Finite elements parameterization of optical tomography with the radiative transfer equation in

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1 Home Search Collection Journal About Contact u My IOPcience Finite element parameterization of optical tomography with the radiative tranfer equation in frequency domain Thi article ha been downloaded from IOPcience. Pleae croll down to ee the full text article J. Phy.: Conf. Ser ( View the table of content for thi iue, or go to the journal homepage for more Download detail: IP Addre: The article wa downloaded on 04/07/2012 at 16:47 Pleae note that term and condition apply.

2 Finite element parameterization of optical tomography with the radiative tranfer equation in frequency domain O. Balima 1,2 and Y. Favennec 3 and F. Dubot 1 and D. Roue 1 1 Indutrial reearch chair in technologie of energy and energy efficiency (t3e). École de Technologie Supérieure, 1100, rue Notre-Dame Ouet, Montral (Qc), Canada H3C 1K3. 2 Département de Science Appliquée. Univerité du Québec à Chicoutimi, QC, Canada G7H 2B1. 3 LTN UMR CNRS 6607 Polytech Nante - La Chantrerie, Rue Chritian Pauc, BP 50609, Nante Cedex 3, France. obalima@uqac.ca Abtract. Optical tomography i a technique of probing emi-tranparent media with the help of light ource. In thi method, the patial ditribution of the optical propertie inide the probed medium i recontructed by minimizing a cot function baed on the error between the meaurement and the prediction of a numerical model of light tranport (alo called forward/direct model) within the medium at the detector location. Optical tomography with finite element method involve generally continuou formulation where the optical propertie are contant per meh element. Thi tudy propoe a numerical analyi in the parameterization of the finite element pace of the optical propertie in order to improve the accuracy and the contrat of the recontruction. Numerical tet with noied data uing the ame algorithm how that continuou finite element pace give better reult than dicontinuou one by allowing a better tranfer of the information between the whole computational node of the inverion. It i een that the reult are more accurate when the number of degree of freedom of the finite element pace of the optical propertie (number of unknown) i lowered. Thi how that reducing the number of unknown decreae the ill-poed nature of the invere problem, thu it i a promiing way of regularizing the inverion. 1. Introduction Among the new imaging modalitie expected to be available in the future, optical tomography i one of the mot promiing although numerou difficultie till exit. It i ued in flow diagnotic, medical imaging, food proceing, etc. Thi laer-baed probing technique may be divided into direct imaging in which the emerging ignal i directly ued for projection and recontruction imaging baed on invere problem alo called optical tomography. For both of them, recent reearch tend to how that the ue of the long term photon, which have travelled for a long time in the whole ample to be probed, generate more information to the image recontruction. Recently, an increaing interet ha been devoted to the finite element formulation when olving the radiative tranfer equation. Thi tendency in favor of the finite element method follow from it implicity, flexibility and property of being able to handle complex geometrie and advection type equation. From a tandard Galerkin formulation [1], a number of tudie Publihed under licence by IOP Publihing Ltd 1

