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APPEARED IN BULLETIN OF THE

arXiv:math/9301215v1 [math.CA] 1 Jan 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 109-115SINGULARITIES OF THE RADON TRANSFORMA. G. Ramm and A. I. ZaslavskyAbstract.

Singularities of the Radon transform of a piecewise smooth functionf(x), x ∈Rn, n ≥2, are calculated.If the singularities of the Radon transformare known, then the equations of the surfaces of discontinuity of f(x) are calculatedby applying the Legendre transform to the functions, which appear in the equationsof the discontinuity surfaces of the Radon transform of f(x); examples are given.Numerical aspects of the problem of finding discontinuities of f(x), given the discon-tinuities of its Radon transform, are discussed.I. IntroductionLet f(x) be a compactly supported function, D be its support, and Γ = ∂D bea union of finitely many C∞hypersurfaces Γ1, .

. .

, Γs in general position, each ofwhich can be written in local coordinates asxn = g(x′),x′ = (x1, . .

. , xn−1), n ≥2,where g(x′) ∈C∞, f(x) ∈C∞(D), f(x)|Γ ≥c > 0.

The discontinuity surface off(x) is Γ, the boundary of D. We assume that the rank of the Hessian gij(x) :=∂2g/∂xi∂xj is constant on each of Γj, 1 ≤j ≤s.Define the Radon transform (RT) of f(x) by the usual formula [GGV] ˆf(p, α) =RRn f(x)δ(p−α·x) dx, where δ is the delta-function. It is well known that ˆf(λp, λα) =|λ|−1 ˆf(p, α), λ ∈R1, λ ̸= 0.

Consider the integral(1)R(p, α; f) :=Zlαpf(x)µ(dx),where lαp is the plane α·x−p = 0, α ∈Rn, p ∈R1, and µ(dx) is the Lebesgue mea-sure on lαp. One has R(p, α; f) = ˆf(p/|α|, α0), α0 := α|α|−1, so that R(p, α; f) =|α| ˆf(p, α), |α| = (α21 + · · · + α2n)1/2.The problems we are interested in are: (P1) Find the singularities of R(p, α; f);and (P2) Find the surface Γ of discontinuity of f(x) given the singularities ofR(p, α; f).No results concerning (P2) were known.

In [N] one can find an estimate of thenorm of ˆf(p, α) in Sobolev spaces. This result does not give information about (P1)and (P2).

In [P] there is a result given without proof, which has a relation to (P1).Our result is more general. In [Q] it is mentioned that the values (α: p), such that1991 Mathematics Subject Classification.

Primary 44A12.Received by the editors March 11, 1992c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2A. G. RAMM AND A. I. ZASLAVSKYlαp is tangent to Γ, play a special role.

This observation is made quantitative inour Theorem 1. Our results are useful for inversion of incomplete tomographic data[R2].The basic results are formulated in §II.

They give solutions of the problems(P1) and (P2).Actually, more general problems are solved; particularly, finitesmoothness of f(x) and Γ is allowed, the role of the intersections of Γj in the studyof the singularities of R(p, α; f) is clarified, etc. In §III proofs are sketched.

In §IVexamples are given. In §V numerical aspects of problem (P2) are discussed.We conclude this introduction by an outline of our ideas.

First, we describe thebehavior of R(p, α; f) in a neighborhood of the set Qf which is the set of singularitiesof R(p, α; f). Second, we prove that, in general, there is an equation of the set Qfwhich is of the form q = h(β), β ∈Rn−1, so that Qf is a hypersurface.

Third,we prove that the function g(x′) (in the equation of Γ) is the Legendre transformof the function h(β) (in the equation of Qf). Fourth, we describe some geometricproperties of Qf.Our results give a theoretical basis for the solution of the practically importantproblem in nondestructive evaluation and remote sensing, the problem of findingthe discontinuities of a function from the knowledge of its RT.II.

Formulation of the resultsThe RT, defined by formula (1), is a function on the projective space RPn, andwe take R(1, 0; f) := 0 for compactly supported f. Let Qf denote the set of thepoints (α: p) in this projective space, which correspond to the planes lαp tangentto Γ = ∂D. We say that lαp is tangent to Γ at a point x ∈Bm := Tmj=1 Γj, if lαp isnot transversal to Bm at the point x.1.

