Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

No v em b er 21, 2018 PHASE RETRIEV AL F OR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES AND RECONSTR UCTION FR OM CO V ARIOGRAMS GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN Abstract. W e propose strongly consisten t algorithms for reconstructing the characteristic function 1 K of an unknown conv ex bo dy K in R n from p ossibly noisy measurements of the mo dulus of its F ourier transform c 1 K . This represen ts a complete theoretical solution to the Phase Retriev al Problem for c haracteristic functions of conv ex b o dies. The approac h is via the closely related problem of reconstructing K from noisy measuremen ts of its cov ariogram, the function giving the v olume of the in tersection of K with its translates. In the man y kno wn situations in whic h the co v ariogram determines a conv ex bo dy , up to reﬂection in the origin and when the position of the b o dy is ﬁxed, our algorithms use O ( k n ) noisy co v ariogram measuremen ts to construct a conv ex p olytop e P k that appro ximates K or its reﬂection − K in the origin. (By recent uniqueness results, this applies to all planar conv ex b o dies, all three- dimensional conv ex p olytopes, and all symmetric and most (in the sense of Baire category) arbitrary conv ex bo dies in all dimensions.) Tw o metho ds are provided, and b oth are shown to b e strongly consistent, in the sense that, almost surely , the minimum of the Hausdorﬀ distance b et w een P k and ± K tends to zero as k tends to inﬁnity . 1. Introduction The Phase R etrieval Pr oblem of F ourier analysis inv olv es determining a function f on R n from the modulus | b f | of its F ourier transform b f . This problem arises naturally and fre- quen tly in v arious areas of science, suc h as X-ray crystallograph y , electron microscop y , optics, astronom y , and remote sensing, in which only the magnitude of the F ourier transform can b e measured and the phase is lost. (Sometimes, as when reconstructing an ob ject from its far-ﬁeld diﬀraction pattern, it is the squared mo dulus | b f | 2 that is directly measured.) In 1984, Rosen blatt [42] wrote that the Phase Retriev al Problem “arises in all exp erimental uses of diﬀracted electromagnetic radiation for determining the in trinsic detailed structure of a diﬀracting ob ject.” T o da y , the word “all” is p erhaps to o strong in view of recen t adv ances in coheren t diﬀraction imaging. In any case, the literature is v ast; see the surveys [32], [34], [36], and [42], as well as the articles [9] and [18] and the references given there. Phase retriev al is fundamen tally under-determined without additional constraints, whic h usually tak e the form of an a priori assumption that f has a particular supp ort or distribution of v alues. An imp ortan t example is when f = 1 K , the c haracteristic function of a con v ex 2010 Mathematics Subje ct Classiﬁc ation. Primary: 42–04, 42B10, 52–04, 52A20; secondary: 52B11, 62H35. Key wor ds and phr ases. Algorithm, auto correlation, conv ex b o dy , conv ex polytop e, cov ariogram, geometric tomograph y , image analysis, least squares, phase retriev al, quasicrystal, set cov ariance. Supp orted in part by U.S. National Science F oundation gran t DMS-0603307. 1 2 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN b o dy K in R n . In this setting, phase retriev al is very closely related to a geometric problem in v olving the c ovario gr am of a conv ex b o dy K in R n . This is the function g K deﬁned b y g K ( x ) = V n ( K ∩ ( K + x )) , for x ∈ R n , where V n denotes n -dimensional Leb esgue measure and K + x is the translate of K b y the vector x . It is also sometimes called the set c ovarianc e and is equal to the auto c orr elation of 1 K , that is, g K = 1 K ∗ 1 − K , where ∗ denotes conv olution and − K is the reﬂection of K in the origin. T aking F ourier transforms, w e obtain the relation (1) c g K = c 1 K d 1 − K = c 1 K c 1 K =   c 1 K   2 . This connects the Phase Retriev al Problem, restricted to characteristic functions of con v ex b o dies, to the problem of determining a conv ex b o dy from its cov ariogram. Both the deﬁnition of co v ariogram and this connection extend to arbitrary measurable sets, but the reason for restricting to con v ex b o dies will b ecome clear. The cov ariogram was in tro duced by Matheron in his b o ok [38] on random sets. He sho w ed that for a ﬁxed u ∈ S n − 1 , the directional deriv ativ es ∂ g K ( tu ) /∂ t , for all t > 0, of the co- v ariogram of a con v ex b o dy K in R n yield the distribution of the lengths of all c hords of K parallel to u . This explains the utility of the co v ariogram in ﬁelds such as stereology , geomet- ric tomograph y , pattern recognition, image analysis, and mathematical morphology , where information ab out an unknown ob ject is to b e retrieved from c hord length measurements; see, for example, [15], [20], and [45]. The co v ariogram has also pla y ed an increasingly imp ortan t role in analytic con v ex geometry . F or example, it w as used by Rogers and Shephard in pro ving their famous diﬀerence b o dy inequalit y (see [46, Theorem 7.3.1]), by Gardner and Zhang [26] in the theory of radial mean b o dies, and b y Tsolomitis [47] in his study of conv olution b o dies, whic h via the work of Sc hm uc k ensc hl¨ ager [44] and W erner [50] allows a cov ariogram-based deﬁnition of the fundamental notion of aﬃne surface area. Here w e eﬀectively solve the following three problems. In each, K is a con v ex b o dy in R n . Problem 1 (Reconstruction from co v ariograms) . Construct an approximation to K from a ﬁnite num ber of noisy (i.e., taken with error) measurements of g K . Problem 2 (Phase retriev al for c haracteristic functions of con v ex b o dies: squared mo dulus) . Construct an appro ximation to K (or, equiv alen tly , to 1 K ) from a ﬁnite num ber of noisy measuremen ts of | c 1 K | 2 . Problem 3 (Phase retriev al for c haracteristic functions of con v ex b o dies: mo d- ulus) . Construct an approximation to K from a ﬁnite num ber of noisy measurements of | c 1 K | . In order to discuss our results, w e m ust ﬁrst address the corresponding uniqueness problems. In view of (1), these are equiv alen t, so we shall focus on the cov ariogram. It is easy to see that g K is in v ariant under translations of K and reﬂection of K in the origin. Matheron [40] PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 3 ask ed the following question, known as the Covario gr am Pr oblem , to which he conjectured an aﬃrmativ e answ er when n = 2. Is a c onvex b o dy in R n determine d, among al l c onvex b o dies and up to tr anslation and r eﬂe ction in the origin, by its c ovario gr am? The fo cus on cov ariograms of con v ex b o dies is natural. One reason is that Mallo ws and Clark [37] constructed non-congruen t conv ex p olygons whose o v erall c hord length distributions (allo wing the directions of the c hords to v ary as w ell) are equal, thereby answ ering a related question of Blaschk e. Thus the information pro vided b y the cov ariogram cannot b e weak ened to o m uc h. Moreo v er, there exist non-congruen t non-conv ex p olygons, even (see [22, p. 394]) horizon tally- and vertically-con v ex p olyominoes, with the same co v ariogram, indicating that the con v exit y assumption also cannot b e signiﬁcantly weak ened. In terest in the Cov ariogram Problem extends far b eyond geometry . F or example, Adler and Pyk e [1] ask whether the distribution of the diﬀerence X − Y of indep enden t random v ariables X and Y , uniformly distributed o v er a conv ex b o dy K , determines K up to translations and reﬂection in the origin. Up to a constan t, the con v olution 1 K ∗ 1 − K = g K is just the probabilit y densit y of X − Y , so the question is equiv alen t to the Cov ariogram Problem. In [2], the Cov ariogram Problem also app ears in deciding the equiv alence of measures induced b y Bro wnian pro cesses for diﬀerent base sets. A detailed historical account of the cov ariogram problem ma y b e found in [4]. The current status is as follo ws, in whic h “determined” alwa ys means determined by the co v ariogram among all conv ex b o dies, up to translation and reﬂection in the origin. Av erk o v and Bianchi [4] sho w ed that planar conv ex b o dies are determined, thereby conﬁrming Matheron’s conjecture. Bianc hi [8] prov ed, b y a long and intricate argument, that three-dimensional con v ex polyhedra are determined. It is easy to see that centrally symmetric conv ex b o dies are determined. (In the symmetric case, con v exit y is not essential; see [22, Prop osition 4.4] for this result, due to Cab o and Jensen.) Go o dey , Sc hneider, and W eil [27] prov ed that most (in the sense of Baire category) con v ex b o dies in R n are determined. Nev ertheless, the Co v ariogram Problem in general has a negative answ er, as Bianc hi [7] demonstrated by constructing conv ex p olytop es in R n , n ≥ 4, that are not determined. It is still unkno wn whether con v ex bo dies in R 3 are determined. None of the ab ov e uniqueness pro ofs provide a metho d for actually reconstructing a con v ex b o dy from its co v ariogram. W e are aw are of only t w o pap ers dealing with the reconstruction problem: Schmitt [43] gives an explicit reconstruction pro cedure for a conv ex p olygon when no pair of its edges are parallel, an assumption remov ed in an algorithm due to Benassi and D’Ercole [6]. In b oth these pap ers, all the exact v alues of the cov ariogram are supp osed to b e a v ailable. In con trast, our ﬁrst set of algorithms take as input only a ﬁnite n um ber of v alues of the co v ariogram of an unknown con v ex bo dy K 0 . Moreov er, these measurements are corrupted b y errors, mo deled b y zero mean random v ariables with uniformly b ounded p th momen ts, where p is at most six and usually four. It is assumed that K 0 is determined b y its co v ariogram, has its centroid at the origin, and is con tained in a kno wn b ounded region of R n , which for con v enience w e take to be the unit cub e C n 0 = [ − 1 / 2 , 1 / 2] n . W e pro vide t wo diﬀeren t metho ds 4 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN for reconstructing, for eac h suitable k ∈ N , a con vex p olytop e P k that appro ximates K 0 or its reﬂection − K 0 . Eac h metho d inv olv es tw o algorithms, an initial algorithm that pro duces suitable outer unit normals to the facets of P k , and a common main algorithm that go es on to actually construct P k . In the ﬁrst metho d, the co v ariogram of K 0 is measured, multiple times , at the origin and at v ectors (1 /k ) u i , i = 1 , . . . , k , where the u i ’s are mutually nonparallel unit v ectors that span R n . F rom these measuremen ts, the initial Algorithm NoisyCo vBlaschk e constructs an o -symmetric conv ex polytop e Q k that approximates ∇ K 0 , the so-called Blaschk e bo dy of K 0 . (See Section 3 for deﬁnitions and notation.) The crucial prop ert y of ∇ K 0 is that when K 0 is a con vex polytop e, each of its facets is parallel to some facet of ∇ K 0 . It follo ws that the outer unit normals to the facets of P k can b e tak en to b e among those of Q k . Algorithm NoisyCovBlasc hke utilizes the kno wn fact that − ∂ g K 0 ( tu ) /∂ t , ev aluated at t = 0, equals the brightness function v alue b K 0 ( u ), that is, the ( n − 1)-dimensional volume of the orthogonal pro jection of K 0 in the direction u . This connection allows most of the w ork to b e done by a v ery eﬃcien t algorithm, Algorithm NoisyBrightLSQ, designed earlier by Gardner and Milanfar (see [24]) for reconstructing a o -symmetric con vex b o dy from ﬁnitely man y noisy measuremen ts of its brightness function. The second method ac hieves the same goal with a quite diﬀeren t approach. This time the co v ariogram of K 0 is measured once at eac h p oin t in a cubic arra y in 2 C n 0 = [ − 1 , 1] n of side length 1 /k . F rom these measuremen ts, the initial Algorithm NoisyCo vDiﬀ( ϕ ) constructs an o -symmetric conv ex p olytop e Q k that approximates D K 0 = K 0 + ( − K 0 ), the diﬀerence b o dy of K 0 . The set D K 0 has precisely the same prop erty as ∇ K 0 , that when K 0 is a con vex p oly- top e, each of its facets is parallel to some facet of D K 0 . F urthermore, D K 0 is just the supp ort of g K 0 . The known prop erty that g 1 /n K 0 is conca v e (a consequence of the Brunn-Mink o wski inequalit y [21, Section 11]) can therefore be combined with tec hniques from m ultiple regres- sion. Algorithm NoisyCovDiﬀ( ϕ ) emplo ys a Gasser-M ¨ uller t yp e k ernel estimator for g K 0 , with suitable k ernel function ϕ , bandwidth, and threshold parameter. The output Q k of either initial algorithm forms part of the input to the main common Al- gorithm NoisyCo vLSQ. The cov ariogram of K 0 is no w measured again , once at each point in a cubic arra y in 2 C n 0 = [ − 1 , 1] n of side length 1 /k . Using these measuremen ts, Algorithm Noisy- Co vLSQ ﬁnds a con vex polytop e P k , eac h of whose facets is parallel to some facet of Q k , whose co v ariogram ﬁts b est the measurements in the least squares sense. Muc h eﬀort is sp ent in pro ving that these algorithms are strongly consistent. Whenev er K 0 is determined among conv ex bo dies, up to translation and reﬂection in the origin, by its co v ariogram, we show that, almost surely , min { δ ( K 0 , P k ) , δ ( − K 0 , P k ) } → 0 as k → ∞ , where δ denotes Hausdorﬀ distance. (If K 0 is not so determined, a rare situation in view of the uniqueness results discussed ab ov e, the algorithms still construct a sequence ( P k ) whose accum ulation p oints exist and hav e the same cov ariogram as K 0 .) F rom a theo- retical p oin t of view, this completely solv es Problem 1. Naturally , the consistency pro of leans hea vily on results and techniques from analytic conv ex geometry , as w ell as a suitable version PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 5 of the Strong Law of Large Numbers. Some eﬀort has b een made to mak e the pro of fairly self-con tained, but some arguments from the pro of from [24] that Algorithm NoisyBrightLSQ is strongly consisten t are used in proving that Algorithm NoisyCo vBlasc hk e is strongly con- sisten t. One suc h argument rests on the Bourgain-Campi-Lindenstrauss stabilit y result for pro jection b o dies. With algorithms for Problem 1 in hand, w e mo v e to Problem 2, assuming that K 0 is an un- kno wn con v ex b o dy satisfying the same conditions as b efore. The basic idea is simple enough: Use (1) and the measurements of | d 1 K 0 | 2 at p oints in a suitable cubic array to appro ximate g K 0 via its F ourier series, and feed the resulting v alues in to the algorithms for Problem 1. How- ev er, t w o ma jor technical obstacles arise. The new estimates of g K 0 are corrupted by noise that now in v olv es dep endent random v ariables, and a new deterministic error app ears as well. A substitute for the Strong Law of Large Numbers m ust b e pro ved, and the deterministic error controlled using F ourier analysis and the fortunate fact that g K 0 is Lipschitz. In the end the basic idea works, assuming that for suitable 1 / 2 < γ < 1, measurements of | d 1 K 0 | 2 are taken at the p oin ts in (1 /k γ ) Z n con tained in the cubic windo w [ − k 1 − γ , k 1 − γ ] n , whose size increases with k at a rate dep ending on the parameter γ . The three resulting algorithms, Al- gorithm NoisyMod 2 LSQ, Algorithm NoisyMo d 2 Blasc hk e, and Algorithm NoisyMo d 2 Diﬀ( ϕ ), are stated in detail and, with suitable restrictions on γ , prov ed to b e strongly consisten t under the same h yp otheses as for Problem 1. Our ﬁnal three algorithms, Algorithm NoisyMo dLSQ, Algorithm NoisyMo dBlaschk e, and Algorithm NoisyMo dDiﬀ( ϕ ) cater for Problem 3. Again there is a basic simple idea, namely , to tak e tw o indep endent measurements at eac h of the p oin ts in the same cubic arra y as in the previous paragraph, m ultiply the tw o, and feed the resulting v alues in to the algorithms for Problem 2. No serious extra tec hnical diﬃculties arise, and we are able to pro v e that the three new algorithms are strongly consistent under the same hypotheses as for Problem 2. This pro vides a complete theoretical solution to the Phase Retriev al Problem for c haracteristic functions of con vex b o dies. T o summarize: • F or Problem 1, ﬁrst use either Algorithm NoisyCovBlasc hke or Algorithm NoisyCovDiﬀ( ϕ ) and then use Algorithm NoisyCovLSQ. • F or Problem 2, ﬁrst use either Algorithm NoisyMo d 2 Blasc hk e or Algorithm NoisyMo d 2 Diﬀ( ϕ ) and then use Algorithm NoisyMo d 2 LSQ. • F or Problem 3, ﬁrst use either Algorithm NoisyModBlaschk e or Algorithm NoisyMo dDiﬀ( ϕ ) and then use Algorithm NoisyMo dLSQ. These results can also be view ed as a con tribution to the literature on the associated unique- ness problems. They sho w that if a conv ex bo dy is determined, up to translation and reﬂection in the origin, by its co v ariogram, then it is also so determined by its v alues at certain countable sets of p oints, even, almost surely , when these v alues are contaminated with noise. Similarly , the c haracteristic function of such a conv ex b o dy is also determined by certain countable sets of noisy v alues of the mo dulus of its F ourier transform. 