3 have been done to improve the accuracy of thee computational method uch a the Streamline Upwind Petrov Galerkin [2, 3], the Leat Square formulation [4, 5] and the Dicontinuou Galerkin formulation [6, 7], to name but a few. Generally, optical tomography with finite element method involve continuou formulation [8, 9, 10, 11]. However, continuou finite element formulation uffer from the lack of local conervativity compared to the dicontinuou formulation which allow the ue of numerical fluxe to achieve local conervativity [12, 13]. In addition, high order accuracy can be achieved with the Dicontinuou Galerkin formulation by uing higher order polynomial approximation than the finite volume method. The Dicontinuou finite element formulation can be viewed a an all-in-one formulation a with the ame formulation, the finite volume method i obtained when contant polynomial element are ued, while uing a continuou function yield the tandard finite element method [14]. Thi tudy preent an analyi of optical tomography where a parameterization of the optical propertie i ued to improve the accuracy of the recontruction. Numerical tet of image recontruction are performed to gauge the recontruction accuracy in relation with the pecific feature of the explored finite element model and the related finite element pace of the optical propertie. 2. Phyical model In optical tomography, the forward model aim at computing the prediction of the boundary reading once the optical propertie are known. Below, we preent the forward model equation baed on the frequency domain radiative tranfer equation and the ued olution method Radiative tranfer in frequency domain In thi tudy, the retained forward model i the Fourier tranform of the tranient radiative tranfer equation which write in a given direction Ω and for each patial poition r D by [15]: ( ) Ω I(r, Ω, iω ω) + c + κ + σ I(r, Ω, ω) = σ I(r, Ω, ω)φ( Ω, Ω)dΩ (1) where i = 1, c i the light peed in the medium, ω i the modulation frequency, I(r, Ω, ω) i the radiant power per unit olid angle per unit area at the patial poition r in direction Ω, κ = κ(r) and σ = σ (r) are repectively the aborption and the cattering coefficient, and Φ( Ω, Ω) i the cattering phae function. Generally, the cattering phae function in tiue i given by the Henyey-Greentein phae function [16]: Φ( Ω, Ω) = 1 g 2 (1 + g 2 2g co(θ)) 3/2 (2) where Θ i the cattering angle between direction Ω and Ω and g i the aniotropic factor. The boundary condition for Eq.(1) i a collimated external radiation that penetrate into the medium with the direction Ω c, at poition r 0. The boundary i taken a non reflecting with the correponding boundary condition: I(r 0, Ω) = q 0 δ(r r 0 )δ( Ω Ω c ) Ω n < 0 r D (3) where q 0 i the total heat flux, n i the outward normal unit vector and δ i the Dirac function, D denote the boundary of the domain D. The radiative intenity i eparated into two component I = I c + I where I c and I are repectively the collimated and cattered intenitie within the medium a treated in [5]. It i 2

4 pointed out in [17], that thi eparation technique remove trong dicontinuitie due to Dirichlettype boundary condition (Eq.(3)) and conequently improve the accuracy of the finite element olution. The collimated component obey the extinction law [16] uch that Ω c I c (r, ω) + where the boundary condition i given by ( iω c + κ + σ ) I c (r, ω) = 0 (4) I c (r 0, ω) = q 0 (r, ω)δ(r r 0 ) Ωc n < 0, r D (5) The olution of the collimated part i ued a a ource term for the cattered component I uch that ( ) Ω I (r, Ω, iω ω) + c + κ + σ I (r, Ω, ω) = σ I (r, Ω, ω)φ( Ω, Ω)dΩ + S c (r, Ω, ω) (6) where S c (r, Ω, ω) i the ource term induced by the cattering of I c within the medium. S c i given by S c (r, Ω, ω) = σ I c (r, ω)δ( Ω Ω c )Φ( Ω, Ω)dΩ = σ I c(r, ω)φ( Ω c, Ω) (7) Let u recall that Eq.(6) ue only a vacuum boundary condition which write I (r, Ω, ω) = 0, Ω n < 0, r D. (8) For optical tomography application, the meaurable quantity ued i the exitance on the boundary obtained by P (r) = I (r, Ω, ω) Ω ndω r D. (9) Ω n>0 Let u point out that here the forward model i a ytem of two equation (Eq. (4) and Eq.(6)) that make poible to handle the collimated direction compared to other tudie [10, 18, 19, 20]. A hown in [9], the collimated ource direction act in the adjoint equation for the computation of the objective function gradient Solution method The olution method i baed on the Dicontinuou Galerkin formulation to olve both the collimated and the cattered Eq. (4) and Eq.(6), wherea the dicrete ordinate method i ued to handle the angular dependency Dicrete ordinate method In the Dicrete Ordinate Method, integral over olid angle are replaced by a numerical quadrature [21]. Thu, Eq.(6) i rewritten a a patial differential equation for each dicrete direction Ω m : ( ) iω Ω m I m (r, ω)+ c + κ + σ I m (r, ω) = σ M m =1 ( ) I m (r, ω)φ Ωm, Ωm w m +Sc m (r, ω) (10) where M i the number of direction of the quadrature, w m and I m are repectively the quadrature weight and the radiative intenity in the direction Ω m. The correponding boundary condition write I m (r, ω) = 0 Ωm n < 0, r D. (11) 3

