On the strength of streak artifacts in limited angle weighted X-ray transform
aa r X i v : . [ m a t h - ph ] D ec ON THE STRENGTH OF STREAK ARTIFACTS IN LIMITEDANGLE WEIGHTED X-RAY TRANSFORM
LINH V. NGUYEN
Abstract.
In this article, we study the limited angle problem for the weightedX-ray transform. We consider the approximate reconstructions by applyingtwo filtered back projection formulas to the limited data. We prove that eachresulted operator can be decomposed into the sum of three Fourier integraloperators whose symbols are of types ( ̺, δ ) = (1 , Introduction
Let us denote by S the unit circle in R and µ ∈ C ∞ ( R × S ) be a strictlypositive function. We consider the weighted X-ray (or Radon) transform R µ f ( θ, s ) = Z x · θ = s µ ( x, θ ) f ( x ) dx, ( θ, s ) ∈ S × R . We are interested in reconstructing f from R µ ( f ).When µ ≡ R µ is the classical X-ray transform R , which appears in computedX-ray tomography (see, e.g., [40]). The function f can be exactly reconstructedfrom R ( f ) by the following filtered back-projection inversion formula(1) f = B f := 14 π R ∗ H dds R f. Here, H : E ′ ( R ) → D ′ ( R ) is the Hilbert transform( Hk )( t ) = 1 π p.v. Z R k ( s ) t − s ds, and R ∗ is the formal adjoint of R , defined by R ∗ g ( x ) = Z S g ( θ, x · θ ) dθ. Since H is a nonlocal operator, in order to compute f at any location x ∈ R ,formula (1) needs the data R ( θ, s ) for all ( θ, s ) ∈ S × R . The research is supported by the NSF grant DMS 1212125.
The following local formula, which is called Lambda reconstruction, provides asimple method to reconstruct the singularities of f from R f (2) Λ f := 14 π R ∗ ( − ∂ s ) R f. A main advantage of Lambda reconstruction is that in order to find Λ f ( x ), one onlyneeds the local data R ( f )( θ, s ) in a neighborhood of the set { ( θ, s ) : x · θ = s } .Discussion about Lamda tomography can be found in, e.g., [53, 52, 10, 9]. Thereader is also referred to [32, 50] for other kinds of local tomography.There is also a large amount of work dealing with the weighted X-ray transform R µ [6, 46, 47, 48, 34]. However, there is no explicit exact reconstruction formulafor a general function µ . Moreover, the function f may not be uniquely determinedfrom R µ f [5]. On the other hand, local uniqueness holds [35]. A special typeof the generalized X-ray transform is called attenuated X-ray transform, whichappears in SPECT (single photon emission computed tomography). The study ofattenuated X-ray transform is well established (e.g., [37, 38, 15, 3, 45, 39, 44, 24,33, 31]). There are two techniques for inverting the attenuated X-ray transform:the complexification method by [45] and the A-analytic method by [3]. The readeris referred to [16, 30] for comprehensive reviews on the attenuated X-ray transform.In this article, we are interested in the singularity reconstruction of f from R µ f .Let ν ∈ C ∞ ( R × S ) be a strictly positive function. We define the followingback-projection type operator R ∗ ν g ( x ) = Z S g ( θ, x · θ ) ν ( x, θ ) dθ, and following analogs of B and L : B f := 14 π R ∗ ν H dds R µ f, Λ f := 14 π R ∗ ν ( − ∂ s ) R µ f. Then, B and Λ are respectively pseudo-differential operators with amplitudes (see,e.g., [32, 27]): a B ( x, y, ξ ) = 12 [ ν ( x, ξ ) µ ( y, ξ ) + ν ( x, − ξ ) µ ( y, − ξ )] , and a Λ ( x, y, ξ ) = 12 | ξ | [ ν ( x, ξ ) µ ( y, ξ ) + ν ( x, − ξ ) µ ( y, − ξ )] . Here, we have extended µ, ν to positively homogeneous functions of degree zerowith respect to ξ . Since a B is homogenous of degree zero and non-vanishing, B f reconstructs all the singularities of f with the exact order. Since a Λ is homogenousof degree one and non-vanishing, Λ f emphasizes all the singularities of f by oneorder (see more details in [32, 31]).We now consider the limited angle problem (see, e.g., [29, 49, 32, 28, 17, 18, 31]): R µ f ( θ, s ) is only known for θ ∈ S V := { (cos φ, sin φ ) : φ < φ < φ } , where < φ < φ < π . RTIFACTS IN GENERALIZED X-RAY TRANSFORM 3
Let us consider κ ∈ C ∞ ( S V ) such that κ > S V . We then extend κ to S byzero and define the limited angle version of B and Λ: B κ f := 14 π R ∗ ν κ H dds R µ f, Λ κ f := 14 π R ∗ ν κ ( − ∂ s ) R µ f. Then, one can write B κ and Λ κ as oscillatory integrals: B κ f ( x ) = 1(2 π ) Z R e i ( x − y ) · ξ a B ,κ ( x, y, ξ ) f ( y ) dξ dy, and Λ κ f ( x ) = 1(2 π ) Z R e i ( x − y ) · ξ a Λ ,κ ( x, y, ξ ) f ( y ) dξ dy. Here, a B ,κ ( x, y, ξ ) = 12 [ ν ( x, ξ ) µ ( y, ξ ) κ ( ξ/ | ξ | ) + ν ( x, − ξ ) µ ( y, − ξ ) κ ( − ξ/ | ξ | )] , and a Λ ,κ ( x, y, ξ ) = 12 | ξ | [ ν ( x, ξ ) µ ( y, ξ ) κ ( ξ/ | ξ | ) + ν ( x, − ξ ) µ ( y, − ξ ) κ ( − ξ/ | ξ | )] . Assume that κ vanishes to infinite order at the end points of S V , then B κ andΛ κ are pseudo-differential operators with principal symbols respectively: σ B ,κ ( x, ξ ) = 12 [ κ ( ξ/ | ξ | ) ν ( x, ξ ) µ ( x, ξ ) + κ ( − ξ/ | ξ | ) ν ( x, − ξ ) µ ( x, − ξ )] , and σ Λ ,κ ( x, ξ ) = 12 | ξ | [ κ ( ξ/ | ξ | ) ν ( x, ξ ) µ ( x, ξ ) + κ ( − ξ/ | ξ | ) ν ( x, − ξ ) µ ( x, − ξ )] . We note that σ B ,κ ( x, ξ ) > σ Λ ,κ ( x, ξ ) > ξ | ξ | ∈ S V or − ξ | ξ | ∈ S V . Therefore,Λ κ and B κ reconstruct all the visible singularities, with the same order as Λ and B do, respectively. We recall that visible singularities are all ( x, ξ ) ∈ WF( f ) such that ξ | ξ | ∈ S V or − ξ | ξ | ∈ S V (see, e.g., [32]). Meanwhile, they do not generate any addedsingularities (i.e., artifacts). The reader is referred to [32] for detailed discussion.Let us now consider the case when κ only vanishes to a finite order k at the endpoints of S V . We denote e = (cos φ , sin φ ) , e = (cos φ , sin φ ). Then, theamplitudes a B ,κ and a Λ ,κ are not smooth with respect to ξ , across the lines ℓ = { ξ ∈ R : ξ = r e : r ∈ R } and ℓ = { ξ ∈ R : ξ = r e : r ∈ R } . Therefore, B κ and Λ κ are no longer pseudo-differential operators in the stan-dard sense. They are, instead, pseudo-differential operators with singular symbols.Moreover, it is observed that B κ f and Λ κ f may contain artifacts (see, e.g. [28, 20]). The goal of this article is to study B κ and Λ κ in this case .In [28], Katsevich obtains the geometric characterization of the artifacts. He alsoanalyzes the strength of the artifacts for the case f only has jump singularities. His That is, κ ( l ) ( θ ( φ )) = 0, 0 ≤ l ≤ k − κ ( k ) ( θ ( φ )) = 0, for φ = φ and φ = φ . LINH V. NGUYEN approach relies on the direct estimates for