6 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN Our noise model is suﬃcien tly general to apply to all the main cases of practical interest: zero mean Gaussian noise, P oisson noise (un biased measuremen ts follo wing a P oisson distribu- tion, sometimes called shot noise), or Poisson noise plus zero mean Gaussian noise. Ho w ev er, the main text of this pap er deals solely with theory . With the exception of Corollary 6.5 and Remark 6.6, where the metho d of pro of leads naturally to rates of con vergence for Al- gorithm NoisyCo vDiﬀ( ϕ ) and hence for the t wo related algorithms for phase retriev al, the fo cus is en tirely on strong consistency . F urther remarks ab out con v ergence rates, sampling designs, and implemen tation issues ha ve b een relegated to the App endix. Muc h remains to b e done. W e b eliev e, ho wev er, that our algorithms will ﬁnd applications. F or example, Baak e and Grimm [5] explain how the problem of ﬁnding the atomic structure of a quasicrystal from its X-ra y diﬀraction image in v olv es reco vering a subset of R n called a windo w from its co v ariogram, and note that this window is in man y cases a con v ex b o dy . W e are grateful to Jim Fienup, David Mason, and Sara v an de Geer for helpful corresp on- dence and to referees for some insightful suggestions that led to signiﬁcant impro v emen ts. 2. Guide to the p aper § 3. Deﬁnitions, notation, and pr eliminary r esults . W e recommend that the reader skip this section and refer back to it when necessary . § 4. The main algorithm for r e c onstruction fr om c ovario gr ams . This presen ts the main ( se c ond stage) Algorithm NoisyCo vLSQ for Problem 1 and its strong consistency , established in Theorem 4.10. § 5. Appr oximating the Blaschke b o dy via the c ovario gr am . The ﬁrst of the tw o ﬁrst-stage algorithms for Problem 1, Algorithm NoisyCov- Blasc hk e, is stated with pro of of strong consistency in Theorem 5.4. The latter requires the assumption that the v ectors u i , i = 1 , . . . , k , are part of an inﬁnite se- quence ( u i ) that is in a sense evenly spread out in S n − 1 , but this is a weak restriction. § 6. Appr oximating the diﬀer enc e b o dy via the c ovario gr am . In this section, the second of the t wo ﬁrst-stage algorithms for Problem 1, Algo- rithm NoisyCo vDiﬀ( ϕ ), is set out and prov ed to b e strongly consisten t in Theorem 6.4. § 7. Phase r etrieval: F r amework and te chnic al lemmas . Necessary material from F ourier analysis is gathered, and the scene is set for results on phase retriev al. This do es not dep end on the previous three sections. § 8. Phase r etrieval fr om the squar e d mo dulus . The algorithms for Problem 2, Algorithm NoisyMo d 2 LSQ, Algo- rithm NoisyMod 2 Blasc hk e, and Algorithm NoisyMo d 2 Diﬀ( ϕ ) are presented and strong consistency theorems for them are prov ed. § 9. Phase r etrieval fr om the mo dulus . The corresp onding algorithms for Problem 3, Algorithm NoisyMo dLSQ, Algo- rithm NoisyMo dBlasc hk e, and Algorithm NoisyMo dDiﬀ( ϕ ), are presen ted and shown to b e strongly consisten t. § 10. App endix . Rates of con vergence and implementation issues are discussed. PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 7 3. Definitions, not a tion, and preliminar y resul ts 3.1. Basic deﬁnitions and notation. As usual, S n − 1 denotes the unit sphere, B n the unit ball, o the origin, and | · | the norm in Euclidean n -space R n . It is assumed throughout that n ≥ 2. W e shall also write C n 0 = [ − 1 / 2 , 1 / 2] n throughout. The standard orthonormal basis for R n will b e denoted by { e 1 , . . . , e n } . A dir e ction is a unit vector, that is, an element of S n − 1 . If u is a direction, then u ⊥ is the ( n − 1)-dimensional subspace orthogonal to u and l u is the line through the origin parallel to u . If x, y ∈ R n , then x · y is the inner pro duct of x and y , and [ x, y ] is the line segment with endp oints x and y . W e denote by ∂ A , int A , diam A , and 1 A the b oundary , interior , diameter , and char acteristic function of a set A , resp ectively . The notation for the usual (orthogonal) pr oje ction of A on a subspace S is A | S . A set is o -symmetric if it is cen trally symmetric, with center at the origin. If X is a metric space and ε > 0, a ﬁnite set { x 1 , . . . , x m } is called an ε -net in X if for ev ery p oin t x in X , there is an i ∈ { 1 , . . . , m } such that x is within a distance ε of x i . W e write V k for k -dimensional Leb esgue measure in R n , where k = 1 , . . . , n , and where w e iden tify V k with k -dimensional Hausdorﬀ measure. If K is a k -dimensional conv ex subset of R n , then V ( K ) is its volume V k ( K ). Deﬁne κ n = V ( B n ). The notation dz will alw a ys mean dV k ( z ) for the appropriate k = 1 , . . . , n . If E and F are sets in R n , then E + F = { x + y : x ∈ E , y ∈ F } denotes their Minkowski sum and (2) E  F = { x ∈ R n : F + x ⊂ E } their Minkowski diﬀer enc e . W e adopt a standard deﬁnition of the F ourier transform b f of a function f on R n , namely b f ( x ) = Z R n f ( y ) e − ix · y dy . If f and g are real-v alued functions on N , then, as usual, f = O ( g ) means that there is a constan t c such that f ( k ) ≤ cg ( k ) for suﬃcien tly large k . The notation f ∼ g will mean that f = O ( g ) and g = O ( f ). 3.2. Con vex geometry. Let K n b e the class of compact con v ex sets in R n , and let K n ( A ) b e the sub class of members of K n con tained in the subset A of R n . A c onvex b o dy in R n is a compact con v ex set with nonempt y in terior. The notation K n ( r , R ) will b e used for the class of con vex b o dies con taining r B n and con tained in R B n , where 0 < r < R . The treatise of Sc hneider [46] is an excellent general reference for conv ex geometry . Figures illustrating man y of the following deﬁnitions can b e found in [20]. If K ∈ K n , then K ∗ = { x ∈ R n : x · y ≤ 1 for all y ∈ K } is the p olar set of K . The function h K ( x ) = max { x · y : y ∈ K } , 8 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN for x ∈ R n , is the supp ort function of K and b K ( u ) = V ( K | u ⊥ ) , for u ∈ S n − 1 , its brightness fun ction . An y K ∈ K n is uniquely determined by its supp ort function. W e can regard h K as a function on S n − 1 , since h K ( x ) = | x | h K ( x/ | x | ) for x 6 = o . The Hausdorﬀ distanc e δ ( K , L ) b etw een t w o sets K, L ∈ K n can then b e con venien tly deﬁned by δ ( K , L ) = k h K − h L k ∞ , where k · k ∞ denotes the suprem um norm on S n − 1 . Equiv alen tly , one can deﬁne δ ( K , L ) = min { ε ≥ 0 : K ⊂ L + εB n , L ⊂ K + εB n } . The surfac e ar e a me asur e S ( K , · ) of a con v ex b o dy K is deﬁned for Borel subsets E of S n − 1 b y S ( K , E ) = V n − 1  g − 1 ( K, E )  , where g − 1 ( K, E ) is the set of p oin ts in ∂ K at which there is an outer unit normal vector in E . Let S ( K ) = S ( K , S n − 1 ). Then S ( K ) is the surfac e ar e a of K . The Blaschke b o dy ∇ K of a con vex b o dy K is the unique o -symmetric conv ex b o dy satisfying (3) S ( ∇ K , · ) = 1 2 S ( K , · ) + 1 2 S ( − K , · ) . The pr oje ction b o dy of K ∈ K n is the o -symmetric set Π K ∈ K n deﬁned b y (4) h Π K = b K . Cauc h y’s pro jection formula states that for an y u ∈ S n − 1 , (5) h Π K ( u ) = b K ( u ) = 1 2 Z S n − 1 | u · v | dS ( K , v ) , and Cauc hy’s surface area formula is (6) S ( K ) = 1 κ n − 1 Z S n − 1 b K ( u ) du ; see [20, (A.45) and (A.49), p. 408]. By (3) and (5), w e hav e (7) b ∇ K = b K , and it can be sho wn (see [20, p. 116]) that ∇ K is the unique o -symmetric conv ex bo dy with this prop ert y . The diﬀer enc e b o dy of K is the o -symmetric conv ex b o dy D K = K + ( − K ). PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 9 3.3. The cov ariogram. The function g K ( x ) = V ( K ∩ ( K + x )) , for x ∈ R n , is called the c ovario gr am of K . Note that g K ( o ) = V ( K ), and that w e hav e g K ( x ) = 0 if and only if x / ∈ in t D K , so the supp ort of g K is D K . Also, g 1 /n K is concav e on its supp ort; see, for example, [26, Lemma 3.2]. Let K be a con vex b o dy in R n and let u ∈ S n − 1 . The (p ar al lel) X-r ay of K in the direction u is the function X u K deﬁned b y X u K ( x ) = Z l u + x 1 K ( y ) dy , for x ∈ u ⊥ . Now deﬁne (8) E K ( t, u ) = { y ∈ u ⊥ : X u K ( y ) ≥ t } and (9) a K ( t, u ) = V  E K ( t, u )  , for t ≥ 0 and u ∈ S n − 1 . Note that if u ∈ S n − 1 , then E K (0 , u ) = K | u ⊥ and a K (0 , u ) = b K ( u ). Let x = tu , where t ≥ 0 and u ∈ S n − 1 , and deﬁne g K ( t, u ) = g K ( tu ). The simple relationship g K ( t, u ) = Z ∞ t a K ( s, u ) ds (10) w as noticed by Matheron [38, p. 86] in the form ∂ g K ( t, u ) ∂ t = − a K ( t, u ) , whic h also yields ∂ g K ( t, u ) ∂ t     t =0 = − b K ( u ) . (Note that the partial deriv ative here is one-sided; g K is not diﬀeren tiable at the origin.) Lemma 3.1. L et r > 0 and let K b e a c onvex b o dy with rB n ⊂ K . If 0 < t ≤ 2 r , then (11)  1 − t 2 r  n − 1 b K ( u ) ≤ g K ( o ) − g K ( tu ) t ≤ b K ( u ) , for al l u ∈ S n − 1 . Pr o of. Let u ∈ S n − 1 . By (10), we ha v e g K ( o ) − g K ( tu ) = Z t 0 a K ( s, u ) ds. F rom this and the fact that a K ( · , u ) is decreasing, we obtain (12) a K ( t, u ) ≤ g K ( o ) − g K ( tu ) t ≤ a K (0 , u ) = b K ( u ) . 10 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN The set M = conv  ( K | u ⊥ ) ∪ [ − r u, r u ]  is generally not a subset of K , but elemen tary geometry using [ − r u, r u ] ⊂ K and (8) gives  1 − t 2 r   K | u ⊥  = E M ( t, u ) ⊂ E K ( t, u ) . T aking the ( n − 1)-dimensional volumes of these sets and using (9) yields  1 − t 2 r  n − 1 b K ( u ) ≤ a K ( t, u ) . The lemma follo ws from the previous inequality and (12).  An inequalit y similar to (11) was derived in [33, Theorem 1] for n = 2. Matheron [40, p. 2] show ed that the cov ariogram of a conv ex b o dy is a Lipsc hitz function. F or the conv enience of the reader, w e provide a pro of of this fact based on [19], whic h yields the optimal Lipsc hitz constant. Prop osition 3.2. If K is a c onvex b o dy in R n and x, y ∈ R n , then | g K ( x ) − g K ( y ) | ≤ max u ∈ S n − 1 b K ( u ) | x − y | . Pr o of. W e hav e ( K ∩ ( K + x )) \ ( K ∩ ( K + y )) ⊂ ( K + x ) \ ( K + y ) . This implies V n ( K ∩ ( K + x )) − V n ( K ∩ ( K + y )) ≤ V n ( K \ ( K + y − x )) = V n ( K ) − V n ( K ∩ ( K + y − x )) . Equiv alently , g K ( x ) − g K ( y ) ≤ g K ( o ) − g K ( y − x ) = g K ( o ) − g K ( x − y ), and in terc hanging x and y yields | g K ( x ) − g K ( y ) | ≤ g K ( o ) − g K ( x − y ) . Using this and the right-hand inequality in (11), we get | g K ( x ) − g K ( y ) | ≤ b K  x − y | x − y |  | x − y | , and the prop osition follo ws immediately .  Corollary 3.3. If K 0 ⊂ C n 0 is a c onvex b o dy, then for al l x, y ∈ R n , | g K 0 ( x ) − g K 0 ( y ) | ≤ √ n | x − y | . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 11 Pr o of. Since K 0 ⊂ C n 0 , Prop osition 3.2 yields | g K ( x ) − g K ( y ) | ≤ max u ∈ S n − 1 b C n 0 ( u ) | x − y | . By Cauc hy’s pro jection formula (5), for u = ( u 1 , u 2 , . . . , u n ) ∈ S n − 1 w e ha ve b C n 0 ( u ) = V  C n 0 | u ⊥  = n X i =1 | u i | , from whic h it is easy to see that b C n 0 ( u ) ≤ √ n .  3.4. Miscellaneous deﬁnitions. Let µ and ν b e ﬁnite nonnegative Borel measures in S n − 1 . Deﬁne (13) d P ( µ, ν ) = inf { ε > 0 : µ ( E ) ≤ ν ( E ε ) + ε, ν ( E ) ≤ µ ( E ε ) + ε, E Borel in S n − 1 } , where E ε = { u ∈ S n − 1 : ∃ v ∈ E : | u − v | < ε } . Then d P is a metric called the Pr ohor ov metric . As S n − 1 is a Polish space, it is enough to tak e the inﬁm um in (13) ov er the class of close d sets. In addition, if µ ( S n − 1 ) = ν ( S n − 1 ), then (14) d P ( µ, ν ) = inf { ε > 0 : µ ( E ) ≤ ν ( E ε ) + ε, E Borel in S n − 1 } ; see [17]. W e need a condition on a sequence ( u i ) in S n − 1 stronger than denseness in S n − 1 . T o this end, for u ∈ S n − 1 and 0 < t ≤ 2, let C t ( u ) = { v ∈ S n − 1 : | u − v | < t } b e the op en spherical cap with center u and radius t . W e call ( u i ) evenly spr e ad if for all 0 < t < 2, there is a constan t c = c ( t ) > 0 and an N = N ( t ) suc h that |{ u 1 , . . . , u k } ∩ C t ( u ) | ≥ ck , for all u ∈ S n − 1 and k ≥ N . Often, we will apply this notion to the symmetrization ( u ∗ i ) = ( u 1 , − u 1 , u 2 , − u 2 , u 3 , − u 3 , . . . ) of a sequence ( u i ). Let p ≥ 1. A family { X α : α ∈ A } of random v ariables has uniformly b ounde d p th absolute moments if there is a constant C such that (15) E ( | X α | p ) ≤ C , for all α ∈ A . Of course, if p is an ev en in teger, w e can and will omit the word “absolute.” If 1 ≤ q ≤ p and (15) holds, then it also holds with p replaced by q and C replaced b y C q /p . T riangular arra ys of random v ariables of the form { X ik : i = 1 , . . . , m k ; k ∈ N } (or, more generally , { X αk : α ∈ A k ; k ∈ N } ) are called r ow-wise indep endent if for each k , the family { X ik : i = 1 , . . . , m k } (or { X αk : α ∈ A k } , resp ectiv ely) is indep enden t. 12 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN 4. The main algorithm for reconstruction fr om co v ariograms W e shall assume throughout that the unkno wn con v ex bo dy K 0 is contained in the cub e C 0 = [ − 1 / 2 , 1 / 2] n , with its centroid at the origin. This assumption can be justiﬁed on both purely theoretical and purely practical grounds. If the measurements are exact, then from the co v ariogram, a con vex p olytop e can b e constructed that contains a translate of K 0 . On the other hand, in practise, an unknown ob ject whose cov ariogram is to b e measured is con tained in some known b ounded region. In either case, one may as w ell suppose that K 0 is con tained in C n 0 , and since in the situations we consider, the co v ariogram determines K 0 up to translation and reﬂection in the origin, we can also ﬁx the centroid at the origin. W e no w state the main, second-stage algorithm. Note that it requires, as part of the in- put, an o -symmetric con vex p olytop e that approximates either the Blaschk e b o dy ∇ K 0 or the diﬀerence b o dy D K 0 of K 0 . These are provided b y the ﬁrst-stage algorithms, Algorithm Noisy- Co vBlasc hke and Algorithm NoisyCovDiﬀ( ϕ ), describ ed in Sections 5 and 6, resp ectiv ely . The reader should b e aw are that here, and throughout the paper, double subscripts in expressions such as x ik , M ik , N ik , etc., represen t triangular arra ys. Th us, for a ﬁxed k , the index i v aries ov er a ﬁnite set of integers that dep ends on k ; and similarly when the ﬁrst index is lab eled b y another letter in expressions such as z j k , X pk , and so on, or is itself represented b y a double index, as in N ij k . Phrases suc h as “the N ik ’s are row-wise indep endent” mean that the corresp onding triangular array is ro w-wise indep enden t, i.e., indep endent for ﬁxed k . Algorithm NoisyCo vLSQ Input: Natural num b ers n ≥ 2 and k ; noisy cov ariogram measurements (16) M ik = g K 0 ( x ik ) + N ik , of an unkno wn con v ex b o dy K 0 ⊂ C n 0 whose cen troid is at the origin, at the p oin ts x ik , i = 1 , . . . , I k = (2 k + 1) n in the cubic array 2 C n 0 ∩ (1 /k ) Z n , where the N ik ’s are row-wise indep enden t zero mean random v ariables with uniformly b ounded third absolute momen ts; an o -symmetric conv ex polytop e Q k in R n , sto c hastically independent of the measurements M ik , that appro ximates either ∇ K 0 or D K 0 , in the sense that, almost surely , lim k →∞ δ ( Q k , ∇ K 0 ) = 0 , or lim k →∞ δ ( Q k , D K 0 ) = 0 . (17) T ask: Construct a con vex p olytop e P k that approximates K 0 , up to reﬂection in the origin. A ction: 1. Compute the outer unit normals {± u j : j = 1 , . . . , s } to the facets of Q k . 