Numerical inversion of 3D geodesic X-ray transform arising from traveltime tomography

Transcription

1 Numercal nverson of D geodesc X-ray transform arsng from traveltme tomography arxv:184.16v1 [math.na] 6 Apr 18 Tak Shng Au Yeung Erc T. Chung Gunther Uhlmann Aprl 7, 18 Abstract In ths paper, we consder the nverse problem of determnng an unknown functon defned n three space dmensons from ts geodesc X-ray transform. The standard X-ray transform s defned on the Eucldean metrc and s gven by the ntegraton of a functon along straght lnes. The geodesc X-ray transform s the generalzaton of the standard X-ray transform n Remannan manfolds and s defned by ntegraton of a functon along geodescs. Ths paper s motvated by Uhlmann and Vasy s theoretcal reconstructon algorthm for geodesc X-ray transform and mathematcal formulaton for traveltme tomography to develop a novel numercal algorthm for the stated goal. Our numercal scheme s based on a Neumann seres approxmaton and a layer strppng approach. In partcular, we wll frst reconstruct the unknown functon by usng a convergent Neumann seres for each small neghborhood near the boundary. Once the soluton s constructed on a layer near the boundary, we repeat the same procedure for the next layer, and contnue ths process untl the unknown functon s recovered on the whole doman. One man advantage of our approach s that the reconstructon s localzed, and s therefore very effcent, compared wth other global approaches for whch the reconstructons are performed on the whole doman. We llustrate the performance of our method by showng some test cases ncludng the Marmous model. Fnally, we apply ths method to a travel tme tomography n D, n whch the nverson of the geodesc X-ray transform s one mportant step, and present several numercal results to valdate the scheme. 1 Introducton We address n ths paper the nverse problem of determnng an unknown functon from ts geodesc X-ray transform. The travel tme nformaton and geodesc X-ray data are encoded n the boundary dstance functon, whch measures the dstance between boundary Department of Mathematcs, The Chnese Unversty of Hong Kong, Hong Kong SAR. Department of Mathematcs, The Chnese Unversty of Hong Kong, Hong Kong SAR. Department of Mathematcs and Insttute of Advanced Study, The Unversty of Scence and Technology of Hong Kong, Hong Kong SAR; and Department of Mathematcs, Unversty of Washngton, Seattle, USA. 1

2 ponts. The geodesc X-ray transform s assocated wth the boundary rgdty problem of determnng a Remannan metrc on a compact manfold from ts boundary dstance functon see [7, ]). In ths paper, we consder the lnearzaton of the boundary rgdty problem n compact Remannan manfold wth boundary n the sotropc case. Recent progress on the nverse problem of nvertng geodesc X-ray transform [5, 15, 4], and the boundary rgdty problem [8, 16,, 1, ] has motvated us to transfer these theoretcal results nto numercal methods for determnng an unknown functon from ts travel tme nformaton and geodesc X-ray data. Ths paper s mostly motvated by Uhlmann and Vasy s theoretcal reconstructon algorthm for geodesc X-ray transform [4] and mathematcal formulaton for traveltme tomography n [, ] to develop a novel numercal reconstructon algorthm. Traveltme tomography handles the problem of determnng the nternal propertes, such as ndex of refracton, of a medum by measurng the traveltmes of waves gong through the medum. It arose n global sesmology n determnng the earth velocty models by measurng the traveltmes of sesmc waves produced by earthquakes measured at dfferent sesmc statons. Compared to other methods such as full wave nverson, traveltme tomography s generally easer to mplement and can be computed much effcently. The standard X-ray transform s defned on the Eucldean metrc and s gven by the ntegraton of a functon along straght lnes. It s the mathematcal model underlyng medcal magng technques such as CT and PET. The geodesc X-ray transform s the generalzaton of standard X-ray transform n Remannan manfolds and s defned by ntegratng a functon along geodescs. Snce the speed of elastc waves ncreases wth depth n the Earth, the rays curve back to the surface and thus geodesc X-ray transform arses n geophyscal magng n determnng the nner structure of the Earth. In two dmensons, the unqueness and stablty for the geodesc X-ray transform was proved by Mukhometov [11] on smple surface and also for a general famly of curves. For smple manfolds, the case of geodescs was generalzed to hgher dmensons n [1]. When the metrc s not smple, the results are proved for some geometres wth some addtonal assumptons. In [5], the authors proved some generc njectvty results and stablty estmates for the X-ray transform for a general famly of curves and weghts wth addtonal assumptons analog to smplcty n the geodesc X-ray transform) of mcrolocal condton whch ncludes real-analytc metrcs for a class of non-smple manfolds n dmenson d. Besdes some njectvty and stablty results, some explct methods for reconstructng the unknown functon have been found. Several methods have been ntroduced to recover the unknown functon from X-ray transforms, ncludng Fourer methods, backprojecton and sngular value decomposton [14]. In [6], the authors ntroduce a reconstructon algorthm to recover functons from ther D X-ray transforms by shearlet and wavelet decompostons. Ths algorthm s effectve and optmal n mean square error rate. On the other hand, under the afolaton condton, Uhlmann and Vasy showed that the geodesc X-ray transform s globally njectve and can be nverted locally n a stable manner [4]. Also, the authors propose a layer strppng type algorthm for recoverng the unknown functon n the form of a convergent Neumann seres. On the numercal sde, the geodesc X-ray transform has been mplemented numercally n two dmensons. In