oscillatory integrals. Recently, Frikeland Quinto [18, 20] provide a general paradigm to study the geometric descriptionfor artifacts arising in limited angle problem of the weighted X-ray and Radontransforms. Their approach relies on the calculus of wave front set for compositions.This article, although having some overlaps with the above mentioned works,has some distinct features. Firstly, we prove a decomposition of T as a sum ofthree Fourier operators with non-classical symbols. This result, being interestingin itself, implies the geometric description of the artifacts. Moreover, it provides thecontinuity of T between Sobolev spaces. Such a continuity has not been obtainedbefore. Secondly, we analyze the strength of the artifacts, when f is an arbitrary compactly supported distribution. Our approach for this goal is a refinement of theprevious work [41].It is worth mentioning that some similar analysis of artifacts for spherical Radontransform has been done in several works [19, 42, 4].To proceed, we will consider a family of operators which contains B κ and Λ κ asspecial cases. Let T be the linear operator whose Schwartz kernel is:(3) K ( x, y ) = 1(2 π ) Z R e i ( x − y ) · ξ a ( x, y, ξ ) χ ( ξ ) dξ, where a ∈ S m (( R × R ) × R ) . Here, χ is the characteristic function of cl( W ),where W = R S V = { ξ = r (cos θ, sin θ ) : φ < θ < φ , r ∈ R } . Obviously, B κ and Λ κ are special cases of T (with m = 0 and m = 1, respectively).In the sequel, we analyze T , and then interpret our results to B κ and Λ κ .Let ∆ ⊂ ( T ∗ R \ × ( T ∗ R \
0) be the diagonal relation∆ = { ( x, ξ ; x, ξ ) : ( x, ξ ) ∈ T ∗ R \ } , and ∆ = { ( x, ξ ; x, ξ ) ∈ ∆ : ξ ∈ cl( W ) } . For j = 1 ,
2, we define C j ⊂ ( T ∗ R \ × ( T ∗ R \
0) by C j = { ( x, γ e j ; x + t e ⊥ j , γ e j ) : x ∈ R , γ, t ∈ R , γ = 0 } . Let us recall the definition of the symbol classes S m̺,δ ( X × R N ) (see, e.g., [26, 51]): Definition 1.1.
Let m, ̺ and δ be real numbers, ≤ ̺, δ ≤ . The class S m̺,δ ( X × R N ) consists of all functions ρ ( x, ξ ) ∈ C ∞ ( X × ( R N \ such that for any multi-indices α, β and any compact set K ⋐ X there exists a constant C = C α,β,K forwhich | ∂ αξ ∂ βx ρ ( x, ξ ) | ≤ C (1 + | ξ | ) m − ̺ | α | + δ | β | , for all ( x, ξ ) ∈ X × R N . An element ρ ∈ S m̺,δ ( X × R N ) is called a symbol of order m and type ( ̺, δ ).We also denote S m ( X × R N ) = S m , ( X × R N ), which are the most common typeof symbol classes (especially in tomography). In this article, however, we needto make use of symbol classes S m̺,δ ( X × R N ) where ( ̺, δ ) = (1 , C be aLagrangian in the cotangent bundle T ∗ X of X . We will denote by I m̺,δ ( C ) theclass of Fourier integral distributions of order m whose symbol is of type ( ̺, δ ) and See the definition of the symbol class S m in Definition 1.1 and the following discussion. RTIFACTS IN GENERALIZED X-RAY TRANSFORM 5 canonical relation is a subset of C . The order of a Fourier integral distribution isnot necessarily the same as the order of its symbol (we will elaborate on this factlater when needed). The reader is referred to [26, 51] for the detailed treatment onFourier integral distributions.Here is the main result of this article: Theorem 1.2.
