Regularization of Diffusion Tensor Maps Using a Non-Gaussian Markov Random Field Approach

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1 Regularization of Diffuion Tenor Map Uing a Non-Gauian Markov Random Field Approach Marco Martín-Fernández, Carlo Alberola-López, Juan Ruiz-Alzola, and Carl-Fredrik Wetin 3 E. T. S. Ingeniero de Telecomunicación. Univeridad de Valladolid 470 Valladolid (SPAIN) {marcma,caralb}@tel.uva.e E. T. S. Ingeniero de Telecomunicación. Univeridad de La Palma 3507 La Palma de Gran Canaria (SPAIN) jruiz@dc.ulpgc.e 3 Brighman and Women Hopital Harvard Medical School, Dept. of Radiology 05 Boton, MA. (USA) wetin@bwh.harvard.edu Abtract. In thi paper we propoe a novel non-gauian MRF for regularization of tenor field for fiber tract enhancement. Two entitie are conidered in the model, namely, the linear component of the tenor, i.e., how much line-like the tenor i, and the angle of the eigenvector aociated to the larget eigenvalue. A novel, to the bet of the author knowledge, angular denity function ha been propoed. Cloed form expreion of the poterior denitie are obtained. Some experiment are alo preented for which color-coded image are viually meaningful. Finally, a quantitative meaure of regularization i alo calculated to validate the achieved reult baed on an averaged meaure of entropy. Introduction Diffuion Tenor (DT) Magnetic Reonance Imaging (MRI) i a volumetric imaging modality in which the quantity aigned to each voxel of the volume canned i not a calar, but a tenor that decribe local water diffuion. Tenor have direct geometric interpretation, and thi erve a a bai to characterize local tructure in different tiue. The procedure by which tenor are obtained can be conulted elewhere [9]. The reult of uch a proce i, ideally peaking, a 3 3 ymmetric poitive-emidefinite (pd) matrix. Tenor upport information of the underlying aniotropy within the data. A a matter of fact, everal meaure of uch aniotropy have been propoed out of tenor to make thing eaier to interpret; ee, for intance, [,9]. However, thee meaure rely of the ideal behavior of the tenor, which may be in ome cae far from reality due to ome ource of noie that may be preent in the imaging proce itelf. A wa pointed out in [7], periodic beat of the cerebro-pinal fluid and partial volume effect may add a non-negligible amount of noie to the data, and the reult i that the hypothei of poitive-emidefinitene may not be valid. Author are aware of thi fact, o ome regularization procedure have been propoed in the pat [5,6,7,9]. To whom correpondence hould be addreed. R.E. Elli and T.M. Peter (Ed.): MICCAI 003, LNCS 879, pp. 9 00, 003. c Springer-Verlag Berlin Heidelberg 003

2 Regularization of Diffuion Tenor Map 93 In thi paper we focu on regularization of DT map uing Markov Random Field (MRF); other regularization philoophie exit (ee, for intance, [8] and [0] and reference therein) although they will not be dicued in the paper. About MRF we are aware of the exitence of other Markovian approache to thi problem [6,7], in which the method preented i called by the author the Spaghetti model. Thee paper propoe an intereting optimization model for data regularization. However, ome iue could be a matter of dicuion. The author build their MRF on the bai of a neighborhood ytem that may change through the optimization proce, a fact that i not theoretically correct, though acceptable practical reult may be obtained. In addition, the tranition model ued by the author doe not eem to have a clear probabilitic interpretation, but, in our opinion, only a functional interpretation. We have recently preented another MRF approach to regularization of DT map [4], which wa baed on Gauian aumption and which operated eparately on each tenor component. Reult, a viually aeed, were acceptable and the optimization proce guaranteed pd olution. However, the price to pay wa the need for dicarding weep in the tochatic optimization proce. In thi paper we decribe an alternative methodology to the one reported in [4] in which, for a -D cae, the pd condition i forced naturally by the model and, for the 3-D cae, the two larget eigenvalue are aured to be poitive. The model i built upon a Bayeian philoophy in which the two term, namely, the prior and the likelihood function, have a clear phyical meaning. Cloed form expreion have been obtained for the poterior, o the reulting model ha a olid probabilitic foundation and mathematical elegance. The maximum a poteriori (MAP) etimator i found by mean of the imulated annealing algorithm [3]. Baic on Tenor Proceing It i well known from matrix theory that a pd tenor, ay A, can be decribed in term of two matrice, pecifically, A = QΛQ T, where Q i the eigenvector matrix, with eigenvector located in every column, and Λ i a diagonal matrix with entrie equal to the correponding eigenvalue. Eigenvalue λ i are known to be real and λ i 0, (i = {,, 3}). The eigenvector are orthonormal, o Q T Q = I, with I the identity matrix. Some calar meaure have been defined to characterize the tenor behavior. One of them [9] i baed on three calar quantitie, namely, the linear, planar and pherical meaure. The linear component, which i the focu of attention of thi paper, i defined by c l = λ λ λ, in which an ordering of the eigenvalue i implicit, i.e., λ λ. Thi meaure hould take on value within the range [0, ] provided that the pd condition i atified. 3 The Model Our regularization cheme purue to highlight fiber tract for further proceing; therefore, we have concentrated on the linear component defined before,