5 The phae function in Eq.(10) i re-normalized in order to avoid normalization error. A in [22, 23], The re-normalized phae function i given by ( Φ Ωm, Ω ) ( m = f m Φ Ωm, Ω ) m (12) where f m i the renormalization factor defined by ( 1 M ( f m = w i Φ Ωm, Ω ) ) 1 i. (13) i= Dicontinuou Galerkin formulation Equation Eq. (4) and Eq.(6) for each dicrete direction m, are advection-type equation that are to be olved. There are many method that can be ued to olve thee equation, among which the finite element method are well uited for handling complex geometrie. Finite element formulation of the radiative tranfer equation ha been ued in optical tomography recently [10, 8, 9]. In thee work, only tandard or continuou finite element are ued. Here we conider the Dicontinuou Galerkin formulation [24]. For an eay preentation of the formulation, let u rewrite the equation of the collimated intenity and the cattered intenity for each direction of the quadrature a β. u + bu = f x D u = h x D (14) where β i the advection direction (β = Ω c or β = Ω m ), u = u(x) i the complex value field, b i the complex extinction coefficient, f the complex ource term, D = {x D, β n < 0} i the inflow boundary according to the advection direction β and h i the inflow boundary complex function that i applied to the ytem. The variational formulation of (14) with the Dicontinuou Galerkin write [17]: Find u h V h uch that B (u h, v h ) = F (v h ) v h V h (15) where V h i the finite element pace where the olution u h i earched, B (u h, v h ) and F (v h ) are repectively the bilinear and linear form given by B (u h, v h ) = k D ( βu h, v h ) k + k D (β.n u h, v h ) k + + k D F (v h ) = (f, v h ) D (β.n h, v h ) D ( β.n u h, v ) h + (bu k \ D h, v h ) D (16) and where u + h and u h are repectively the value of u h in cell k and in the neighboring cell. In the following, the pace of piecewie linear dicontinuou element i choen for the olution of the collimated and the cattered component of the light intenity. The olution method of the forward model equation i done by olving firt the collimated component olution. The reulting ource term (Eq. (7)) i ued for the olution of the cattered part through an iterative procedure where the initial cattered field i null. The iterative algorithm top when the maximum abolute relative error between the current iteration and the previou one on the cattered field i lower than a uer-defined value, i.e. when max i,m I m,k 1,i I m,k,i I m,k,i 10 6 (17) 4

