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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 8, AUGUST 2008 1295

A Variational Method for Geometric Regularizationof Vascular Segmentation in Medical Images

Ali Gooya, Student Member, IEEE, Hongen Liao, Member, IEEE, Kiyoshi Matsumiya, Ken Masamune,Yoshitaka Masutani, Member, IEEE, and Takeyoshi Dohi

Abstract—In this paper, a level-set-based geometric regulariza-tion method is proposed which has the ability to estimate the localorientation of the evolving front and utilize it as shape inducedinformation for anisotropic propagation. We show that preservinganisotropic fronts can improve elongations of the extracted struc-tures, while minimizing the risk of leakage. To that end, for anevolving front using its shape-offset level-set representation, anovel energy functional is defined. It is shown that constrainedoptimization of this functional results in an anisotropic expansionflow which is usefull for vessel segmentation. We have validatedour method using synthetic data sets, 2-D retinal angiogramimages and magnetic resonance angiography volumetric datasets. A comparison has been made with two state-of-the-art vesselsegmentation methods. Quantitative results, as well as qualitativecomparisons of segmentations, indicate that our regularizationmethod is a promissing tool to improve the efficiency of bothtechniques.

Index Terms—Anisotropic propagation, blood vessel segmenta-tion, energy optimization, shape analysis, surface evolution.

I. INTRODUCTION

V ESSEL segmentation is one of demanding applicationsthat has received a considerable attention [1]. It is impor-

tant for evaluation of vascular abnormalities (such as stenosesand plaques) and also has applications for surgical planning.

Manuscript received May 31, 2007; revised March 25, 2008. First publishedJune 24, 2008; last published July 11, 2008 (projected). This work was sup-ported in part by the Special Coordination Funds for Promoting Science andTechnology of the Ministry of Education, Culture, Sports, Science and Tech-nology (MEXT) in Japan; in part by the Grant-in-Aid for Scientific Research(17680037) of MEXT; in part by the Grant for Industrial Technology Research(07C46050) of the New Energy and Industrial Technology Development Or-ganization, Japan (H. Liao), and in part by the Grant-in-Aid for Scientific Re-search (17100008) of the Japan Society for the Promotion of Science (T. Dohi).This paper was presented in part at the IEEE 4th International Symposium onBiomedical Imaging: From Nano to Macro. The associate editor coordinatingthe review of this manuscript and approving it for publication was Dr. ThomasS. Denney, Jr.

A. Gooya is with the Graduate School of Engineering, The University ofTokyo, Tokyo 113-8656, Japan (e-mail: [email protected]).

H. Liao is with the Department of Bioengineering, Graduate School of En-gineering, and the Translational Systems Biology and Medicine Initiative, TheUniversity of Tokyo, Tokyo 113-8656 (e-mail: [email protected]).

K. Matsumiya is with the Department of Biomedical Engineering, OkayamaUniversity of Science, Okayama, Ridaichou 1-1, 700-0005, Japan (e-mail:[email protected]).

K. Masamune and T. Dohi are with the Department of Mechano-Informatics,Graduate School of Information Science and Technology, The University ofTokyo, Tokyo 113-8656, Japan (e-mail: [email protected]; [email protected]).

Y. Masutani is with the Division of Radiology, The University of Tokyo Hos-pital, and also with the Department of Biomedical Engineering, Graduate Schoolof Medicine, The University of Tokyo, Tokyo 113-8655, Japan (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2008.925378

The vast majority of methodologies for robust extraction ofvascular structures are mainly based on features extractedfrom image content. Multiscale analysis of second derivativesis widely used for enhancement or detection of curvilinearstructures in 2-D and 3-D medical images [2]–[8]. Althoughthese have proved to be useful for enhancement of line features,the final output of such procedures is not a direct segmenta-tion of the input image. Skeleton-based methods [9], [10] arealgorithms in which subsequent 2-D slices of vessels are re-solved using tubular shape priors for ridge detection. Recentlyanother tracking methodology was proposed by Tyrrell et al.[11] using 3-D cylindroidal superellipsoids and local regionalstatistics to extract topological information from microvascula-ture networks. These methods are shown to be robust againstnoise, however, their explicit parametrical shape priors are tooexclusive, in case of complex vessel boundaries. Another seriesof publications using statistical mixture modeling coupled withexpectation-maximization algorithm include [12]–[14]. Theseare histogram based and, therefore, need an accurate paramet-rical estimation or nonparametric modeling [15] of involvedprobability density functions.