Our first result is the following theorem in which the description of the singu-larities of R(p, α; f) is given.Let lαp be tangent to Γ at the point x. We claim that if α is generic, then theset Qf is a smooth hypersurface in a neighborhood U of (α: p).

If A is a symmetricmatrix with real-valued entries, then its inertia index (inerdex) is defined to be thenumber of its negative eigenvalues. Consider first the case when Γ consists of onesurface.Theorem 1.

There exists an equation ζ(α: p) = 0, ∇ζ ̸= 0 in U, which definesQf in U, and two C∞functions r1 and r2 in U such that(2)R(p, α; f) =(ζ(n−1)/2+r1 + r2,ifIn is even,ζ(n−1)/2(ln |ζ|)r1 + r2,ifIn is odd.Here I is the inerdex of the matrix zkj, where zkj is the Hessian of the functionz = (α · x −p)/|α| on Γ at the point x and z+ = max(z, 0).If x ∈Bm and (α: p) is generic, then the following result holds.Theorem 1′. There exists ζ(α: p), ∇ζ ̸= 0 in U, such that the equation ζ(α: p) =0 is the equation of Qf in U, and two C∞functions r1 and r2 in U, such that(2′)R(p, α; f) =(ζ(n+m−2)/2+r1 + r2,if I(n + m −1) is even,ζ(n+m−2)/2(ln |ζ|)r1 + r2,if I(n + m −1) is odd.

SINGULARITIES OF THE RADON TRANSFORM3In [RZ1] the constant r1(α: p) is calculated. In [RZ2] this result is used fora derivation of the asymptotics of the Fourier transform of a piecewise smoothfunction.2.

Let us define the Legendre transform of a function g(y), y ∈Rn−1 in a neighbor-hood Uy of a point y at which the matrix gij(y) := ∂2g/∂yi∂yj is nondegenerate,i.e., det gij(y) ̸= 0 in Uy. Define Lg := h(β) := β ·y −g(y), where the dot stands forthe inner product and y = y(β) is the unique solution of the equation β = ∇g(y)in a neighborhood Uβ of the point β = ∇g(y).

One can prove that if g ∈Cl(Uy),l ≥2, and det gij(y) ̸= 0 in Uy, then h(β) ∈Cl(Uβ).It is known that under our assumptions Lh = g(y), i.e., the Legendre transform isinvolutive: g(y) = β·y−h(β), where β = β(y) is the unique solution to the equationy = ∇h(β), β ∈Uβ. One can prove that det hij(β) ̸= 0 in Uβ if det gij(y) ̸= 0 in Uy;moreover, the matrix hij(β) is inverse to gij(y), where β = β(y).

Recall that Γ isa union of hypersurfaces Γj, 1 ≤j ≤s, Γ1, . .

. , Γs are C∞and in general position.Denote bBm := Γ1,...,m the set of (α: p) ∈RPn such that lαp is tangent to Bm.

Theset bBm ⊂RPn may not be a hypersurface (see Theorem 3); however, as Theorem1′ claims, it is indeed a smooth hypersurface outside a set of (n −1)-dimensionalLebesgue’s measure zero.3. Our second result gives the relation between the discontinuity surfaces for R(p, α; f)and those for f(x); namely, the function g(x′) in the local equation of Γ, xn = g(x′),is the Legendre transform of the function h(β) which gives the equation of Qf,q = h(β).Assume that q = h(β), β ∈Uβ, where Uβ is a neighborhood of a point β,q = h(β), and det hij(β) ̸= 0 in Uβ, where hij := ∂2h/∂βi∂βj.

Let x′ = ∇h(β).Theorem 2. If h(β) ∈Cl(Uβ), l ≥2, then Lh = g(x′), and g(x′) ∈Cl(Ux′).This result allows one to recover the surfaces of discontinuity of f(x) given thesurfaces of discontinuity of R(p, α; f).4.

Examples show that the Legendre transform h(β) = Lg of a function g(x′),x′ ∈Rn−1, may have domain of definition of dimension less than n−1. Since Qf isa union of several varieties of codimension one in RPn (called components below),the question arises: which of the components of Qf and which of their intersectionsprovide, after applying the generalized Legendre transform defined in [RZ1], partsof Γ = ∂D which have codimension one in Rn.