3 [1], Monard develops a numercal mplementaton of geodesc X-ray transform based on the fan-beam geometry and the Pestov-Uhlmann reconstructon formulas [15]. In ths paper we consder the nverse problem of computng the unknown functon f defned n a doman Ω from ts geodesc X-ray transform n three dmensons. As far as we know, ths s the frst numercal mplementaton n D. Our numercal method s based on the theoretcal results n Uhlmann and Vasy [4], whch shows that the unknown functon can be represented by a convergent Neumann seres and can be recovered by a layer strppng type algorthm. Based on ths theoretcal foundaton, we develop a numercal approach usng a truncated Neumann seres and layer strppng. A related method s developed for photoacoustc tomography n the work by Qan, Stefanov, Uhlmann, and Zhao [17]. The key deas of our approach are dscussed as follows. Frst, we use a layer strppng algorthm to separate the doman nto small regons and then reconstruct the unknown functon on each small regon by usng a truncated Neumann seres and the gven geodesc X-ray data restrcted to the regon. We start ths reconstructon process wth the outermost regon. Combnng the above methodologes, the resultng algorthm can undergo the reconstructon layer by layer so to prevent full nverson n the whole doman. Thus, our approach s very effcent and consumes much less computatonal tme and memory. Numercal examples ncludng synthetc sotropc metrcs and the Marmous model are shown to valdate our new numercal method. Our proposed numercal approach for geodesc X-ray transform s a foundaton of a new numercal scheme for traveltme tomography n D. Our new method s based on the Stefanov-Uhlmann dentty whch lnks two Remannan metrcs wth ther travel tme nformaton. The method frst uses those geodescs that produce smaller msmatch wth the travel tme measurements and contnues on n the sprt of layer strppng. We then apply the above reconstructon method to the Stefanov-Uhlmann dentty n order to solve the nverse problem for reconstructng the ndex of refracton of a medum. The new method s motvated by the methods n [, ], but we use the above numercal X-ray transform for solvng an ntegral equaton nvolved n the nverson process. We wll present some numercal test cases ncludng the Marmous model to show the performance of the nverson. The paper s organzed as follows. In Secton, we wll present some background materals and the reconstructon method, and n Secton, we present n detal the numercal algorthm and mplementaton. Numercal results are shown n Secton 4 to demonstrate the performance of our method. In Secton 5, we present our nverson method for traveltme tomography and numercal results. The paper ends wth a concluson. Problem descrpton In ths secton, we wll present the mathematcal formulaton of the nverse problem of reconstructng an unknown functon usng ts geodesc X-ray transform, that s, ntegrals of the functon along geodescs. We wll then present the numercal reconstructon algorthm based on the theoretcal foundaton n [4]. The algorthm s based on a representaton of the unknown functon by a convergent Neumann-seres.

4 .1 Geodescs We frst ntroduce the noton of geodescs. Let Ω be a strctly convex bounded doman n R d and let g j ) be a Remannan metrc defned on t. In the case of sotropc medum, g j = 1 c δ j, where c s a gven functon defned n R d. Followng [,, 4], we defne the Hamltonan H g by H g x, ξ) = 1 d g j x)ξ ξ j 1),j=1 for each x Ω and ξ R d, where g j ) = g j ) 1 s the nverse of g j ). Let X ) = x), ξ)) be a gven ntal condton, where x) Ω and ξ) R d, such that the followng nflow condtons hold: H g x), ξ)) = 1, d g j x))ξ ν j x)) <,j=1 where νx) s the unt outward normal vector of Ω at the pont x. We defne X g s, X ) ) = xs), ξs)) by the soluton to the hamltonan system defned by dx ds = H g ξ wth the ntal condton x), ξ)) = X )., dξ ds = H g x The soluton X g defnes a geodesc n the physcal space, parametrcally va xs) wth the co-tangent vector ξs) at any pont xs). The parameter s denotes travel tme. Thus, we denote the set of all geodescs X g, whch are contaned n Ω wth endponts on Ω, by M Ω.. The reconstructon method Let f be a smooth functon from Ω to R. The am of ths paper s to develop a numercal scheme to determne the unknown functon f x) from the geodesc X-ray transform of f x). Our method s based on a truncaton of a convergent Neumann seres. Frst, we defne the geodesc X-ray transform. Gven a functon f defned on Ω, ts geodesc X-ray transform s the collecton {I f )X g )} of ntegrals of f along geodescs X g M Ω, where I f )X g ) := f s) ds, where X g s) = xs), ξs)). We note that ths s the measurement data, and we use ths data to reconstruct f x). Let Λ be the adjont of the operator I. Then Uhlmann and Vasy [4] show that there s an operator R such that xs) RΛI f ) = f K f, 4 1)

5 where the error operator K s small n the sense of K < 1 for an approprate norm. Hence, the unknown functon f can be represented by the followng convergent Neumann seres f = K n RΛI f ). ) n= We remark that the operator R s the nverse of the operator Λ I. Moreover, results from [4] suggest that the unknown functon f can be reconstructed locally n a layer by layer fashon. In partcular, one can frst reconstruct the unknown functon f usng ) n small neghborhoods near the boundary of the doman, and then repeat the procedure n the next nner layer of the doman, and so on. The challenges n the numercal computatons of the unknown functon f usng the above representaton ) le n the fact that computng the operators Λ and R as well as the mplementaton of the layer strppng algorthm needs some careful constructons. It s the purpose of the rest of the paper to develop numercal procedures to compute these operators and mplement the layer strppng algorthm.. The numercal procedure We wll gve a general overvew of our numercal scheme, and present the mplementaton detals n Secton 5.. We wll assume that the set of geodesc X-ray transforms {I f )X g )} s gven as a set of measurement data) and s a fnte set. Let Z be a set of grd ponts, denoted as {z }, n the doman Ω. We wll determne the values of the unknown functon f x) at these grd ponts usng the gven data set {I f )X g )}. Next, we wll present our approach to numercally construct the operators Λ and R. For a gven pont x Ω, we defne M Ω x) as the set of all geodescs passng through the pont x. We use the notaton M Ω x) to represent the number of elements n the set M Ω x). Then, numercally, we can defne the acton of the operator Λ I as the average of the lne ntegrals back-projecton) by ΛI f ) := M Ω x) j=1 I f )X j g) M Ω x), ) where we use the notatons {X j g}, j = 1,,, M Ω x), to denote the elements n the set M Ω x). Notce that the above formula defnes a functon wth doman Ω usng the gven geodesc X-ray transform data I f )X g ). For standard X-ray transform, the above operator Λ gves a good approxmaton to the unknown functon f x), as for the standard back-projecton operator. For geodesc X-ray transform, ths operator provdes an ntal approxmaton, whch s the ntal term n a convergent Neumann seres representaton of f x). Motvated by [5, 4], we wll use an operator A to model the acton of the operator R presented above. In partcular, we wll construct an operator A such that A A) 1 Λ I = Id K, 4) 5