6 where I m,k,i i the dicretized cattered component olution in direction m at iteration k and i i a degree of freedom of the finite element olution. In all computation, linear equation are olved with the SuperLu olver [25]. The criterion Eq. (17) i choen in order to make ure that both the global convergence and the local convergence are achieved. The accuracy of the method i given in [17]. 3. Inverion In the inverion procedure, the aim i to recover the optical propertie of the medium through the minimization of an objective function. Here, a gradient-type optimization i ued where the objective function gradient i deduced through an additional adjoint tate formulation Objective function and gradient computation The objective function to be minimized in the inverion procedure i baed on the error between the meaurement and the prediction at the detector poition with a numerical forward model. Here, our model i baed on a Dicontinuou Galerkin formulation which lead u to ue an integral form of objective function in order to be conitent with dicontinuou field a the value of the olution i not well defined at boundarie due to the dicontinuity of the finite element pace but their integral can be computed. Hence, the objective function write J (θ) = 1 2 N N d n=1 d D d P (r, ω n, I ) M(r d, ω n ) 2 d (18) where D d i the urface of the d th detector uch that D d D, D being the boundary of the domain whoe propertie are to be recovered. θ = (κ, σ) i the vector of parameter, i.e. the aborption and the cattering coefficient. The complex value P (r, ω n, I ) and M(r d, ω n ) are the prediction and the meaurement whom difference are integrated over all the detector for all collimated ource. Next, N i the number of collimated ource, and N d i the number of detector, and z 2 = z z z C where z i i the conjugate of z i. We uppoe that κ and σ belong to the ame finite element pace uch that θ = (κ, σ) R 2Nc where N c i the number of degree of freedom of the choen finite element pace. In the following, the pace of piecewie polynomial function of degree 1 i choen for κ and σ field. The gradient of the objective function i computed with the adjoint method with the following expreion [8]: J (θ) δκ = (λ I δκ) + (λ c I c δκ) c ( J (θ) ( δσ = (λ I δσ ) + (λ c I c δσ c 1 λ I (r, Ω ) ) ), ω Φ( Ω, Ω)dΩ δσ where (..) and (..) c are inner product aociated to the olution pace repectively of I and I c and λ = λ (r, Ω, ω), λ c = λ c (r, ω) are the correponding complex vector which repreent the olution of the following adjoint equation ytem: [ Ω iωc ] + κ a + σ λ (r, Ω, ω) σ λ (r, Ω, ω)φ( Ω, Ω)dΩ = 0 (20) and [ Ω iωc ] + κ a + σ λ c (r, ω) σ λ (r, ( Ω, ω)φ Ω, ) Ω c dω = 0 (21) (19) 5

7 where λ c = 0 and Ω J (θ) nλ + = 0 for Ω I n > 0, r D. We refer to reference [8, 9, 26, 27, 28] for more detail on how to deduce the gradient of the objective function through a Lagrangian formulation Recontruction algorithm Gradient-baed algorithm have hown to be efficient for large cale optimization problem in optical tomography [18, 29]. Here, our recontruction cheme i baed on the limited-memory verion of the BFGS method [30] where a caling of both the objective function and it gradient are ued to handle round-off error and the low level of the boundary meaurement due to the high extinction in the medium [9]. 4. Tet and reult 4.1. Tet decription A 2 cm 2 cm domain which contain two incluion i choen where the heterogeneity in the optical propertie of the medium i repreented by two incluion. The optical propertie of the medium are given in Table 1. It i aumed that the medium i forward-cattering where the phae function i given by the Henyey-Greentein function with an aniotropic factor g= 0.9. Four collimated ource with zero-phaed, modulated at the frequency of 600 MHz are placed at mid-centre of each ide of the quare and the meaurement are done with eight detector of 0.8 cm of extenion regularly placed around the boundarie beyond the ource poition. The experiment meaurement data i computed numerically. The above forward model i ued with a regular fine triangular meh of 8142 element and 24 angular dicrete ordinate (S 6 quadrature). The invere problem i baed on a coarer meh of 3786 element coupled to 24 dicrete direction. In thi tudy, the parameter (κ and σ ) and the tate variable (I c and I ) are computed on the ame meh, but they belong to different finite element pace for parameterization purpoe. The obtained complex-valued intenitie with the original propertie (Table 1) are noied with a Gauian ditribution uch that the level of the noie in db unit i given by ( ) Md SNR = 10 log 10 (22) σ Md where σ Md i the tandard deviation. The inverion computation are performed with the limited memory quai-newton method of Brodyen-Fletcher-Goldfarb-Shanno [30, 31]. The quality of the recontruction i meaured with error ɛ 1 and ɛ 2 defined by: ɛ 1 = 1 N c N c i=1 ( θ r i θi o ) 2 (, ɛ 2 = θ o i D 1/2 (θi r θi o ) 2 dx/ (θi o ) dx) 2 (23) D where N c i the number of degree of freedom related to the parameter finite element pace, and upercript r and o refer to the recontructed and original image, repectively. ɛ 1 repreent the mean quadratic error per degree of freedom and ɛ 2 i the relative error of the recontruction with repect to the original ditribution. Table 1: Optical propertie of the tet medium Background Bottom incluion Top incluion κ 0.25 cm cm cm 1 σ 20 cm 1 30 cm 1 10 cm 1 6