The answer is given in Theorem3. This theorem describes Qf in terms of differential geometry of Γ.

Recall thatthe principal curvatures of a hypersurface S ⊂Rn, which is the graph of a functionxn = g(x′), are the eigenvalues of the matrix (gij)·(δij +gigj)−1·(1+Pn−1i=1 g2i )−1/2,gi = ∂g/∂xi. One can prove that if k, k ≥1, principal curvatures of a hypersurfaceS vanish identically, then for every point P ∈S there exists an affine k-dimensionalspace LP such that P ∈LP ⊂S.Theorem 3.

(a) Assume that Bm is nonempty. Then m principal curvatures ofbBm vanish identically;(b) If k principal curvatures of Γ1 vanish identically, then bΓ1 has codimensionk + 1 in RPn.Every point of bΓ1 is a vertex of a cone K, which belongs to Γ1j, where Γ1 ∩Γj ̸=∅.

The directrix of K is (k −1)-dimensional, and this directrix can be described as

4A. G. RAMM AND A. I. ZASLAVSKYfollows: Take an arbitrary point P ∈Γ1, and let Lk(P) ⊆Γ1 be a k-dimensionalaffine space containing P, which exists since k principal curvatures of Γ1 vanishidentically.

Let dP := {(α: p): lαp be tangent to Γj at the points of LK(P) ∩Γj},and let lα0p0 be tangent to Γ1 at the point P. The vertex of K is the point (α0 : p0).The directrix of K is the set dP .The set Qf is a union of the sets bΓi1···ik, Qf = S bΓi1···ik where the union is takenover all combinations of indices 1 ≤ik ≤s. Theorem 3 gives a recipe to select thecomponents of Qf which yield after the Legendre transform the components of Γof codimension 1, i.e., hypersurfaces Γj which are parts of Γ, Γ = Ssj=1 Γj.

Notethat if a component of Qf has some principal curvatures vanishing identically, thenits preimage in Rn has codimension greater than one.Therefore, if one wishesto recover hypersurface-type components of Γ, then one should apply the Legendretransform to those components of Qf, which do not have principal curvatures whichvanish identically. Those hypersurfaces Γj which have identically vanishing princi-pal curvatures are reconstructed by applying the generalized Legendre transform,which was introduced in [RZ1], to high-codimension parts of Qf described in The-orem 3(b).

The generalized Legendre transform was applied in [Z] to the study ofdual varieties in algebraic geometry.It is well known that the Radon transform may be considered as a Fourier integraloperator, so it makes sense to study its action on the wave front set of f. In [RZ1]we study a relation of the wave front of f and the set Qf.III. Proofs of Theorems 1 and 2We sketch the proofs in the simplest case m = 1, n = 2, but the ideas are similarin the general case.First we prove that if D ⊂C∞and f ∈C∞, then R(p, α; f) ∈C∞on the setVf := RPn\Qf.

Thus, the singularities of f are in the set Qf. Second, we provethat, generically, Qf is a C∞hypersurface in RPn and find the equation of thishypersurface.Third, we prove that there exists a neighborhood U of a generic point (α: p) andan equation ζ(α: p) = 0, ∇ζ ̸= 0 in U, such that (2) holds.

(a) Let us start with the second claim and prove also Theorem 2 for n ≥2. Letα · x −p = 0 be a tangent plane lαp to Γ at a point x ∈Γ.

Assume that αn ̸= 0,and write xn = β · x′ −q, βi := −αi/αn, q := −p/αn, x′ = (x1, . .

. , xn−1).

Letxn = g(x′) be the equation of Γ in a neighborhood U of x, and det gij(x′) ̸= 0.Then ∇g(x′) = β, q = β·x′−g(x′). Thus q = h(β) := Lg.

The equation q = h(β) isthe equation of Qf in the inhomogeneous coordinates (β, q). One can prove that ifq ∈Cs(U), s ≥2, and det gij(x′) ̸= 0, then h ∈Cs(U), where U is a neighborhoodof the point (β, q), ∇g(x′) = β, q = β · x′ −g(x′).

Since L is involutive, g = Lh.Theorem 2 is proved. (b) Let us prove the first claim for n = 2.