6 where K s an error operator wth K < 1 for some approprate norm, and Id s the dentty operator. From [5, 4], we know that the operator A s essentally ntegrals along geodescs, and the operator A s the adjont operator of A. Usng the above, we can wrte the followng Neumann seres f = K n A A) 1 ΛI f ) n= for the unknown functon f x). Note that I f s the gven data. We remark that, n computatons, the nverse of A A s not the actual nverse of the operator Λ I, but gves a knd of approxmaton to the operator R. In the followng, we wll explan a numercal approxmaton to ths operator A and ts adjont A. We remark that both the operators I and A correspond to ntegratons along geodescs, but we use dfferent notatons snce they wll be defned on two dfferent grds, whch wll be explaned n the next secton. The stuaton s smlar for the operators Λ and A, whch correspond to takng averages of ntegrals on geodescs. Numercal mplementatons In the prevous secton, we presented a general overvew of our numercal procedure and ts correspondng theoretcal motvatons. In ths secton, we gve detals of the mplementatons of the numercal reconstructon algorthm. There are three mportant ngredents n our numercal algorthm. They are 1. Gven the data {I f )X g )}, compute ΛI f ).. Compute the acton of Λ I.. Compute the acton of K := Id A A) 1 Λ I)..1 Implementaton detals Frst, we dscuss the computaton of ΛI f ) usng the gven data set {I f )X g )}. In fact, we wll use the formula ) for ths purpose. We emphasze that, snce we wll only recover the unknown functon f on the set of ponts Z, the output ΛI f ) s also defned only on the same set of grd ponts Z. One dffculty s that one, n general, cannot fnd the geodesc that s passng through a gven pont z Z snce there are only fntely many geodescs n computatons. We wll choose a small neghborhood of z and fnd all the geodescs passng through the ɛ-neghborhood of z. Then, we can apply the formula ) usng ths set of geodescs for each z. Fgure 1 shows how to choose the set of geodescs. The black geodescs shown n Fgure 1 are sad to be the geodesc that s passng through the ɛ-neghborhood of z but the red geodesc s not. Next, we wll dscuss the acton of the operator Λ I. Notce that ths s needed n the computaton of the error operator K. We wll dscuss the computaton of the operator I. The acton of Λ s smlar to the above. For a gven functon f whose values are defned only on the set of grd ponts Z, we wll evaluate I f. Thus, we need to compute the 6

7 X g ε z Fgure 1: An example of choosng the geodescs passng through a gven grd pont z. ntegral of f on a geodesc X g. To compute a geodesc X g, we wll solve the ray equaton 1) n the phase space startng from a partcular ntal pont X ) by the 4th order Runge- Kutta Method. Then we wll obtan a set of ponts {xs )} defnng the geodesc n the physcal space. Next, we use a verson of the trapezodal rule to compute the lne ntegral I f )X g ) of the functon f along the geodesc X g. In partcular, we have I f )X g ) f xs )) x s )) s s 1 ). We remark that, n the above formula, s s 1 ) s the step sze used n the 4th order Runge-Kutta Method. In addton, the expresson x s ) can be computed usng x = H g ξ. The term f xs )) s not well defned snce f s only defned on the set of grd ponts Z and the pont xs ) may not be one of the grd ponts. To overcome ths ssue, we replace f xs )) by f xs )), whch s the lnear nterpolaton of f usng the grd ponts near xs ). So, we wll use the followng formula to compute the acton of f : I f )X g ) f xs )) x s )) s s 1 ). 5) Fnally, we dscuss the acton of the error operator K. In ths part, the man ngredent s to compute the operator A A) 1, whch approxmates the acton of the operator R n ). From [5, 4], we know that the operator A s essentally ntegrals along geodescs, and the operator A performs average of lne ntegrals passng through a gven pont. Notce that, the acton of A s smlar to that of I, and the acton of A s smlar to the acton of Λ. In order to obtan a good approxmaton to the operator R and hence a good reconstructed f, we wll perform the acton of A and A on a fner grd Z f, whch s a refnement of the grd Z. Fgure llustrates the grds n D. In partcular, the grd ponts 7