3 94 M. Martín-Fernández et al. together with the angle that define the direction of the eigenvector aociated to the larget eigenvalue of the tenor (apart from the ambiguity in π radian inherent in every eigenvector). For implicity, the model for the calar meaure will be denoted by amplitude modeling, and the model for the angle argument modeling. A for the amplitude modeling, the model will force moothne in the olution component according to neighboring voxel, and will elect a value that lie within the range [0, ]. Thi aure that the two larget eigenvalue are nonnegative. A for the angle, moothne in the olution component will alo be guaranteed together with the appropriate range of poible value to account for the above mentioned ambiguity in π radian. The model, in it current tage, i two dimenional o only one angle with value within ( π/,π/] will be dealt with. However, a it will be obviou hortly, the model extend traightforwardly to three dimenion. 3. Amplitude Modeling The prior: The linear component of the tenor will be probabilitically modeled be mean of a truncated Gauian prior. Thi i to force that value in the prior model lie within the interval [0, ] and, in addition, it provide the model with mathematical tractability. Formally, denoting by x the linear component of a tenor in a pixel located at ite, and denoting by δ() the neighbor of ite, the probability denity function (pdf) will be f(x /x u,u δ()) = e (x A) σ () β σ π for 0 x, and zero otherwie. The dependence of ite with it neighbor δ() i through the ditribution parameter A = A (x u,u δ()) and σ = σ (x u,u δ()). Thi dependence, however, will not be explicitly tated for implicity. If erfc tand for the complementary error function, the proportionality factor that aure that the area under thi denity i unity can be hown to be β = ( ( ) ( )) A A erfc erfc () σ σ Thi ditribution ha the mean and variance that follow [ ] η = A + σ e A σ e ( A) σ (3) π β ς = σ + [ σ (A )e ( A) σ π β σ A e A ] (η A ) (4) Due to the Gauian-like expreion it can be eaily een that the equation ytem for the maximum likelihood etimation (MLE) of the parameter of the

4 Regularization of Diffuion Tenor Map 95 ditribution, A and σ, are the ame a equation (3) and (4) with η and ς ubtituted by the ample mean and the ample variance repectively. The olution to thee equation i found through a imple and fat iterative method which ha been derived by the author. The tranition model: Even though the eigenvalue of a pd tenor are nonnegative, noie in the acquiition proce may caue the tenor not to atify thi retriction, and, conequently, the linear component i not guaranteed to lie within the interval [0, ] anymore. For intance, if λ > 0 and λ < 0 then c l >. In thi ituation, the cloer λ to zero, the cloer c l to. On the other hand, if both eigenvalue are negative, c l i negative, and, again, the cloer λ to zero, the cloer c l to. A proviion for thi can be eaily derived by mean of a tranition noie model. We have reorted to a imple Gauian noie model, in which the noie variance ha to be inferred from the data. Depite it implicity, reult that will be hown eem atifactory and, in addition, a cloed-form expreion for the poterior can be eaily calculated. Denoting by y the oberved tenor linear component at ite, the pdf of thi variable conditioned to a realization of the prior (and it neighborhood) i a Gauian pdf, the mean of which equal the realization of the prior, and the variance σ n will be inferred from the data. Formally f(y /x ; x u,u δ()) = e (y x) σ n (5) σ n π How the parameter σn i etimated will be decribed in ection 4. The Poterior: A it i well-known, Baye theorem give a relation between the three probabilitic entitie involved in our modeling cenario, i.e., f(x /y ; x u,u δ()) f(y /x ; x u,u δ())f(x /x u,u δ()) (6) Once again, the fact that equation () i a truncated Gauian and that equation (5) i a pure Gauian make the analytical determination of the poterior particularly imple. After ome algebra one can arrive at f(x /y ; x u,u δ()) = e (x A /y ) σ /y (7) β /y σ /y π with 0 <x and zero otherwie, and A /y = y σ + A σn σ + σn ; σ/y = σ σn σ + σn (8) In order for thi function to be a truncated Gauian the proportionality factor β /y hould be a that expreed in equation () with the updated parameter a in equation (8).