We have a) For any n > , we can write K = K ,n + K ,n + K ,n , where K ,n ∈ I m n , (∆) , K ,n ∈ I m + n − , n ( C ) , K ,n ∈ I m + n − , n ( C ) . Moreover, the symbol σ ( x, ξ ) of K ,n satisfies i) σ ( x, ξ ) − a ( x, x, ξ ) ∈ S m − n n , ( R × V ) , for any closed conic set V ⊂ W ;and ii) σ ( x, ξ ) ∈ S −∞ n , ( R × V ) , for any closed conic set V ⊂ R \ cl( W ) . b) Assume that a ( x, y, ξ ) vanishes to order k across the line ℓ i . Then, near C j \ ∆ , K is microlocally in the space I m − k − / ( C j ) . Let T ,n , T ,n , and T ,n be the operators whose Schwartz kernels are K ,n , K ,n ,and K ,n , respectively. We obtain from Theorem 1.2 a) the following continuity:(C.1) Since K ,n ∈ I m n , (∆), T ,n is a continuous map from H scomp ( R ) to H s − mloc ( R )(see, e.g., [51, Theorem 7.1]), and(C.2) Since K ,n ∈ I m + n − , n ( C ), and K ,n ∈ I m + n − , n ( C ), T ,n and T ,n arecontinuous maps from H scomp ( R ) to H s − m − n loc ( R ) (see, e.g., [26, Theorem4.3.2]). It should be noticed here the difference between the order m + n − of K ,n , K ,n and the aforementioned mapping property. This comes fromthe fact that the canonical relations C and C are not local graph. Theirleft and right projections are fibered of dimension one.(C.1) and (C.2), in particular, imply the continuity of T from H scomp ( R ) to H s − m − n loc ( R ), for any n >
0. To the best of our knowledge, this continuity has notbeen proved anywhere. We note here that if a ( x, y, ξ ) vanishes to infinite order atthe boundary of W , then from the standard theory of pseudo-differential operator, T is a continuous map from H scomp ( R ) to H s − mloc ( R ). It is interesting to seewhether such optimal bound also holds for the case a ( x, y, ξ ) only vanishes to finiteorder, as being considered in this article.Let us interpret Theorem 1.2 further as properties of B κ and Λ κ (where m = 0and m = 1, respectively). From Theorem 1.2 a), we obtainWF( K ) ⊂ ∆ ∪ C ∪ C . This result was obtained in [28, 18] by other methods. It, in particular, describes thegeometry of the artifacts introduced in B κ f and Λ κ f . The artifacts are generatedby the “edge” singularities via the canonical relations C and C , as follows. An“edge” singularity is an element ( x, ξ ) ∈ WF( f ) such that ξ k e j ( j = 1 , An artifact in T f is a singularity ( x, ξ ) ∈ WF( T f ) such that ( x, ξ ) WF( f ). LINH V. NGUYEN generate the artifacts at other elements ( y, ξ ) = ( x, ξ ) satisfying ( y, ξ ; x, ξ ) ∈ C j .That is, the artifacts generated by ( x, ξ ) are located along the line passing through x and orthogonal to e j . The same phenomenon in the limited angle problem of thestandard X-ray transform was presented in [17, 41].From (C.1), we obtain that the reconstructed singularities are at most m order(s)stronger than the original ones. Moreover, let us recall that the amplitudes a B ,κ and a Λ ,κ (of B κ and Λ κ , respectively) are nonvanishing on W . The formula forthe symbol of K ,n in Theorem 1.2 a) shows that the singularity ( x, ξ ) ∈ WF( f )satisfying ξ ∈ W (i.e., ( x, ξ ) is a visible singularity) is reconstructed and thereconstructed singularity is exactly m order stronger than the original singularity.On the other hand, the singularity ( x, ξ ) ∈ WF( f ) such that ξ W (i.e., ( x, ξ ) isan invisible singularity) is completely smoothened out by B κ and Λ κ . This is dueto the fact that the symbol of K ,n vanishes for ξ W (see Theorem 1.2 a) ii)). Letus mention that the descriptions presented in this paragraph have been obtainedpreviously in several works [28, 17, 41] by other methods.Theorem 1.2 b) is a generalization of [41, Theorem 3.1 b)], where the standardX-ray transform was considered. It provides explicit bounds for the