8 n the set Z are represented by { }, and the grd ponts n the set Z f are represented by both {, }. For the computaton of A, we use the same formula as n 5) but wth the grd Z f. For the computaton of A, we use the same formula as n ) but wth the grd Z f. Thus, the operator A A) and ts nverse A A) 1 take nputs as functons defned on the grd Z f, and gve outputs also as functons defned on the grd Z f. Fgure : Illustraton of the two grds. All the { } form the grd Z and all {, } form the fner grd Z f. To complete the steps, we need a projecton operator P, whch maps functons defned on the fner grd Z f to functons defned on the grd Z. In fact, the operator P s the restrcton operator. Then the operator P maps functons defned on the grd Z to functons defned on the fner grd Z f. Now, we can wrte down the reconstructon formula: where f = K n PA A) 1 P ΛI f ) n= K = Id PA A) 1 P Λ I). To construct f, we wll use a truncated verson of the above nfnte sum. Notce that, to regularze the problem, we wll replace the above sum by f = K n PA A δ ) 1 P ΛI f ) 6) n= where δ > s a regularzaton parameter and s the Laplace operator. Fgure : An example of dvdng the frst and second layers of D doman nto dsks. 8

9 . A layer strppng algorthm for D models The algorthm can be mplement on a D doman. We consder domans wth compact closure Ω equpped wth a functon ρ : Ω [, ) whose level sets Σ t = ρ 1 t), t < T, are strctly convex, wth Σ = X and X \ t [,T) Σ t = ρ 1 [T, )) ether havng measure of havng empty nteror. In general, one can take a fnte partton of [, T] by such ntervals [t, t +1 ), = 1,..., k and proceed nductvely to recover f on t [,T) Σ t. We consder the sphere as our D doman. Frst, we dvde the D doman nto k layers by the functon ρ. We set t = 1, = 1,..., k + 1. Then {σ = ρ 1 [t, t +1 ))} k =1 s partton of D doman. We then separate the layers nto m small domans whch s smlar to a D dsc by the functon ρ j. Smlarly, {σ j = ρ 1 j [t, t +1 ))} m j=1 s the set of the small doman whch the unon of the set returns the layer {ρ 1 [t, t +1 ))}. Fgure shows an example of dvdng the frst and second layers of D doman nto small dsks by yellow lnes. The colored layers are dfferent D dsks. Ths separaton s done for several drectons.the heght of ths dsc s the grd sze h so each dsc contans about one layer of grd ponts, that s, k = r/h. Second, we choose dfferent ntal ponts X ) of the geodescs accordng to the dscs. These ntal ponts set at the mddle of the boundary of the dsc and also form a crcle. Then we solve the system 1) to get the geodesc and calculate the lne ntegral of f for each geodesc X j g. Thrd, for each small domans, we use the regularzed verson of the formula f = n= Kn A A) 1 Λ f to calculate the approxmaton value of f at each fne grd pont z. Also, for each small domans, the calculaton of f n the nner layer use the result of that of outer layers. Fnally, we add up all the approxmaton values of f of each small domans. If there are some common ponts between dfferent small domans, then we take the average of approxmaton values. The whole algorthm s llustrated n the flow chart n Fgure 4. 4 Numercal experments In ths secton we demonstrate the performance of our method usng several test examples. The doman Ω s a sphere wth center.5,.5,.5) and radus. To solve the system 1) to get the geodesc curves, we appled the classcal Runge-Kutta method of 4th order. Also, for the calculatons of the error operator K n 6), the regularzaton parameter s chosen as δ =. 4.1 Accuracy tests In ths secton, we llustrate the performance of our reconstructon method usng several unknown functons f x). The reconstructon s defned on a fne grd wth grd sze h =.. We assume that the speed c s chosen as cx, y, z) = 1 +. cos x.5) + y.5) + z.5) ). We consder the followng fve test cases for ths experment: f 1 =.1 + snπx + y + z)/1), 9

10 Dvde doman nto k layers Set =1 Dvde th layer nto small dsks end For each small dsk Solve ODE to get a set of geodescs Calculate lne ntegral of f Calculate approxmaton f = n= Kn A A) 1 Λ f Sum up all approxmaton f for common ponts Check f =k no update by +1 yes Approxmaton f s found Fgure 4: The layer strppng algorthm. f =.1 + snπx + y)/1) + cosπz/), f = x + y + z /, f 4 = 1 + 6x + 4y + 9z + snπx + z)) + cosπy), f 5 = x + e y+z/. We wll apply our algorthm to reconstruct each of these fve functons. Table 1 shows the L -norm relatve errors of these fve test cases. For these cases, we compute the reconstructons usng up to the frst 5 terms n the seres 6). We see clearly that the results are mproved as we add more terms n the Neumann seres 6). Fgures 5 and 6 show the exact and reconstructed solutons of these test functons wth n = 4. We observe that our scheme s able to produce very good reconstructons. 1

11 relatve error f 1 f f f 4 f 5 n= 47.8% 47.45% 46.8% 46.97% 47.18% n=1.89% 4.1%.4%.61%.76% n= 1.9% 1.6% 1.% 1.% 1.% n= 8.5% 8.5% 9.% 9.48% 8.5% n=4 6.99% 6.74% 8.71% 8.8% 7.14% Table 1: Relatve errors for reconstructng the functons f, = 1,, Accuracy tests wth nosy data Next, we show some numercal tests usng nose contamnated data Λ ɛ g. We use the same speed cx, y, z) and grd sze h =. as n Secton 4.1 and perform ths test by reconstructng the functon gx, y, z) =.1 snπx + y + z)/1). The measurement data has been contamnated by unformly dstrbuted nose ɛ, Λ ɛ g := Λg + ɛ wth relatve error ɛ / Λg =.5.e. 5% nose), where ɛ s a random functon. In ths smulaton, we use n = 5, that s, we use 6 terms n the Neumann seres 6). Fgure 7 shows the exact and reconstructed solutons usng the exact data and nosy data. By usng the exact data that s, the data wth no nose), we obtan a reconstructed soluton wth 6.7% error, whle usng the nosy data, we obtan a reconstructed soluton wth 8.5% error. We observe that our scheme s qute robust wth respect to some nose n the data. 4. Experments wth dfferent speeds Fnally, we compare the performance of the reconstructon usng dfferent choces of sound speeds. We use grd sze h =. as above and perform ths test by reconstructng the followng functon f x, y, z) =.1 snπx + y + z)/1). We wll test the performance of our algorthm usng three choces of sound speeds. The frst speed n our test s defned as c 1 x, y, z) = 1 + snπx) snπy) snπz). The second and the thrd tests are related to the well-known benchmark problem: the Marmous model, shown n Fgure 8. In our smulatons, we take two sphercal sectons of the Marmous model, called c and c, as shown n Fgure 8. Fgure 9 shows the exact and reconstructed solutons usng the three speeds and n = 4, that s, fve terms n the Neumann seres. From these fgures, we observe very good 11