Recently, a number of surface evolution level-set-based algo-rithms for vessel segmentation have been developed. A combi-national method is proposed by Gazit et al. [16], which uses Har-alik edge detector, Chan–Vese minimal variance functional andgeodesic active contours. Also, capillary active contours was in-vented by Yan et al. [17], a method that is based on the capillaryforce acting on the free fluid surface. Another level-set-basedmethod introduced in [18], is an algorithm for artery–vein sep-aration with a couple of level set functions, where the frontspeed is a heuristic composition of three different factors de-termined by image gradient, histogram and a vessel enhance-ment filter’s response. A front evolution technique is proposedby Jackowski et al. [19] that is based on wave propagation inoriented domains.

Some other methods are concerned with basic challengesusing surface evolution for vessel segmentation. Lorigo et al.[20] proposed the use of active contours with co-dimensiontwo utilizing the curvature of the underlying 3-D curve [21] forsmoothing. From the practical point of view, this is computedas minimum magnitude surface principal curvature of -levelset. This is still too restrictive to allow extended elongations:in a common seeding or thresholding initialization scenario,evolution starts from the inside of vascular structures andextends over the thin lower contrast parts. In such a procedure,the “tips” of the evolving surface experience higher smoothnessconstraint since the magnitude of surface minimum principalcurvature is high exactly at such structures. In fact, one basic

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1296 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 8, AUGUST 2008

Fig. 1. (a) Portion of a TOF-MRA 1.5 Tesla data set, the arrow indicates adamaged pattern of a thin vessel. (b) Same portion in 3 Tesla indicates a smoothvascular pattern.

limitation of free curvature-based evolution is that the surfacealways shrinks to zero. Other state-of-the-art vessel segmenta-tion techniques include flux maximizing flows [22], [23]. Theyoffer a mutliscale computing scheme for robust estimation ofimage laplacian which allows handling structures with varioussizes. However, these are curvature free evolutionary schemes,and rely on initial smoothing of data. The drawback is that it isnot clear how much smoothing is required prior to segmenta-tion. Geometric regularization is essential for obtaining smoothsegmenting fronts without much altering initial data.

In order to reduce the risk of front leakage to the back-ground, a number of methods use shape constraints. Forexample, topology constrained surface evolution has been pro-posed in [24], a method that uses 3-D skeletons of the front torefine the spurious branches for iterative bifurcation and vesselsegmentation. Also, a soft prior has been introduced in [25]for minimizing the leakage from noisy edges using a ball-filterwhich penalizes the deviations from tubular structures. Ob-viously, any geometric regularization of the segmentationprocess, with the ability to minimize the leakage is importantfor vessel segmentation.

Fig. 1 illustrates another difficulty arising in most of vesselsegmentation algorithms. The arrow in panel (a) indicates thepinching of a thin vessel caused by noise. However, in a highermagnetic field, the smooth intensity pattern from the samevessel clearly shows its extension. Such noise “speckles” maystop the front evolution towards the thiner part, and conse-quently multiple distinct vessel fragments may be obtained.Masutani et al. [26] addressed this issue, using a mathematicalmorphology region-growing method. However, the vesselstructures remained un-interpolated between discontinuities.