artifacts, asfollow (see [41] for the detailed explanation): • The artifacts are at most ( m − k ) order(s) stronger than their strongestgenerating singularities if m > k . The artifacts are at most as strong astheir strongest generating singularities if m = k . The artifacts are at least ( k − m ) order(s) smoother than their strongest generating singularities if k > m . • Assume that the artifact ( x, ξ ) ∈ WF( T f ) has finitely many generatingsingularities ( y, ξ ) ∈ WF( f ), each of them is conormal to a curve S havingnon-vanishing curvature at y . Then, the artifact is at most ( m − k − )order(s) stronger than its strongest generating singularity if m > k − . Itis at most as strong as its strongest generating singularity if m = k − .It is at least ( k − m − ) order(s) smoother than its strongest generatingsingularity if k > m − .These bounds for the artifacts are better than what can be obtained from (C.2).However, it should be noted Theorem 1.2 b) does not explain the strength of ( x, ξ ) ∈ WF( T f ) if ( x, ξ ) is an “edge” singularity of f (i.e., ( x, ξ ) ∈ WF( f ) and ξ k e j for j = 1 or j = 2). Meanwhile, (C.2) implies that such singularity ( x, ξ ) ∈ WF( T f )is at most m + n order(s) stronger than ( x, ξ ) ∈ WF( f ), for any n .Let us briefly discuss the main ideas for the proof of Theorem 1.2. The proof ofTheorem 1.2 a) uses a nonlinear cutoff for the phase variable ξ . It is a modificationof the well known parabolic cutoff, first used by Boutet de Monvel [7] to deal withhypoelliptic operators, and later by Guillemin to deal with the pseudo-differentialoperators with singular symbol (see the discussion in [22]). The proof of Theo-rem 1.2 b) is similar to that of [41, Theorem 3.1]. Namely, it makes use of someproper integrations by parts. However, it is worth mentioning that our argumentin this article is cleaner than that in [41], where the special case - standard X-raytransform - is studied.A deep theory of pseudo-differential operators with singular symbols was de-veloped by Guillemin, Melrose, Uhlmann and others (see, e.g., [36, 25, 2]). Theuse of that theory to analyze the X-ray transform, when the canonical relation isnot a local canonical graph, was pioneered by Greenleaf and Uhlmann [21, 23]. RTIFACTS IN GENERALIZED X-RAY TRANSFORM 7
It has been then exploited intensively to analyze other imaging scenarios (e.g.,[14, 43, 11, 13, 8, 12, 1]). Although we do not make use of it explicitly, our analysisis inspired by that theory.In the next section, we will present the proof of Theorem 1.2. We will start bystudying a model distribution defined by an oscillatory integral, whose amplitudeis in the class S m when ξ is away from a straight line through the origin. Weshow that it can be decomposed into two parts. One belongs to I m n , (∆) and theother to I m + n − , n ( C ), where C is a Lagrangian defined later. Moreover, the modeldistribution is microlocally in the space I m − k − ( C ) near C\ ∆. With all these resultsin hands, we then prove Theorem 1.2 by a simple partition of unity argument.2. Proof of Theorem 1.2
Model oscillatory integrals.
Let a ( x, y, ξ ) ∈ S m (( R × R ) × R ) such thatfor any ( x, y ) ∈ R × R there is M x,y > a ( x, y, . ) ⊂ { ξ : | ξ | ≤ M x,y | ξ |} . We consider the oscillatory integral K ± ( x, y ) = 1(2 π ) Z R e i ( x − y ) · ξ a ( x, y, ξ ) H ( ± ξ ) dξ. (5)Here, H is the Heaviside function, defined by H ( s ) = ( , s ≥ , , s < . We also recall the diagonal canonical relation in ( T ∗ R \ × ( T ∗ R \ { ( x, ξ ; x, ξ ) : ( x, ξ ) ∈ T ∗ R \ } , and define C ⊂ ( T ∗ R \ × ( T ∗ R \
0) by C = { ( x, ξ ; y, ξ ) ∈ ( T ∗ R \ × ( T ∗ R \
0) : x − y = 0 , ξ = 0 } . The following results characterize the distribution K ± : Proposition 2.1.