5 96 M. Martín-Fernández et al. 3. Eigenvector Argument Modeling The prior: The argument of the eigenvector aociated to the larget eigenvalue will be modeled by mean of an unreported, to the bet of the author knowledge, angular ditribution, which turn out to be periodic with period π. The pdf can be expreed a f Θ (θ /θ u,u δ()) = co(θ ϕ) ( )e λ (9) πi 0 λ for π <θ π and zero otherwie. I 0 (x) tand for the modified Beel function of order 0 []. The dependence of ite with it neighbor δ() i through the ditribution parameter ϕ = ϕ (θ u,u δ()) (π, π] and λ = λ (θ u,u δ()) > 0. A before, thi dependence will not be explicitly tated. Clearly thi ditribution i well-behaved at the extreme of the interval of allowable value ince f Θ ( π) =f Θ (π). Thi hould be o ince both argument are phyically the ame vector pointing direction. Due to non-linear and non Gauian nature of the pdf for variable θ, the expreion for the MLE of the two parameter involved are not imple data average. After ome algebra one can arrive at ( N ) ˆϕ ML = e jθi ; ˆλML = ( ) (0) N i= B co(θ i ˆϕ ML ) N i= with θ i, i = {,...,N} the obervation and B(x) = d dx ln I 0(x), which i a monotonically growing function, and thu invertible. Thi function i particularly imple to invert due to well-known approximation of function I 0 (x) which are valid for mot of the domain on which thi function i defined []. It i obviou that eigenvector have an ambiguity of π radian, i.e., if v i an eigenvector, o i vector v. Conequently, the pdf in equation (9) hould have an interval range of π radian, a oppoed to π. Thi i eaily accomplihed by defining a new angle, ϑ, which i related to θ by mean of the tranformation ϑ = θ /. Conequently, the prior for the argument will be f(ϑ )= f Θ(ϑ ) with φ = ϕ. Thi expreion i valid within π <ϑ π = co(ϑ φ) ( )e λ () πi 0 λ and zero otherwie. The tranition model: The teering direction of the eigenvector aociated to the larget eigenvalue may be affected by noie uperimpoed in the acquiition proce. Thi can be eaily incorporated in the model by mean of a tranition function of the ame type a the prior to connect obervation with prior knowledge by mean of a noie model. Formally f(ψ /ϑ ; ϑ u,u δ()) = co(ψ ϑ) ( )e λn () πi 0 λ n

6 Regularization of Diffuion Tenor Map 97 The Poterior: Once again, Baye theorem allow u to write f(ϑ /ψ ; ϑ u,u δ()) f(ψ /ϑ ; ϑ u,u δ())f(ϑ /ϑ u,u δ()) (3) Recalling equation () and () we can write f(ϑ /ψ ; ϑ u,u δ()) e co(ψ ϑ) + co(ϑ φ) λn λ (4) Doing ome algebra, parallel to the analytical characterization of a complex envelope of a narrow band ignal, one can arrive at for π <ϑ π f(ϑ /ψ,ϑ u,u δ()) = πi 0 ( and zero otherwie, with )e λ /ψ co(ϑ φ /ψ ) λ /ψ (5) φ /ψ = ( λ e jψ + λ n e jφ) (6) λ /ψ = λ n λ λ + λ n +λ λ n co(ψ φ ) (7) 4 Implementation Detail In the experiment only one angle ha been modeled, i.e., the approach i -D. The traightforward model extenion to 3-D jut need to model a econd angle in the ame way a we do here. Thee two angle would uniquely determine (but for the ambiguity in π radian) the direction of the eigenvector in the 3-D pace. The noie level σn for the amplitude i etimated once. Thi parameter i to be elected within the interval σn [min σy, σy ], where the latter expree the arithmetic mean of the value σy in every ite of the oberved field. σy i the unbiaed ample variance calculated with N = 0 neighbor of ite. A imilar procedure ha been ued for parameter λ n of the tranition model for the argument. In thi cae, λ n [min λ y, λ y ], with λ y the MLE of thi parameter (equation (0)) uing N = 0 neighbor of ite. For the optimization proce to find the MAP olution of the field we have ued the imulated annealing procedure [3] with a partially parallel viit chedule and a logarithmic cooling cheme. The number of weep ha been, in all the experiment, 5. Subequent etimation of the poterior parameter (a the optimization proce evolve) ha been done with neighbor, excluding the pixel under analyi. In order to quantify the degree of regularization we have calculated a meaure of cro-entropy, a follow: R = N Rcl m= n= N ϑ p mn log p mn (8)

