The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes
TThe modulus of the Fourier transform on asphere determines -dimensional convexpolytopes Konrad Engel ∗ and Bastian Laasch † University of Rostock, Institute for Mathematics, 18057 Rostock,GermanySeptember 23, 2020
Abstract
Let P and P (cid:48) be -dimensional convex polytopes in R and S ⊆ R bea non-empty intersection of an open set with a sphere. As a consequence ofa somewhat more general result it is proved that P and P (cid:48) coincide up totranslation and/or reflection in a point if | (cid:82) P e − i s · x dx | = | (cid:82) P (cid:48) e − i s · x dx | for all s ∈ S . This can be applied to the field of crystallography regardingthe question whether a nanoparticle modelled as a convex polytope isuniquely determined by the intensities of its X-ray diffraction pattern onthe Ewald sphere. Keywords
Fourier transform; convex polytope; modulus; covari-ogram; Ewald sphere; rationally parameterisable hypersurface
In this paper, we prove the statement of the title: The modulus of theFourier transform on a sphere determines -dimensional convex polytopes.In small-angle X-ray scattering and partly also in wide-angle X-ray scatter-ing of nanoparticles the modulus of the Fourier transform of the reflectedbeam wave vectors can be “approximately” measured on the Ewald sphere ,see e.g. [21], [2], [23]. Hence, uniqueness questions are of special interestand by our result, at least theoretically, the polytopes can be reconstructedfrom the measurements.The statement can be formulated in a more general form and thereforewe need several definitions and notations first. ∗ [email protected] † [email protected] a r X i v : . [ m a t h . M G ] S e p he Fourier transform F f of an L -integrable function f : R n → R isdefined by F f ( s ) = (cid:90) R n f ( x ) e − i s · x dx , s ∈ R n , where the product · is the standard scalar product, see e.g. [19]. If f = χ P is the characteristic function of an n -dimensional compact set P ⊆ R n ,then F f ( s ) = (cid:90) P e − i s · x dx is the Fourier transform of P . We denote it briefly by F P ( s ) .An important question is the following: Assume that P and P (cid:48) are n -dimensional compact sets in R n and that |F P ( s ) | = |F P (cid:48) ( s ) | for all s belonging to a sparse subset of R n . Is it true that P and P (cid:48) have the same“structure”?We say that P and P (cid:48) are strongly congruent , denoted by P ∼ = s P (cid:48) , ifthere is some vector v and an (cid:15) ∈ {− , } such that P (cid:48) = (cid:15) P + v , (1)i.e., P and P (cid:48) coincide up to translation and/or reflection in a point . Oneeasily obtains from (1) that for all s ∈ R n F P (cid:48) ( s ) = (cid:40) e i s · v F P ( s ) if (cid:15) = 1 ,e i s · v F P ( s ) if (cid:15) = − , which implies |F P (cid:48) ( s ) | = |F P ( s ) | . Hence, two strongly congruent sets cannot be distinguished by themodulus of their Fourier transform and the above question has to beformulated more accurately, e.g. in form of a problem as follows:
Problem 1.1
