A Robust Parametric Method for Bias Field Estimation and Segmentation of MR Images

Transcription

1 A Robust Parametrc Method for Bas Feld Estmaton and Segmentaton of MR Images Chunmng L, Chrs Gatenby,LWang 2, John C. Gore Vanderblt Unversty of Imagng Scence, Nashvlle, TN 37232, USA 2 Nanjng Unversty of Scence and Technology, Nanjng, Chna lchunmng@gmal.com Abstract Ths paper proposes a new energy mnmzaton framework for smultaneous estmaton of the bas feld and segmentaton of tssues for magnetc resonance mages. The bas feld s modeled as a lnear combnaton of a set of bass functons, and thereby parameterzed by the coeffcents of the bass functons. We defne an energy that depends on the coeffcents of the bass functons, the membershp functons of the tssues n the mage, and the constants approxmatng the true sgnal from the correspondng tssues. Ths energy s convex n each of ts varables. Bas feld estmaton and mage segmentaton are smultaneously acheved as the result of mnmzng ths energy. We provde an effcent teratve algorthm for energy mnmzaton, whch converges to the optmal soluton at a fast rate. A salent advantage of our method s that ts result s ndependent of ntalzaton, whch allows robust and fully automated applcaton. The proposed method has been successfully appled to 3-Tesla MR mages wth desrable results. Comparsons wth other approaches demonstrate the superor performance of ths algorthm.. Introducton In quanttatve processng and analyss of magnetc resonance (MR) mages, major dffcultes arse from varatons n ntensty due to B and B0 feld nhomogenetes. Such ntensty non unformtes cause ntensty varatons even for a sngle tssue, whch may mslead many mage analyss algorthms, such as segmentaton and regstraton. Therefore, correcton for such ntensty nhomogenetes s often a mandatory step before quanttatve analyss of the mage data. Bas correcton methods can be broadly categorzed nto two classes: prospectve methods [9, 6] and retrospectve methods [8, 6, 4,, 8]. Prospectve methods try to avod ntensty nhomogenety n the acquston process by usng specal hardware or specfc sequences. These methods are able to correct some of the ntensty nhomogenety caused by the MR scanner. However, they cannot correct for sources of nhomogenety that are patent dependant, whch makes them of lmted value for clncal use [0]. In contrast to the prospectve methods, retrospectve methods rely exclusvely on the nformaton wthn the acqured mage and thus can be appled to remove patent dependant effects. One of the most popular types of methods for bas feld correcton are segmentaton based approaches [8, 7, 4,, 8, 9]. In these methods, the tasks of bas feld correcton and segmentaton are nterleaved n an teratve process such that they beneft from each other to yeld better results. In [8], Wells et al. developed an approach based on an expectaton-maxmzaton (EM) algorthm for nterleaved bas feld correcton and segmentaton. Ths method was later mproved by Gullemaud and Brady n [ 4]. However, methods based on the EM algorthm requre a good ntalzaton for ether the bas feld or for the classfcaton estmate [5]. They typcally requre manual selectons of representatve ponts for each tssue class to perform ntalzaton. Such ntalzatons are subjectve and often rreproducble [7]. Moreover, the fnal correcton and segmentaton are senstve to the specfc choces of ntal condtons [, 5, 7]. Based on the EM framework n [8], Leemput et al. [7] proposed an explct parametrc model of the bas feld. Instead of manual nterventon, ther method used a dgtal bran atlas that provdes a pror probablty maps for whte matter (WM), gray matter (GM), and cerebrospnal flud (CSF). Although ths method s clamed to be more robust than the method of Wells et al., the ntalzaton of the parameters remans crtcal [5]. In [], Pham and Prnce proposed an energy mnmzaton approach for segmentaton and bas feld estmaton n whch a fuzzy C-means (FCM) algorthm was used for segmentaton. In ther proposed energy functon, a term was ntroduced to ensure the smoothness of the computed bas feld. The coeffcent of the smoothng term s, however, sometmes dffcult to ad /09/$ IEEE 28