7 98 M. Martín-Fernández et al. with N Rcl the number of dicrete value conidered for the ratio of linear component between two neighboring ite, and N ϑ the number of dicrete value conidered for the difference of the argument in thee two ite. The former ha been forced to lie within the interval (0, ) by uing the larget of the two linear component in the denominator. The angle difference i unwrapped o it lie within (0,π). The number of neighbor ued to calculate thi 0 = 8 ince we have ued the nearet neighbor in the 0-ite neighborhood that have not been ued in the optimization proce referred to above. 5 Reult Figure how both the original (a) and the regularized field (b) with the parameter σ n and λ n elected in the mid point of the repective interval decribed above. The image are a color-coded compoition in which every pixel (i.e. every field ite) ha an RGB coordinate, with R = c l R(ϑ ), G = c l G(ϑ ) and B = c l B(ϑ ). The R( ), G( ) and B( ) function are periodic with period π. Conequently every argument ha a color aociated, and the darker the pixel, the cloer to zero the linear component of the tenor in that ite. (a) (b) Fig.. (a) Color-coded image in original field. (b) Regularized field.

8 Regularization of Diffuion Tenor Map 99 The value of the entropie, a defined above, are for the original image and for the proceed image. Viual aement of the regularized field matche numerical reult. 6 Concluion In thi paper we have decribed a novel probabilitic Bayeian model for the regularization of DT map. We have only taken into account the linear tenor component together with the direction of the eigenvector aociated to the larget eigenvalue, an information which hould be ufficient for a tractography application; we are aware of the lack of tability in DT-MRI tractography baed only on thi direction. However, the linear component add information about how much the argument of the eigenvector hould be truted in uch an application. A direct extenion of the method for the three dimenional cae, a pointed out in the paper, i fairly traightforward. However, taking the other component (planar and pherical) into account i a matter of further reearch. Acknowledgment. The author acknowledge the Comiión Interminiterial de Ciencia y Tecnología for reearch grant TIC C0, NIH grant P4- RR38 and CIMIT. Reference. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Function with Formula, Graph, and Mathematical Table, Dover, New York, 97.. P. Baer, C. Pierpaoli, Microtructural and Phyiological Feature of Tiue Elucidated by Quantitative-Diffuion-Tenor MRI, Journal of Magnetic Reonance, Ser. B, Vol., No. 3, June 996, pp S. Geman, D. Geman, Stochatic Relaxation, Gibb Ditribution and the Bayeian Retoration of Image, IEEE Tran. on PAMI, Vol. 6, No. 6, Nov. 984, pp M. Martín-Fernández, R. San Joé Etépar, C. F. Wetin, C. Alberola-López, A Novel Gau-Markov Random Field Approach for Regularization of Diffuion Tenor Map, Proc. of the NeuroImaging Workhop, Eurocat 003, La Palma de Gran Canaria, Feb. 003, pp G. J. M. Parker, J. A. Schnabel, M. R. Symm, D. J. Werring, G. J. Barker, Nonlinear Smoothing for Reduction of Sytematic and Random Error in Diffuion Tenor Imaging, Journal of Magnetic Reonance Imaging, Vol., No. 6, 000, pp C. Poupon, J. F. Mangin, V. Frouin, J. Regi, F. Poupon, M. Pachot-Clouard, D. Le Bihan, I. Bloch, Regularization of MR Diffuion Tenor Map for Tracking Brain White Matter Bundle, in Lecture Note in Computer Science, W. M. Well, A. Colcheter, S. Delp, Ed., Vol. 946, Oct. 998, pp C. Poupon, C. A. Clark, V. Frouin, J. Regi, D. Le Bihan, I. Bloch, J. F. Mangin, Regularization Diffuion-Baed Direction Map for the Tracking of Brain White Matter Facicle, NeuroImage, Vol., 000, pp

9 00 M. Martín-Fernández et al. 8. D. Tchumperlé, R. Deriche, DT-MRI Image: Etimation, Regularization and Application, Proc. of the NeuroImaging Workhop, Eurocat 003, La Palma de Gran Canaria, Feb. 003, pp C. F. Wetin, S. E. Maier, H. Mamata, A. Nabavi, F. A. Jolez, R. Kikini, Proceing and Viualization for Diffuion Tenor MRI, Medical Image Analyi, Vol. 6, No., June 00. pp C. F. Wetin, H. Knutton, Tenor Field Regularization uing Normalized Convolution, Proc. of the NeuroImaging Workhop, Eurocat 003, La Palma de Gran Canaria, Feb. 003, pp