In this paper, it is assumed that vessels are curvelinear struc-tures, this is the basic assumption as in [2]–[8] for enhancingthe vascular structures. Based on this general assumption, foreffective geometrical regularization of tube-like structures,minimization of a shape functional is proposed. Our regular-ization method preserves cylindrical structures by imposinganisotropic front constraint in the level-set variational frame-work and has three important basic properties. 1) Unlike theprevious curvature flows, the solution is not always a shrinkingsurface, but instead it can extend toward local “meaningful”surface features. 2) Since leakage develops isotropic structures,by applying anisotropic front constraint, the risk of leakage isminimized. 3) It is a curvature dependent flow with smoothing

effect. Therefore, it can basically overcome some limitationsencountered in previous methods. We also utilize the idea ofthe ball-filter to extract information about the local segmentedstructures, however, our methodology is basically different from[25] since it provides a robust image-independent estimationof the vessel direction for expansion. In fact, as the evolutionmay stop to extract the entire vessel, extracted structures showsome elongations. One option is to utilize this “shape-induced”information to propagate the surface. This can be useful whenthe image information is not reliable due to noise or intensityambiguities. In that case, our model enforces the surface toexpand anisotropically in the main surface orientation so that itcan pass over small noise speckels. The outlined framework canbe combined with existing level-set-based vessel segmentationfunctionals. Our idea was first proposed in [27]. In this paper,two pioneer methods introduced in [20] and [22] are chosen forcomparison purposes. These methods are especially preferredsince they require a minimum set of parameters and simplifythe comparison tasks.

The rest of the paper is organized as follows: illustration ofthe basic idea and the definition of the local anisotropy measureare included in II. Section III describes our shape functional en-ergy minimization strategy. In Section IV, this shape functionalis used to regularize the CURVES and flux maximizing flowfunctionals [20], [22] and Section V contains the result. Con-cluding remarks and discussion is included in Section VI.

II. DESCRIPTION OF THE ANISOTROPY CONSTRAINT

We assume that for a given open region specified byits border , the evolving surface is represented as thezero level of the level-set function where forinside of the object, and for outside. is theHeaviside function such that if , otherwise

. Also, is the Dirac deltafunction. Throughout this paper, the phrases such as: front,surface (3-D) or contour (2-D) are used interchangeably withimplied dimensions.

A. Estimation of the Surface Local Structure

Robust estimation of local structure can be achieved using thecorrelation matrix of the image gradients (see Weickert et al. in[28]). Similarly, the local structure of the evolving contour atpoint can be expressed by the correlation matrix of its normalvectors inside a scale-selectable neighborhood. Since we aremainly interested in the orientation of underlying skeleton, theiso-level contours can be considered as “shape offsets” of themost internal iso-level and the orientation matrix is resolvedonly using inner contours. In fact, because the iso-levels tendto be spherical as they get farther from the surface, exclusion ofouter region helps resolving the orientation ambiguity.

As illustrated in [29], in the level-set framework normal vec-tors to evolving surface can be computed as .Therefore, assuming that level-set function holds the shapeoffset representation with , we define the followingmatrix:

(1)

GOOYA et al.: VARIATIONAL METHOD FOR GEOMETRIC REGULARIZATION OF VASCULAR SEGMENTATION 1297

Fig. 2. Evolving contour and its iso-levels are indicated in dark bold andgrey levels, respectively. On every point, such as and lying on the , localorientation matrix is resolved using gradient vectors of level-set function inintersection of a scale selectable neighborhood (indicated in dot circles)and inside of contour (specified in grey).

where is the transpose of the gradient vector .is the neighborhood function with the general property

of: . In this paper for the sake of simplicitywe define: if lies inside the neighborhood ofsize pixels around . Therefore, corresponds to thecorrelation matrix of gradient vectors of the surface signed dis-tance transform lying in the intersection area of the neighbor-hood of and inside of the contour as shown in grey in Fig. 2.We note that is a positive semi-definite matrix.