2. F or an y v ector a = ( a + 1 , a − 1 , a + 2 , a − 2 , . . . , a + s , a − s ), where a + j , a − j ≥ 0, j = 1 , . . . , s , such that P s j =1 ( a + j − a − j ) u j = o , let P ( a ) = P ( a + 1 , a − 1 , a + 2 , a − 2 , . . . , a + s , a − s ) b e the conv ex p olytop e with cen troid at the origin, facet outer unit normals in {± u j : j = 1 , . . . , s } and such that the facet with normal u j (or − u j ) has ( n − 1)-dimensional measure a + j (or a − j , resp ectively), j = 1 , . . . , s . Solv e the following least squares problem: PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 13 (18) min I k X i =1  M ik − g P ( a ) ∩ C n 0 ( x ik )  2 o v er the v ariables a + 1 , a − 1 , a + 2 , a − 2 , . . . , a + s , a − s , sub ject to the constrain ts s X j =1 ( a + j − a − j ) u j = o and a + j , a − j ≥ 0 , j = 1 , . . . , s. These constrain ts guarantee that the output will corresp ond to a conv ex p olytop e. 3. Let a set of optimal v alues b e ˆ a + 1 , ˆ a − 1 , ˆ a + 2 , ˆ a − 2 , . . . , ˆ a + s , ˆ a − s , and call the corresp onding p olytop e P ( ˆ a ). Then the output polytop e P k is the translate of P ( ˆ a ) ∩ C n 0 that has its cen troid at the origin. Note that in this case − P k also corresp onds to a set of optimal v alues obtained b y switching a + j and a − j , j = 1 , . . . , s . Lemma 4.1. L et 0 < r < R and let Q ∈ K n ( r , R ) b e an o -symmetric c onvex p olytop e. Then ther e ar e fac ets of Q with outer unit normals u 1 , . . . , u n such that (19) | det( u 1 , . . . , u n ) | > ( r /R ) n ( n − 1) / 2 . Pr o of. The p olar b o dy Q ∗ of Q is con tained in K n (1 /R, 1 /r ) and has its vertices in the directions of the outer unit normals to the facets of Q , so it suﬃces to pro ve that there are v ertices v 1 , . . . , v n of Q ∗ suc h that with u i = v i / | v i | , (19) holds. The proof will b e by induction on n . Let n = 2. W e ma y assume that Q ∗ has a vertex, v 1 sa y , on the p ositive x 2 -axis. Since Q ∗ ∈ K 2 (1 /R, 1 /r ), there m ust be another v ertex v 2 of Q ∗ with distance at least 1 /R from the x 2 -axis, and b y the symmetry of Q ∗ , such that also v 2 · e 2 ≥ 0. If α is the angle b et w een v 1 and v 2 , w e m ust then hav e θ ≤ α ≤ π / 2, where θ is the angle b et w een the v ectors (0 , 1 /r ) and  1 /R, p (1 /r 2 ) − (1 /R 2 )  . Then, if u i = v i / | v i | for i = 1 , 2, w e ha v e | det( u 1 , u 2 ) | = sin α ≥ sin θ = r /R , whic h pro ves (19) for n = 2. Supp ose that (19) holds with n replaced b y n − 1 and let Q ∗ ∈ K n (1 /R, 1 /r ). W e ma y assume that Q ∗ has a vertex, v 1 sa y , on the p ositiv e x n -axis, so that v 1 / | v 1 | = e n . Since Q ∗ | e ⊥ n ∈ K n − 1 (1 /R, 1 /r ) (where w e are iden tifying e ⊥ n with R n − 1 ), b y the inductiv e h yp othesis, there are v ertices w 2 , . . . , w n of Q ∗ | e ⊥ n suc h that if z i = w i / | w i | , i = 2 , . . . , n , then (20) | det( z 2 , . . . , z n ) | ≥ ( r /R ) ( n − 1)( n − 2) / 2 . Let v i b e a vertex of Q ∗ suc h that v i | e ⊥ n = w i , i = 2 , . . . , n , and let u i = v i / | v i | , i = 1 , . . . , n . By the symmetry of Q ∗ , we may also assume that v i · e n ≥ 0 for i = 2 , . . . , n . Let α i b e the angle b et w een v i and w i , for i = 2 , . . . , n . Using the fact that Q ∗ | e ⊥ n ∈ K n − 1 (1 /R, 1 /r ), we see that 14 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN eac h v i , i = 2 , . . . , n has distance at least 1 /R from the x n -axis. Therefore cos α i ≥ sin θ = r /R for i = 2 , . . . , n . Then, using (20) and noting that u 1 = e n and u i = u i | e ⊥ n + ( u i · e n ) e n for i = 2 , . . . , n , w e obtain | det( u 1 , . . . , u n ) | = | det( u 2 | e ⊥ n , . . . , u n | e ⊥ n ) | = | det( z 2 , . . . , z n ) | n Y i =2 cos α i ≥ ( r /R ) ( n − 1)( n − 2) / 2 ( r /R ) n − 1 = ( r /R ) n ( n − 1) / 2 .  Lemma 4.2. L et K ∈ K n ( r , R ) , let 0 < ε < κ n − 1 r n − 1 / 2 , and let L b e a c onvex b o dy c ontaining the origin in R n such that (21) d P ( S ( K , · ) , S ( L, · )) < ε. Then ther e is a c onstant a 1 dep ending only on ε , r , and R such that L ⊂ a 1 B n . If L is o - symmetric, ther e is also a c onstant a 0 > 0 dep ending only on ε , r , and R such that a 0 B n ⊂ L . Pr o of. Using (4) and (5), w e obtain (22) | h Π K ( u ) − h Π L ( u ) | = | b K ( u ) − b L ( u ) | ≤ d D ( S ( K , · ) , S ( L, · )) . Here d D is the Dudley metric, deﬁned by d D ( µ, ν ) = sup      Z S n − 1 f d ( µ − ν )     : k f k B L ≤ 1  , where for an y real-v alued function f on S n − 1 w e deﬁne k f k L = sup u 6 = v | f ( u ) − f ( v ) | | u − v | and k f k B L = k f k ∞ + k f k L . (Note that for an y u ∈ S n − 1 , the function f ( v ) = | u · v | / 2, v ∈ S n − 1 satisﬁes k f k B L = 1.) By [17, Corollary 2], we hav e the relation (23) d D ( µ, ν ) ≤ 2 d P ( µ, ν ) , for ﬁnite nonnegativ e Borel measures µ and ν in S n − 1 . Now (22), (23), and (21) yield | h Π K ( u ) − h Π L ( u ) | ≤ 2 d P ( S ( K , · ) , S ( L, · )) < 2 ε, for eac h u ∈ S n − 1 . Since K ∈ K n ( r , R ), w e ha ve Π K ∈ K n ( κ n − 1 r n − 1 , κ n − 1 R n − 1 ), so Π L ∈ K n ( κ n − 1 r n − 1 − 2 ε, κ n − 1 R n − 1 + 2 ε ). No w exactly the same argument as in the pro of of Lemma 4.2 of [25], b eginning with formula (16) in that pap er, yields the existence of a 1 and a 0 . (The assumption of o -symmetry made in [25] is only needed for the latter. Explicit v alues for a 0 and a 1 can b e giv en in terms of ε , r , and R , but w e do not need them here.)  PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 15 Lemma 4.3. L et K b e a c onvex b o dy in R n . Then ther e is an ε 0 > 0 such that for al l 0 < ε < ε 0 , if Q is an o -symmetric c onvex p olytop e in R n such that either (24) d P ( S ( ∇ K , · ) , S ( Q, · )) < ε or (25) d P ( S ( D K , · ) , S ( Q, · )) < ε, then ther e is a c onstant c 1 > 0 dep ending only on K and a c onvex p olytop e J whose fac ets ar e e ach p ar al lel to some fac et of Q , such that (26) d P ( S ( K , · ) , S ( J, · )) < c 1 ε. Pr o of. W e choose ε 0 > 0 so that Lemma 4.2 holds when ε is replaced b y ε 0 and K is replaced b y either ∇ K or D K , as appropriate. Let 0 < ε < ε 0 . Let ± u 1 , . . . , ± u s b e the outer unit normals to the facets of Q and for i = s + 1 , . . . , 2 s , let u i = − u i − s . Set I = { 1 , . . . , 2 s } . Supp ose that (24) holds. By (13), S ( ∇ K , E ) < S ( Q, E ε ) + ε for eac h Borel subset E of S n − 1 . If E ε ∩ ∪ i ∈ I { u i } = ∅ , w e hav e S ( Q, E ε ) = 0. This implies that S ( ∇ K , E ) < ε and so b y (3), (27) S ( K , E ) < 2 ε. If instead (25) holds, then (13) implies that S ( D K , E ) < S ( Q, E ε ) + ε for eac h Borel subset E of S n − 1 . Then, if E ε ∩ ∪ i ∈ I { u i } = ∅ , w e ha v e S ( D K , E ) < ε . By [46, (5.1.17), p. 275], S ( D K , E ) = S ( K + ( − K ) , E ) = S ( K , E ) + n − 1 X j =1  n − 1 j  S ( K , n − 1 − j ; − K, j, E ) , where S ( K, n − 1 − j ; − K, j , · ) denotes the mixed area measure of n − 1 − j copies of K and j copies of − K . Since all these terms are nonnegativ e, we obtain S ( K , E ) < ε and so (27) holds again. F or i ∈ I , let V i = { u ∈ S n − 1 : | u − u i | ≤ | u − u j | for eac h j ∈ I , j 6 = i } b e the V oronoi cell in S n − 1 con taining u i . Cho ose Borel sets W i suc h that relin t V i ⊂ W i ⊂ V i for eac h i and W i ∩ W j = ∅ for i 6 = j , so that { W i : i ∈ I } forms a partition of S n − 1 . Let a i = S ( K , W i ) and let w = P i ∈ I a i u i . Since S ( K , · ) is balanced, i.e., Z S n − 1 u dS ( K , u ) = o, w e ha ve w = X i ∈ I a i u i = X i ∈ I u i Z W i dS ( K , u ) − Z S n − 1 u dS ( K , u ) = X i ∈ I Z W i ( u i − u ) dS ( K , u ) . 16 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN F or each u ∈ S n − 1 and t > 0, let C t ( u ) = { v ∈ S n − 1 : | u − v | ≤ t } . Let W = ∪ i ∈ I ( W i \ C ε ( u i )). Then u i 6∈ W ε for i ∈ I , so (27) implies that S ( K , W ) < 2 ε . Using this, we obtain | w | =      X i ∈ I Z W i ∩ C ε ( u i ) ( u i − u ) dS ( K , u ) + X i ∈ I Z W i \ C ε ( u i ) ( u i − u ) dS ( K , u )      ≤ X i ∈ I Z W i ∩ C ε ( u i ) | u i − u | dS ( K , u ) + 2 Z W dS ( K , u ) < εS ( K , S n − 1 ) + 4 ε = ( S ( K ) + 4) ε. (28) Since Q is o -symmetric, we can apply Lemma 4.2 (with K and L replaced by ∇ K (or D K ) and Q , resp ectively) and Lemma 4.1 to conclude that there exist outer unit normals u i 1 , . . . , u i n to facets of Q such that | det( u i 1 , . . . , u i n ) | > c 2 , where c 2 dep ends only on K . In particular, u i 1 , . . . , u i n forms a basis for R n , so there exist real num b ers b i 1 , . . . , b i n suc h that − w = n X j =1 b i j u i j . Replacing u i j b y − u i j , if necessary , we ma y assume that b i j > 0 for j = 1 , . . . , n . By Cramer’s rule, we obtain b i j ≤ | w | / | det( u i 1 , . . . , u i n ) | < | w | /c 2 , for j = 1 , . . . , n . Deﬁne b i = 0 for eac h i ∈ I such that i 6∈ { i 1 , . . . , i n } . Then, by (28), (29) X i ∈ I b i ≤ n | w | /c 2 < c 3 ε, where c 3 dep ends only on K . Let µ 0 = X i ∈ I a i δ u i and µ 1 = X i ∈ I b i δ u i , and let µ = µ 0 + µ 1 . Then the supp ort of µ is not con tained in a great sphere, and since Z S n − 1 u dµ ( u ) = X i ∈ I ( a i + b i ) u i = w − w = o, µ is balanced. By Minko wski’s existence theorem [20, Theorem A.3.2], there is a conv ex p olytop e J such that S ( J, · ) = µ . By its deﬁnition, eac h facet of J is parallel to a facet of Q . It remains to prov e (26). Using (29), we obtain d P ( S ( J, · ) , S ( K, · )) = d P ( µ 0 + µ 1 , S ( K , · )) ≤ d P ( µ 0 + µ 1 , µ 0 ) + d P ( µ 0 , S ( K , · )) = d P ( µ 1 , 0) + d P ( µ 0 , S ( K , · )) < c 3 ε + d P ( µ 0 , S ( K , · )) , where 0 is the zero measure in S n − 1 . In view of µ 0 ( S n − 1 ) = S ( K , S n − 1 ) and (14), it is therefore enough to ﬁnd a constant c 4 , dep ending only on K , such that (30) µ 0 ( E ) < S ( K , E c 4 ε ) + c 4 ε, PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 17 for an y Borel set E in S n − 1 . Let X = ∪{ W i : u i ∈ E } \ E ε . W e hav e S ( K , E ε ) ≥ S ( K , E ε ∩ ( ∪{ W i : u i ∈ E } )) = X { S ( K , W i ) : u i ∈ E } − S ( K , X ) = µ 0 ( E ) − S ( K , X ) . (31) If x ∈ X , then for some i with u i ∈ E we ha v e x ∈ W i , and so | x − u i | ≥ ε since x 6∈ E ε . Moreo v er, if j 6 = i , then | u j − x | ≥ | u i − x | ≥ ε . Hence ∪ i ∈ I { u i } ∩ X ε = ∅ , and b y (27), we ha v e S ( K , X ) < 2 ε . No w (31) implies that (30) holds with c 4 = 2.  F or a ﬁxed ﬁnite set z 1 , . . . , z q of p oin ts in R n , deﬁne a pseudonorm | · | q b y (32) | f | q = 1 q q X i =1 f ( z i ) 2 ! 1 / 2 , where f is any real-v alued function on R n . F or a conv ex bo dy K contained in C n 0 , vector z q = ( z 1 , . . . , z q ) of the p oin ts z 1 , . . . , z q in R n , and vector X q = ( X 1 , . . . , X q ) of random v ariables X 1 , . . . , X q , let (33) Ψ( K, z q , X q ) = 1 q q X i =1 g K ( z i ) X i . Lemma 4.4. L et k ∈ N and let K 0 ⊂ C n 0 b e a c onvex b o dy with its c entr oid at the origin. Supp ose that P k is an output fr om A lgorithm NoisyCovLSQ as state d ab ove. L et P ( a ) b e any c onvex p olytop e admissible for the minimization pr oblem (18). Then (34) | g K 0 − g P k | 2 I k ≤ 2Ψ( P k , x I k , N I k ) − 2Ψ( P ( a ) ∩ C n 0 , x I k , N I k ) +   g K 0 − g P ( a ) ∩ C n 0   2 I k , wher e for e ach k ∈ N , | · | I k and Ψ( K , x I k , N I k ) ar e deﬁne d by (32) and (33), r esp e ctively, with q = I k , x I k = ( x 1 k , . . . , x I k k ) , and N I k = ( N 1 k , . . . , N I k k ) . Pr o of. If P ( ˆ a ) ∩ C n 0 is a solution of (18), then since g P k = g P ( ˆ a ) ∩ C n 0 , w e obtain I k X i =1 ( M ik − g P k ( x ik )) 2 ≤ I k X i =1  M ik − g P ( a ) ∩ C n 0 ( x ik )  2 , Substituting for M ik from (16) and rearranging, we obtain I k X i =1 ( g K 0 ( x ik ) − g P k ( x ik )) 2 ≤ 2 I k X i =1 g P k ( x ik ) N ik − 2 I k X i =1 g P ( a ) ∩ C n 0 ( x ik ) N ik + + I k X i =1  g K 0 ( x ik ) − g P ( a ) ∩ C n 0 ( x ik )  2 . In view of (32) and (33), this is the required inequality .  18 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN Let K b e an y con vex b o dy in R n and let ε > 0. The inner p ar al lel b o dy K  εB n is the Mink o wski diﬀerence of K and εB n as deﬁned in (2). Then K  εB n = \ y ∈ εB n ( K − y ) , so the inner parallel b o dy is con vex. (It ma y b e empt y .) F or further prop erties, see [46, pp. 133–137]. The follo wing prop osition is an immediate consequence of the fact that if K is a con vex b o dy in R n , then (35) V ( K ) − V ( K  εB n ) < S ( K ) ε. This follows directly from either an inequality of Sangwine-Y ager or one of Brannen; see The- orem 1 or Corollary 2 of [13], respectively . The estimate (35) both generalizes and strengthens [23, Lemma 4.2], whic h concerns the case n = 2. The authors of the latter paper w ere una w are that an ev en stronger estimate for n = 2 was found earlier by Matheron [39]. Prop osition 4.5. If K ⊂ C n 0 is a c onvex b o dy and ε > 0 , then V ( K ) − V ( K  εB n ) < 2 nε. Let G b e the class of all nonnegativ e functions g on R n with supp ort in 2 C n 0 that are the co v ariogram of some con v ex b o dy contained in C n 0 , together with the function on R n that is iden tically zero. Note that for eac h g ∈ G and x ∈ R n , g ( x ) ≤ g C n 0 ( x ) ≤ V ( C n 0 ) = 1. Lemma 4.6. L et 0 < ε < 1 b e given. Then ther e is a ﬁnite set { ( g L j , g U j ) : j = 1 , . . . , m } of p airs of functions in G such that (i) k g U j − g L j k 1 ≤ ε for j = 1 , . . . , m and (ii) for e ach g ∈ G , ther e is an j ∈ { 1 , . . . , m } such that g L j ≤ g ≤ g U j . Pr o of. Let 0 < ε < 1 and let c 5 = c 5 ( n ) ≥ 1 b e a constan t, to b e c hosen later. Since K n ( C n 0 ) with the Hausdorﬀ metric is compact, there is an ε/c 5 -net { K 1 , . . . , K m } in K n ( C n 0 ). F or eac h j = 1 , . . . , m , let K U j = ( K j + ( ε/c 5 ) B n ) ∩ C n 0 and K L j = K j  ( ε/c 5 ) B n . Deﬁne g U j = g K U j and g L j = g K L j , j = 1 , . . . , m . Both g U j and g L j b elong to G , j = 1 , . . . , m . W e ﬁrst prov e (ii). Let g ∈ G . There is a K ∈ K n ( C n 0 ) such that g = g K . Cho ose j ∈ { 1 , . . . , m } suc h that δ ( K , K j ) ≤ ε/c 5 . Since K ⊂ C n 0 and K ⊂ K j + ( ε/c 5 ) B n , w e ha v e K ⊂ ( K j + ( ε/c 5 ) B n ) ∩ C n 0 = K U j . Also, we ha v e ( K j  ( ε/c 5 ) B n ) + ( ε/c 5 ) B n ⊂ K j ⊂ K + ( ε/c 5 ) B n , yielding K L j = K j  ( ε/c 5 ) B n ⊂ K . These facts imply that g L j ≤ g ≤ g U j , as required. It remains to prov e (i). It is easy to prov e (see, for example, [46, p. 411]) that for an y con v ex b o dy L in R n , Z DL g L ( x ) dx = V ( L ) 2 . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 19 Applying this, Steiner’s form ula with quermassintegrals (see [20, (A.30), p. 404], basic prop- erties of mixed v olumes (see [20, (A.16) and (A.18), p. 399]) together with K j ⊂ C n 0 ⊂ ( n/ 4) 1 / 2 B n and c 5 ≥ 1, and Prop osition 4.5 with ε replaced by ε/c 5 , w e obtain k g U j − g L j k 1 = Z 2 C n 0  g U j ( x ) − g L j ( x )  dx = V  K U j  2 − V  K L j  2 ≤ 2  V  K U j  − V  K L j  ≤ 2  V  K j + ε c 5 B n  − V ( K j )  +  V ( K j ) − V  K j  ε c 5 B n  ≤ 2 κ n n X i =1  n i   n 4  ( n − i ) / 2 + 2 n !  ε c 5  < ε, pro vided that c 5 is c hosen suﬃciently large.  By analogy with [48, Deﬁnition 2.2], w e refer to a ﬁnite set { ( g L j , g U j ) : j = 1 , . . . , m } of pairs of functions in G satisfying (i) and (ii) of Lemma 4.6 as an ε -net with br acketing for the class G . The follo wing prop osition is a version of the strong law of large n umbers that applies to a triangular family , rather than a sequence, of random v ariables. A version with the assumptions of full independence and uniformly b ounded fourth moments is prov ed in detail in [23, Lemma 4.4], with m k = k . The stronger statement below follows directly from [30, Corollary 1] (with p = 1 and n = m k there); in fact, it is enough to assume the uniform b oundedness of p th absolute moments where p = 2 + ε for some ε > 0, but we prefer to a v oid this extra parameter in the sequel. Prop osition 4.7. L et X ik , k ∈ N , i = 1 , . . . , m k , wher e m k ≥ k , b e a triangular arr ay of r ow-wise indep endent zer o me an r andom variables. If the arr ay has uniformly b ounde d thir d absolute moments, then, almost sur ely, (36) 1 m k m k X i =1 X ik → 0 as k → ∞ . Lemma 4.8. F or every k ∈ N , let x ik , i = 1 , . . . , I k , b e the p oints in the cubic arr ay 2 C n 0 ∩ (1 /k ) Z n . L et N ik , k ∈ N , i = 1 , . . . , I k , b e r ow-wise indep endent zer o me an r andom variables with uniformly b ounde d thir d absolute moments. Then, almost sur ely, sup K ∈K n ( C n 0 ) Ψ( K, x I k , N I k ) → 0 as k → ∞ , wher e for e ach k ∈ N , Ψ( K, x I k , N I k ) is deﬁne d by (33) with q = I k , x I k = ( x 1 k , . . . , x I k k ) , and N I k = ( N 1 k , . . . , N I k k ) . Pr o of. Let 0 < ε < 1 and let { ( g L j , g U j ) : j = 1 , . . . , m } be an ε -net with brack eting for G , as pro vided b y Lemma 4.6. Let K ∈ K n ( C n 0 ) and let g = g K ∈ G . Cho ose j ∈ { 1 , . . . , m } suc h 20 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN that g L j ≤ g ≤ g U j . Deﬁne N + ik = max { N ik , 0 } and N − ik = N + ik − N ik for k ∈ N and i = 1 , . . . , I k . Then for k ∈ N , w e ha v e Ψ( K, x I k , N I k ) = 1 I k I k X i =1 g ( x ik ) N + ik − 1 I k I k X i =1 g ( x ik ) N − ik ≤ 1 I k I k X i =1 g U j ( x ik ) N + ik − 1 I k I k X i =1 g L j ( x ik ) N − ik ≤ W k ( ε ) , where (37) W k ( ε ) = max j =1 ,...,m ( 1 I k I k X i =1 g U j ( x ik ) N + ik − 1 I k I k X i =1 g L j ( x ik ) N − ik ) is indep enden t of K . Consequen tly , (38) sup K ∈K n ( C n 0 ) Ψ( K, x I k , N I k ) ≤ W k ( ε ) , for all 0 < ε < 1. Fix j ∈ { 1 , . . . , m } , and let X ik = g U j ( x ik ) N + ik − g U j ( x ik ) E ( N + ik ) , for k ∈ N and i = 1 , . . . , I k . Since g U j ( x ik ) ≤ 1, it is easy to c heck that the random v ariables X ik satisfy the h yp otheses of Prop osition 4.7. By (36) with m k = I k , we obtain, almost surely , lim sup k →∞ 1 I k I k X i =1 g U j ( x ik ) N + ik = lim sup k →∞ 1 I k I k X i =1 g U j ( x ik ) E ( N + ik ) . The same argumen t, with limits sup erior replaced b y limits inferior, applies when X ik is deﬁned b y X ik = g L j ( x ik ) N − ik − g L j ( x ik ) E ( N − ik ). Our moment assumption on the random v ariables N ik implies that there is a constant C such that E ( N + ik ) = E ( N − ik ) = 1 2 E ( | N ik | ) ≤ C . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 21 Also, by Lemma 4.6(i) we ha ve k g U j − g L j k 1 ≤ ε and by Lemma 4.6(ii) w e may assume that g U j − g L j ≥ 0, for i = 1 , . . . , m . Therefore, almost surely , lim k →∞ W k ( ε ) = max j =1 ,...,m ( lim sup k →∞ 1 I k I k X i =1 g U j ( x ik ) E ( N + ik ) − lim inf k →∞ 1 I k I k X i =1 g L j ( x ik ) E ( N − ik ) ) ≤ max j =1 ,...,m ( C lim sup k →∞ 1 I k I k X i =1 g U j ( x ik ) − lim inf k →∞ 1 I k I k X i =1 g L j ( x ik ) !) ≤ max j =1 ,...,m ( C 2 n Z 2 C n 0  g U j ( x ) − g L j ( x )  dx ) ≤ C ε 2 n . This and (38) complete the pro of.  Lemma 4.9. L et K 0 ⊂ C n 0 b e a c onvex b o dy with its c entr oid at the origin. Supp ose that P k is an output fr om A lgorithm NoisyCovLSQ as state d ab ove. Then, almost sur ely, (39) lim k →∞ | g K 0 − g P k | I k = 0 . Pr o of. Let Q k b e the o -symmetric p olytop e from the input of Algorithm NoisyCo vLSQ that satisﬁes, almost surely , (17). Fix a realization for which (17) holds. W e ma y assume that lim k →∞ δ ( Q k , ∇ K 0 ) = 0 , as the other case is completely analogous. By [46, Theorem 4.2.1], S ( Q k , · ) con v erges w eakly to S ( ∇ K 0 , · ) as k → ∞ . By [10, Theorem 6.8], w eak conv ergence is equiv alent to conv ergence in the Prohoro v metric, so S ( Q k , · ) con verges in the Prohorov metric to S ( ∇ K 0 , · ) as k → ∞ . No w Lemma 4.3 ensures that if J k is the con vex p olytop e corresp onding to Q k in that lemma, then S ( J k , · ) con verges in the Prohoro v metric to S ( K 0 , · ) as k → ∞ . W e may assume that the cen troid of J k is at the origin for each k . By Lemma 4.2 (with K and L replaced b y K 0 and J k , resp ectively), there are constan ts a 1 and k 0 ∈ N , dep ending only on K 0 , such that J k ⊂ a 1 B n for all k ≥ k 0 . By Blasc hk e’s selection theorem and the fact that a con vex b o dy is determined up to translation b y its surface area measure, the sequence ( J k ) has an accum ulation p oint and every suc h accumulation p oin t must b e a translate of K 0 . But J k and K 0 ha v e their centroids at the origin and K 0 ⊂ C n 0 , so lim k →∞ δ ( K 0 , J k ∩ C n 0 ) = lim k →∞ δ ( K 0 , J k ) = 0 . (This consequence of the fact that d P ( S ( J k , · ) , S ( K 0 , · )) → 0 as k → ∞ can also b e derived from a stability estimate of Hug and Schneider [31, Theorem 3.1], but w e do not need the full force of that result here.) It follows from the con tin uit y of v olume that k g K 0 − g J k ∩ C n 0 k ∞ → 0 as k → ∞ and hence that (40) lim k →∞   g K 0 − g J k ∩ C n 0   I k = 0 . 22 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN Next, w e observe that J k can serve as the P ( a ) in Lemma 4.4. By its deﬁnition, a translate of P k is contained in C n 0 , and the quan tity Ψ( P k , x I k , N I k ) is unaﬀected by this translation. F rom Lemma 4.8 we obtain (41) lim k →∞ Ψ( P k , x I k , N I k ) = 0 and lim k →∞ Ψ( J k ∩ C n 0 , x I k , N I k ) = 0 . No w (39) follows directly from (34) (with P ( a ) replaced by J k ), (40), and (41).  Theorem 4.10. Supp ose that K 0 ⊂ C n 0 is a c onvex b o dy with its c entr oid at the origin. Sup- p ose also that K 0 is determine d, up to tr anslation and r eﬂe ction in the origin, among al l c onvex b o dies in R n , by its c ovario gr am. If P k , k ∈ N , is an output fr om A lgorithm NoisyCovLSQ as state d ab ove, then, almost sur ely, (42) min { δ ( K 0 , P k ) , δ ( − K 0 , P k ) } → 0 as k → ∞ . Pr o of. By Lemma 4.9, almost surely , (43) | g K 0 − g P k | I k → 0 , as k → ∞ . Fix a realization for whic h this statemen t holds. F or eac h k , P k has its centroid at the origin and is a translate of a subset of C n 0 , so P k ⊂ 2 C n 0 and by Blaschk e’s selection theorem, ( P k ) has an accum ulation p oint, L , say . Note that L m ust also hav e its centroid at the origin and b e a translate of a subset of C n 0 . Let ( P k 0 ) b e a subsequence con verging to L . Then since g K 0 − g P k 0 con v erges uniformly to g K 0 − g L as k 0 → ∞ , w e hav e   g K 0 − g P k 0   2 I k 0 → 1 2 n Z 2 C n 0 ( g K 0 ( x ) − g L ( x )) 2 dx, as k 0 → ∞ . F rom this and (43), we obtain k g K 0 − g L k L 2 (2 C n 0 ) = 0, and hence, since cov ari- ograms are clearly con tinuous, g K 0 = g L on 2 C n 0 . As the supp orts of g K 0 and g L are contained in 2 C n 0 , we hav e g K 0 = g L in R n . The h yp othesis on K 0 no w implies that L = ± K 0 . Since L w as an arbitrary accumulation p oint of ( P k ), w e obtain (42).  5. Appro xima ting the Blaschke body via the cov ariogram Algorithm NoisyCo vBlasc hke Input: Natural n umbers n ≥ 2 and k ; m utually nonparallel v ectors u i ∈ S n − 1 , i = 1 , . . . , k that span R n ; noisy co v ariogram measurements M (1) ij k = g K 0 ( o ) + N (1) ij k and M (2) ij k = g K 0 ((1 /k ) u i ) + N (2) ij k , for i = 1 , . . . , k and j = 1 , . . . , k 2 , of an unknown conv ex bo dy K 0 ⊂ C n 0 whose cen troid is at the origin, where the N ( m ) ij k ’s are row-wise indep endent (i.e., indep endent for ﬁxed k ) zero mean random v ariables with uniformly b ounded sixth moments. PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 23 T ask: Construct an o -symmetric con vex p olytop e Q k that approximates the Blaschk e b o dy ∇ K 0 . A ction: 1. F or i = 1 , . . . , k and j = 1 , . . . , k 2 , let y ik = 1 k 2 k 2 X j =1 k ( M (1) ij k − M (2) ij k ) . 2. With the natural n um b ers n ≥ 2 and k , and v ectors u i ∈ S n − 1 , i = 1 , . . . , k use the sample means y ik instead of noisy measurements of the brigh tness function b K ( u i ) as input to Algorithm NoisyBrigh tLSQ (see [24, p. 1352]). The output of the latter algorithm is Q k . F or a ﬁxed ﬁnite set u 1 , . . . , u q of p oin ts in S n − 1 , deﬁne a pseudonorm | · | q b y (44) | f | q = 1 q q X i =1 f ( u i ) 2 ! 1 / 2 , where f is an y real-v alued function on S n − 1 . F or a conv ex bo dy K contained in C n 0 , a sequence ( u i ) in S n − 1 , and a vector X k = ( X 1 k , . . . , X kk ) of random v ariables, let Ψ( K, ( u i ) , X k ) = 1 k k X i =1 b K ( u i ) X ik . The same notations were used for a technic ally diﬀerent pseudonorm and function Ψ in the previous section, but this should cause no confusion. Lemma 5.1. L et K 0 b e a c onvex b o dy in R n with c entr oid at the origin and such that r B n ⊂ K 0 ⊂ C n 0 for some r > 0 . L et ( u i ) b e a se quenc e in S n − 1 . If Q k is an output fr om A lgorithm NoisyCovBlaschke as state d ab ove, then, almost sur ely, ther e is a c onstant c 6 = c 6 ( n, r ) such that (45) | b K 0 − b Q k | 2 k ≤ 2Ψ( Q k , ( u i ) , X k ) − 2Ψ( K 0 , ( u i ) , X k ) + c 6 k | b K 0 − b Q k | k , for al l k ∈ N . Her e X k = ( X 1 k , . . . , X kk ) , with X ik = 1 k k 2 X j =1 ( N (1) ij k − N (2) ij k ) , for i = 1 , . . . , k . Pr o of. F or i = 1 , . . . , k , w e hav e y ik = g K 0 ( o ) − g K 0 ((1 /k ) u i ) 1 /k + 1 k k 2 X j =1 ( N (1) ij k − N (2) ij k ) = µ ik + X ik , 24 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN where the X ik ’s are row-wise indep endent zero mean random v ariables. Note that the y ik ’s are also row-wise indep endent. F urthermore, by Khinchine’s inequality (see, for example, [29, (4.32.1), p. 307] with α = 6), there is a constan t C suc h that E  | X ik | 6  ≤ C k 2 k 2 X j =1 E     N (1) ij k − N (2) ij k    6  , from whic h we see that the X ik ’s also ha ve uniformly bounded sixth momen ts. By Lemma 3.1, lim k →∞ µ ik = b K 0 ( u i ) . In fact, the conv ergence is uniform. This is b ecause for each u ∈ S n − 1 , w e hav e b K 0 ( u ) ≤ b C n 0 ( u ) ≤ b ( √ n/ 2) B n ( u ) = ( n/ 4) ( n − 1) / 2 κ n − 1 and (46) 0 ≤ b K 0 ( u ) − µ ik ≤ 1 −  1 − 1 2 r k  n − 1 ! b K 0 ( u ) ≤ n − 1 2 r k b K 0 ( u ) , k ≥ 1 / (2 r ) , b y Lemma 3.1, so there is a constant c 7 = c 7 ( n, r ) suc h that (47) 0 ≤ b K 0 ( u i ) − µ ik ≤ c 7 k , for all k ∈ N and i = 1 , . . . , k . By the form ulation of Algorithms NoisyCovBlasc hke and NoisyBrigh tLSQ (see [24, p. 1352] and tak e [24, Prop osition 2.1] into account), Q k minimizes (48) k X i =1 ( b K ( u i ) − y ik ) 2 o v er the class of all o -symmetric conv ex b o dies K in R n . By (7), for each conv ex bo dy there is an o -symmetric conv ex bo dy with the same brightness function. F rom this it follo ws that Q k is actually a minimizer ov er the class of all conv ex b o dies K in R n . Substituting K = Q k and K = K 0 in (48), w e obtain k X i =1 ( b Q k ( u i ) − µ ik − X ik ) 2 ≤ k X i =1 ( b K 0 ( u i ) − µ ik − X ik ) 2 . Rearranging and using (44), we obtain | b K 0 − b Q k | 2 k ≤ 2 k k X i =1 ( b Q k ( u i ) − b K 0 ( u i )) ( X ik − ( b K 0 ( u i ) − µ ik )) . The deﬁnition of Ψ and Cauch y-Sch w arz inequalit y yields | b K 0 − b Q k | 2 k ≤ 2Ψ( Q k , ( u i ) , X k ) − 2Ψ( K 0 , ( u i ) , X k ) + 2 | b K 0 − b Q k | k 1 k k X i =1 ( b K 0 ( u i ) − µ ik ) 2 ! 1 / 2 . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 25 In view of (47), this prov es (45) with c 6 = 2 c 7 .  Lemma 5.2. Supp ose that the assumptions of L emma 5.1 ar e satisﬁe d with a se quenc e ( u i ) such that ( u ∗ i ) is evenly spr e ad. Supp ose also that the se c ond moments of the X ik ’s ar e uniformly b ounde d by a c onstant C > 0 . Then, almost sur ely, ther e ar e c onstants c 8 = c 8 ( C, n, r, ( u i )) and N 1 = N 1 (( X ik ) , ( u i )) such that (49) S ( Q k ) ≤ c 8 , for al l k ≥ N 1 . Pr o of. By the Cauc h y-Sc h w arz inequalit y , Ψ( Q k , ( u i ) , X k ) − Ψ( K 0 , ( u i ) , X k ) ≤ | b K 0 − b Q k | k 1 k k X i =1 X 2 ik ! 1 / 2 . This and (45) imply that | b K 0 − b Q k | k ≤ 2 1 k k X i =1 X 2 ik ! 1 / 2 + c 6 k , for all k ∈ N . Since the X ik ’s ha ve uniformly b ounded sixth momen ts, w e can apply Prop o- sition 4.7 with m k and X ik replaced by k and X 2 ik − E ( X 2 ik ), resp ectiv ely , to conclude that the ﬁrst term on the righ t-hand side is b ounded, almost surely . Thus, almost surely , there are constan ts c 9 = c 9 ( C, n, r ) and N 2 = N 2 (( X ik ) , ( u i )) suc h that (50) | b K 0 − b Q k | k ≤ c 9 , for all k ≥ N 2 . As ( u ∗ i ) is evenly spread, we can apply [24, Lemma 7.1] with K and L replaced b y Π K 0 and Π Q k , resp ectively . Using this, the fact that Π K 0 ⊂ Π C n 0 = 2 C n 0 ⊂ √ nB n (see [20, p. 145]), and (4), we ﬁnd that there are constan ts c 10 = c 10 (( u i )) and N 3 = N 3 (( u i )) suc h that (51) b Q k ≤ c 10 | b K 0 − b Q k | k + 2 √ n, for k ≥ N 3 . Finally , (49) follows directly from (50), (51), and (6).  Lemma 5.3. Supp ose that the assumptions of L emma 5.1 ar e satisﬁe d with a se quenc e ( u i ) such that ( u ∗ i ) is evenly spr e ad. Then, almost sur ely, (52) lim k →∞ | b K 0 − b Q k | k = 0 . Pr o of. Cho ose a constant C 1 suc h that E ( | X ik | 2 ) ≤ C 1 for all i and k . Due to (45) and (50), there is, almost surely , a constant c 11 = c 11 ( C 1 , n, r ) suc h that (53) | b Q k − b K 0 | 2 k ≤ 2Ψ( Q k , ( u i ) , X k ) − 2Ψ( K 0 , ( u i ) , X k ) + c 11 k , for all k ≥ N 2 . By Prop osition 4.7 with m k = k and X ik replaced b y b K 0 ( u i ) X ik , the v ariable Ψ( K 0 , ( u i ) , X k ) con verges to zero, almost surely , as k → ∞ . 