2 just for desrable results [7]. In addton, ths method s also computatonally ntensve [2], due to the ntroducton of the bas feld smoothng term. In ths paper, we propose a new energy mnmzaton approach for jont bas feld estmaton and tssue segmentaton. A bas feld s modeled as a lnear combnaton of smooth bass functons, and hence parameterzed as the coeffcents of the bass functons. We defne an energy that depends on the coeffcents of the bass functons, the membershp functons of the tssues n the mage, and the constants approxmatng the true sgnal from the correspondng tssues. Ths energy s convex n each of ts varables. Bas feld estmaton and mage segmentaton are smultaneously acheved as the result of mnmzng ths energy. A salent advantage of our method s that ts result s ndependent of ntalzaton, whch allows robust and fully automated applcaton. 2. Problem Formulaton and an Energy Mnmzaton Method The ntensty nhomogenety n an MR mage can be modeled as a multplcatve component of an observed mage descrbed by I(x) = b(x) J(x)+n(x) () where I(x) s the measured mage ntensty at locaton (voxel) x, J s the true sgnal to be restored, b s an unknown bas feld, n s addtve nose. Ideally, the true sgnal J from each tssue should take a specfc value of the physcal property beng measured (e.g. the proton densty for MR mages). Therefore, t can be assumed that the true sgnal J s pecewse approxmately constant. More specfcally, we assume that there are N tssues n the regon of nterest, denoted by, and that the true sgnals J orgnated from the -th tssue are approxmately a constant c. The bas feld b s commonly assumed to be slowly varyng. It s our goal to estmate the unknown bas feld b and true sgnal J from the measured mage I. In ths work, the bas feld s estmated by a lnear combnaton of a set of bass functons. Let g,,g M be a set of bass functons defned on. We estmate the bas feld by a lnear combnaton of the bass functons b(x) = M w k g k (x) (2) k= where w k R, k =,,M, are the combnaton coeffcents. Theoretcally, any functon can be approxmated by a lnear combnaton of a set of bass functons up to arbtrary accuracy [2], gven a suffcently large number of bass functons. In our current mplementaton, we use orthogonal polynomals as the bass functons,.e. the bass functons g,,g M satsfy g (x)g j (x)dx = δ j (3) where δ j =0for j and δ j =for = j. The segmentaton s gven by an optmal estmaton of the true sgnal J by a pecewse constant map J, whch takes a constant value c n the regon of the -th tssue. These regons { } N of N tssues form a partton of the mage doman n the sense that N =and j = for j. Thus, the pecewse constant map J can be wrtten as J(x) = c u (x). (4) where u (x) s the membershp functon of the regon such that {, x u (x) = (5) 0, else. and u (x) =. (6) 2.. Energy Formulaton We formulate the problem of segmentaton and bas feld estmaton as a task of seekng the best pecewse constant map J and bas feld b such that ther product b J best fts the measured ntensty mage I. The pecewse constant map J can be expressed as J = N c u as n Eq. (4), whle the bas feld b s modeled as a lnear combnaton b = M k= w kg k as n Eq. (2). Therefore, we defne the fttng error F = M I(x) [ w k g k (x)][ c u (x)] 2 dx (7) k= Ths fttng error s the proposed energy n terms of the constants c,,c N, the membershp functons u,,u N, and the coeffcents w,,w M. The mnmzaton of ths energy gves the optmal membershp functons u,,u N as a segmentaton result, whle the optmal coeffcents w,,w M of the bass functons defne the estmated bas feld. The scalar constants c,,c N and w,,w M, and the functons u,,u N and g,,g M can be represented n the form of column vectors,.e. c =(c,..., c N ) T, w = (w,..., w M ) T, U(x) = (u (x),..., u N (x)) T, and G(x) =(g (x),,g M (x)) T. Thus, the above energy F can be rewrtten as F (U, c, w) = I(x) (w T G(x))(c T U(x)) 2 dx (8) 29