Application of anisotropy constraint is achieved by evaluatingthe availability of a major local orientation and propagating thesurface at that direction. This is accomplished by analysing theeigenvalues and vectors of . The following theorem stan-dard in linear algebra will be illustrative.

Theorem 1: Let to be the eigenvaluesof the correlation matrix and , to be their corre-sponding eigenvectors. Then

Proof: It is easy to see that the right-hand side of the aboveequation is equal to . By taking its derivative with re-spect to , we have . Therefore, should be an eigenvector of , i.e., we should have and in that casethe minimum value is the smallest eigenvalue: .

According to this theorem, for a given point placed on theevolving zero-level surface if , then one possibility is tohave the normal to all the gradient vectors in its neigh-borhood, i.e., . This happens when the gradientvectors are placed within a plane perpendicular to and thelocal structure has a cylindrical from.

Fig. 2 is a 2-D illustration of this idea using some samplepoints. We observe that for a point such as where the localstructure (shaded in grey) is rather ambiguous and isotropic,normal vectors to inner contours span every orientation, and,therefore, there is no preferred direction minimizing the right-hand side in Theorem 1. On the contrary, in the neighborhoodof point where the local structure is anisotropic, a large por-tion of normal vectors to inner iso-levels are in parallel and aminimizing local orientation can be identified by the medial ofzero-level contour . Therefore, lower value of is expected.

B. Approximating the Local Anisotropy

From the discussion in the previous section, we realize thatand implies a cylindrical local sur-

face which should remain stable by further evolution. Therefore,such a condition must be considered as a contour-stopping cri-teria, resembling the behaviour of the edge in the geodesic activecontour (GAC) model [30], [31]. In other words, we should de-vise a measure that selects large values for and

, similar to the gradient strength in GAC. Possible anisotropicmeasures similar to those suggested in [3]–[8] that are propor-tional to and are not good choices, since they arenot analytically differentiable. Our solution is to “preserve” thecylindrical property all over the front by assuming that the ini-tial structures are tube-like structures and is sufficiently largeso that . By this assumption, we only need toenforce to be a very small value all over the surface. A dif-ferentiable form that takes large values for is the traceof the inverse of the matrix

(2)

which is a good approximation to and, hence, the localanisotropy for elongated structures. It should be reminded thatthe proposed measure, can be infinite for noncylindrical struc-tures as well. In general, it approaches infinity if: 1) ap-proaches zero, i.e., is placed on a cylindrical surface locally;2) and approaches zero, i.e., is placed on a planner sur-face locally; 3) all values approach zero, i.e., a surface whichis shrinking. As we will see in the next section, (2) is integratedwithin a shape functional to enforce large values on .This means that the final solution to our energy minimizationscheme depends on the initial shape and will be either a tube likestructure (planes are considered as tubes with infinite radius),or a shrinking surface. The later is also an interesting propertywhich enables the method to remove small noise like structures.

We note that one may consider an alternative measure as. Though this measure reduces com-

plexity of derivation, but from computational point of view, itis not as sensitive as (2) in a sense that large values of maymoderate small values of .

III. SHAPE-DEPENDENT FUNCTIONAL

Since our shape functional is based on the geometric [30] andgeodesic active contours models [31], a brief review is provided.

A. Geodesic Active Contour Model

Geodesic active contour model proposed by Casselles et al.[31], is a curvature-dependent front propagation technique thathas had outstanding applications for image segmentation. Givenan image and a non-negative decreasing function , its min-imization energy functional in the level-set framework can bedescribed as

(3)


According to (3), the minimum occurs when the evolving con-tour identified by , lies on the minimum cost pathspecified by . Since is a decreasing function of gra-dient magnitude, the method can detect the object border if theinitial contour is placed close enough to the edge. Minimizationis achieved using gradient-descent method and it can be shownthat the corresponding Euler–Lagrange (EL) equation of (3) is

(4)

Similar to [32], since we are interested in steady state solutionof (4), the can be replaced with . Therefore, (4) canbe written as

(5)

where is the embedding level-set function. This later formu-lation tends to keep the level-set representation as a signed dis-tance function since evolution is applied on every level. Theterm is the mean curvature of the evolving frontcomputed directly from level-set function and provides smooth-ness. is the advection term that pulls the curve intoareas with lower values of , i.e., object’s borders.