Determine sufficient conditions for P , P (cid:48) , S ⊆ R n such that ∀ s ∈ S |F P ( s ) | = |F P (cid:48) ( s ) | = ⇒ P ∼ = s P (cid:48) . (2)We will see that in the case S = R n this problem is equivalent to theso-called covariogram problem , introduced by Matheron in [17].The covariogram of an n -dimensional compact set P ⊆ R n is a function g P : R n → R defined by g P ( x ) = λ n ( P ∩ ( P + x )) , x ∈ R n , where λ n denotes the n -dimensional Lebesgue measure, see [16]. Thus g P associates with each x the volume of the intersection of P with the Minkowski sum P + x . It is again easy to see that P ∼ = s P (cid:48) implies g P ( x ) = g P (cid:48) ( x ) for all x ∈ R n , i.e., the covariogram is invariant withrespect to translation and reflection in a point. roblem 1.2 (covariogram problem) Determine sufficient conditionsfor P , P (cid:48) ⊆ R n such that ∀ x ∈ R n g P ( x ) = g P (cid:48) ( x ) = ⇒ P ∼ = s P (cid:48) . (3)The equivalence of both problems (see e.g. [5]) follows from the fol-lowing theorem which we reprove in Section 2 in order to make the paperself-contained. Theorem 1.1
We have ∀ s ∈ R n |F P ( s ) | = |F P (cid:48) ( s ) | ⇐⇒ ∀ x ∈ R n g P ( x ) = g P (cid:48) ( x ) . (4)Fortunately, there exist deep results in the literature concerning Prob-lem 1.2. The following conditions are sufficient for (3) (see also Theorem 1of the survey paper [12]): • P , P (cid:48) ⊆ R are -dimensional convex polygons [18, Theorem 3.1]. • P , P (cid:48) ⊆ R are -dimensional convex and compact sets [1, Theorem1.1]. • P , P (cid:48) ⊆ R are -dimensional convex polytopes [5, Theorem 1.1]. • P , P (cid:48) ⊆ R n are n -dimensional ( n ≥ ) convex simplicial polytopes(i.e., each facet is a simplex and each polytope is in general relativeposition to its reflection, see [10, Corollary to Theorem 2] and [24]).Note that, for example, in the -dimensional case this is fulfilled fora regular tetrahedron but not for a regular icosahedron.In other words, these bodies are determined by their covariogram,respectively by the modulus of their Fourier transform, over R n up totranslation and reflection in a point. Note that for arbitrary convex setsin R the question is still open and for dimension n ≥ counterexamplescan be constructed, e.g. for special convex polytopes (see [4, Theorem 1.2]and [5, Remark 9.3]).The more general problem of the reconstruction of an arbitrary function f from the modulus of its Fourier transform over R n is called phase retrievalproblem and was intensively studied in the past (see e.g. [13, 14, 22] andthe references given in [6]). Usually, phase retrieval is under-determinedwithout any additional constraints. Hence, a priori assumptions that thefunction has a particular form, e.g. f is the characteristic function of an n -dimensional polytope, are needed.The problem will become relevant for physical applications if we assumethe identity |F P ( s ) | = |F P (cid:48) ( s ) | not over R n , like in (4), but only over asubset S ⊆ R n , like in (2). We can use the above listed results to solveProblem 1.1 if we have a solution of the following one: Problem 1.3
Determine sufficient conditions for P , P (cid:48) , S ⊆ R n such that ∀ s ∈ S |F P ( s ) | = |F P (cid:48) ( s ) | = ⇒ ∀ s ∈ R n |F P ( s ) | = |F P (cid:48) ( s ) | . (5)Section 3 is devoted to this problem. In Section 4 we discuss thephysical application for X-ray scattering and, in particular, derive thefollowing corollary as an answer to the title of this paper: orollary 1.1 If P and P (cid:48) are 3-dimensional convex polytopes in R and S ⊆ R is a non-empty intersection of an open set with a sphere, then ∀ s ∈ S |F P ( s ) | = |F P (cid:48) ( s ) | = ⇒ P ∼ = s P (cid:48) . Finally, Section 5 contains a generalization to a further class of n -dimensionalconvex polytopes in addition to the already mentioned simplicial polytopes. Recall that the convolution of two functions f, g : R n → R with regard tothe measure λ n is defined by ( f ∗ g )( x ) = (cid:90) R n f ( y ) g ( x − y ) d λ n ( y ) , x ∈ R n . Hence, we have ( χ P ∗ χ −P )( x ) = (cid:90) R n χ P ( y ) χ −P ( x − y ) d λ n ( y )= (cid:90) R n χ P ( y ) χ P ( y − x ) d λ n ( y )= (cid:90) R n χ P ( y ) χ P + x ( y ) d λ n ( y )= (cid:90) P∩ ( P + x ) d λ n ( y ) = λ n ( P ∩ ( P + x )) , and consequently g P ( x ) = ( χ P ∗ χ −P )( x ) ∀ x ∈ R n . (6)Now well-known properties of the Fourier transform imply that thefollowing statements are equivalent: g P ( x ) = g P (cid:48) ( x ) ∀ x ∈ R n ⇐⇒ F g P ( s ) = F g P(cid:48) ( s ) ∀ s ∈ R n ⇐⇒ F P ( s ) F −P ( s ) = F P (cid:48) ( s ) F −P (cid:48) ( s ) ∀ s ∈ R n ⇐⇒ F P ( s ) F P ( s ) = F P (cid:48) ( s ) F P (cid:48) ( s ) ∀ s ∈ R n ⇐⇒ |F P ( s ) | = |F P (cid:48) ( s ) | ∀ s ∈ R n . The first equivalence holds because of the injectivity of the Fourier trans-form (see [19]). The second one follows from (6) and the identity betweenthe Fourier transform of two convoluted functions with the product oftheir Fourier transforms. For the last equivalence we used | z | = zz for z ∈ C . (cid:3) In this paper, we not only allow the restriction of the Fourier transform toa sphere, but more generally to rationally parameterisable hypersurfaces atisfying two general conditions. Therefore, we need some definitionsfrom [9].A rationally parameterisable hypersurface (briefly rp-hypersurface ) is aset S of points in R n of the form S = { σ ( t ) : t ∈ D } , (7)where σ ( t ) = σ ( t , . . . , t n − ) ... σ n ( t , . . . , t n − ) , the functions σ j are rational functions, j = 1 , . . . , n , and D ⊆ R n − is thedomain of S .Using spherical coordinates and the standard substitution t = tan( α ) ,which implies cos( α ) = − t t and sin( α ) = t t , one obtains that the -dimensional unit sphere (cid:26)(cid:18) t t − t t , t t t t , − t t (cid:19) : t ∈ R + , t ∈ R (cid:27) with the missing segment ( −√ − λ , , λ ) , λ ∈ [ − , , is an rp-hypersurface.Since affine transformations do not violate the rationality this is also truefor any sphere in R .With the function σ : R n − → R n we associate its normalized function ,i.e., the function ˆ σ : R n − → R n − defined by ˆ σ ( t ) = σ ( t ,...,t n − ) σ ( t ,...,t n − ) ... σ n ( t ,...,t n − ) σ ( t ,...,t n − ) . Note that ˆ σ ( t ) is only defined if t ∈ D \ σ − (0) , where σ − (0) is the zeroset of σ . For a set O ⊆ D \ σ − (0) let ˆ σ ( O ) = { ˆ σ ( t ) : t ∈ O } . In the following we need that, for an rp-hypersurface S given by (7),an open subset O of D \ σ − (0) in R n − satisfies two conditions: Hyperplane condition : σ ( O ) is not contained in a hyperplane. Inner point condition : There is a t ∈ O such that ˆ σ ( t ) is an innerpoint of ˆ σ ( O ) in R n − .The proof of the following Theorem 3.2 is based on a result on E-functions of [9, Theorem 2.3 together with Theorem 2.2. and Lemma 2.2].A function F : R n → C is called an E-function of degree d if it has theform F ( s ) = m (cid:88) k =1 P k ( s ) e − i v k · s with distinct points v k , k = 1 , . . . , m , and homogeneous rational functions P k ( s ) of degree d . heorem 3.1 (see [9]) Let S = { σ ( t ) : t ∈ D } be an rp-hypersurfaceand let O ⊆ D \ σ − (0) be an open subset of R n − that satisfies thehyperplane and the inner point condition. Let F ( s ) be an E-function ofdegree d . If F ( s ) = 0 ∀ s ∈ σ ( O ) , then all coefficients of the exponential functions and hence also F are thezero function (up to the cases where some denominator equals 0). Now we are able to present an answer to Problem 1.3.