We have a) For any n ≥ , we can write K ± = K + K where K ∈ I m n , (∆) and K ∈ I m + n − , n ( C ) . Moreover, the symbol σ ( x, ξ ) of K satisfies: i) σ ( x, ξ ) − a ( x, x, ξ ) ∈ S m − n n , ( R × V ) for any conic closed conic set V ⊂ { ξ ∈ R : ± ξ > } , and ii) σ ( x, ξ ) ∈ S −∞ n , ( R × V ) for any conic closed conic set V ⊂ { ξ ∈ R : ± ξ < } . b) Assume that a ( x, y, ξ ) vanishes up to order k at ξ = 0 . Then, K ± ismicrolocally in the space I m − k − ( C ) near C \ ∆ .Proof. We only consider K + , since the proof for K − is similar. LINH V. NGUYEN
We first prove a). Let c ∈ C ∞ ( R ) be a cut-off function satisfying: c ( τ ) = 1 for | τ | ≤ c ( τ ) = 0 for | τ | ≥
2. Let us define K ( x, y ) = 1(2 π ) Z R e i ( x − y ) · ξ h − c (cid:0) ξ n / h ξ i (cid:1)i a ( x, y, ξ ) H ( ξ ) dξ, and K ( x, y ) = 1(2 π ) Z R e i ( x − y ) · ξ c (cid:0) ξ n / h ξ i (cid:1) a ( x, y, ξ ) H ( ξ ) dξ. Then, K + = K + K . We now prove that K and K satisfy the properties statedin Proposition 2.1 a).To analyze K , let us denote b ( x, y, ξ ) = h − c (cid:0) ξ n / h ξ i (cid:1)i a ( x, y, ξ ) H ( ξ ) . Then, b ∈ C ∞ (( R × R ) × ( R \ α = ( α , α ): ∂ αξ b ( x, y, ξ ) = α X l =0 α X k =0 ∂ lξ ∂ kξ (cid:2) − c (cid:0) ξ n / h ξ i (cid:1)(cid:3) ∂ α − lξ ∂ α − kξ a ( x, y, ξ ) H ( ξ ) . Since c ( τ ) = 1 for | τ | ≤ c ( τ ) = 0 for | τ | ≥ (cid:12)(cid:12) ∂ lξ ∂ kξ (cid:2) − c (cid:0) ξ n / h ξ i (cid:1)(cid:3) (cid:12)(cid:12) ≤ C h ξ i − ln − k . Noting that a ∈ S m (( R × R ) × R ), we obtain | ∂ αξ b ( x, y, ξ ) | ≤ C h ξ i m − | α | n . It is, hence, straight forward to show that b ∈ S m n , (( R × R ) × R ). Therefore, K ∈ I m n , (∆).Let V ⊂ { ξ ∈ R : ξ > } be a closed conic set. Then, there is ε > | ξ | ≥ ε | ξ | for all ξ ∈ V . Hence, for big enough R > | ξ | n ≥ | ξ | , if ξ ∈ V and | ξ | ≥ R. That is, b ( x, y, ξ ) = a ( x, y, ξ ) , of ξ ∈ V and | ξ | ≥ R. Thus, b − a ∈ S −∞ n , ( R × V ) . On the other hand, using the formula of symbol (e..g, [26, (2.1.4)]), we obtain: σ ( x, ξ ) − b ( x, x, ξ ) ∈ S m − n n , ( R × R ) . Therefore, σ ( x, ξ ) − a ( x, x, ξ ) ∈ S m − n n , ( R × V ) . This proves i). On the other hand, assume that V ⊂ { ξ ∈ R : ξ < } is a closedconic set. Then b ( x, y, ξ ) = 0 , ( x, ξ ) ∈ R × V. This implies σ ( x, ξ ) ∈ S −∞ n , ( R × V ) . Here, we use the Japanese bracket h . i convention: h ξ i = (1 + | ξ | ) / . RTIFACTS IN GENERALIZED X-RAY TRANSFORM 9