26 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN F or m ∈ N , let H m = { K ∈ K n : S ( K ) ≤ m } . If we can show that for all m ∈ N , almost surely , (54) lim k →∞ sup K ∈H m | Ψ( K, ( u i ) , X k ) | = 0 , then b y (49), almost surely , lim k →∞ Ψ( Q k , ( u i ) , X k ) = 0 . This and (53) will yield (52), completing the pro of. T o prov e (54), note ﬁrst that b y (5), we hav e | Ψ( K, ( u i ) , X k ) | =      1 k k X i =1 b K ( u i ) X ik      ≤ 1 2 Z S n − 1      1 k k X i =1 | u i · v | X ik      dS ( K , v ) . Since S ( K ) = S ( K , S n − 1 ) ≤ m for K ∈ H m , it is enough to prov e that, almost surely , (55) lim k →∞ sup v ∈ S n − 1      1 k k X i =1 | u i · v | X ik      = 0 . This follows essen tially from the uniform contin uity of the function | u i · v | , v ∈ S n − 1 , and the fact that S n − 1 is compact. Indeed, suppose that (55) do es not hold almost surely . Cho ose a constan t C 2 suc h that E ( | X ik | ) ≤ C 2 for all i and k . Then there is a δ > 0 such that (56) lim sup k →∞ sup v ∈ S n − 1 1 k k X i =1 | u i · v | X ik > δ C 2 with p ositive probability . Let { w 1 , . . . , w m } b e a δ / 2-net in S n − 1 . F or any realization and any k ∈ N , there is a v k ∈ S n − 1 suc h that (57) 1 k k X i =1 | u i · v k | X ik = sup v ∈ S n − 1 1 k k X i =1 | u i · v | X ik . Let A j denote the set of all ev ents suc h that an accum ulation p oin t of ( v k ) has distance at most δ / 2 from w j , j = 1 , . . . , m . F or a realization in A j and any subsequence ( k 0 ) of ( k ) suc h that | v k 0 − w j | ≤ δ holds for suﬃcien tly large k , w e hav e, almost surely , lim sup k 0 →∞      1 k 0 k 0 X i =1 | u i · v k 0 | X ik 0 − 1 k 0 k 0 X i =1 | u i · w j | X ik 0      ≤ δ lim sup k 0 →∞ 1 k 0 k 0 X i =1 | X ik 0 | ≤ δ C 2 , b y Prop osition 4.7 with m k and X ik replaced b y k 0 and | X ik 0 | − E ( | X ik 0 | ), resp ectively . But Prop osition 4.7, with m k and X ik replaced by k 0 and | u i · w j | X ik 0 , resp ectiv ely , also implies that, almost surely , the second term on the left-hand side con v erges to zero, as k 0 → ∞ . In view of (57), this yields lim sup k 0 →∞ sup v ∈ S n − 1 1 k 0 k 0 X i =1 | u i · v | X ik 0 ≤ δ C 2 , PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 27 for almost all even ts in A j . As any sequence in S n − 1 has at least one accumulation p oin t, the latter inequalit y holds, almost surely , con tradicting (56).  Theorem 5.4. L et K 0 ⊂ C n 0 b e a c onvex b o dy with its c entr oid at the origin. L et ( u i ) b e a se quenc e in S n − 1 such that ( u ∗ i ) is evenly spr e ad. If Q k is an output fr om A lgorithm Noisy- CovBlaschke as state d ab ove, then, almost sur ely, (58) lim k →∞ δ ( ∇ K 0 , Q k ) = 0 . Pr o of. W e hav e o ∈ int K 0 , so there is an r > 0 such that r B n ⊂ K 0 . By Lemmas 5.2 and 5.3, w e can ﬁx a realization for which b oth (49) and (52) are true. Using (4), we observ e that (52) is equiv alent to (59) lim k →∞ | h Π K 0 − h Π Q k | k = 0 . W e also ha ve h Π Q k = b Q k ≤ S ( Q k ), so b y (49), the sets Π Q k are uniformly b ounded. With these observ ations and the fact that ( u 1 , − u 1 , u 2 , − u 2 , . . . ) is evenly spread, we can follow the pro of of [24, Theorem 6.1]), from the fourth line, with K and ˆ P k replaced b y Π K 0 and Π Q k , resp ectiv ely , to conclude that (60) lim k →∞ δ (Π K 0 , Π Q k ) = 0 . No w r B n ⊂ K 0 ⊂ C n 0 yields sB n ⊂ Π K 0 ⊂ tB n with s = κ n − 1 r n − 1 and t = √ n . Moreov er, (4) and (7) give Π( ∇ K 0 ) = Π K 0 . Hence (60) implies that s 2 B n ⊂ Π( ∇ K 0 ) , Π Q k ⊂ 3 t 2 B n , for suﬃcien tly large k , where s and t dep end only on n and r . Exactly as in the proof from (48) to (49) of [24, Theorem 7.2] (whic h in turn follo ws the pro of of [25, Lemma 4.2]), this leads to r 0 B n ⊂ ∇ K 0 , Q k ⊂ R 0 B n , for suﬃciently large k , where r 0 > 0 and R 0 dep end only on n and r . Then (58) follows from (60) and the Bourgain-Campi-Lindenstrauss stabilit y result for pro jection b o dies (see [11] and [16], or [20, Remark 4.3.13]).  6. Appro xima ting the difference body via the cov ariogram Throughout this section, ϕ will b e a nonnegativ e b ounded measurable function on R n with supp ort in C n 0 , suc h that R R n ϕ ( x ) dx = 1. Algorithm NoisyCo vDiﬀ( ϕ ) Input: Natural num b ers n ≥ 2 and k ; p ositive reals δ k and ε k ; noisy co v ariogram measure- men ts (61) M ik = g K 0 ( x ik ) + N ik , 28 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN of an unknown con vex b o dy K 0 ⊂ C n 0 at the p oints x ik , i = 1 , . . . , I k in the cubic arra y 2 C n 0 ∩ (1 /k ) Z n , where the N ik ’s are row-wise independent zero mean random v ariables with uniformly b ounded fourth momen ts. T ask: Construct an o -symmetric conv ex p olytop e Q k in R n that appro ximates the diﬀerence b o dy D K 0 . A ction: 1. Let ϕ ε k ( x ) = ε − n k ϕ ( x/ε k ) for x ∈ R n , and let (62) g k ( x ) = I k X i =1 M ik Z (1 /k ) C n 0 + x ik ϕ ε k ( x − z ) dz = I k X i =1 M ik 1 (1 /k ) C n 0 + x ik ! ∗ ϕ ε k ( x ) . 2. Deﬁne the ﬁnite set (63) S k = { x ∈ 2 C n 0 ∩ (1 /k ) Z n : g k ( x ) ≥ δ k } . The output is the conv ex p olytop e Q k = (1 / 2)(con v S k + ( − con v S k )). The input δ k in the algorithm is a threshold parameter. The function g k ( x ) is a Gasser- M ¨ uller type k ernel estimator for g K 0 with kernel function ϕ and bandwidth ε k . As the design p oin ts x ik are deterministic, g k is a m ultiv ariate ﬁxed design kernel estimator. Such estimators are common in multiv ariate regression and are discussed in detail b y Ahmad and Lin [3]. Among other things, strong p oin t wise consistency and a b ound for the rate of w eak p oint wise con v ergence are giv en there. W e shall need uniform b ounds and establish them in the next t w o lemmas. By [3, Theorem 1], for an y x ∈ R n , g k ( x ) is an asymptotically un biased estimator for g K 0 ( x ), if ε k → 0 as k → ∞ . W e shall sho w that this holds uniformly in x . Lemma 6.1. Supp ose that K 0 , ε k , and g k ar e as in Algorithm NoisyCovDiﬀ( ϕ ). F or e ach k ∈ N and x ∈ R n , | E ( g k ( x )) − g K 0 ( x ) | ≤ n ( ε k + 1 /k ) . Conse quently, g k is uniformly asymptotic al ly unbiase d whenever lim k →∞ ε k = 0 . Pr o of. Using (61), (62), and the deﬁnition of ϕ ε k , w e obtain (64) | E ( g k ( x )) − g K 0 ( x ) | ≤ I k X i =1 | g K 0 ( x ik ) − g K 0 ( x ) | Z (1 /k ) C n 0 + x ik ϕ ε k ( x − z ) dz , for all x ∈ R n . The supp ort of ϕ ε k is con tained in ε k C n 0 , so for ﬁxed x , the supp ort of the in tegrand ϕ ε k ( x − z ) is contained in ε k C n 0 + x . Now if x ik 6∈ ( ε k + 1 /k ) C n 0 + x , then ε k C n 0 + x and (1 /k ) C n 0 + x ik are disjoint, so the corresp onding summand in (64) v anishes. Moreo ver, for x ik ∈ ( ε k + 1 /k ) C n 0 + x , Corollary 3.3 and the fact that the diameter of C n 0 is √ n imply that | g K 0 ( x ik ) − g K 0 ( x ) | ≤ n ( ε k + 1 /k ) . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 29 Consequen tly , | E ( g k ( x )) − g K 0 ( x ) | ≤ n ( ε k + 1 /k ) I k X i =1 Z (1 /k ) C n 0 + x ik ϕ ε k ( x − z ) dz ≤ n ( ε k + 1 /k ) Z R n ϕ ε k ( x − z ) dz = n ( ε k + 1 /k ) , as required.  In [3, Lemma 1], a p olynomial rate of con vergence result in the weak sense is established for indep enden t identically distributed measuremen t errors with p olynomial tails. In contrast, w e assume only uniformly b ounded fourth moments and obtain a con vergence rate that holds uniformly , using the Lipschitz con tin uit y of the co v ariogram. Lemma 6.2. Supp ose that K 0 , ε k , and g k ar e as in A lgorithm NoisyCovDiﬀ( ϕ ) and let δ > 0 and lim k →∞ ε k = 0 . Then ther e ar e c onstants c 12 = c 12 ( ϕ ) and N 4 = N 4 (( ε k ) , n ) ∈ N such that (65) Pr ( | g k ( x ) − g K 0 ( x ) | > δ ) ≤ c 12 (2 k + 1) n δ − 4 ( k ε k ) − 3 n , for al l k ≥ N 4 and al l x ∈ R n . Pr o of. Let x ∈ R n and k ∈ N b e ﬁxed and deﬁne (66) β ik = β ik ( x ) = Z (1 /k ) C n 0 + x ik ϕ ε k ( x − z ) dz , for i = 1 , . . . , I k . Then (67) β ik ≤ k ϕ ε k k ∞ V ((1 /k ) C n 0 ) = k ϕ k ∞ ( k ε k ) − n and (68) I k X i =1 β ik ≤ Z R n ϕ ε k ( x − z ) dz = 1 . In view of (61), (62), and (66), g k ( x ) − E ( g k ( x )) = I k X i =1 β ik N ik is a sum of zero mean indep enden t random v ariables. The assumption that the N ik ’s ha ve uniformly b ounded fourth moments implies that E ( | N ik | 4 ) ≤ C for some constant C and all i and k . No w, using Marko v’s inequality , Khinchine’s inequalit y (see, for example, [29, (4.32.1), 30 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN p. 307] with α = 4), (67), and (68), w e obtain Pr ( | g k ( x ) − E ( g k ( x )) | ≥ δ / 2) ≤ ( δ / 2) − 4 E        I k X i =1 β ik N ik      4   ≤ cδ − 4 I k I k X i =1 E  | β ik N ik | 4  ≤ cC δ − 4 I k I k X i =1 β 4 ik ≤ cC δ − 4 I k  k ϕ k ∞ ( k ε k ) − n  3 I k X i =1 β ik ≤ c 12 (2 k + 1) n δ − 4 ( k ε k ) − 3 n , (69) for all δ > 0, where c is a constant and c 12 = cC k ϕ k 3 ∞ . By Lemma 6.1, there is a constan t N 4 = N 4 (( ε k ) , n ) ∈ N such that for all k ≥ N 4 and x ∈ R n , we ha ve | E ( g k ( x )) − g K 0 ( x ) | ≤ δ / 2 and therefore Pr ( | g k ( x ) − g K 0 ( x ) | > δ ) ≤ Pr ( | g k ( x ) − E ( g k ( x )) | + | E ( g k ( x )) − g K 0 ( x ) | > δ ) ≤ Pr ( | g k ( x ) − E ( g k ( x )) | > δ / 2) . No w (65) follows from this and (69).  F or a con vex b o dy K in R n and δ > 0, let K ( δ ) = { x ∈ R n : g K ( x ) ≥ δ } . Since g 1 /n K is conca v e on its supp ort, K ( δ ) is a compact con vex set, sometimes called a conv olution b o dy of K . References to results on con volution b o dies can b e found in [20, p. 378]. Lemma 6.3. L et K b e a c onvex b o dy in R n . If 0 < δ < V ( K ) , then  1 − δ 1 /n V ( K ) 1 /n  D K ⊂ K ( δ ) . Pr o of. Let t = ( δ /V ( K )) 1 /n and let x ∈ (1 − t ) D K . Since D K is the supp ort of g K , there is a y in the supp ort of g K suc h that x = (1 − t ) y + to . As g 1 /n K is conca v e on its supp ort, we hav e g K ( x ) 1 /n ≥ (1 − t ) g K ( y ) 1 /n + tg K ( o ) 1 /n ≥ tV ( K ) 1 /n = δ 1 /n . It follo ws that x ∈ K ( δ ).  Theorem 6.4. Supp ose that K 0 , δ k , ε k , and g k ar e as in Algorithm NoisyCovDiﬀ( ϕ ). Assume that lim k →∞ ε k = lim k →∞ δ k = 0 and that (70) lim inf k →∞ δ 4 k ε 3 n k k n − 3 / 2 > 0 . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 31 L et c 13 > √ n (2 /V ( K 0 )) 1 /n . If Q k is an output fr om Algorithm NoisyCovDiﬀ( ϕ ) as state d ab ove, then, almost sur ely, (71) δ ( D K 0 , Q k ) ≤ c 13 δ 1 /n k , for suﬃciently lar ge k . In p articular, almost sur ely, Q k c onver ges to DK 0 , as k → ∞ . Pr o of. Let a k = max x ∈ 2 C n 0 ∩ (1 /k ) Z n | g k ( x ) − g K 0 ( x ) | . By Lemma 6.2 and (70), we hav e Pr ( a k ≥ δ k ) ≤ X x ∈ 2 C n 0 ∩ (1 /k ) Z n Pr ( | g k ( x ) − g K 0 ( x ) | ≥ δ k ) ≤ c 12 (2 k + 1) 2 n δ − 4 k ( k ε k ) − 3 n = O  k − 3 / 2  . Therefore, by the Borel-Cantelli lemma, we see that, almost surely , a k < δ k for suﬃcien tly large k . Fix a realization and a k ∈ N such that a k < δ k and  2 δ k V ( K 0 )  1 /n + 3 s ( K 0 ) k ≤ 1 , (72) where s ( K 0 ) = max { ρ ≥ 0 : ρC n 0 ⊂ D K 0 } . As a k < δ k , the deﬁnition (63) of S k implies K 0 (2 δ k ) ∩ 1 k Z n ⊂ S k ⊂ D K 0 . The set on the left is o -symmetric, and D K 0 is con vex and o -symmetric, so con v  K 0 (2 δ k ) ∩ 1 k Z n  ⊂ Q k ⊂ D K 0 . (73) W e claim that K 0 (2 δ k )  3 k C n 0 ⊂ con v  K 0 (2 δ k ) ∩ 1 k Z n  , (74) where Minko wski diﬀerence  is deﬁned by (2). Indeed, let x ∈ K 0 (2 δ k )  (3 /k ) C n 0 . As { y + (1 /k ) C n 0 : y ∈ (1 /k ) Z n } is a co v ering of R n , there is a y ∈ (1 /k ) Z n with x ∈ (1 /k ) C n 0 + y and hence y ∈ (1 /k ) C n 0 + x . It follows that x ∈ 1 k (2 C n 0 ) + y ⊂ 3 k C n 0 + x ⊂ K 0 (2 δ k ) . As the vertices of (1 /k )(2 C n 0 ) + y are in (1 /k ) Z n , we ha ve x ∈ con v ( K 0 (2 δ k ) ∩ (1 /k ) Z n ), pro ving the claim. Let t k = (2 δ k /V ( K 0 )) 1 /n . The fact that D K 0 is conv ex and con tains the origin, (72), Lemma 6.3 (with δ = 2 δ k ), and the deﬁnition of s ( K 0 ) imply that  1 −  t k + 3 s ( K 0 ) k  D K 0 = (1 − t k ) D K 0   3 s ( K 0 ) k D K 0  ⊂ K 0 (2 δ k )  3 k C n 0 . 32 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN F rom this, (74), and (73), we obtain  1 −  t k + 3 s ( K 0 ) k  D K 0 ⊂ Q k ⊂ D K 0 . As D K 0 ⊂ √ nB n , this yields δ ( D K 0 , Q k ) ≤ √ n  t k + 3 s ( K 0 ) k  = √ n  2 V ( K 0 )  1 /n + 3 √ n s ( K 0 ) k δ 1 /n k ! δ 1 /n k . By (70), k δ 1 /n k → ∞ as k → ∞ , and (71) follo ws.  The estimate (71) rev eals that the rate of conv ergence of Q k to D K 0 dep ends on the asymptotic b ehavior of the threshold parameter δ k , whic h is linked to the bandwidth ε k b y (70). If we assume that V ( K 0 ) is b ounded from b elow by a known constant, then c 13 in the statement of Theorem 6.4 can be c hosen independent of K 0 . W e note the resulting rate of con vergence as a corollary , where w e c ho ose ε k and δ k as appropriate pow ers of k . In particular, it sho ws that a con vergence rate of k − p can b e attained, where p is arbitrarily close to 1 / 4 − 3 / (8 n ). Corollary 6.5. Supp ose that K 0 , δ k , ε k , and g k ar e as in A lgorithm NoisyCovDiﬀ( ϕ ). L et 0 < b < V ( K 0 ) , let δ k = k − ( n − 3 αn − 3 / 2) / 4 , and let ε k = k − α , for some 0 < α < 1 / 3 − 1 / (2 n ) . If Q k is an output fr om A lgorithm NoisyCovDiﬀ( ϕ ) as state d ab ove, then, almost sur ely, δ ( Q k , D K 0 ) ≤ √ n  2 b  1 /n k − (1 − 3 α − 3 / (2 n )) / 4 , for suﬃciently lar ge k . Remark 6.6. Here w e outline ho w a stronger assumption, but one that still applies to all the noise mo dels of practical interest, on the random v ariables in Algorithm NoisyCo vDiﬀ( ϕ ) leads to a b etter conv ergence rate in Corollary 6.5. Consider a family { X α : α ∈ A } of zero mean random v ariables with v ariances σ 2 α that satisfy the h yp othesis of Bernstein’s inequality (see [14, Theorem 5.2, p. 27] or [49, Lemma 2.2.11]), that is, (75) | E ( X m α ) | ≤ m ! 2 σ 2 α H m − 2 , for some H > 0 and all α ∈ A and m = 2 , 3 , . . . , and also hav e uniformly b ounded v ariances, that is, (76) σ 2 α ≤ σ 2 , sa y , for all α ∈ A . If the family { X 1 , . . . , X r } of indep endent zero mean random v ariables satisﬁes (75) with A = { 1 , . . . , r } , then Bernstein’s inequality states that Pr      r X i =1 X i      ≥ δ ! ≤ 2 exp  − δ 2 2 ( δ H + P r i =1 σ 2 i )  , PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 33 for all δ > 0. Supp ose that the random v ariables N ik in Algorithm NoisyCovDiﬀ( ϕ ) are ro w-wise inde- p enden t, zero mean, and satisfy (75) and (76). Then Bernstein’s inequalit y can b e applied in the pro of of Lemma 6.2, together with (67) and (68), to show that (77) Pr( | g k ( x ) − E ( g k ( x )) | ≥ δ / 2) ≤ 2 exp  − δ 2 ( k ε k ) n 4 k ϕ k ∞ ( δ H + 2 σ 2 )  , for all δ > 0. (Compare the weak er upp er b ound in (69).) As at the end of the pro of of Lemma 6.2, this results in the same upp er b ound for Pr ( | g k ( x ) − g K 0 ( x ) | > δ ). The impro v ed b ound (77), com bined with the argument of Theorem 6.4, leads to the assumption (78) lim inf k →∞ δ 2 k ( k ε k ) n log k > c 14 ( n + 2) , where c 14 = 12 k ϕ k ∞ σ 2 , instead of (70). In Corollary 6.5 w e tak e instead δ k = k − n (1 − α ) / 2 log k and ε k = k − α , for some 0 < α < 1. The ﬁnal conclusion is that if Q k is an output from Algorithm NoisyCo vDiﬀ( ϕ ), then, almost surely , δ ( Q k , D K 0 ) ≤ √ n  2 b  1 /n k − (1 − α ) / 2 (log k ) 1 /n , for suﬃciently large k . In particular, a conv ergence rate of k − p can b e attained, where p is arbitrarily close to 1 / 2. Note that families of zero mean Gaussian and cen tered P oisson random v ariables satisfy (75) and (76). Also, if t wo indep endent families with the same index set satisfy (75) and (76), the same is true for their sums (with p ossibly diﬀerent constants H and σ 2 ). 