B. Definition of the Shape Functional

In definition of our shape functional, we are inspired byGAC. The main idea is to replace the edge indicator with ourpreviously defined anisotropy approximation. In this way, thegeodesic minimal path is affected by contour evolution. Let

, be the level-set function, and bea non-negative decreasing function. Consider the followingfunctional:

(6)

The important point is that since estimation of local structure in(1) is based on the fact that the underlying level-set function is ashape offset representation, minimization should be performedin the constraint space of signed distance functions. In otherwords, our constrained minimization problem is

(7)

C. Minimization

As explained in [32], the constraint in (7) can be fullfilled bydifferent methodologies. One approach is to specify the speedfunction values exactly on the zero level contour, and then ex-tend it over other levels. In this method, off-the-border speedvalues are replaced by the values from the closest points on theborder. But in our application since we deal with sharp corners,finding the closest point on the border around theses cornerscan introduce inaccuracy. As we will see, a more natural ex-tention of speed values over off-front area can be achieved byusing the smeared out version of delta function, i.e., , ob-tained directly from derivation of minimization equation. This is

followed by reinitialization of the level-set function into signeddistance function to ensure a true shape offset representation ofthe evolving contour.

Therefore, the solution of the constrained probelm in (7) isobtained by first considering the unconstrained problem and de-riving the corresponding EL equation. We have used the Fréchetderivative [33] in a similar variational level-set framework out-lined in [32]. For the sake of completeness of this paper, thedefinition of Fréchet derivative and its basic properties are cov-ered in Appendix A. The following theorem holds.

Theorem 2: The level-set solution of (7) can be achieved bythe following gradient-descent evolutionary equation:

(8)

where the dependency on is implied and the matrix isdefined as follows:

(9)

in which denotes the derivative of .Proof: See Appendix B.

It is interesting to look at the properties of (8) in further de-tail. The right-hand side of evolution consists of three differentterms:

Smoothing: is a mean curvaturedependent smoothing term. Note that by multiplication of ,the curvature remains effective if the local structure is isotropic;therefore, annihilation of narrow structures with lower values of

is limited.Advection: The second term is an advection term that attracts

the object’s border toward lower values of . It reduces the ori-entation ambiguity and also minimizes the leakage from spu-rious noisy edges.

Propagation: To analyse this term, we note that for agiven point , the matrix is the weighted average ofpositive semi-definite matrices of its neighborhooddetermined by . Hence, is a robust estima-tion of a bigger local structure. Since , we have

, i.e., this term propagates the surfaceoutward. The important point is that depending on the orien-tations of and eigenvectors of the expansion isanisotropic. For segmentation of tubular structures, providedthat the size of neighborhood ( in Fig. 2) is large enough,

maintains its main component in the axial orientation. Asa result, propagation may only appear at the endings of thosestructures.

D. Implementation

The central-differencing scheme was used for computingand defined in (1) and (9), respectively. Having the values ofcost function , advection term: is calculated by simple


Fig. 3. Evolution of a cylinder; the speed of propagation on the surface is colorcoded by its corresponding color bar in right. (a) Top: initial cylinder; middle:after 100 iterations; bottom: after 200 iterations, elongation occurs only in axialorientation. (b) Minimization of the shape energy.

up-winding scheme. The first order forward-Euler method isused for digitization over time and Heaviside and delta func-tions are evaluated using their smeared-out versions and

with pixel, as explained in [34]

(10)

In this paper, we have set where is asmall positive number to prevent singularity of di-vision by zero. This setting is optional but, practically we ob-tained better results using this definition. The algorithm is im-plemented using a fast narrow band level-set method [35]. Wenote that, in (9), computation of matrix is expensive interms of CPU cycles. As an optimization, at each iteration theevolving front is compared to its status in previous iteration and

is computed at places where the front has displacements.By this means, a large portion of the surface that remains intactdoes not require updating of structural matrix and the programexecutes much faster.