8 Eurotherm Conference No. 95: Computational Thermal Radiation in Participating Media IV IOP Publihing (a) (b) Figure 1: Original ditribution of the optical propertie. (a) i the aborption coefficient, (b) i the reduced cattering coefficient (a) (b) Figure 2: Etimated ditribution where the optical propertie (κ and σ) belong to the piecewie contant finite element pace (P 0). (a) i the etimated aborption coefficient, (b) i the etimated reduced cattering coefficient. Table 2: Comparative accuracy of the recontruction. 1 and 2 are the correponding error, ndof i the number of degree of freedom of the finite element pace. ndof 1 2 1,κ 1,σ 2,κ 2,σ P P 1dc P P

9 Eurotherm Conference No. 95: Computational Thermal Radiation in Participating Media IV IOP Publihing (a) (b) Figure 3: Etimated ditribution where the optical propertie (κ and σ) belong to piecewie linear dicontinuou finite element pace (P 1dc). (a) i the etimated aborption coefficient, (b) i the etimated reduced cattering coefficient (a) (b) Figure 4: Etimated ditribution where the optical propertie (κ and σ) belong to piecewie linear continuou finite element pace (P 1). (a) i the etimated aborption coefficient, (b) i the etimated reduced cattering coefficient 4.2. Reult and analyi A comparative tet i carried out with different finite element pace for the optical propertie where the intenity field are uppoed to be in the pace of linear dicontinuou element (P 1dc). The original ditribution of the optical propertie are preented in Fig. 1 where the reduced 0 cattering coefficient σ = (1 g)σ i ued. A comparative etimation i carried out with the computed noied data. The recontruction i done by taking the parameter repectively in the pace of piecewie contant (P 0), piecewie linear dicontinuou (P 1dc), piecewie linear continuou (P 1) and piecewie quadratic (P 2) element. The recovered ditribution are reported 8

10 (a) (b) Figure 5: Etimated ditribution where the optical propertie (κ and σ) belong to piecewie quadratic finite element pace (P 2). (a) i the etimated aborption coefficient, (b) i the etimated reduced cattering coefficient for piecewie contant (Fig.2), piecewie linear dicontinuou (Fig.3), piecewie linear continuou (Fig.4) and piecewie quadratic (Fig.5) element. It i een that continuou finite element pace give better reult than the dicontinuou one. The reult how that continuou finite element approximation of the optical propertie introduce ome implicit regularization on the inverion by moothing the reult. For each type of finite element pace (continuou or dicontinuou), it i een that increaing the number of degree of freedom do not improve the quality of the recontruction. The mallet error defining the quality of the recontruction i given by the P 1 finite element pace whoe number of degree i the lowet (ee ɛ 2 in Table 2). The error given by ɛ 1 take into account the number of degree of freedom and how that increaing the number of degree of freedom lower the error ɛ 1 a expected. However, taking into account the total number of degree of freedom, the piecewie linear element (P 1) olution remain the more accurate. From the invere analyi point of view, high order parameterization may increae the illpoed nature of the invere problem uch a the exitence of correlation, the number of poible olution and the enitivity to round error and noie. And one may end-up with le accurate olution. Then, reducing the number of unknown i a way of reducing the number of poible olution i.e a way of regularizing the inverion. Phyically, with continuou parameterization, information i well conducted among the whole computational node which improve the recontruction a een in the reult. 5. Concluion In thi tudy, numerical tet of recontruction of optical propertie of cattering and aborbing medium are done. The forward model i a finite element model baed on the dicrete ordinate method and the Dicontinuou Galerkin method. A comparative tudy of the parameterization of the finite element pace of the optical propertie i done with piecewie contant, linear (continuou and dicontinuou), and quadratic element. The reult how that better reult are obtained when the finite element pace of the unknown coefficient have a low number of degree of freedom through a decreae of the ill-poed nature of the invere problem a a regularization technique. The analyi how that continuou finite element parameterization 9