7. Phase retriev al: Framework and technical lemmas In this section we set the scene for our results on phase retriev al, b eginning with the nec- essary material from F ourier analysis. Let g be a contin uous function on R n whose supp ort is contained in [ − 1 , 1] n and let L ≥ 1. By the classical theory , the F ourier series of g is X z ∈ Z n c z e iπ z · x/L , for x ∈ [ − L, L ] n , where c z = 1 (2 L ) n Z [ − L,L ] n g ( t ) e − iπ z · t/L dt = 1 (2 L ) n Z R n g ( t ) e − iπ z · t/L dt = 1 (2 L ) n b g ( π z /L ) . Let Z n k = { z ∈ Z n : z = ( z 1 , . . . , z n ) , | z j | ≤ k , j = 1 , . . . , n } . 34 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN If g is also Lipschitz, then b y [35, Theorem 3], the square partial sums P z ∈ Z n k c z e iπ z · x/L of the F ourier series of g con verge uniformly to g . Therefore, if g is also an ev en function, we can write (79) g ( x ) = 1 (2 L ) n X z ∈ Z n b g ( π z /L ) e iπ z · x/L = 1 (2 L ) n X z ∈ Z n b g ( π z /L ) cos π z · x L , for all x ∈ [ − L, L ] n , where equalit y is in the sense of uniform con vergence of square partial sums. Let Z n k (+) b e a subset of Z n k suc h that (80) Z n k (+) ∩ ( − Z n k (+)) = ∅ and Z n k = { o } ∪ Z n k (+) ∪ ( − Z n k (+)) . Supp ose that g is ev en and for some ﬁxed 0 < γ < 1 and eac h k ∈ N , we can obtain noisy measuremen ts (81) e g z ,k = b g ( z /k γ ) + X z ,k , of b g , for z ∈ { o } ∪ Z n k (+), where the X z ,k ’s are row-wise indep enden t (i.e., indep enden t for ﬁxed k ) zero mean random v ariables. Deﬁne X z ,k = X − z ,k , for z ∈ ( − Z n k (+)) and note that then X z ,k = X − z ,k for all z ∈ Z n k . Since g is ev en, b g is also ev en, and we ha v e e g z ,k = e g − z ,k for z ∈ Z n k . Using these facts, (79) with L = π k γ , and (81), we obtain (82) 1 (2 π k γ ) n X z ∈ Z n k e g z ,k cos z · x k γ = g ( x ) + 1 (2 π k γ ) n   X z ∈ Z n k X z ,k cos z · x k γ − X z ∈ Z n \ Z n k b g  z k γ  cos z · x k γ   , for all x ∈ [ − π k γ , π k γ ] n . Here the left-hand side is an estimate of g ( x ) and the second and third terms on the righ t-hand side are a random error and a deterministic error, resp ectiv ely . Since it has all the required prop erties, we can apply the previous equation to the cov ari- ogram g = g K 0 of a conv ex b o dy K 0 con tained in C n 0 , in which case d g K 0 = | d 1 K 0 | 2 . In order to mo v e closer to the notation used earlier, we no w use i as an index and again list the p oin ts in [ − 1 , 1] n ∩ (1 /k ) Z n = (1 /k ) Z n k , but this time a little diﬀeren tly . W e let x 0 k = o , list the p oin ts in (1 /k ) Z n k (+) as x ik , i = 1 , . . . , I 0 k = ((2 k + 1) n − 1) / 2, and then let x ik = − x ( − i ) k for i = − I 0 k , . . . , − 1. Now let z ik = k 1 − γ x ik , so that (1 /k γ ) Z n k = { z ik : i = − I 0 k , . . . , I 0 k } . Setting e g j k = g g K 0 z j k ,k and X j k = X z j k ,k , w e use (81) to rewrite (82) as (83) M k ( x ) = g K 0 ( x ) + N k ( x ) − d k ( x ) , where (84) M k ( x ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · x ) e g j k PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 35 is an estimate of g K 0 , (85) N k ( x ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · x ) X j k is a random v ariable, and (86) d k ( x ) = 1 (2 π k γ ) n X z ∈ Z n \ Z n k cos  z · x k γ  d g K 0 ( z /k γ ) is a deterministic error. W e shall need three technical lemmas. The ﬁrst of these provides a con trol on the deter- ministic error. Lemma 7.1. L et d k = sup {| d k ( x ) | : x ∈ R n } . Then d k = O ( k γ − 1 (log k ) n ) as k → ∞ . Pr o of. F rom (86), the fact that d g K 0 = | d 1 K 0 | 2 is nonnegativ e, and (79) with g = g K 0 and L = π k γ , w e hav e (87) d k ≤ 1 (2 π k γ ) n X z ∈ Z n \ Z n k d g K 0 ( z /k γ ) = g K 0 (0) − 1 (2 π k γ ) n X z ∈ Z n k d g K 0 ( z /k γ ) . F or t ∈ R , let D k ( t ) = k X l = − k e ilt = sin(( k + 1 / 2) t ) sin( t/ 2) b e the Diric hlet k ernel. Note that for x = ( x 1 , . . . , x n ) ∈ R n , w e hav e X z ∈ Z n k e iz · x = n Y l =1 k X l = − k e ilx l ! = n Y l =1 D k ( x l ) . Using this and the fact that g K 0 is ev en, with supp ort in [ − 1 , 1] n , w e obtain 1 (2 π k γ ) n X z ∈ Z n k d g K 0 ( z /k γ ) = 1 (2 π k γ ) n X z ∈ Z n k Z [ − π k γ ,π k γ ] n g K 0 ( x ) e − iz · x/k γ dx = 1 (2 π k γ ) n Z [ − π k γ ,π k γ ] n g K 0 ( x ) n Y l =1 D k ( − x l /k γ ) dx = 1 (2 π ) n Z [ − 1 , 1] n g K 0 ( y k γ ) n Y l =1 D k ( y l ) dy . (88) Since R π − π D k ( t ) dt = 2 π , we hav e (89) g K 0 (0) = 1 (2 π ) n Z [ − π ,π ] n g K 0 (0) n Y l =1 D k ( y l ) dy . 36 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN Th us, b y (87), (88), and (89), d k ≤      1 (2 π ) n Z [ − 1 , 1] n ( g K 0 (0) − g K 0 ( y k γ )) n Y l =1 D k ( y l ) dy      + + g K 0 (0)      1 (2 π ) n Z [ − π ,π ] n \ [ − 1 , 1] n n Y l =1 D k ( y l ) dy      . (90) By Prop osition 3.2, g K 0 is Lipschitz and hence the Lipsc hitz norm of g K 0 ( y k γ ) is O ( k γ ). No w [35, Theorem 1] implies that (91)      1 (2 π ) n Z [ − 1 , 1] n ( g K 0 (0) − g K 0 ( y k γ )) n Y l =1 D k ( y l ) dy      ≤ c 15 k γ − 1 n − 1 X l =0 (log k ) n − l , for some constan t c 15 indep enden t of k . (In the statemen t of [35, Theorem 1], D j ( Y ) should b e D J ( Y ). In that theorem w e are taking α = 1 and J = ( k , k , . . . , k ) ∈ Z n .) In view of (90) and (91), the pro of will b e complete if we show that (92) Z [ − π ,π ] n \ [ − 1 , 1] n n Y l =1 D k ( x l ) dx = O (1 /k ) , as k → ∞ . T o this end, observ e that, by trigonometric addition formulas and integration by parts, Z − 1 − π D k ( t ) dt = Z π 1 D k ( t ) dt = Z π 1 sin( k t ) cos( t/ 2) sin( t/ 2) dt + Z π 1 cos( k t ) dt = cos k cot(1 / 2) k + Z π 1 cos( k t ) k d dt (cot( t/ 2)) dt − sin k k = O (1 /k ) . (93) No w [ − π , π ] n \ [ − 1 , 1] n = ∪ n i =1 ( A i ∪ B i ) , where A i = { ( x 1 , . . . , x n ) : − 1 ≤ x j ≤ 1 for j < i , 1 ≤ x i ≤ π , − π ≤ x j ≤ π for j > i } and B i = − A i . By (93), we ha v e, for eac h i , Z A i n Y l =1 D k ( x l ) dx =  Z 1 − 1 D k ( t ) dt  i − 1 Z π 1 D k ( t ) dt  Z π − π D k ( t ) dt  n − i = (2 π − O (1 /k )) i − 1 O (1 /k ) (2 π ) n − i . Since int ( A i ) ∩ int ( A j ) = ∅ , for eac h i, j with i 6 = j , in t ( A i ) ∩ int ( B j ) = ∅ , for eac h i, j , and Q n l =1 D k ( x l ) is ev en, the previous estimate prov es (92).  PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 37 It is p ossible that the previous lemma could also b e obtained via some estimates pro v ed in [12] for the rate of decay of R S n − 1 | d 1 K 0 ( r u ) | 2 du as r → ∞ . The next tw o lemmas will allow us to circumv ent Prop osition 4.7, the version of the Strong La w of Large Numbers used earlier. Lemma 7.2. L et Y j k , j = 1 , . . . , m k , k ∈ N b e a triangular arr ay of r ow-wise indep endent zer o me an r andom variables with uniformly b ounde d fourth moments, wher e m k ∼ k n as k → ∞ . L et ν and a pq k , p, q = 1 , . . . , m k b e c onstants such that | a pq k | = O ( k ν ) as k → ∞ uniformly in p and q , wher e 2 n − 4 nγ + 2 ν < − 1 . Then, almost sur ely, Z k = 1 (2 π k γ ) 2 n m k X p,q =1 a pq k Y pk Y q k → 0 , as k → ∞ . Pr o of. Note that E ( Y pk Y q k ) = E ( Y pk ) E ( Y q k ) = 0 unless p = q . Therefore E ( Z k ) = 1 (2 π k γ ) 2 n m k X p,q =1 a pq k E ( Y pk Y q k ) = 1 (2 π k γ ) 2 n m k X p =1 a ppk E ( Y 2 pk ) . Since the Y pk ’s ha ve uniformly bounded second moments, | E ( Z k ) | = O ( k n − 2 nγ + ν ) and hence E ( Z k ) con verges to zero as k → ∞ . Let v ( k ) pq rs = co v ( Y pk Y q k , Y rk Y sk ) = E ( Y pk Y q k Y rk Y sk ) − E ( Y pk Y q k ) E ( Y rk Y sk ) . If the cardinalit y of the set { p, q , r, s } is 3 or 4, then at least one of the indices, sa y p , is diﬀeren t from all the others and v ( k ) pq rs = E ( Y pk ) E ( Y q k Y rk Y sk ) − E ( Y pk ) E ( Y q k ) E ( Y rk Y sk ) = 0 − 0 = 0 . If the cardinalit y of the set { p, q , r , s } is 1, then v ( k ) pq rs = v ( k ) pppp = E ( Y 4 pk ) − E ( Y 2 pk ) 2 . If the cardinalit y of the set { p, q , r , s } is 2, then either p = q , r = s and p 6 = r , and v ( k ) pq rs = v ( k ) pprr = E ( Y 2 pk Y 2 rk ) − E ( Y 2 pk ) E ( Y 2 rk ) = 0 , or p = r , q = s and p 6 = q , and v ( k ) pq rs = v ( k ) pq pq = E ( Y 2 pk Y 2 q k ) − E ( Y pk Y q k ) 2 = E ( Y 2 pk ) E ( Y 2 q k ) − E ( Y pk ) 2 E ( Y q k ) 2 , or p = s , q = r and p 6 = q , and v ( k ) pq rs = v ( k ) pq q p = E ( Y 2 pk Y 2 q k ) − E ( Y pk Y q k ) 2 = E ( Y 2 pk ) E ( Y 2 q k ) − E ( Y pk ) 2 E ( Y q k ) 2 . 38 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN In view of the fact that the Y j k ’s hav e uniformly b ounded fourth momen ts, the co v ariances v ( k ) pq rs are also uniformly b ounded, and hence v ar ( Z k ) = 1 (2 π k γ ) 4 n m k X p,q ,r ,s =1 a pq k a rsk v ( k ) pq rs = 1 (2 π k γ ) 4 n m k X p =1 a 2 ppk v ( k ) pppp + 1 (2 π k γ ) 4 n m k X p 6 = q =1 a 2 pq k v ( k ) pq pq + m k X p 6 = q =1 a pq k a q pk v ( k ) pq q p ! = O  k 2 n − 4 nγ +2 ν  . Let ε > 0. F or suﬃciently large k , we ha ve ε − E ( Z k ) > 0, and for suc h k , by Cheb yshev’s inequalit y , Pr( Z k > ε ) = Pr  Z k − E ( Z k ) > ε − E ( Z k )  ≤ v ar ( Z k ) ( ε − E ( Z k )) 2 = O  k 2 n − 4 nγ +2 ν  . Our h yp othesis and the Borel-Can telli Lemma imply that, almost surely , Z k con v erges to zero, as k → ∞ .  Lemma 7.3. L et Y ( r ) j k , j = 1 , . . . , m k , r = 1 , 2 , k ∈ N , b e a triangular arr ay of r ow-wise indep endent (i.e., indep endent for ﬁxe d k ) zer o me an r andom variables with uniformly b ounde d fourth moments, wher e m k ∼ k n as k → ∞ . L et ν and a pq k , p, q = 1 , . . . , m k b e c onstants such that | a pq k | = O ( k ν ) as k → ∞ uniformly in p and q , wher e 2 n − 4 nγ + 2 ν < − 1 . Then, almost sur ely, Z k = 1 (2 π k γ ) 2 n m k X p,q =1 a pq k Y (1) pk Y (2) pk Y (1) q k Y (2) q k → 0 , as k → ∞ . Pr o of. As in the pro of of Lemma 7.2, w e ha v e E ( Z k ) = 1 (2 π k γ ) 2 n m k X p =1 a ppk E   Y (1) pk  2  E   Y (2) pk  2  , so | E ( Z k ) | = O ( k n − 2 nγ + ν ) and hence E ( Z k ) con verges to zero as k → ∞ . Let w ( k ) pq rs = co v  Y (1) pk Y (2) pk Y (1) q k Y (2) q k , Y (1) rk Y (2) rk Y (1) sk Y (2) sk  . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 39 Straigh tforw ard mo diﬁcations to the pro of of Lemma 7.2 and the assumption of uniformly b ounded fourth momen ts yield v ar ( Z k ) = 1 (2 π k γ ) 4 n m k X p,q ,r ,s =1 a pq k a rsk w ( k ) pq rs = 1 (2 π k γ ) 4 n m k X p =1 a 2 ppk w ( k ) pppp + 1 (2 π k γ ) 4 n m k X p 6 = q =1 a 2 pq k w ( k ) pq pq + m k X p 6 = q =1 a pq k a q pk w ( k ) pq q p ! = O  k 2 n − 4 nγ +2 ν  . The pro of is concluded as in Lemma 7.2.  8. Phase retriev al from the squared modulus This section addresses Problem 2 in the introduction. Algorithm NoisyMo d 2 LSQ Input: Natural num b ers n ≥ 2 and k ; a real n umber γ such that 0 < γ < 1; noisy measuremen ts (94) e g ik = | d 1 K 0 ( z ik ) | 2 + X ik , of the squared mo dulus of the F ourier transform of the c haracteristic function of an unkno wn con v ex b o dy K 0 ⊂ C n 0 whose cen troid is at the origin, at the p oints in { z ik : i = 0 , 1 , . . . , I 0 k } = { o } ∪ (1 /k γ ) Z n k (+) , where Z n k (+) satisﬁes (80) and where the X ik ’s are ro w-wise indep enden t zero mean random v ariables with uniformly bounded fourth momen ts; an o -symmetric con vex polytop e Q k in R n , sto c hastically indep endent of the measuremen ts e g ik , that appro ximates either ∇ K 0 or D K , in the sense that, almost surely , lim k →∞ δ ( Q k , ∇ K 0 ) = 0 , or lim k →∞ δ ( Q k , D K 0 ) = 0 . T ask: Construct a con vex p olytop e P k that approximates K 0 , up to reﬂection in the origin. A ction: 1. Let e g ik = e g ( − i ) k , for i = − I 0 k , . . . , − 1, let x ik = k γ − 1 z ik , i = − I 0 k , . . . , I 0 k b e the p oints in the cubic arra y 2 C n 0 ∩ (1 /k ) Z n , and let (95) M k ( x ik ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · x ik ) e g j k , for i = − I 0 k , . . . , I 0 k . 2. Run Algorithm NoisyCovLSQ with inputs n , k , Q k , and with M ik replaced b y M k ( x ik ), for i = − I 0 k , . . . , I 0 k and with the ob vious re-indexing in i . The resulting output P k of that algorithm is also the output of the present one. 40 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN The main result in this section corresponds to Theorem 4.10 abov e. W e ﬁrst state it, and then show that it can b e pro v ed by suitable mo diﬁcations to the pro of of Theorem 4.10 if in addition γ > 1 / 2 + 1 / (4 n ). Theorem 8.1. Supp ose that K 0 ⊂ C n 0 is a c onvex b o dy with its c entr oid at the origin. Supp ose also that K 0 is determine d, up to tr anslation and r eﬂe ction in the origin, among al l c onvex b o dies in R n , by its c ovario gr am. L et (96) 1 / 2 + 1 / (4 n ) < γ < 1 . If P k , k ∈ N , is an output fr om Algorithm NoisyMo d 2 LSQ as state d ab ove, then, almost sur ely, min { δ ( K 0 , P k ) , δ ( − K 0 , P k ) } → 0 as k → ∞ . As w e shall now show, the pro of of this theorem basically follows the analysis giv en in Sec- tion 4. Of course, alterations must b e made, since the measurements M ik in Algorithm Noisy- Co vLSQ ha ve been replaced b y the new measuremen ts M k ( x ik ) deﬁned b y (95) or equiv alently b y (84) with x = x ik . In view of (83), w e ha v e M k ( x ik ) = g K 0 ( x ik ) + N k ( x ik ) − d k ( x ik ) , i = − I 0 k , . . . , I 0 k , where N k ( x ik ) and d k ( x ik ) are giv en by (85) and (86), resp ectively , with x = x ik . W e b egin with a lemma. Note that I k = 2 I 0 k + 1, so the expression in the lemma is the sample mean. Also, recall that b y their deﬁnition, the random v ariables X ik ha v e uniformly b ounded fourth moments, and X pk and X q k are indep enden t unless p = ± q , in which case they are equal. Lemma 8.2. L et N k ( x ik ) + = max { N k ( x ik ) , 0 } for al l i and k . If (96) holds, then, almost sur ely, 1 I k I 0 k X i = − I 0 k N k ( x ik ) + → 0 , as k → ∞ . Pr o of. Note ﬁrstly that 1 I k I 0 k X i = − I 0 k N k ( x ik ) + ≤ 1 I k I 0 k X i = − I 0 k | N k ( x ik ) | ≤   1 I k I 0 k X i = − I 0 k N k ( x ik ) 2   1 / 2 . Th us it suﬃces to prov e that, almost surely , S k = 1 I k I 0 k X i = − I 0 k N k ( x ik ) 2 → 0 , as k → ∞ . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 41 W e hav e S k = 1 I k I 0 k X i = − I 0 k   1 (2 π k γ ) n I 0 k X p = − I 0 k cos( z pk · x ik ) X pk   2 = 1 (2 π k γ ) 2 n I 0 k X p,q = − I 0 k   1 I k I 0 k X i = − I 0 k cos( z pk · x ik ) cos( z q k · x ik )   X pk X q k = 1 (2 π k γ ) 2 n I 0 k X p,q = − I 0 k c pq k X pk X q k , sa y . Since c ( − p ) q k = c p ( − q ) k = c pq k , it is clearly enough to show that, almost surely , 1 (2 π k γ ) 2 n I 0 k X p,q =1 c pq k X pk X q k → 0 , as k → ∞ . In view of (96) and the fact that | c pq k | = O (1), this follo ws from Lemma 7.2 with Y j k = X j k , m k = I 0 k , a pq k = c pq k for all p , q , and k , and ν = 0.  