 0, r sin 0) = 0 for at most a finite number of values of r < bo. x 2 )dxdx 2 , and f is said to be determined by its moments if it is the only image function that has the moment sequence (n,m).
If F(u,,u 2 ) is the Fourier transform off and F(nm) (ul, u 2 ) = anamiaunaumF(ui,u 2 ), then, whenever it exists, I-n,m = F(nm)(0,0) (i27r)-n-m. Let (DI(u,v) = log F(-u/27r, -u 2 /27r). The cumulants Knm of f are the quantities (i-n-m)c(D(nm)(0,0), and, just as in the 1-D case, the cumulants of f uniquely determine its moments and vice versa. Thus f is determined by its moments if and only if it is determined by its cumulants. And if f(X) is determined by its cumulants, so too is f(X + C) for any constant C = (C1, C 2 ), since the cumulants of the latter are the same as those of f(X) except for K,O, which becomes c 1 , 0 -c, and K0,1, which becomes K0,i -C 2 . So if f(X) is determined by its moments, f(X + C) is also. If f and g are images that have the same triple correlation, the proof of Theorem 1' shows that, for some C, G(U) = exp(i27rC -U)F(U) in a neighborhood of the origin, so g(X) has the same moments as f(X + C). Consequently, if f is an image that is uniquely determined by its moments, then f is uniquely determined up to translation by its triple correlation. We state this as Theorem 5' below. Shohat and Tamarkin' 7 discuss the general problem of characterizing the nonnegative real functions on R 2 that are uniquely determined by their moments. We mention only one sufficient condition, which is analogous to the earlier condition Eq. (25). For that purpose, we define a sequence {Mn: n = 1,2,.. .} by Mn = fR2IXinf(X)dX where XI = (xi 2 + X 2 2 )11 2 , and show that f is determined by its moments if lim supIMn/n!I"'n = A < oo.
(40) n -x Theorem 5'. If f is an integrable image that is uniquely determined by its moments [for example, if f satisfies condition (40)] and g is another image with a 2 ,g = a 2 , f, then g(X) = f(X + C) for some constant C. Proof In view of the preceding discussion we need show only that Eq. (40) guarantees that f is determined by its moments. We know that, for some constant C, G(U) = exp(i27rC U)F(U) in a neighborhood of the origin, and as usual we assume without loss of generality that F(O) = 1. Then F is the characteristic function of a random vector X = (xi, X 2 ), and for every fixed value of U = (u 1 , U 2 ) and any real number As in the proof of Theorem , the function exp(i27rtC U)(Iu(t) is uniquely determined by analytic continuation for all real t (in particular, for t = 1) by its values in a neighborhood of t = 0 (i.e., by its derivatives at 0) if the moments of U X are finite and satisfy the condition lim sup vn/n!I 1/n = < 00.
n-x It remains to be shown that Eq. (40) implies that this condition is satisfied. Starting with n || 2(U X)Yf(X)dX f U Xjnf(X)dX and writing U = p(cos a, sin a) and X = q(cos 13, sin 1), we have U X = pq(cos a -B) and U XI" s (pq)n = JUIIXjn. Thus Ivlin!l ""< UI Mn/n!nI , so, from the assumption of the theorem, lim supIvn/n!"n c U|A = p < o. does not depend on dimensionality, and the proofs of Theorems 3'-5' rely on reducing 2-D statements to 1-D ones in ways that will work as well for n dimensions. Thus Theorem 1', for example, shows that nonnegative integrable functions of time and space with bounded support in R 3 are determined up to an arbitrary translation by their triple correlations, and condition (c) of Theorem 3' shows that this remains true of such functions after they are subjected to low-pass filtering.

C. Uniqueness for Periodic 2-D Images
Examples (26) and (27) in Subsection 2.E show that for every k there are 2-D periodic images that are not translations of each other but that have identical kth-order ACF's. Consequently, the only completely general uniqueness result that one can hope for here is that every periodic image is determined up to translation by its entire set of ACF's of all orders. As in the 1-D periodic case, we do not know whether that is true, 19 so we prove instead a weaker result analogous to the earlier Theorem 6: Theorem 6'. Suppose that f is a periodic 2-D image with period p and F(1/p, 0) and F(O, l/p) are both nonzero. Then dkg = ak,f for all k for another image g if and only if g(X) = f(X + C) for some constant C.