3 From Eq. (5), we have c T U(x) =c for x. Thus, the energy F can be rewrtten as F (U, c, w) = = = I(x) (w T G(x))(c T U(x)) 2 dx I(x) (w T G(x))c 2 dx (9) I(x) (w T G(x))c 2 u (x)dx From the expresson of F n the last lne, t s easy to mnmze wth respect to the membershp functons u,,u N Energy Mnmzaton It s worth notng that the energy F (U, c, w) s convex n each of ts varables. The energy F (U, c, w) can be mnmzed by an teratve process of nterleaved mnmzaton wth respect to each varable. The mnmzer of F (U, c, w) n each varable, U, c, orw, s gven below. Mnmzaton wth respect to U. For fxed c and w,we mnmze F (U, c, w) wth respect to U under a constrant that U =(u,,u N ) T satsfes Eq. (6). It can be shown that the mnmzer Û =(û,, û N ) T s gven by {, = mn (x); û (x) =, =,N. (0) 0, mn (x). where mn (x) =argmn { I(x) (w T G(x))c 2 }. Mnmzaton wth respect to c. For fxed U and w, there s a unque mnmzer of the functon F (U, c, w) n the varable c. Ths unque mnmzer, denoted by ĉ = (ĉ,, ĉ N ) T, s gven by ĉ = I(x)b(x)u (x)dx, =,,N. () b2 (x)u (x)dx Mnmzaton wth respect to w. For fxed U and c, to mnmze F (U, c, w) wth respect to w, we take dervatve of F wth respect to w, we get where and F = 2v +2Aw w v = A = I(x)G(x)J(x)dx, G(x)G(x) T J 2 (x)dx, (2) wth J(x) =c T U. Note that A s an M M matrx, wth M beng the number of the bass functons. It can be shown that the above matrx A s nonsngular (see below). Therefore, the lnear equaton F w = 2v +2Aw = 0 has a unque soluton ŵ = A v (3) The entre procedure of mnmzaton of the energy F (U, c, w) s descrbed as below: Step. Intalzaton of c, w and U; Step 2. Update c to be ĉ gven by Eq. (); Step 3. Update w to be ŵ gven by Eq. (3); Step 4. Update U to be Û gven by Eq. (0); Step 5. Check convergence crteron. If convergence has been reached, stop the teraton, otherwse, go to Step Matrx Analyss for Numercal Issues The non-sngularty of matrx A s verfed as the followng. We frst defne h m (x) g m (x) N c2 u (x). Thus, the (m, k) entry of A can be expressed as the nner product of h m and h k gven by h m,h k = h m (x)h k (x)dx. Therefore, the matrx A s the Graman matrx of h,,h M. By lnear algebra [5], the Graman matrx of h,,h M s non-sngular f and only f they are lnearly ndependent. It s easy to see that the above defned functons h,,h M are lnearly ndependent, whch mples the non-sngularty of A. Numercal stablty s an mportant ssue n computng the nverse matrx A n Eq. (3), and may have a sgnfcant mpact on the accuracy of the fnal result of our method. In general, the numercal stablty of computng an nverse matrx A can be characterzed by the condton number [3] of the matrx A. The condton number of a postvedefnte matrx A s gven by κ(a) =λ max (A)/λ mn (A), where λ mn (A) and λ max (A) are the mnmal and maxmal egenvalues of matrx A, respectvely. The stablty s ensured when the condton number s bounded by a reasonably small number. For the above defned matrx A n Eq. (2) wth the bass functons g,,g M satsfyng the orthogonalty condton n Eq. (3), we are able to prove that 0 < mn{c 2 } λ mn (A) λ max (A) max{c 2 } 220

4 Therefore, an upper bound of the condton number of A s provded by max{c 2 } κ(a) mn{c 2 }. (4) For example, f max {c } = 250 and mn {c } =50,by the nequalty (4), we have κ(a) =25.Wehave 2 observed that the condton numbers are less than 26.0 for all the experments n ths paper, whch s suffcent to ensure the numercal stablty of the nverson operaton n our algorthm. 3. Results In ths secton, we demonstrate the effectveness of the proposed method, especally ts robustness to ntalzaton. To verfy the robustness of our method, we randomly generate dfferent ntalzatons of all the varables U, c, and w n usng our method. Expermental results have confrmed that our algorthm converges to the same result from dfferent ntalzatons that are generated randomly. Moreover, the convergence s reached after a small number of teratons. The typcal number of teratons needed for convergence s between 0 and 20 for most of mages. Our method has been appled to 3-tesla MR mages, whch have sgnfcant ntensty nhomogenetes. We frst show the results of our method for the 3-tesla MR mages n the frst column of Fg.. The estmated bas felds, the segmentaton results, and bas corrected mages are shown n the second, thrd, and fourth columns, respectvely. As can be seen from thrd column, the tssue segmentaton results are qute consstent wth the bran anatomy. In the bas corrected mages n the fourth column, the ntenstes wthn each tssue become qute homogeneous. Fgure. Results of our method on 3T bran MR mages. Column : Orgnal mages; Column 2: Estmated bas felds; Column 3: Segmentaton results. Column 4: Bas corrected mages. We compared our method wth the methods of Wells et al. [8], Leemput et al. [7]. We tested these three methods on a synthetc mage of crcles wth ntensty