We, therefore, have proved ii) and finished the analysis of K .We now analyze K . Let us write K ( x, y ) = 1(2 π ) Z R e i ( x − y ) ξ b ( x, y, ξ ) dξ , where b ( x, y, ξ ) = 1(2 π ) Z R e i ( x − y ) ξ c (cid:0) ξ n / h ξ i (cid:1) a ( x, y, ξ ) H ( ± ξ ) dξ . Since c ( τ ) = 0 for | τ | ≥
2, there is M = M ( n ) such that for all ξ satisfying c (cid:0) ξ n / h ξ i (cid:1) = 0: | ξ | ≤ M h ξ i n . Therefore, since a ∈ S m (( R × R ) × R ), for any compact set K ⋐ R × R : | b ( x, y, ξ ) | ≤ C M h ξ i n Z h ξ i m dξ ≤ C h ξ i m + n , ∀ ( x, y ) ∈ K. Similarly, we obtain | ∂ αx ∂ βy ∂ lξ b ( x, y, ξ ) | ≤ C h ξ i m + n + α n + β n − l . Therefore, b ∈ S m + n , n (( R × R ) × R ) . We arrive to K ∈ S m + n − , n ( C ). We note here that the difference between theorders of b and K comes the following formula (see, e.g., [26]):(6) order of K = order of b + ( N − n ) / , where n = 2 is the dimension of the spatial variables x, y and N = 1 is the dimensionof the phase variable ξ . The proof of a) is finished.Let us now prove b). Let ( x ∗ , ξ ∗ ; y ∗ , ξ ∗ ) ∈ C \ ∆, then x ∗ = y ∗ . Let O ⊂ R × R be an open set containing ( x ∗ , y ∗ ) such that x = y for any ( x, y ) ∈ O . It sufficesto prove that K + | O ∈ I m − k − ( C ). Let us write(7) K + ( x, y ) = 1(2 π ) Z R e i ( x − y ) ξ a ( x, y, ξ ) dξ , where a ( x, y, ξ ) = Z R e i ( x − y ) ξ a ( x, y, ξ ) H ( ξ ) dξ = ∞ Z e i ( x − y ) ξ a ( x, y, ξ ) dξ . Let us note that, due to (4), the above integral is, in fact, over a finite interval.Taking integration by parts ( k + 1) times with respect to ξ , we obtain: a ( x, y, ξ ) = 1[ i ( y − x )] k +1 h ∂ kξ a ( x, y, ξ ,
0) + ∞ Z e i ( x − y ) ξ ∂ k +1 ξ a ( x, y, ξ ) dξ i . Since a ∈ S m ( O × R ) the first term on the right hand side belongs S m − k ( O × R ).Due to Lemma 2.2 below, the second term belongs to S m − k ( O × R ) . Therefore, wearrive to a ∈ S m − k ( O × R ). Therefore, microlocal near C \ ∆, K + ∈ I m − k − ( C ).Here, we have used a formula similar to (6) to determine the order of K + . Thisfinishes the proof for b). (cid:3) Lemma 2.2.