12 n = n = 1 n = n = n = 4 relatve error for test speed c %.88% 14.19% 11.6% 11.47% relatve error for test speed c 48.% 5.9% 15.71% 1.4% 11.8% relatve error for test speed c 58.18% 5.41%.78% 16.5% 1.9% Table : Relatve errors for usng the test speeds wth grd sze h =.. qualty of reconstructons. The errors of the reconstructons usng the frst fve terms of the Neumann seres are shown n Table 4. We observe that the errors decrease as more terms n the Neumann seres are used. Ths confrms the theoretcal fndngs. Lastly, we perform a smlar test for the Marmous model wth the reconstructon of f 5 = x + e y+z/, whch s the same as the functon n prevous secton 4.1. We also consder the performance wth nosy data and the use of a fner grd sze h =.1. The correspondng results usng the frst 6 terms n the Neumann seres are shown n Table. We observe that the qualty of approxmaton s mproved when more terms are used n the Neumann seres. We also observe that our method s qute robust wth respect to 5% nose n the data. On the other hand, the performance of the reconstructons of f and f 5 s qute smlar. n = n = 1 n = n = n = 4 n = 5 for functon f 5.18% 8.95% 17.5% 1.5% 1% 9.4% for functon f and 5% nosy data 5.8% 9% 19.47% 14.15% 11.75% 18% for functon f % 8.95% 17.64% 1.57% 1.56% 9.84% Table : Relatve errors usng the Marmous model wth fner grd sze h =.1. 5 A new phase space method for traveltme tomography Our new method s related to the frst part of the work and the Stefanov-Uhlmann dentty whch lnks two Remannan metrcs wth ther travel tme nformaton. The method frst uses those geodescs that produce smaller msmatch wth the travel tme measurements and contnues on n the sprt of layer strppng. We then apply the above reconstructon method to the Stefanov-Uhlmann dentty n order to solve the nverse problem for reconstructng the ndex of refracton of a medum. The new method s related to the methods n [, ]. However, we wll use the above new method for X-ray transform n solvng an ntegral equaton nvolved n the nverson process. We wll dscuss the method n detal n the followng secton. 5.1 Mathematcal formulaton for traveltme tomography Let Ω be a strctly convex bounded doman wth smooth boundary Γ = Ω n R d and let gx) = g j x)) be a Remannan metrc defned on t. Let d g x, y) denote the geodesc 1

13 dstance between x and y for any x, y Ω. The nverse problem s whether we can determne the Remannan metrc g up to the natural obstructon of a dffeomorphsm by knowng d g x, y) for any x, y Γ. Whle theoretcally there are a lot of works addressng ths queston, we explore the possblty of recoverng the Remannan metrc numercally. see [1, 4, 9, 11, 1, 16, 18]) We use phase space formulaton so that multpathng n physcal space s allowed. Rather than the boundary dstance functon we look at the scatterng relaton whch measures the pont and drecton of ext of a geodesc plus the travel tme f we know the pont and drecton of entrance of the geodesc. The notaton mostly follow the prevous sectons and the prevous work []. The contnuous dependence on the ntal data of the soluton of the Hamltonan system s characterzed by the Jacoban, J gj s) = J gj s, X ) ) := X g j X ) s, X) ) = x x ) ξ x ) x ξ ) ξ ξ ) It can be shown easly from the defnton of J and the correspondng Hamltonan system the detal wll be derved n the Appendx) that J gj, j = 1,, satsfes dj ds = MJ, J) = I, where, n terms of H = H gj j = 1, ), the matrx M s defned by Hξ,x H M = ξ,ξ H x,x H x,ξ Followng [], we use the lnearzed Stefanov-Uhlmann dentty the detal wll be derved n the Appendx) : X g1 t, X ) ) X g t, X ) ) t ). J g t s, Xg s, X ) ) ) g V g g 1 g ) X g s, X ) ) ) ds, whch s the foundaton for our numercal procedure. By the group property of Hamltonan flows the Jacoban matrx s equal to J g t s, Xg s, X )) = J g t, X ) ) J g s, X ) ) 1 ). 7) In the case of an sotropc medum, g j = 1 c δ j, where c s a functon from R d to R. Then V gk = c k ξ, c k)c k ξ ). 1