Assume that (α: p) ∈Vf, i.e., lαp isnot tangential to Γ. Write R(p, α; f) asJ :=Z a2(q,β)a1(q,β)f(x1, βx1 −q) dx1,where ai := ai(q, β) are the points of intersection of lαp with Γ.

The integral J is asum of the integrals over the intervals (a1, b), (b, c), (c, a2), where a1 < b < c < a2

SINGULARITIES OF THE RADON TRANSFORM5and b, c do not depend on q, β. Obviously the integral over (b, c) is a Cl function ofβ and q if f ∈Cl, l ≥0.

The integrals over (a1, b) and (c, a2) are treated similarly.Let us prove that the integral over (a1, b) is Cl function of q, β if Γ, f ∈Cl,l ≥2, and lαp is transversal to Γ, that is, β ̸= g′(a1). It is sufficient to prove thata1(q, β) ∈Cl.

The function a1(q, β) is the root of the equation q = βa1 −g(a1).By the transversality condition β −g′(a1) ̸= 0. Thus, the implicit function theoremimplies that the root a1(q, β) ∈Cl if g ∈Cl.

The first claim is proved. (c) Let us prove the last claim.

Let (α: p) ∈Qf and (β, q) be the correspondingnonhomogeneous coordinates. For a generic (α: p) the condition g′′(x1) ̸= 0 followsfrom the equation g′(x1) = β and Sard’s theorem.

We can assume therefore thatg′′(x1) ̸= 0. Consequently, the point x1 is a Morse-type (nondegenerate) criticalpoint of the function z := α · x −p on Γ ∩U, i.e., of the function −βx1 + g(x1) + q.The part of integral (1) taken over the complement to U is a C∞-function of(α: p) according to (b).

It gives r2 in formula (2). By the Morse lemma, thereare coordinates u1, u2 such that the equation of Γ in these coordinates is u1 = 0,the region D ∩U is described by the inequality u1 ≥0, and z = u1 + u22 in U. Tostudy the singularity of R(p, α; f), take a curve γ which intersects Qf transversally,for instance, γ = {(α: p): α = α}.

Parameter p gives the position of a point on γ.On lαp one has α · x −p = 0 and z = α · x −p, so z = p −p on lαp. Thus, z canbe used as a parameter which determines the position of a point on γ; therefore,the domain of integration in (1) can be described by the inequality u1 ≥0 and theequation z −u1 −u22 = 0.

Thus, z −u22 = u1, so −z1/2+≤u2 ≤z1/2+since z = z+ ≥0in the integration region. We haveR(p, α; f) =Zlαpf(x)µ(dx) =Zlf1(u1, u2)µ1(du) =Z z1/2+−z1/2+f2(u2, z+) du2,where f2(u2, z) is a C∞-function, l is the curve given by the equation z−u1−u22 = 0and u1 ≥0, µ1(du) comes from µ(dx) via the Morse lemma change of variables,and the last integral comes after an elimination of u1.From this formula onederives (2).

Indeed, write f2(u2, z) as a sum of even fe and odd fo functions of u2,fe(u2, z) ∈C∞, fo(u2, z) ∈C∞. Then the integralZ z1/2+−z1/2+fe(u2, z+) du2 = z1/2+ r1andZ z1/2+−z1/2+fo(u2, z+) du2 = 0,where r1 ∈C∞.

The function r2 in formula (2) vanishes if Γ is strictly convex sothat lαp intersects Γ at two points only.IV. Examples1.

Let f(x) = 1, |x| ≤a, f(x) = 0, |x| > a, x ∈Rn, n ≥2, ˆf(p, α0) = 2pa2 −p2,α0 = α|α|−1. Thus p2/|α|2 = a2 is the equation of Qf.

In (β, q) coordinates theequation of Qf is q = ±ap1 + β2, β ∈Rn−1.Thus h(β) = ±ap1 + β2.ByTheorem 2 the equation xn = g(x′) of the surface of discontinuity of f(x) is givenby g(x′) = Lh = ∓√a2 −x′2. The equation xn = ±√a2 −x′2 defines the sphere|x| = a.2.

Let f(x) = 1, b ≤|x| ≤a, f(x) = 0, |x| < b or |x| > a, 0 < b < a, n ≥2. Thenˆf(p, α0) = 2pa2 −p2, b ≤p ≤a; ˆf(p, α0) = 2(pa2 −p2 −pb2 −p2), 0 ≤p ≤b;

6A. G. RAMM AND A. I. ZASLAVSKYˆf(p, α0) = 0, p > a, |α0| = 1.