Pr o of of The or em 8.1 . W e shall indicate the modiﬁcations needed in Section 4. No c hanges are required in the lemmas b efore Lemma 4.4. F or the latter, we shall use the same notation as b efore, with the understanding that the indexing has changed and the new random v ariables N k ( x ik ) replace the random v ariables N ik of Section 4. Th us w e write | f | I k =   1 I k I 0 k X i = − I 0 k f ( z i ) 2   1 / 2 , with corresp onding changes in indexing in the deﬁnitions of x I k , N I k , and Ψ. With the same pro of as Lemma 4.4, we now hav e the inequalit y | g K 0 − g P k | 2 I k ≤ 2Ψ( P k , x I k , N I k ) − 2Ψ( P ( a ) ∩ C n 0 , x I k , N I k ) +   g K 0 − g P ( a ) ∩ C n 0   2 I k + + 2 I k I 0 k X i = − I 0 k  g P ( a ) ∩ C n 0 ( x ik ) − g P k ( x ik )  d k ( x ik ) , (97) instead of (34). Prop osition 4.5 and Lemma 4.6 are unchanged. W e do not require Prop osition 4.7 in order to conclude as in Lemma 4.8 that, almost surely , (98) sup K ∈K n ( C n 0 ) Ψ( K, x I k , N I k ) → 0 , 42 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN as k → ∞ . Indeed, it is enough to show that, almost surely , the new expression corresp onding to (37), namely , W k ( ε ) = max j =1 ,...,m    1 I k I 0 k X i = − I 0 k g U j ( x ik ) N k ( x ik ) + − 1 I k I 0 k X i = − I 0 k g L j ( x ik ) N k ( x ik ) −    , con v erges to zero, as k → ∞ . This follo ws from Lemma 8.2, b ecause the co eﬃcients g U j ( x ik ) and g U j ( x ik ) are uniformly b ounded by 1 and Lemma 8.2 holds b oth when such coeﬃcients are inserted and when N k ( x ik ) + is replaced by N k ( x ik ) − = N k ( x ik ) − N k ( x ik ) + = max {− N k ( x ik ) , 0 } . All this is enough to ensure that Lemma 4.9 still holds. Indeed, since a translate of P k is con tained in C n 0 , and Ψ( P k , x I k , N I k ) is unc hanged b y such a translation, w e kno w from (98) that, almost surely , the ﬁrst and second terms on the righ t-hand side of (97) con v erge to zero, as k → ∞ . W e hav e g P ( a ) ∩ C n 0 ( x ik ) ≤ 1 and g P k ( x ik ) ≤ V (2 C n 0 ), since P k ⊂ 2 C n 0 , and then Lemma 7.1 implies that the new fourth term on the right-hand side of (97) conv erges to zero as k → ∞ . The rest of the pro of of Lemma 4.9 pro ceeds as b efore. The pro of of the main Theorem 4.10 now applies without change.  The user of Algorithm NoisyMo d 2 LSQ must supply as input an o -symmetric con vex p oly- top e Q k in R n that approximates either ∇ K 0 or D K . F or this purpose w e provide t w o algo- rithms that do the work of Algorithm NoisyCovBlasc hke and Algorithm NoisyCo vDiﬀ( ϕ ). Algorithm NoisyMo d 2 Blasc hk e Input: Natural n um b ers n ≥ 2 and k ; a p ositiv e real num b er h k ; mutually nonparallel v ectors u i ∈ S n − 1 , i = 1 , . . . , k that span R n ; noisy measuremen ts (99) e g ik = | d 1 K 0 ( z ik ) | 2 + X ik , of the squared mo dulus of the F ourier transform of the c haracteristic function of an unkno wn con v ex b o dy K 0 ⊂ C n 0 whose cen troid is at the origin, at the p oints in { z ik : i = 0 , 1 , . . . , I 0 k } = { o } ∪ (1 /k γ ) Z n k (+) , where Z n k (+) satisﬁes (80) and where the X ik ’s are ro w-wise indep enden t zero mean random v ariables with uniformly b ounded fourth moments. T ask: Construct an o -symmetric con vex p olytop e Q k that approximates the Blaschk e b o dy ∇ K 0 . A ction: 1. Let e g ik = e g ( − i ) k , for i = − I 0 k , . . . , − 1, and let M k ( o ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k e g j k and M k ( h k u i ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · h k u i ) e g j k , PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 43 for i = 1 , . . . , k . Then for i = 1 , . . . , k , let (100) y ik = M k ( o ) − M k ( h k u i ) h k . 2. With the natural n um b ers n ≥ 2 and k , and v ectors u i ∈ S n − 1 , i = 1 , . . . , k use the quan tities y ik instead of noisy measurements of the brigh tness function b K ( u i ) as input to Algorithm NoisyBrigh tLSQ (see [24, p. 1352]). The output of the latter algorithm is Q k . W e shall show that the argumen t of Section 5 can b e mo diﬁed to yield a conv ergence result corresponding to Theorem 5.4. It is clear that any such result must require the input h k to satisfy h k → 0 as k → ∞ , but w e need a stronger condition phrased in terms of parameters ε and γ that satisfy (101). Since the second inequalit y in (101) is equiv alen t to γ > (2 n + 5 − 4 ε ) / (4 n + 4), which decreases as n increases and equals (9 − 4 ε ) / 12 when n = 2, it is p ossible to choose γ and ε so that (101) is satisﬁed. Sp eciﬁcally , one can choose 3 / 4 ≤ γ < 1 and 0 < ε < 1 − γ . Note also that (101) implies (96). There is considerable ﬂexibility in the choice of the parameter h k , and it would b e p ossible to introduce a further parameter q k b y working with input v ectors u i ∈ S n − 1 , i = 1 , . . . , q k , where q k → ∞ as k → ∞ . T o av oid o v ercomplicating the exp osition, how ev er, we shall not discuss this an y further. Theorem 8.3. L et K 0 ⊂ C n 0 b e a c onvex b o dy with its c entr oid at the origin. L et ( u i ) b e a se quenc e in S n − 1 such that ( u ∗ i ) is evenly spr e ad. Supp ose that h k ∼ k γ − 1+ ε , k ∈ N , wher e ε and γ satisfy (101) 0 < ε < 1 − γ and 2 n − 4 nγ + 4(1 − γ − ε ) < − 1 . If Q k is an output fr om A lgorithm NoisyMo d 2 Blaschke as state d ab ove, then, almost sur ely, lim k →∞ δ ( ∇ K 0 , Q k ) = 0 . Pr o of. W e shall indicate the c hanges needed in Section 5. Note that by (100), and (83) with x = o and x = h k u i , w e hav e y ik = M k ( o ) − M k ( h k u i ) h k = g K 0 ( o ) − g K 0 ( h k u i ) h k + N k ( o ) − N k ( h k u i ) h k − d k ( o ) − d k ( h k u i ) h k , for i = 1 , . . . , k , where N k ( o ), d k ( o ), N k ( h k u i ), and d k ( h k u i ) are given by (85) and (86) with x = o or x = h k u i , as appropriate. Lemma 3.1 is unchanged. T urning to the pro of of Lemma 5.1, we no w ha v e y ik = ζ ik + T ik , where (102) ζ ik = g K 0 ( o ) − g K 0 ( h k u i ) h k − d k ( o ) − d k ( h k u i ) h k and T ik = N k ( o ) − N k ( h k u i ) h k , 44 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN for i = 1 , . . . , k . Since h k ∼ k γ − 1+ ε for 0 < ε < 1 − γ , the second term in the previous expression for ζ ik con v erges to zero as k → ∞ , by Lemma 7.1, and hence ζ ik → b K 0 ( u i ) as k → ∞ , as b efore, for i = 1 , . . . , k . Moreo v er, b K 0 ( u i ) − ζ ik =  b K 0 ( u i ) − g K 0 ( o ) − g K 0 ( h k u i ) h k  + d k ( o ) − d k ( h k u i ) h k , so arguing as in the pro of of Lemma 5.1, w e use Lemma 3.1 with t = h k to obtain (46) with t = h k , that is, 0 ≤ b K 0 ( u i ) − g K 0 ( o ) − g K 0 ( h k u i ) h k ≤ ( n − 1) h k 2 r b K 0 ( u i ) , if h k ≤ 2 r . W e also hav e d k ( o ) − d k ( h k u i ) h k = O ( k − ε ) , b y Lemma 7.1, so there is a constant c 16 = c 16 ( n, r ) suc h that | b K 0 ( u i ) − ζ ik | ≤ c 16 k − β , for β = min { ε, 1 − γ + ε } , and all k ∈ N and i = 1 , . . . , k . The rest of the pro of of Lemma 5.1 can b e follo w ed, yielding that, almost surely , there is a constan t c 17 = c 17 ( n, r ) suc h that (103) | b K 0 − b Q k | 2 k ≤ 2Ψ( Q k , ( u i ) , T k ) − 2Ψ( K 0 , ( u i ) , T k ) + c 17 k β | b K 0 − b Q k | k , for all k ∈ N . (Again, we assume that the ob vious c hanges are made in the notation.) The next task is to c heck that Lemma 5.2 still holds. With (103) in hand, this rests on pro ving that, almost surely , V k = 1 k k X i =1 T 2 ik is b ounded. In fact we claim that, almost surely , V k → 0 as k → ∞ . T o see this, note that V k = 1 k k X i =1  N k ( o ) − N k ( h k u i ) h k  2 = 1 k k X i =1   1 (2 π k γ ) n I 0 k X j = − I 0 k  1 − cos( z j k · h k u i ) h k  X j k   2 = 1 (2 π k γ ) 2 n I 0 k X p,q = − I 0 k a pq k X pk X q k , where (104) a pq k = 1 k h 2 k k X i =1  1 − cos( z pk · h k u i )  1 − cos( z q k · h k u i )  PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 45 and hence | a pq k | ≤ 4 /h 2 k . As in the pro of of Lemma 8.2, w e may tak e the indices p, q from 1 to I 0 k , and then, by (101), the claim follo ws from Lemma 7.2 with m k = I 0 k and ν = 2(1 − γ − ε ). A t this stage the w ork for Lemma 5.3 is already done. Indeed, by the Cauc hy-Sc hw arz inequalit y , Ψ( Q k , ( u i ) , T k ) − Ψ( K 0 , ( u i ) , T k ) ≤ | b K 0 − b Q k | k 1 k k X i =1 T 2 ik ! 1 / 2 = | b K 0 − b Q k | k V 1 / 2 k . Using this and (103) we see that, almost surely , | b K 0 − b Q k | k ≤ 2 V 1 / 2 k + c 17 k β → 0 , as k → ∞ . Finally , the pro of of Theorem 5.4 can b e applied without change.  The next algorithm corresp onds to Algorithm NoisyCovDiﬀ( ϕ ). As for that algorithm, ϕ is a nonnegativ e b ounded measurable function on R n with supp ort in C n 0 , such that R R n ϕ ( x ) dx = 1. Algorithm NoisyMo d 2 Diﬀ( ϕ ) Input: Natural num b ers n ≥ 2 and k ; p ositive reals δ k and ε k ; a real num b er γ satisfying 0 < γ < 1; noisy measurements (105) e g ik = | d 1 K 0 ( z ik ) | 2 + X ik , of the squared mo dulus of the F ourier transform of the c haracteristic function of an unkno wn con v ex b o dy K 0 ⊂ C n 0 whose cen troid is at the origin, at the p oints in { z ik : i = 0 , 1 , . . . , I 0 k } = { o } ∪ (1 /k γ ) Z n k (+) , where Z n k (+) satisﬁes (80) and where the X ik ’s are ro w-wise indep enden t zero mean random v ariables with uniformly b ounded fourth moments. T ask: Construct an o -symmetric conv ex p olytop e Q k in R n that appro ximates the diﬀerence b o dy D K 0 . A ction: 1. Let e g ik = e g ( − i ) k , for i = − I 0 k , . . . , − 1, let x ik = k γ − 1 z ik , i = − I 0 k , . . . , I 0 k b e the p oints in the cubic arra y 2 C n 0 ∩ (1 /k ) Z n , and let (106) M k ( x ik ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · x ik ) e g j k , for i = − I 0 k , . . . , I 0 k . 2. Run Algorithm NoisyCovDiﬀ( ϕ ) with inputs n , k , δ k , ε k , and M ik replaced b y M k ( x ik ), for i = − I 0 k , . . . , I 0 k and with the obvious re-indexing in i . The output Q k of that algorithm is also the output of the present one. 46 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN W e shall sho w that the argument in Section 6 used to prov e Theorem 6.4 can be modiﬁed to yield the following conv ergence result. Theorem 8.4. Supp ose that K 0 , δ k , ε k , and g k ar e as in Algorithm NoisyMo d 2 Diﬀ( ϕ ). As- sume that lim k →∞ ε k = lim k →∞ δ k = 0 and that (107) lim inf k →∞ δ 4 k k 4 γ n − 3 n − 3 / 2 > 0 , wher e γ > 3(1 + 1 / (2 n )) / 4 . If Q k is an output fr om A lgorithm NoisyMo d 2 Diﬀ( ϕ ) as state d ab ove, then, almost sur ely, δ ( D K 0 , Q k ) ≤ c 13 δ 1 /n k , for suﬃciently lar ge k . In p articular, almost sur ely, Q k c onver ges to DK 0 as k → ∞ . Pr o of. Algorithm NoisyMo d 2 Diﬀ( ϕ ) can b e regarded formally as Algorithm NoisyCo vDiﬀ( ϕ ) with M ik and N ik replaced by M k ( x ik ) deﬁned by (106) and N k ( x ik ) − d k ( x ik ) deﬁned by (85) and (86) with x = x ik , resp ectiv ely . W e follow the arguments of Section 6 with this substitution in mind. F or Lemma 6.1, w e note ﬁrst that b y (85), E ( N k ( x ik )) = 0 for all i and k . The same calculations as in the pro of of Lemma 6.1 lead to | E ( g k ( x )) − g K 0 ( x ) | ≤ n ( ε k + 1 /k ) + d k , where d k is as in Lemma 7.1. By that lemma, d k → 0 as k → ∞ and hence the second statemen t in Lemma 6.1 still holds. Next, for Lemma 6.2, recall the deﬁnition (66) of β ik ( x ). Then we ha v e, b y (85), g k ( x ) − E ( g k ( x )) = I 0 k X i = − I 0 k β ik ( x ) N k ( x ik ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k   I 0 k X i = − I 0 k β ik ( x ) cos( z j k · x ik )   X j k = 1 (2 π k γ ) n I 0 k X j = − I 0 k ξ j k ( x ) X j k , sa y . This is a w eighted sum of indep enden t random v ariables, so w e can apply Khinc hine’s inequalit y (see, for example, [29, (4.32.1), p. 307] with α = 4) to obtain E         I 0 k X i = − I 0 k β ik ( x ) N k ( x ik )       4   ≤ c (2 k + 1) n (2 π k γ ) 4 n I 0 k X j = − I 0 k E | ξ j k ( x ) X j k | 4 . PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 47 for some constan t c > 0. Also, | ξ j k ( x ) | 4 ≤   I 0 k X i = − I 0 k β ik ( x )   4 ≤ 1 , b y (68). The same argumen t as in the pro of of Lemma 6.2 no w leads to the conclusion that there are constan ts c 18 = c 18 ( ϕ ) and N 5 = N 5 (( ε k ) , n ) ∈ N suc h that if δ > 0, then (108) Pr( | g k ( x ) − g K 0 ( x ) | > δ ) ≤ c 18 (2 k + 1) 2 n k − 4 γ n δ − 4 , for all k ≥ N 5 and all x ∈ R n . (Compare (65).) Lemma 6.3 is unc hanged. With (107) instead of the hypothesis (70) of Theorem 6.4, and the new estimate (108), we arrive in the pro of of Theorem 6.4 at the estimate Pr( a k ≥ δ k ) ≤ c 18 (2 k + 1) 3 n k − 4 γ n δ − 4 k = O ( k − 3 / 2 ) , so the Borel-Can telli lemma can b e used as b efore. This is all that is required to allo w the pro of of Theorem 6.4 to go through un til near the end, when w e use the fact that k δ 1 /n k → ∞ as k → ∞ . By (107) and the fact that γ < 1, this still holds. Then the conclusion is the same, namely that, almost surely , δ ( D K 0 , Q k ) ≤ c 13 δ 1 /n k , for suﬃcien tly large k .  Concerning Corollary 6.5, by using γ > 3(1 + 1 / (2 n )) / 4 and (107) instead of (70), we can achiev e a con vergence rate arbitrarily close to k − 1 / 4+3 / (8 n ) , the same as b efore. If we assume instead that the random v ariables X ik in Algorithm NoisyMo d 2 Diﬀ( ϕ ) are row-wise indep enden t, zero mean, and satisfy (75) and (76), that γ > 1 / 2, and that (109) lim inf k →∞ δ 2 k k n (2 γ − 1) log k > c 19 ( n + 2) , where c 19 = c 19 ( n, σ ) = (3 n +2 σ 2 ) / ((2 π ) 2 n ), then a rate arbitrarily close to k − 1 / 2 can be ob- tained b y the metho ds outlined in Remark 6.6. 9. Phase retriev al from the modulus This section addresses Problem 3 in the in tro duction. A simple trick con verts Problem 3 in to one very closely related to Problem 2, considered in the previous section. Supp ose, more generally , that noisy measurements are tak en of p b g , where g is an ev en con tin uous real-v alued function on R n with supp ort in [ − 1 , 1] n . The just-mentioned tric k is to tak e tw o indep enden t measuremen ts at each p oin t, m ultiply the t w o, and use the resulting quan tities in place of the measuremen ts of b g considered earlier. Th us instead of (81) ab ov e w e ha ve, for r = 1 , 2, measurements g ( r ) z ,k = p b g ( z /k γ ) + X ( r ) z ,k , 48 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN of p b g , for z ∈ { o } ∪ Z n k (+), where Z n k (+) satisﬁes (80) and where the X ( r ) z ,k ’s are row-wise in- dep enden t (i.e., indep endent for ﬁxed k ) zero mean random v ariables with uniformly b ounded fourth momen ts. Then w e replace ˜ g z ,k in (81) b y (110) g z ,k = g (1) z ,k g (2) z ,k = b g ( z /k γ ) + p b g ( z /k γ )  X (1) z ,k + X (2) z ,k  + X (1) z ,k X (2) z ,k . Setting g j k = g K 0 z j k ,k and X j k = X z j k ,k , the same notation and analysis that gav e (83), but no w using (82) and (110), leads instead to M k ( x ) = g K 0 ( x ) + N k ( x ) − d k ( x ) , where (111) M k ( x ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · x ) g j k is an estimate of g K 0 ( x ), (112) N k ( x ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k q d g K 0 ( z j k /k γ ) cos( z j k · x )  X (1) j k + X (2) j k  + 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · x ) X (1) j k X (2) j k is a random v ariable, and the deterministic error d k ( x ) is giv en as b efore by (86). F or our analysis it will b e conv enien t to let (113) N k 1 ( x ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k q d g K 0 ( z j k /k γ ) cos( z j k · x )  X (1) j k + X (2) j k  and (114) N k 2 ( x ) = 1 (2 π k γ ) n I 0 k X j = − I 0 k cos( z j k · x ) X (1) j k X (2) j k , so that N k ( x ) = N k 1 ( x ) + N k 2 ( x ). T o k eep the exp osition brief, we shall not give a formal presentation of our algorithms, called Algorithm NoisyMo dLSQ , Algorithm NoisyMo dBlasc hke , and Algorithm NoisyMo dDiﬀ( ϕ ) , since they are very similar to Algorithm NoisyMo d 2 LSQ, Algorithm NoisyMo d 2 Blasc hk e, and Algorithm NoisyMo d 2 Diﬀ( ϕ ), resp ectiv ely . In eac h case the input is as b efore, except that instead of (94), (99), and (105), we now ha v e measuremen ts g ( r ) ik = | d 1 K 0 ( z ik ) | + X ( r ) ik , for r = 1 , 2, of the mo dulus of the F ourier transform of the c haracteristic function of K 0 , where the X ( r ) ik ’s are ro w-wise indep enden t zero mean random v ariables with uniformly b ounded fourth moments. The task is the same in each case. F or the actions, we ﬁrst let g ik = g (1) ik g (2) ik and then follo w the actions of the appropriate algorithms in the previous section, replacing e g PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 49 b y g . Th us in the action of each algorithm, we replace M k ( x ) by M k ( x ) deﬁned by (111), for the appropriate x . Theorem 9.1. The or em 8.1 holds when A lgorithm NoisyMo d 2 LSQ is r eplac e d by Algorithm NoisyMo dLSQ. Pr o of. In the action of Algorithm NoisyModLSQ, the measurements used in Algorithm Noisy- Co vLSQ are now M k ( x ik ), i = − I 0 k , . . . , I 0 k , where M k ( x ik ) is given by (111) with x = x ik . Th us w e hav e M k ( x ik ) = g K 0 ( x ik ) + N k ( x ik ) − d k ( x ik ) , i = − I 0 k , . . . , I 0 k , where N k ( x ik ) and d k ( x ik ) are giv en by (112) and (86), resp ectively , with x = x ik . W e claim that Lemma 8.2 holds when N k ( x ik ) is replaced by N k ( x ik ). T o see this, use the triangle inequalit y to obtain 1 I k I 0 k X i = − I 0 k N k ( x ik ) + ≤   1 I k I 0 k X i = − I 0 k N k ( x ik ) 2   1 / 2 ≤   1 I k I 0 k X i = − I 0 k N k 1 ( x ik ) 2   1 / 2 +   1 I k I 0 k X i = − I 0 k N k 2 ( x ik ) 2   1 / 2 , where N k 1 ( x ik ) and N k 2 ( x ik ) are giv en b y (113) and (114), respectively , with x = x ik . Since d g K 0 is b ounded, the same analysis as in the pro of of Lemma 8.2, up to a constant, applies to the ﬁrst of the tw o sums in the previous expression. So it suﬃces to prov e that, almost surely , S k = 1 I k I 0 k X i = − I 0 k N k 2 ( x ik ) 2 → 0 , as k → ∞ . As in the pro of of Lemma 8.2, it is enough to show that, almost surely , 1 (2 π k γ ) 2 n I 0 k X p,q =1 c pq k X (1) pk X (2) pk X (1) q k X (2) q k → 0 , as k → ∞ . This follows from Lemma 7.3 and pro v es the claim. With this in hand, we can conclude exactly as in the pro of of Theorem 8.1 that Algorithm NoisyCo vLSQ w orks with the new measurements under the same hypotheses.  