Theorem 3.2
Let S = { σ ( t ) : t ∈ D } be an rp-hypersurface in R n andlet O ⊆ D \ σ − (0) be an open subset of R n − that satisfies the hyperplaneand the inner point condition. If P and P (cid:48) are n -dimensional convexpolytopes in R n and S = σ ( O ) , then (5) is true. Proof
For an n -dimensional convex polytope P let V P ⊆ R n be itsvertex set and let E P = { v − v : v , v ∈ V P } . For the Fourier transformof a convex polytope P it is known (see [3, 7, 8, 15, 20]) that F P ( s ) = (cid:88) v ∈ V P Q P , v ( s ) e − i v · s ∀ s ∈ R n \ Z P , (8)where each coefficient is a rational function of the form Q P , v ( s ) = (cid:88) I ∈I P λ P , v ,I (cid:81) e ∈ I e · s , which is not the zero function, see [11, Lemma 2]. The numerators λ P , v ,I are real numbers and I P is a family of n -element linearly independentsubsets of E P . The set Z P contains only vectors s for which the scalarproduct e · s vanishes. Therefore, (8) is an E-function of degree − n andsince | z | = zz for z ∈ C we have for the squared modulus of the Fouriertransform |F P ( s ) | = (cid:88) v ∈ V P Q P , v ( s ) e − i · s + (cid:88) v i , v j ∈ V P v i (cid:54) = v j Q P , v i ( s ) Q P , v j ( s ) e − i ( v i − v j ) · s . (9)It is possible that the exponents of the exponential functions in the secondsum of (9) are not distinct. In such cases we can merge the correspondingterms and see that (9) is an E-function of degree − n . Note that a linearcombination of E-functions of degree d is again an E-function of degree d .Therefore, |F P ( s ) | − |F P (cid:48) ( s ) | is also an E-function of degree − n .The assumption in (5), respectively (2), implies |F P ( s ) | − |F P (cid:48) ( s ) | = 0 ∀ s ∈ σ ( O ) (10)and by Theorem 3.1 and the continuity of the Fourier transform in s |F P ( s ) | − |F P (cid:48) ( s ) | = 0 ∀ s ∈ R n . (cid:3) Proof of Corollary 1.1 and a physicalapplication
Proof
In [9] the following theorem is proved for quadratic hypersurfaces,i.e., zero sets of an equation of the form s T A s + b T s + c = 0 , where A is a symmetric matrix different from the zero matrix. Theorem 4.1 If S is a quadratic hypersurface that does not contain aline but at least two points, then, up to an exceptional set of hypersurfacemeasure zero, it is an rp-hypersurface with some parameter domain D andevery open subset O of D \ σ − (0) in R n − satisfies the hyperplane andthe inner point condition. Clearly, this theorem can be applied to a sphere since a sphere is aquadratic hypersurface that does not contain a line but at least two points.We may assume that the sphere is the unit sphere and that the non-empty intersection S of the open set with the sphere does not containpoints from the missing segment ( −√ − λ , , λ ) , λ ∈ [ − , , in theparametrization of the unit sphere and no points with vanishing firstcoordinate. Obviously, the set O = (cid:26) ( t , t ) : (cid:18) t t − t t , t t t t , − t t (cid:19) ∈ S (cid:27) has the properties required in Theorem 4.1 and thus Corollary 1.1 followsfrom Theorem 3.2, Theorem 1.1 and the fact that 3-dimensional convexpolytopes are positive examples for Problem 1.2. (cid:3) Now we apply our results to the field of crystallography. An interest-ing questions is whether the -dimensional structure of nanoparticles isdetermined by their single-shot X-ray diffraction pattern (see [21], [2] and,for a current list of references, [23]).To this end we consider the following -dimensional setup (see Figure1) – the -dimensional case follows analogously. The incoming normalizedX-ray beam wave vector k in = (0 , T with wavelength illuminates ananoparticle P modelled as a convex polygon. The occurring diffractionevent causes scattered wave vectors k out in different directions. The set ofall such vectors constitutes the so-called diffraction pattern. We assumethat during the diffraction process there is no energy gained or lost. Hence, (cid:107) k out (cid:107) = (cid:107) k in (cid:107) = 1 , i.e., the wavelengths of the diffracted beams are also . Therefore, in the -dimensional case the vectors k out form a unit circle, respectively in the -dimensional case a unit sphere, the Ewald sphere. Note that we assumethat the diffracted wave vectors k out are originated in the center of theEwald sphere since the particle P is infinitesimally small. The intensity I of a scattered wave is proportional to (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) P e − i ( k out − k in ) · x dx (cid:12)(cid:12)(cid:12)(cid:12) , (11) x − − • •• Ewald “sphere” P k in = (cid:18) (cid:19) k out ϕ • I ( ϕ ) Figure 1: The -dimensional single-shot scattering model for a convex polygon,respectively nanoparticle, P with normalized wavelengths for the incoming andreflected X-ray beam wave vectors k in and k out . The solid arc marks the areaof the ( -dimensional) Ewald “sphere” where the scattering intensities I can bemeasured during experiments. 