nhomogenety (top left n Fg. 2) and an MR bran mage wth nose %, ntensty non-unformty (INU) 80% (top rght Fg. 2), whch was obtaned from McGll Bran Web []. Snce the algorthms of Wells et al. and Leemput et al. are senstve to the ntalzaton of the parameters (mean, varances and a pror probablty for each tssue), proper ntalzaton of these parameters s necessary n usng ther methods. Unless otherwse specfed, the ntalzaton of the parameters n ther methods are obtaned from a prelmnary segmentaton and estmaton of the parameters usng the K-means algorthm. When such ntalzatons are used, the segmentaton results of Wells et al. s method (Fgs. 2(c) and 2(e)) and Leemput et al. s method (Fgs. 2(g) and 2()) are acceptable. However, when the ntalzaton of these parameters are changed, the segmentaton results of these two methods vary sgnfcantly, as shown n the two results n Fgs. 2(d), 2(f), 2(h), 2(j). The results of our method for two randomly generated ntalzatons are shown n the bottom row of Fg. 2. No vsble dfference can be seen from the results of our method for the two randomly generated ntalzatons. Fgs. 3 and 4 show the comparson results for two synthetc mages wth severe ntensty nhomogenetes. The results of the methods by Wells et al., Leemput et al., and the proposed one are shown n the frst, second and thrd rows, respectvely. The estmated bas feld, segmentaton results, and bas corrected mage are shown n every row. Whle t s dffcult to vsually compare the bas corrected mages, the segmentaton results of our method are more accurate than the other two methods, especally n the lower part of the mage. To quanttatvely evaluate the performance of the algorthms, we use Jaccard smlarty (JS) [3] as an ndcator of the segmentaton accuracy. The JS between two regons S and S 2 s defned as the rato between the areas of the ntersecton and the unon of them, namely, J(S,S 2 )= S S 2 S S 2. To evaluate the accuracy of segmentaton, we compute the JS between the segmented regon S by the algorthm and the correspondng regon S 2 gven by the ground truth. The closer the JS values to, the better the segmentaton and bas correcton. We tested the three methods on 30 mages from McGll bran data. For the methods of Wells et al. and Leemput et al., we used 20 dfferent ntalzatons of the means, whle the varances and a pror probablty are ntalzed properly by prelmnary estmaton of them. The JS values for WM and GM of the results obtaned by the three methods are shown n Fg. 5. The JS values of the Wells et al., the Leemput et al. and our method are plotted wth red squares, green crcles and blue damonds respectvely. The JS values of our method for the 20 dfferent ntalzatons show 22

5 (a) (b) (c) (d) (e) (f) (g) (h) () (j) (k) (l) (m) (n) Fgure 3. Comparson results for a synthetc mage shown n the fst column. The estmated bas felds, segmentaton results, and bas corrected mages are shown n the second, thrd, and fourth columns, respectvely. The results of the methods of Wells et al., Leemput et al., and the proposed method are shown n rows,2,3, respectvely. Fgure 2. Results of the method of Wells et al., the method of Leemput et al. and our method wth two dfferent ntalzatons. The segmentaton results wth good guess are shown n columns and 3; the segmentaton results wthout good guess are shown n columns 2 and 4. The results of the methods of Wells et al., Leemput et al., and the proposed method are shown n rows 2,3,4, respectvely. no vsble dfference n Fg. 5, whch demonstrate the ndependence of ntalzaton of our method. By contrast, there s large varablty n the JS values for the results obtaned by the methods of Wells et al. and Leemput et al. for 20 dfferent ntalzatons, as can be seen n Fg. 5. We have appled our method to 3D MR mages. Fg. 6 shows the result of our method for a 3-tesla 3D MR mage. To vsualze the results, we select four sagttal slces as shown n the frst row of Fg. 6. The correspondng estmated bas felds, fnal segmentaton results and corrected mages are shown n the second, thrd, and fourth rows respectvely. The ntenstes wth each tssue become qute homogeneous n the bas corrected mages. Meanwhle, the segmentaton results show hgh agreement wth the bran anatomy. 4. Concluson In ths paper, we have presented a new energy mnmzaton framework for smultaneous estmaton of the bas feld and segmentaton of tssues for magnetc resonance mages. Fgure 4. Comparson results for a synthetc mage shown n the fst column. The estmated bas felds, segmentaton results, and bas corrected mages are shown n the second, thrd, and fourth columns, respectvely. The results of the methods of Wells et al., Leemput et al., and the proposed method are shown n rows,2,3, respectvely. We provde an effcent teratve algorthm for energy mnmzaton, whch converges to the optmal soluton at a fast rate. A salent advantage of our method s that ts result s ndependent of ntalzaton, whch allows robust and fully 222