Thus p2 = |α|2a2 and p2 = |α|2b2 are the equationsof Qf.Taking Legendre’s transform yields the surfaces |x| = a and |x| = b ofdiscontinuity of f(x).3. Consider f(x) = 0 outside of the region D bounded by Γ, where Γ is the unionof the curves x2 = 0 and x2 = x21 −1, and let f(x) = 1, x ∈D.

The R(p, α; f)is a function whose support is bounded by the curves q = β, q = −β from below,q = 14β2 + 1 in the interval −2 ≤β ≤2, and q = β, q = −β for |β| ≥2 from above.One can check that on the lines q = ±β, −∞< β < ∞, the function R(p, α; f) hasa singularity of the type |z| and on the parabola q = 14β2 + 1 it has the singularityof the type z1/2+ . Applying Legendre’s transform first to the function q = 14β2 + 1,−2 ≤β ≤2, yields the parabola x2 = x21 −1, −1 ≤x1 ≤1; and secondly, applyingit to the functions q = ±β yields two points x1 = ±1, x2 = 0.

By Theorem 3, thestraignt line joining these two points also belongs to Γ. Thus Γ is recovered.V.

Numerical aspectsThe RT of f(x) is usually given with an error. Hence, the first numerical problemis to calculate the function h(β) which gives the equation of the set Qf of thesingularities of RT given the noisy measurements of the RT.

The second numericalproblem is to calculate Lh = g(x′). Calculation of the Legendre transform of afunction h(β) known with errors is a well-posed problem, at least in the case whendet gij(x′) ̸= 0.It is proved in [RSZ] that if a function gδ(x′) given such that|gδ(x′) −g(x′)| < δ, gδ(x′) is not necessarily in C2 but is continuous, then one cancalculate Lg with the accuracy O(δ) as δ →0.

This means that a stable method isgiven in [RSZ] for calculating the Legendre transform of noisy data. See also [R5].Our result in part 3 of §II has an interesting connection with the envelopes theory[T, Zl].AcknowledgmentsA.

G. Ramm thanks ONR, NSF, and USIEF for support. The research of A. I.Zaslavsky was supported in part by a grant from the Ministry of Science and the“Ma-agara”-special project for absorption of new immigrants, in the Departmentof Mathematics, Technion.References[GGV]I. M. Gelfand, M. I. Graev and N. Ya.

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Soc., Providence, RI, 1990,pp. 123–150.[Q]E.

Quinto, Tomographic reconstructions from incomplete data—numerical inversion ofthe exterior Radon transform, Inverse Problems 4 (1988), 867–876.[R1]A. G. Ramm, Random fields estimation theory, Longman, New York, 1990.

[R2], Inversion of limited-angle tomographic data, Comp. and Math.

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[R3], On numerical differentiation, Izvestiya Vuzov Math. 11 (1968), 131–135.

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102(1984), 244–250; (with T. Miller) 110 (1985), 429–435. [R5], Multidimensional inverse scattering problems, Longman, New York, 1992 (ex-panded Russian edition will be published by Mir, Moscow, 1993).

SINGULARITIES OF THE RADON TRANSFORM7[RZ1]A. G. Ramm and A. Zaslavsky, Reconstructing singularities of a function from its Radontransform, Technion, preprint 1992. [RZ2], Asymptotic behavior of the Fourier transform of a piecewise smooth function,Technion, preprint 1992.[RSZ]A.

G. Ramm, A. Steinberg and A. Zaslavsky, Stable calculation of the Legendre transform,Technion, J. Math.

Anal. Appl.

(to appear).[T]R. Thom, Sur la th´eorie des enveloppes, J. de Math.

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I. Zaslavsky, Dual varieties and Legendre transforms, submitted.[Zl]V. A. Zalgaller, The theory of envelopes, Nauka, Moscow, 1975.

(Russian)Mathematics Department, Kansas State University, Manhattan, Kansas 66506-2602E-mail address: ramm@ksuvm.ksu.eduDepartment of Mathematics, Technion-Israel Institute of Technology, 32000Haifa, IsraelE-mail address: mar9315@technion.technion.ac.il

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