Assume that ρ ( x, y, ξ ) ∈ S m (( R × R ) × R ) , and for any ( x, y ) ∈ R × R there is M x,y > : supp ρ ( x, y, . ) ⊂ { ξ : | ξ | ≤ M x,y | ξ |} . Let
O ⊂ R × R such that x = y for all ( x, y ) ∈ O . Define the function R ( x, y, ξ ) = ∞ Z e i ( x − y ) ξ ρ ( x, y, ξ ) dξ . Then, R ( x, y, ξ ) ∈ S m +1 ( O × R ) .Proof. We need to prove that for any multi-indices α, β , any integer l ≥
0, andcompact set K ⋐ O , there is C = C K,α,β,l > | ∂ αx ∂ βy ∂ lξ R ( x, y, ξ ) | ≤ C h ξ i m − l +1 , for all ( x, y ) ∈ K. We observe that ∂ αx ∂ βy ∂ lξ R ( x, y, ξ ) is equal to α X k =0 β X k =0 ∞ Z e i ( x − y ) ξ ( iξ ) k ( − iξ ) k ∂ α − k x ∂ β − k y ∂ α x ∂ β y ∂ lξ ρ ( x, y, ξ ) d ξ = α X k =0 β X k =0 I k ,k ( x, y, ξ ) . Taking integration by parts with respect to ξ , we obtain I k ,k ( x, y, ξ ) = ∞ Z e i ( x − y ) ξ ( iξ ) k ( − iξ ) k ∂ α − k x ∂ β − k y ∂ α x ∂ β y ∂ lξ ρ ( x, y, ξ ) d ξ = 1[ i ( x − y )] k + k ∞ Z ∂ k + k ξ [ e i ( x − y ) ξ ] ( iξ ) k ( − iξ ) k ∂ α − k x ∂ β − k y ∂ α x ∂ β y ∂ lξ ρ ( x, y, ξ ) d ξ = ( − k + k [ i ( x − y )] k + k ∞ Z e i ( x − y ) ξ ∂ k + k ξ (cid:2) ( iξ ) k ( − iξ ) k ∂ α − k x ∂ β − k y ∂ α x ∂ β y ∂ lξ ρ ( x, y, ξ ) (cid:3) d ξ . The amplitude in the above integral is in S m − l ( O × R ). Therefore,(9) | I k ,k ( x, y, ξ ) | ≤ C M x,y | ξ | Z h ξ i m − l d ξ ≤ C h ξ i m − l +1 , for all ( x, y ) ∈ K. This proves (8) and finishes the proof. (cid:3) Using integration by parts once and apply Lemma 2.2, one can show that this term is, in fact,in S m − k − ( O × R ). RTIFACTS IN GENERALIZED X-RAY TRANSFORM 11
We now consider a generalization of K ± . Namely, let e ∈ R be a unit vector andlet us consider the distributions(10) K ± e ( x, y ) = 1(2 π ) Z R e i ( x − y ) · ξ a e ( x, y, ξ ) H ( ± e ⊥ · ξ ) dξ, and the canonical relation: C e = { ( x, s e ; x + t e ⊥ , s e ) : x ∈ R , t, s ∈ R , γ = 0 } . Proposition 2.3.
We have a) For any n > , K ± e ∈ I m n , (∆) + I m + n − , n ( C e ) . b) Assume that a e ( x, y, ξ ) vanishes to order k across the line ℓ = { ξ = r e : r ∈ R } . Then, K ± e is microlocally in the space I m − k − ( C e ) near C e \ ∆ . Proposition 2.3 can be easily derived from Proposition 2.1, by a rotation argu-ment. We skip it for the sake of brevity.2.2.
Proof of Theorem 1.2.
For j = 1 ,
2, let ρ j ∈ C ∞ ( R \
0) be homogeneousof degree zero such that ρ j = 1 in a (small) conic neighborhood of ℓ j \
0. Moreover, ρ j is supported inside a small conic neighborhood of ℓ j .We can write: K ( x, y ) = 1(2 π ) Z R e i ( x − y ) · ξ a ( x, y, ξ ) χ ( ξ ) dξ = 1(2 π ) Z R e i ( x − y ) · ξ h − X j =1 ρ j ( ξ ) i a ( x, y, ξ ) χ ( ξ ) dξ + X j =1 π ) Z R e i ( x − y ) · ξ ρ j ( ξ ) a ( x, y, ξ ) χ ( ξ ) dξ = K ( x, y ) + X j =1 K j ( x, y ) . We notice that K ∈ I m (∆) with the amplitude function h − X j =1 ρ j ( ξ ) i a ( x, y, ξ ) χ ( ξ ) ∈ S m (( R × R ) × R ) . Applying the Proposition 2.3 for K and K , we finish the proof. Acknowledgements
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