8 he difference k out − k in of the reflected and the incoming wave vector iscalled scattering vector, i.e, the intensity is proportional to the modulusof the Fourier transform of the scattering vector.In experiments we can measure these intensities on a semicircle of theEwald “sphere” with exception of the points (1 , T , (0 , T and ( − , T (see the solid arc in Figure 1). Representing k out in polar coordinatesyields k out = (cid:18) cos( ϕ )sin( ϕ ) (cid:19) , ϕ ∈ (0 , π ) \ (cid:110) π (cid:111) , and together with (11) we get I ( ϕ ) ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) P e − i (cos( ϕ ) x +(sin( ϕ ) − x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) F P (cid:18)(cid:18) cos( ϕ )sin( ϕ ) − (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (12)Hence, the set S mentioned in the assumptions of Problem 1.1 and Problem1.3 is given by S = (cid:26)(cid:18) cos( ϕ )sin( ϕ ) − (cid:19) : ϕ ∈ (0 , π ) \ (cid:110) π (cid:111)(cid:27) , i.e., a semicircle shifted in the second coordinate.Analogously to (12) we can compute the intensities for the -dimensionalcase on a set S corresponding to a (translated) hemisphere of the Ewaldsphere with some excluded points in spherical coordinates I ( ϕ, θ ) ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) P e − i (sin( θ ) cos( ϕ ) x +sin( θ ) sin( ϕ ) x +(cos( θ ) − x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F P sin( θ ) cos( ϕ )sin( θ ) sin( ϕ )cos( θ ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By Corollary 1.1 we know that two -dimensional convex polytopes withthe same diffraction pattern on the hemisphere are strongly congruent. Sothe measured intensities on the Ewald sphere of scattered X-ray beamsdetermine the underlying object uniquely up to translation and reflectionin a point. Note that the experimental setup delivers “only” finitelymany approximated intensities and we assumed infinitely many exactmeasurements. n -dimensional convex polytopes If P and P (cid:48) are convex polytopes of arbitrary dimension with vertex sets V P and V P (cid:48) , then clearly V P ∼ = s V P (cid:48) = ⇒ P ∼ = s P (cid:48) . (13) hus, we are lead to a discrete variant.The points of a finite set V in R n are in general position if each linethrough the origin contains, in addition to the origin, at most one pair ± w of the multiset V − V . The following theorem is proved in [25, Theorem1]: Theorem 5.1
Let V and V (cid:48) be finite subsets of R n , each with points ingeneral position. Then V − V = V (cid:48) − V (cid:48) = ⇒ V ∼ = s V (cid:48) . This theorem enables us to present another answer to Problem 1.1 for the n -dimensional case. Theorem 5.2
Let S = { σ ( t ) : t ∈ D } be an rp-hypersurface in R n andlet O ⊆ D \ σ − (0) be an open subset of R n − that satisfies the hyperplaneand the inner point condition. Further let P and P (cid:48) be n -dimensionalconvex polytopes in R n such that the vertices of P as well of P (cid:48) are ingeneral position. Finally let S = σ ( O ) . Then ∀ s ∈ S |F P ( s ) | = |F P (cid:48) ( s ) | = ⇒ P ∼ = s P (cid:48) . Proof
We use again (9) and (10). Since the coefficients of the exponentialfunctions are not the zero function we obtain that V P − V P = V P (cid:48) − V P (cid:48) .Now the assertion follows immediately from Theorem 5.1 and (13). (cid:3) Note that this theorem can be applied in particular to simplices becausetheir vertex sets are in general position.
In this paper we used a series of results of discrete geometry, namelysolutions for the covariogram problem, to answer the question whether theintensities of an X-ray diffraction pattern on the Ewald sphere determinethe underlying object uniquely. For physical application the -dimensionalcase is of interest and if the illuminated nanoparticle is modelled as a convexpolytope the reconstruction is unique up to translation and reflection ina point. However, our theoretically result assumes infinitely many exactmeasurements what is not the case in experimental setups. Therefore, thequestion remains open whether the result is true under the condition offinitely many given intensities. Acknowledgement
We would like to thank Thomas Fennel and Stefan Scheel for presentingthe motivation of this study.This work was partly supported by the European Social Fund (ESF) andthe Ministry of Education, Science and Culture of Mecklenburg-WesternPomerania (Germany) within the project NEISS – Neural Extraction ofInformation, Structure and Symmetry in Images under grant no ESF/14-BM-A55-0006/19. eferences [1] Gennadiy Averkov and Gabriele Bianchi. Confirmation of Matheron’sconjecture on the covariogram of a planar convex body. Journal ofthe European Mathematical Society , 11(6):1187–1202, 2009.[2] Ingo Barke, Hannes Hartmann, Daniela Rupp, Leonie Flückiger,Mario Sauppe, Marcus Adolph, Sebastian Schorb, Christoph Bostedt,Rolf Treusch, Christian Peltz, Stephan Bartling, Thomas Fennel,Karl-Heinz Meiwes-Broer, and Thomas Möller. The 3D-architectureof individual free silver nanoparticles captured by X-ray scattering.