14 Hence the dervatve of V wth respect to g, g V g λ) s gven by g V g λ) = cλξ, c λ + λ c) ξ ). In the case of an sotropc medum wth the metrc g j = 1 c δ j, we have M = cξ c) T c)c ξ c) c) T ξ c) I c c)ξ T 5. New phase space method usng reconstructon of X-ray transform The numercal method s an teratve algorthm based on the lnearzed Stefanov-Uhlmann dentty usng a hybrd approach. The metrc g s defned on an underlyng Euleran grd n the physcal doman. The ntegral equaton 8) s recovered by the geodesc X-ray data along a ray for each X ) n phase space. On the ray, values of g are computed by nterpolaton from grd pont values. Hence, each ntegral equaton along a partcular ray yelds a lnear equaton for grd values of g n the neghborhood of the ray n the physcal doman. Here s the teratve algorthm for fndng g. Let Z be the set of all grd pont z j, j = 1,,..., p. Let X ) ntal locatons and drectons of those m measurements X g t, X ) ext tme correspondng to the th geodesc startng at X ) of the metrc g, we construct a sequence g n as follows. Defne the msmatch vector and an operator Î along the th geodesc by Î g g n ) := t d n = X g t, X ) ) X g nt, X ) ) ). S, = 1,,..., m, be the ) S + where t s the. Startng wth an ntal guess J g n t s, Xg ns, X ) ) ) g nv g ng g n ) X g ns, X ) ) ) ds Note that both d n and Î g g n ) depend on X ). Then we use the above reconstructon method to recover λ := g g n at each grd ponts by the msmatch vector. Thus, we defne a new msmatch vector d ˆ n from Î ˆ d n := Î d n Also, we defne an operator I based on 8) along the th geodesc Iλ)X g t, X ) )) = Î Î λ). Thus, for each n, we use the reconstructon formula λ = K n PA A δ ) 1 P ΛIλ), n= 14

15 where P, A s defned as the same as the prevous secton and Hence we then defne, K = Id PA A) 1 P Λ I). g n+1 = g n + λ. We remark that more detals can be found n the Appendx. 5. Numercal mplementatons In ths secton, we wll brefly explan the detals of the numercal mplementatons of phase space method. Frst, we wll dscuss the detal of constructng the msmatch vector d n. Then, we wll explan the calculaton of the lne ntegrals,.e. the operator Î. Fnally, the detal of the reconstructon formula and how the metrc s updated wll be explaned Setup of msmatch vector The msmatch vector s defned by d n = X g t, X ) ) X g nt, X ) ). Hence, we need a set of the ntal locatons and drectons {X ) }. For our settngs, we wll dvde the D doman nto dfferent layers and thus dvde the layer nto many small dsks. For each dsk, we wll choose 9 ntal locatons and drectons around the boundary of the dsks evenly. From ths set of data, we wll derve a set of msmatch vectors usng the guess g n and also the observed data X g t, X ) ). Also, snce the method of choosng the ntal locatons and drectons cannot promse the geodesc wll reman n the layer. We wll elmnate the geodesc whch does not reman n the same layer. Then, the remanng set of the msmatch vector wll be the data we used to derve the method. 5.. Calculaton of the lne ntegrals The operator Î s defned to calculate the lne ntegrals, whch s defned by Î g g n ) := t J g n t s, X g ns, X ) ) ) g nv g ng g n ) X g ns, X ) ) ) ds. Snce we have the dscrete phase space of each geodesc,.e. X g ns j, X ) ) for any s j t, then we can approxmate the operator by Î g g n ) s j J g n t s j, X g ns j, X ) ) ) g nv g ng g n ) X g ns j, X ) ) ) X g ns j, X ) )) s j s j 1 ). Hence, the operator can be approxmated by a matrx and thus the adjont operator Î. 15

16 5.. Update of the metrc After we construct the lne ntegral operators, we wll apply the reconstructon formula to compute the update of the metrc. Based on the reconstructon formula, t s the nfnte sum of Neumann seres. In computaton, we need to choose some terms of the nfnte sum to represent the whole term. Here, we choose the frst fve terms, snce ths terms represent the man part of the sum. Next, ths update of the metrc wll be completed on each dsk. To ensure the metrc can recover, we do the update fve tmes whch make the successve error to be small. After dong the update for each dsks n the same layer, we wll compute the fnal metrc of ths layer and move on to the next layer. When we compute the fnal metrc of ths layer, there are some overlappng regons for dfferent dsks. Then we take the mean of ths values to calculate the fnal metrc. 5.4 Numercal experments In ths secton we demonstrate the performance of our method usng several test examples. The doman Ω s a sphere wth center.5,.5,.5) and radus. To solve the system to get the geodesc curves, we appled the classcal Runge-Kutta method of 4th order. Also, for the calculatons of the error operator K, the regularzaton parameter s chosen as δ = Constant case and the lnear case To test our algorthm, we test t on dfferent speeds. Frst, we test the constant case and the lnear case. For these two test cases, we dvde the D doman nto layers and we llustrate the performance n the frst two outermost layers. For the constant case g = 1, we have the relatve error.4% for frst layer and.5% for the second layer. Thus, the speeds are well recovered n ths two layer. Also, refer to the Table 4, we can note that the relatve error grows after addng more layers. For the lnear case g = x + y + z), we have the relatve error.77% for frst layer and.599% for the second layer. For ths case, we have the smlar result as the constant case. By observng Table 4, we also note that the frst two layers are recovered almost exactly and the errors grow after a few layers. The fact that there are larger errors n nner layers s because there are less data avalable for those regons. 1 st layer nd layer rd layer 4 th layer 5 th layer relatve error for constant case.4%.5%.164%.5194% 1.88% relatve error for lnear case.77%.599%.647%.676% 14.1% Table 4: Relatve errors for dfferent cases. 16