Proof "If" is property (ii') in Subsection 3.A. To show "only if," we take p = 1 for convenience. From a 2 , 5 = a 2 ,j and Eq. (34) we have F(U) = G(U) for all U. Since F(1, 0) • 0, neither F(-1, 0) nor G(-1, 0) is zero, and we can write  For condition (a) of Theorem 3, suppose that f and g have the same triple correlation and that the transform F never vanishes. Then from Eq. (8), G(u)l = F(u)l for all u, so G is also nonvanishing, and the phase relationship [Eq. (9)] holds for all u and u The continuity argument given in the proof of Theorem 1 then shows that N(u, v) = 0 for all u,v. Consequently, functional Eq. (10) is valid for all u,v and PhaG(u) = PhaF(u) + 2cu for all u, so G(u) = exp(i2,r-cu)F(u) is valid for all u and g(x) = f(x + c).
To prove condition (b), suppose that f is a band-limited image whose transform is nonvanishing within its bandwidth; i.e., F(u) 0 O for ul 2 b, and F(u) 0 for ul < b.
To prove condition (c), suppose that f is a band-limited image whose transform F(u) 0 for ul 2 b, with F(u) = 0 in the interval (-b,b) at u = +Z1,±Z2,...,±ZM, where 0 < Z < Z 2 1 < ... zml = b. If a 2 , 5 = a 2 , for another image g, the transform G has the same zeros as F The proof of (b) showed that, for some constant c, PhaG(u) = PhaF(u) + 27rcu for all u in the interval (-zl, zj). We need to show that, for the same c, PhaG(u) = PhaF(u) + 27rcu + 2rN (Al) for every u at which F(u) is nonvanishing, with N an arbitrary integer. Suppose first that M = 2, i.e., that there is only one zero z 1 below the cutoff b. If b s 2z 1 , then for all u E (zi, b) we have u/2 < z 1 . If we write u = u/2 + u/2, Eq. (9) implies that Eq. (Al) holds for z 1 < u < 2z 1 . If b > 2z 1 , then at 2z 1 we write 2z 1 = 2z 1 -e + E, and Eq. (9) shows that Eq. (8) holds at u = 2z 1 . Then Eq. (Al) can be extended to 2z < u < max{3z,, b} by writing u = 2z 1 + pzl, < p < 1. If we continue in this way, Eq. (Al) can eventually be shown to hold for all u in the interval (z 1 , b), so G(u) = exp(-2vcu)F(u) for all u. Now suppose that M is arbitrary. The argument above shows that Eq. (Al) holds in the interval (Z 1 , Z 2 ). We assume that it holds for (zn-1 , zn) and show that this implies that it holds for (zn, zn+ 1 ). Pick an integer N large enough that z/N < z,, -z 1 -, and write u = Zn -z/N + pz 1 , with 1/N < p < 1. Then z < u < zn + z(l -1/N), Repeating this process will eventually extend Eq. (Al) over the entire interval (zn, z,+i). Thus PhaG(u) = PhaF(u) + 2cu + 27TN for every 0 u < b at which F(u) is not zero, and the same is true for -b < u 0 because PhaG is odd. So G(u) = exp(i27rcu)F(u) for all u, and g(x) = f(x + c). Finally, for condition (d), suppose that F has infinite support but at most a finite number of zeros in any finite interval and that a 2 , 5 = a 2 , for some g. Then, for any u, either F(u) = G(u) = 0, so that G(U) = exp(i2ircu)F(u) trivially, or F(u) • 0; and for 0 ' u' < u, F(u') either never vanishes or vanishes at a finite set of points Z 1 < 2 < ... < Zn < u. The first case is equivalent to condition (b), and in the second case the induction argument used to prove condition (c) can be used to show that PhaG(u) = PhaF(u) + 2rcu + 2rN, where the constant c is the same for all u at which F is nonvanishing. So again G(u) = exp(i2'ncu)F(u) for all u, and g(x) = f(x + c).