11 give better reult becaue the information i well conducted among the whole computational node which improve the recontruction a een in the reult. In a future tep, we hope to peed up the forward and the adjoint model by uing multiproceing olver in order to ue high order quadrature. Alo, new optimization cheme baed on the finite element flexibilitie will be worked to improve the recontruction. Acknowledgment Thi work wa upported by the National Science and Engineering Reearch Council of Canada (NSERC). Reference [1] An W, Ruan L, Qi H and Liu L 2005 Journal of Quantitative Spectrocopy & Radiative Tranfer [2] Brook A N and Hughe T J R 1982 Computer Method in Applied Mechanic and Engineering ISSN [3] Tarvainen T, Vauhkonen M and Arridge S 2008 Journal of Quantitative Spectrocopy & Radiative Tranfer [4] Pontaza J and Reddy J 2004 Journal of Quantitative Spectrocopy & Radiative Tranfer [5] Balima O, Pierre T, Charette A and Marceau D 2010 Journal of Quantitative Spectrocopy and Radiative Tranfer [6] Ciu X and Li B Q 2005 Journal of Quantitative Spectrocopy & Radiative Tranfer [7] Gao H and Zhao H 2009 Tranport Theory and Statitical Phyic [8] Balima O, Boulanger J, Charette A and Marceau D 2011 Journal of Quantitative Spectrocopy and Radiative Tranfer [9] Balima O, Boulanger J, Charette A and Marceau D 2011 Journal of Quantitative Spectrocopy and Radiative Tranfer [10] Tarvaimen T, Vauhkonen M and Arridge S R 2008 Journal of Quantitative Spectrocopy and Radiative Tranfer [11] Mohan P, Tarvainen T, Schweiger M, Pulkkinen A and Arridge S 2011 Journal of Computational Phyic [12] Hughe T J R, Engel G, Mazzei L and Laron M 2000 A comparion of dicontinuou and continuou Galerkin method baed on error etimate, conervation, robutne and efficiency (Springer-Verlag) [13] Chung T J 2010 Computational fluid dynamic (Cambridge Univerity pre) [14] Li B Q 2006 Dicontinuou Finite Element in Fluid Dynamic and Heat Tranfer (Springer-Verlag) [15] Arridge S R 1999 Invere Problem 15 R41 R93 [16] Modet M F 2003 Radiative Heat Tranfer 2nd ed (Academic Pre) [17] Balima O, Charette A and Marceau D 2010 Journal of Computational and Applied Mathematic [18] Charette A, Boulanger J and Kim H K 2008 Journal of Quantitative Spectrocopy and Radiative Tranfer [19] Ren K, Ball G and Hielcher A H 2006 Society for Indutrial and Applied Mathematic [20] Kim H K and Charette A 2007 Journal of Quantitative Spectrocopy and Radiative Tranfer [21] Fiveland W 1984 Journal of Heat Tranfer [22] Kim T and Lee H 1989 Journal of Quantitative Spectrocopy & Radiative Tranfer [23] Liu L H, Ruan L M and Tan H P 2002 International Journal of Heat and Ma Tranfer [24] Cockburn B and Shu C 1989 Mathematic of Computation [25] Demmel J W, Eientat S C, Gilbert J R, Li X S and Liu J W H 1999 SIAM J. Matrix Analyi and Application [26] Favennec Y, Labbé V and Bay F 2003 International Journal of Computational Phyic [27] Rouizi Y, Favennec Y, Ventura J and Petit D 2009 J. Comput. Phy ISSN [28] Favennec Y, Le Maon P and Jarny Y 2011 Optimization for function etimation Thermal Meaurement and Invere technique : Eurotherm CNRS advanced pring hool ed of Nante U URL [29] Hielcher A H, Kloe A and Hanon K M 1999 IEEE Tranaction on Medical Imaging [30] Liu D and Nocedal J 1989 Mathematical Programming: Serie A and B [31] Cardona A and Idelohn S 1986 International Journal for Numerical Method in Engineering