17 5.4. Marmous model Next, we test the performance usng the Marmous model. We dvde the D doman nto 1 layers and we recover the model n the frst few outermost layers. Then we have the relatve error 8.88% for frst layer, % for the second layer,9.6% for the thrd layer and % for the forth layer. Fgure 1-1, show the graphs of true and approxmate soluton of frst,second, thrd and forth layers of standard Marmous model. We observe that the recovered solutons are n good agreement wth the exact solutons. 1 st layer nd layer rd layer 4 th layer 5 th layer relatve error 8.88% % 9.6% % 1.91% Table 5: Relatve errors for recoverng the Marmous model. Concluson In ths paper, we develop a novel numercal scheme for the reconstructon of an unknown functon usng ts geodesc X-ray transform. Our scheme s based on a convergent Neumann seres and a layer strppng algorthm. In our reconstructon process, we wll frst compute the unknown functon on each small neghborhood near the boundary of the doman. When the unknown functon s reconstructed on a layer near the boundary, we wll reconstruct the unknown functon one layer nsde the doman, and contnue ths process untl the functon s reconstructed for the whole doman. Our method s very effcent, snce all computatons are performed locally. We present a few test cases, ncludng the Marmous model, to show the performance of the scheme. We observe that the qualty of approxmaton wll be mproved as more terms n the Neumann seres are used. We also observe that the method s qute robust to some nose n the data. Fnally, we apply ths method to a travel tme tomography n D, n whch the nverson of the geodesc X-ray transform s one mportant step, and present several numercal results to valdate the scheme. Acknowledgement The research of Erc Chung s partally supported by the Hong Kong RGC General Research Fund Project: 14114) and CUHK Drect Grant for Research The research of Gunther Uhlmann s partally supported by NSF and S-Yuan Professorshp at IAS, HKUST. References [1] G. Beylkn 198 Stablty and unqueness of the soluton of the nverse knematc problem of sesmology n hgher dmensons J. Sov. Math

18 [] Chung E, Qan J, Uhlmann G and Zhao H K 7 A new phase space method for recoverng ndex of refracton from travel tmes Inverse Problems 9-9 [] Chung E, Qan J, Uhlmann G and Zhao H K 11 An adaptve phase space method wth applcaton to reflecton traveltme tomography Inverse Problems [4] C. Croke, N. S. Darbekov and V. A. Sharafutdnov Local boundary rgdty of a compact Remannan manfold wth curvature bounded above Trans. Am. Math. Soc [5] B. Frgyk, P. Stefanov and G. Uhlmann 8 The x-ray transform for a generc famly of curves and weghts J. Geom. Anal [6] K. Guo and D. Labate 1 Optmal recovery of D X-ray tomographc data va sherbet decomposton Advances n Computatonal Mathematcs 9) 7-55 [7] S. Ivanov 1 Volume comparson vs boundary dstances Proceedngs of the Internatonal Congress of Mathematcans New Delh [8] Y. Kurley, M. Lassas and G. Uhlmann 1 Rgdty of broken geodescs flow and nverse problems Am. J. Math [9] R. Mchel 1981 Sur la rgdte mposee par la longueur des geodesques On the rgdty mposed by the length of geodescs) French) Invent. Math [1] F. Monard 14 Numercal mplementaton of two-dmensonal geodesc X-ray transforms and ther nverson SIAM J. Imagng Scences 7 no [11] R. G. Mukhometov 1977 The reconstructon problem of a two-dmensonal Remannan metrc, and ntegral geometry Russan) Dokl. Akad. Hauk SSSR no. 1-5 [1] R. G. Mukhometov 198 On a problem of reconstructng Remannan metrcs Sberan Math. J. 4- [1] R. G. Mukhometov and V. G. Romanov 1978 On the problem of fndng an sotropc Remannan metrc n an n-dmensonal space Russan) Dokl. Akad. Hauk SSSR 4 no [14] F. Natterer and F. Wubbelng 1 Mathematcal methods n mage reconstructon SIAM Monographs on Mathematcal Modelng and Computaton SIAM Phladelpha [15] L. Pestov and G. Uhlmann 4 On characterzaton of the range and nverson formulas for the geodesc x-ray transform Int. Math. Res. Not [16] L. Pestov and G. Uhlmann 5 Two-dmensonal compact smple Remannan manfolds are boundary dstance rgd Ann. Math [17] J. Qan, P. Stefanov, G. Uhlmann, and H. Zhao 11 An effcent neumann seresbased algorthm for thermoacoustc and photoacoustc tomography wth varable sound speed SIAM J. Imagng Scences

19 [18] V. A. Sharafutdnov 1994 Integral Geometry of Tensor Felds Utrecht:VSP) [19] P. Stefanov and G. Uhlmann 1998 Rgdty for metrcs wth the same lengths of geodescs Math. Res. Lett [] P. Stefanov and G. Uhlmann 4 Stablty estmates for the x-ray transform of tensor felds and boundary rgdty Duke Math. J [1] P. Stefanov and G. Uhlmann 5 Boundary rgdty and stablty for generc smple metrcs J. Am. Math. Soc [] P. Stefanov and G. Uhlmann 5 Recent progress on the boundary rgdty problem Electron. Res. Announc. Am. Math. Soc [] P. Stefanov and G. Uhlmann 8 Boundary and lens rgdty, tensor tomography and analytc mcrolocal analyss Algebrac Analyss of Dfferental Equatons, Festschrft n Honor of Takahro Kawa edted by T. Aok, H. Majma, Y. Kate and N. Tose 75-9 [4] G. Uhlmann and A. Vasy 15 The nverse problem for the local geodesc ray transform Invent. math. DOI 1.17/s A Appendx A.1 Formaton of the Hamltonan system The contnuous dependence on the ntal data of the soluton of the Hamltonan system s characterzed by the Jacoban, J gj s) = J gj s, X ) ) := X g j X ) s, X) ) = x x ) ξ x ) x ξ ) ξ ξ ) ). 19