6 JS for WM Method by Wells et al. Method by Leemput et al. Proposed method Image JS for GM Method by Wells et al. Method by Leemput et al. Proposed method Image Fgure 5. Test of senstvty to ntalzaton for our method and the methods of Wells et al. and Leemput et al. The x-axs represents 30 mages, and the y-axs represents the JS values for WM (left) and GM (rght) of the three methods for 20 dfferent ntalzatons. Fgure 6. Sagttal vew of the 3D segmentaton and bas correcton results. Row : Orgnal mages. Row 2: Estmated bas felds. Row 3: Segmentaton results. Row 4: Bas corrected mages. automated applcaton. The proposed method has been successfully appled to 3-Tesla MR mages wth desrable results. Comparsons wth other approaches demonstrate the superor performance of ths algorthm. References [] 4 [2] M. Ahmed, S. Yamany, N. Mohamed, A. Farag, and T. Morarty. A modfed fuzzy c-means algorthm for bas feld estmaton and segmentaton of MRI data. IEEE Trans. Med. Imagng, 2(3):93 99, March [3] G. Golub and C. V. Loan. Matrx Computatons. The Johns Hopkns Unversty Press, 3rd edton, [4] R. Gullemaud and J. Brady. Estmatng the bas feld of MR mages. IEEE Trans. Med. Imag., 6(3):238 25, June 997. [5] R. Horn and C. Johnson. Matrx Analyss. Cambrdge Unversty Press, Cambrdge, [6] B. Johnston, M. S. Atkns, B. Mackewch, and M. Anderson. Segmentaton of multple scleross lesons n ntensty corrected multspectral MRI. IEEE Trans. Med. Imag., 5(2):54 69, Aprl 996. [7] V. Leemput, K. Maes, D. Vandermeulen, and P. Suetens. Automated model-based bas feld correcton of MR mages of the bran. IEEE Trans. Med. Imag., 8(0): , October 999., 4 [8] C. L, R. Huang, Z. Dng, C. Gatenby, D. Metaxas, and J. Gore. A varatonal level set approach to segmentaton and bas correcton of medcal mages wth ntensty nhomogenety. In Proceedngs of Medcal Image Computng and Computer Aded Interventon (MICCAI), volume LNCS 5242, Part II, pages , [9] C. L, C. Xu, A. Anderson, and J. Gore. MRI tssue classfcaton and bas feld estmaton based on coherent local ntensty clusterng: A unfed energy mnmzaton framework. In Proceedngs of Informaton Processng n Medcal Imagng (IPMI), Wllamsburg, VA on July 5-0, [0] B. Lkar, M. Vergever, and F. Pernus. Retrospectve correcton of mr ntensty nhomogenety by nformaton mnmzaton. IEEE Trans. Med. Imag., 20(2):398 40, December 200. [] D. Pham and J. Prnce. Adaptve fuzzy segmentaton of magnetc resonance mages. IEEE Trans. Med. Imag., 8(9): , September 999. [2] M. J. D. Powell. Approxmaton Theory and Methods. Cambrdge Unversty Press, Cambrdge, [3] D. W. Shattuck, S. R. Sandor-Leahy, K. A. Schaper, D. A. Rottenberg, and R. M. Leahy. Magnetc resonance mage tssue classfcaton usng a partal volume model. Neuromage, 3: , May [4] J. Sled, A. Zjdenbos, and A. Evans. A nonparametrc method for automatc correcton of ntensty nonunformty n MRI data. IEEE Trans. Med. Imagng, 7():87 97, January 998. [5] M. Styner, C. Brechbuhler, G. Szekely, and G. Gerg. Parametrc estmate of ntensty nhomogenetes appled to MRI. IEEE Trans. Med. Imag., 9(3):53 65, March [6] M. Tncher, C. R. Meyer, R. Gupta, and D. M. Wllams. Polynomal modelng and reducton of rf body col spatal nhomogenety n MRI. IEEE Trans. Med. Imag., 2(2):36 365, 993. [7] U. Vovk, F. Pernus, and B. Lkar. A revew of methods for correcton of ntensty nhomogenety n MRI. IEEE Trans. Med. Imag., 26(3):405 42, March 2007., 2 [8] W. Wells, E. Grmson, R. Kkns, and F. Jolesz. Adaptve segmentaton of MRI data. IEEE Trans. Med. Imag., 5(4): , 996., 4 [9] D. A. G. Wcks, G. J. Barker, and P. S. Tofts. Correcton of ntensty nonunformty n MR mages of any orentaton. Magn. Reson. Imag., (2):83 96,