W e remark that the computation of E ( Z k ) in Lemma 7.3 sho ws wh y we tak e t wo inde- p enden t measuremen ts of p d g K 0 and m ultiply , rather than taking a single measuremen t and squaring it. In the latter case w e w ould b e led to E ( Z k ) = 1 (2 π k γ ) 2 n m k X p,q =1 a pq k E ( Y 2 pk ) E ( Y 2 q k ) = O ( k 2 n − 2 nγ + ν ) , 50 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN whic h ma y b e unbounded as k → ∞ . Theorem 9.2. The or em 8.3 holds when Algorithm NoisyMo d 2 Blaschke is r eplac e d by Algo- rithm NoisyMo dBlaschke. Pr o of. W e now hav e y ik = ζ ik + T ik , where ζ ik is as in (102) and (115) T ik = N k ( o ) − N k ( h k u i ) h k = N k 1 ( o ) − N k 1 ( h k u i ) h k + N k 2 ( o ) − N k 2 ( h k u i ) h k , for i = 1 , . . . , k , where N k 1 and N k 2 are giv en b y (113) and (114). The pro of of Theorem 8.3 can b e follo w ed, except that for Lemma 5.2, one now shows that, almost surely , V k = 1 k k X i =1 T 2 ik → 0 as k → ∞ . Using the fact that the earlier analysis applies to N k 1 , and using also the triangle inequalit y , as we did in the proof of Theorem 9.1, with (115), we see that it suﬃces to examine 1 (2 π k γ ) 2 n I 0 k X p,q =1 a pq k X (1) pk X (2) pk X (1) q k X (2) q k , where a pq k is giv en by (104). Then Lemma 7.3 shows that it is p ossible to c ho ose γ and ε exactly as in Theorem 8.3 to ensure that Lemma 5.2 holds. No further changes are required, so Algorithm NoisyCo vBlaschk e w orks with the new measurements under the same h yp otheses as in Theorem 8.3.  Theorem 9.3. The or em 8.4 holds when Algorithm NoisyMo d 2 Diﬀ( ϕ ) is r eplac e d by A lgorithm NoisyMo dDiﬀ( ϕ ). Pr o of. Note that by (112), w e ha v e E ( N k ( x ik )) = 0 for all i and k . Therefore the same calculations as in the pro of of Theorem 8.4 sho w that the second statemen t in Lemma 6.1 still holds. In Lemma 6.2, it is enough in view of the pro of of Theorem 8.4 to consider the con tribution to g k ( x ) − E ( g k ( x )) from N k 2 ( x ik ), namely , 1 (2 π k γ ) n I 0 k X j = − I 0 k I 0 k X i = − I 0 k β ik ( x ) cos( z j k · x ik ) X (1) j k X (2) j k . This allo ws the same estimate as b efore, up to a constant. No further c hanges are required, so Algorithm NoisyCovDiﬀ( ϕ ) works with the new measuremen ts under the same h yp otheses as in Theorem 8.4.  PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 51 The previous result provides a conv ergence rate for Algorithm NoisyMo dDiﬀ( ϕ ) arbitrarily close to k − 1 / 4+3 / (8 n ) , as was noted for Algorithm NoisyMod 2 Diﬀ( ϕ ) after Theorem 8.4. If we assume instead that the random v ariables X ik in Algorithm NoisyMo dDiﬀ( ϕ ) are row-wise indep enden t, zero mean, and satisfy (75) and (76), that γ > 1 / 2, and that (109) holds, then a rate arbitrarily close to k − 1 / 2 can b e obtained b y the metho ds outlined in Remark 6.6. 10. Appendix 10.1. Con vergence rates. Rates of con v ergence for Algorithm NoisyCo vDiﬀ( ϕ ), and hence for the tw o related algorithms for phase retriev al, are pro vided in Corollary 6.5 and Re- mark 6.6. F or the other algorithms, ho wev er, rates of conv ergence are more diﬃcult to obtain. T o explain why , it will b e necessary to describ e some results from [24], where conv ergence rates w ere obtained for algorithms for reconstructing conv ex b o dies from ﬁnitely many noisy measuremen ts of either their supp ort functions or their brightness functions. The algorithms are called Algorithm NoisySupp ortLSQ and Algorithm NoisyBrightnessLSQ, resp ectively . In [24], an unkno wn con vex b o dy K is assumed to b e con tained in a kno wn ball RB n , R > 0, in R n . An inﬁnite sequence ( u i ) in S n − 1 is selected, and one of the algorithms is run with noisy measurements from the ﬁrst k directions in the sequence as input. The noise is mo deled b y Gaussian N (0 , σ 2 ) random v ariables. With an assumption on ( u i ) slightly stronger than the condition that it is evenly spread (but still mild and satisﬁed by man y natural sequences), and another unimp ortant assumption on the relation b etw een R and σ , it is pro ved in [24, Theorem 6.2] that if P k is the corresp onding output from Algorithm NoisySupp ortLSQ, then, almost surely , there are constants C = C ( n, ( u i )) and N = N ( σ, n, R, ( u i )) suc h that (116) δ 2 ( K, P k ) ≤ C σ 4 / ( n +3) R ( n − 1) / ( n +3) k − 2 / ( n +3) , for k ≥ N , provided that the dimension n ≤ 4. Here δ 2 is the L 2 metric, so that δ 2 ( K, P k ) = k h K − h P k k 2 , where k · k 2 denotes the L 2 norm on S n − 1 . Con v ergence rates for the Hausdorﬀ metric are then obtained by using the known relations b etw een the t w o metrics. It is an artifact of the metho d that while conv ergence rates can also b e obtained for n ≥ 5, neither these nor those for the Hausdorﬀ metric are expected to be optimal. In con trast, it has recen tly b een pro ved b y Gun tub o yina [28] that the rate giv en in (116) for n ≤ 4 is the b est p ossible in the minimax sense. With the additional assumption that K is o -symmetric, corre- sp onding rates for Algorithm NoisyBrigh tLSQ are obtained in [24, Theorem 7.6] from those for Algorithm NoisySupportLSQ b y exploiting (4) and the Bourgain-Campi-Lindenstrauss stabilit y theorem for pro jection b o dies. There are tw o principal ingredients in the pro of of (116). The ﬁrst is [24, Corollary 4.2], a corollary of a deep result of v an de Geer [48, Theorem 9.1]. This corollary provides con v ergence rates for least squares estimators of an unknown function in a class G , based on ﬁnitely man y noisy measuremen ts of its v alues, where the noise is uniformly sub-Gaussian. The result and the rates dep end on having a suitable estimate for the size of G in terms of its ε -entrop y with resp ect to a suitable pseudo-metric. The second ingredient is a kno wn estimate (see [24, 52 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN Prop osition 5.4]) of the ε -en trop y of the class of supp ort functions of compact con vex sets con tained in B n , with resp ect to the L ∞ metric. It should b e p ossible to apply this metho d to obtain con vergence rates for Algorithm Noisy- Co vBlasc hke and the t wo related algorithms for phase retriev al. With Gaussian noise, or more generally uniformly sub-Gaussian noise, this requires a modiﬁcation to [48, Theorem 9.1] that, in our situation, allo ws (53) to b e used instead of the same inequality without the term c 11 /k . (Compare [48, (9.1), p. 148].) This would yield the same conv ergence rates given in [24, The- orem 7.6] for Algorithm NoisyBrightLSQ. T o cov er the case of Poisson noise, how ev er, one can mak e the general assumption that the random v ariables are ro w-wise indep endent, zero mean, and satisfy (75) and (76), as in Remark 6.6. This creates considerable further technical diﬃculties. It may w ell b e p ossible to o v ercome these, using the mac hinery b ehind another result of v an de Geer [48, Theorem 9.2]. But, as v an de Geer p oints out in [48, p. 134], there is a price to pay: One no w requires a uniform bound on the class G of functions, as well as estimates of ε -entrop y “with brac keting.” The former condition migh t b e dealt with b y (49), whic h implies that the sets Π Q k are uniformly b ounded for an y ﬁxed realization. It should also b e p ossible to obtain the latter, by combining suitable modiﬁcations of the brack eting argumen t of Lemma 4.6 and of the pro of in [24, Theorem 7.3] of the ε -en tropy estimate for the class of zonoids contained in B n . But w e hav e not carried out a complete in vestigation into con v ergence rates for Algo- rithm NoisyCo vBlaschk e and the related algorithms for phase retriev al, despite ha ving a strat- egy for doing so, describ ed in the previous paragraph. The main reason is that there are more serious tec hnical obstacles in ac hieving conv ergence rates for Algorithm NoisyCo vLSQ, ev en for the case of Gaussian noise. In principal, the metho d outlined ab o ve could b e applied b y taking G to b e the class of cov ariograms of compact con vex subsets of the unit ball in R n . Ho w ev er, an estimate would b e required of the ε -entrop y of this class with resp ect to the L ∞ metric or some other suitable pseudo-metric. Even if this were a v ailable, an application of the theory of empirical pro cesses as describ ed ab o v e would yield con vergence rates not for δ 2 ( K, P k ) but rather for k g K − g P k k 2 . T o obtain rates for δ 2 ( K, P k ), one would then also need suitable stabilit y versions of the uniqueness results for the Co v ariogram Problem describ ed in the In tro duction. In view of the diﬃculty of these uniqueness results, pro ving such stability v ersions will presumably b e very challenging. In summary , a full study of con vergence rates for the other algorithms prop osed here must remain a pro ject for future study . 10.2. Implemen tation issues. The study undertaken in this paper is a theoretical one. Although we prop ose algorithms in enough detail to allow implemen tation, the lab orious task of writing all the necessary programs, carrying out numerical exp eriments, and comparing with other algorithms, largely lies ahead. A t the present time we only ha ve a rudimentary implemen tation of Algorithms Noisy- Co vBlasc hke and NoisyCovLSQ. The programs w ere written, mainly in Matlab, b y Mic hael Sterling-Go ens while he w as an undergraduate studen t at W estern W ashington Univ ersity , and are conﬁned to the planar case. Algorithm NoisyCovBlasc hke seems to be v ery fast; this is PHASE RETRIEV AL FOR CHARA CTERISTIC FUNCTIONS OF CONVEX BODIES 53 to b e exp ected, since it is based on Algorithm NoisyBrightnessLSQ, which is also fast even in three dimensions. Behind b oth of these latter t wo algorithms is a linear least squares problem (cf. [25, (18) and (19)]). In contrast, the least squares problem (18) in Algorithm NoisyCovLSQ is nonlinear. Preliminary exp erimen ts indicate that reasonably go o d reconstructions, such as those depicted in Figures 1 – 4 (based on Gaussian N (0 , σ 2 ) noise, k = 60 equally spaced direc- tions in Algorithm NoisyCo vBlaschk e and k = 8 in Algorithms NoisyCovLSQ), can usually b e obtained in a reasonable time in the planar case. Occasionally , ho wev er, reconstructions can b e considerably w orse, particularly for regular m -gons for very small m . Better and faster reconstructions, also in higher dimensions, will probably require bringing to bear the usual arra y of techniques for nonlinear optimization, such as simulated annealing. Figure 1. Pen tagon, no noise Figure 2. Pen tagon, σ = 0 . 01 Figure 3. Ellipse, no noise Figure 4. Ellipse, σ = 0 . 01 Since the least squares problem (18) is nonlinear, it is imp ortan t to control the n um b er of v ariables, that is, the num b er of facets of the appro ximation Q k to the Blaschk e b o dy ∇ K 0 of K 0 . T o a large exten t, Algorithm NoisyCo vBlaschk e already do es this; the p otential O ( k n − 1 ) v ariables that w ould otherwise b e required (see [24, p. 1335]) is, as experiments sho w, 54 GABRIELE BIANCHI, RICHARD J. GARDNER, AND MARKUS KIDERLEN considerably reduced. In fact, if there is little or no noise, a linear programming version of the brigh tness function reconstruction program due to Kiderlen (see [25, p. 289], where it is stated for measuremen ts without noise) is not only eve n faster, but also pro duces appro ximations Q k to ∇ K 0 with at most 2 k facets. Beyond this, there is the p ossibility of using the pruning tec hniques discussed in [41, Section 3.3]. There is also the p ossibility of c hanging the v ariables in the least squares problem (18). A con v ex p olytop e P whose facet outer unit normals are a subset of a prescrib ed set {± u j : j = 1 , . . . , s } of directions can b e sp eciﬁed by the vector h = ( h + 1 , h − 1 , . . . , h + s , h − s ) suc h that P = P ( h ) = { x ∈ R n : − h − j ≤ x · u j ≤ h + j , j = 1 , . . . , s } . The p ossible adv antage in using these v ariables arises from the fact that, b y the Brunn- Mink o wski inequality (cf. [21, Section 11]), the cov ariogram g P ( h ) ( x ) turns out to b e (1 /n )- conca v e (i.e., g P ( h ) ( x ) 1 /n is concav e) on its supp ort in the combined v ariable ( h, x ). One may therefore try solving the problem (117) min I k X i =1  M ik − g P ( h ) ( x ik )  2 o v er the v ariables h + 1 , h − 1 , . . . , h + s , h − s . By expanding the square in (117), appro ximating the sums b y integrals, and using the Pr´ ekopa-Leindler inequality [21, Section 7], the ob jective function can b e seen as an approximation to the diﬀerence of tw o log-conca v e functions. These admittedly weak concavit y prop erties may help. Regularization is often used to improv e F ourier inv ersion in the presence of noise. W e exp ect this to b e of b eneﬁt in implemen ting the phase retriev al algorithms, where preliminary in v estigations indicate that regularization will allow the restriction on the parameter γ to b e considerably relaxed. Corresp onding to the t w o basic approac hes to reconstruction—one via the Blasc hk e b o dy and one via the diﬀerence b o dy—there are t wo diﬀerent sampling designs. F or the former, measuremen ts are made ﬁrst at the origin and at p oints in a small sphere centered at the origin, and then again at p oints in a cubic arra y . 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Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

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