20 It can be shown easly from the defnton of J and the correspondng Hamltonan system that J gj, j = 1,, satsfes ) dj ds = d Xgj ds X ) s, X) ) ) = d ds = = x x ) ξ x ξ ) ξ x ) ξ ) d x d x ds x ) ds ξ ) d ξ d ξ ds x ) ds ξ ) dx dx x ) ds ξ ) ds = = = MJ dξ x ) ds H ξ x ) H x x ) H ξ x H x x dξ ξ ) ds H ξ ξ ) H x ξ ) ) ) x + H ξ ξ x ) ξ x ) x + H x ) x ξ ξ x ) ) H ξ x H x x x + H ξ ξ ξ ) ξ ξ ) x + H ξ ) x ξ ξ ξ ) ) Then, we have the followng equaton: dj ds = MJ, J) = I, where, n terms of H = H gj j = 1, ), the matrx M s defned by Hξ,x H M = ξ,ξ H x,x H x,ξ A. Lnearzng the Stefanov-Uhlmann dentty Gven boundary measurements for g 1, we are nterested n recoverng the metrc g 1. We lnk two metrcs by ntroducng the functon ). Fs) := X g t s, Xg1 s, X ) ) ),

21 where t = maxt g1, t g ) and t g = t g X ) ) s the length of the geodesc ssued from X ) wth the endpont on Γ. Then we frst have F) = X g t, Xg1, X ) ) ) = X g t, X ) ) Ft) = X g t t, Xg1 t, X ) ) ) = X g, Xg1 t, X ) ) ) = X g1 t, X ) ) Consequently, we have t F s)ds = Ft) F) = X g1 t, X ) ) X g t, X ) ). On the other hand, we can denote V gj := ) H gj ξ, H g j x and then we get F s) = d ds X g t s, Xg1 s, X ) ) ) Then we clam that = dx g t s, Xg1 s, X ) ) ) + X g t s, ds X ) Xg1 s, X ) ) ) dx g 1 ds s, X) ) = V g X g t s, Xg1 s, X ) ) )) + X g t s, X ) Xg1 s, X ) ) ) ) V g1 X g1 s, X ) ) V g X g t s, Xg1 s, X ) ) )) = X g t s, X ) Xg1 s, X ) ) ) ) V g X g1 s, X ) ) To prove the clam, we frst need to notce that d XT s, Xs, X ) )) ds s= = d XT, X ) ) ds s= = Also, we have d XT s, Xs, X ) )) ds s= =VXT, X ) )) + X X ) T, X) )VX ) ), T 1

22 By settng T = t s and X ) = X g1 s, X ) ), thus we prove the clam. Hence, the tme ntegral on the left-hand sde s equal to the followng [19], t F s)ds = = t t X g X ) t s, Xg1 s, X ) ) ) V g1 V g ) Xg1 s, X ) ) ) ds J g t s, Xg1 s, X ) ) ) V g1 V g ) Xg1 s, X ) ) ) ds, where V gj = ) ) Hgj ξ, H g j = g 1 ξ, 1 x xg 1 ξ) ξ. Ths s the so-called Stefanov-Uhlmann dentty. We lnearze the above dentty about g by followng a full underlyng patn g, t F s)ds t J g t s, Xg1 s, X ) ) ) g V g g 1 g ) X g s, X ) ) ) ds, where g V g λ) s the dervatve of V g wth respect to g at λ. Thus, we have the followng formula, X g1 t, X ) ) X g t, X ) ) t J g t s, Xg s, X ) ) ) g V g g 1 g ) X g s, X ) ) ) ds, 8)

23 a) exact soluton for f 1 b) approxmate soluton for f 1 c) exact soluton for f d) approxmate soluton for f e) exact soluton for f f) approxmate soluton for f Fgure 5: Graphs of true and reconstructed solutons for functons f 1, f, f.

24 a) exact soluton for f 4 b) approxmate soluton for f 4 c) exact soluton for f 5 d) approxmate soluton for f 5 Fgure 6: Graphs of exact and reconstructed solutons for functons f 4, f 5. 4

25 a) exact soluton b) approxmate soluton for g 1 c) approxmate soluton for g Fgure 7: Graphs of exact and reconstructed solutons usng the exact data g 1 error = 6.7%) and the nosy data g error = 8.5%). 5

26 a) The D Marmous model. b) A sphercal secton of the Marmous model as the test speed c. c) A sphercal secton of the Marmous model as the test speed c. Fgure 8: The D Marmous model. 6

27 a) exact soluton b) approxmate soluton for c 1 c) approxmate soluton for c d) approxmate soluton for c Fgure 9: Graphs of exact and approxmate solutons reconstructng from speed c 1 and the Marmous models c and c. 7

28 a) exact soluton for frst layer front) b) approx. soluton for frst layer front) c) exact soluton for frst layer back) d) approx. soluton for frst layer back) Fgure 1: Graphs of true and approxmate soluton of frst layer of standard marmous model wth fewer layers 8

29 a) exact soluton for second layer front) b) approx. soluton for second layer front) c) exact soluton for second layer back) d) approx. soluton for second layer back) Fgure 11: Graphs of true and approxmate soluton of second layer of standard marmous model wth fewer layers 9

30 a) exact soluton for thrd layer front) b) approx. soluton for thrd layer front) c) exact soluton for thrd layer back) d) approx. soluton for thrd layer back) Fgure 1: Graphs of true and approxmate soluton of thrd layer of standard marmous model wth fewer layers

31 a) exact soluton for forth layer front) b) approx. soluton for forth layer front) c) exact soluton for forth layer back) d) approx. soluton for forth layer back) Fgure 1: Graphs of true and approxmate soluton of forth layer of standard marmous model wth fewer layers 1