Nature communications , 6(1):1–7, 2015.[3] Aleksandr I Barvinok. Computation of exponential integrals.
Journalof Mathematical Sciences , 70(4):1934–1943, 1994.[4] Gabriele Bianchi. Matheron’s conjecture for the covariogram problem.
Journal of the London Mathematical Society , 71(1):203–220, 2005.[5] Gabriele Bianchi. The covariogram determines three-dimensionalconvex polytopes.
Advances in Mathematics , 220(6):1771–1808, 2009.[6] Gabriele Bianchi, Fausto Segala, Aljoša Volčič, et al. The solutionof the covariogram problem for plane C convex bodies. Journal ofDifferential Geometry , 60(2):177–198, 2002.[7] Michel Brion. Points entiers dans les polyedres convexes. In
Annalesscientifiques de l’Ecole Normale Superieure , volume 21, pages 653–663,1988.[8] Philip J Davis. Triangle formulas in the complex plane.
Mathematicsof Computation , 18(88):569–577, 1964.[9] Konrad Engel. An identity theorem for the Fourier transform of poly-topes on rationally parameterisable hypersurfaces. arXiv:2008.00935,2020.[10] Paul Goodey, Rolf Schneider, and Wolfgang Weil. On the determina-tion of convex bodies by projection functions.
Bulletin of the LondonMathematical Society , 29(1):82–88, 1997.[11] Nick Gravin, Jean Lasserre, Dmitrii V. Pasechnik, and Sinai Robins.The inverse moment problem for convex polytopes.
Discrete andComputational Geometry , 48(3):596–621, 2012.[12] Ákos G Horváth. On convex bodies that are characterizable by volumefunction.
Arnold Mathematical Journal , 6:1–20, 2020.[13] Norman E Hurt.
Phase Retrieval and Zero Crossings: MathematicalMethods in Image Reconstruction , volume 52. Springer Science &Business Media, 1989.[14] Michael V Klibanov, Paul E Sacks, and Alexander V Tikhonravov.The phase retrieval problem.
Inverse problems , 11(1):1, 1995.[15] Jim Lawrence. Polytope volume computation.
Mathematics of Com-putation , 57(195):259–271, 1991.[16] Georges Matheron.
Random Sets and Integral Geometry . Wiley Seriesin Probability and Mathematical Statistics. Wiley, [1974, C1975],1974.
17] Georges Matheron. Le covariogramme géometrique des compactsconvexes de R Technical ReportN-2/86/G, Centre de Géostatistique,Ecole Nationale Supérieure des Mines de Paris , 1986.[18] Werner Nagel. Orientation-dependent chord length distributionscharacterize convex polygons.
Journal of applied probability , 30(3):730–736, 1993.[19] Gerlind Plonka, Daniel Potts, Gabriele Steidl, and Manfred Tasche.
Numerical Fourier Analysis . Springer, 2018.[20] Aleksandr Valentinovich Pukhlikov and Askold Georgievich Khovan-skii. The Riemann-Roch theorem for integrals and sums of quasipoly-nomials on virtual polytopes.
Algebra i analiz , 4(4):188–216, 1992.[21] Kevin S Raines, Sara Salha, Richard L Sandberg, Huaidong Jiang,Jose A Rodríguez, Benjamin P Fahimian, Henry C Kapteyn, JinchengDu, and Jianwei Miao. Three-dimensional structure determinationfrom a single view.
Nature , 463(7278):214–217, 2010.[22] Joseph Rosenblatt. Phase retrieval.
Communications in mathematicalphysics , 95(3):317–343, 1984.[23] Jörg Rossbach, Jochen R. Schneider, and Wilfried Wurth. 10 years ofpioneering X-ray science at the free-electron laser FLASH at DESY.
Physics Reports , 808:1–74, 2019.[24] Rolf Schneider. On the determination of convex bodies by projectionand girth functions.
Results in Mathematics , 33(1-2):155–160, 1998.[25] Marjorie Senechal. A point set puzzle revisited.
European Journal ofCombinatorics , 29:1933–1944, 2008., 29:1933–1944, 2008.