As described in Section III-B, in our algorithm, reinitializa-tion is crucial. For that purpose, we follow the method describedin [36] by solving the following equation:

(11)

where . To minimize numerical errors in oursimulated examples, discretization over spatial coordinates isachieved using Hamilton–Jacobi WENO scheme [29].

Fig. 4. Evolution of a ring with a few embedded gaps. Left from top to bottom:initial surface, after 20 and 60 iterations under the surface minimum curvature,respectively, the surface shrinks to zero by further iterations. Right from top tobottom: after 20, 30, and 50 iterations using our proposed shape regularizationmethod, anisotropic expansion appears exclusively on the “tips” so that the ringcloses itself.

E. Sample Evolutions

Fig. 3 is the evolution of a cylinder. For this experiment, theneighborhood radius size is set to four pixels, i.e., the sameradius of the cylinder. Initial shape is indicated on the top, andat every iteration the right-hand side of (8) is color mapped ontothe evolving front. Note that the expansion occurs exclusively inthe axial orientation, and, as the evolution proceeds, both endsturn into needle like structures with anisotropy. Fig. 3(b) showsthat decreases quickly, which corresponds to rapid deforma-tion of both ends in the beginning. Also note that inconsistencyin the tube radius (or tapering) is a necessary property for pro-ducing locally anisotropic heads and, hence, decreasing of .Basically, by extra iteration, the s development of sharper headsis possible; however, it is practically limited to the resolution ofthe implementation grid.

Fig. 4 is a comparison between our proposed regularizationmethod and surface minimum curvature method [20]. For bothmethods, the top left image is the initial surface. Note thatthe surface remains smooth and the anisotropic expansionappearing exclusively on axial orientations closes the ring. Thisshows that, using our method while preserving smoothness,improvement of vessel segmentation is possible, even if theimage information is noisy or not adequate.

IV. VASCULAR SEGMENTATION

In this section, the proposed energy functional is used for ge-ometrical regularization of level-set-based vessel segmentationtechniques such as [22] and [20]. Though these state-of-the-art


methods are well proved, here the main emphasis is given to theimprovement of the performance upon using our proposed reg-ularization method. We briefly review the concept of both algo-rithms. To keep consistency with the original works, the samefunction definitions and implementation issues have been fol-lowed and the common factor arising from minimizationof energy functionals, similar to [32], is replaced by when-ever required.

A. Regularization of the CURVES Algorithm

Lorigo et al. in [20] proposed the CURVES, a vessel seg-mentation method based on GAC model. To prevent annihila-tion of narrow structures, mean curvature in (5) was replacedwith surface minimal curvature, that was an approximationto the curvature of underlying 1-D curve. Ambrosio and Sonerin [21] proved that such curvature can be directly computedusing level-set function , if we consider the minimum mag-nitude eigenvalue of projection of Hessian matrix into thetangential direction of the underlying curve, i.e.,

(12)

where the projection operator is defined as follows:

(13)

Therefore, the level-set update equation of CURVES becomes

(14)

where is the edge detector function described in Sec-tion III-A and, as in the original work, is set to .

Incorporation of is heuristic and required for vesselsegmentation. This term encourages the alignment of and

vectors by maximizing their inner product, a notion whichwas also further considered in [22] as the basic flux term. Toenforce anisotropy using CURVES algorithm, a general combi-national energy functional is proposed

(15)

where is a user defined constant. The final level-set equa-tion with implied dependency on is given by

(16)

where the has the same definition as in Section III. Notethat, in order to yield a consistent formulation with CURVES,the total minimizing equation has been divided by , so thatby , (16) reduces to the standard CURVES equation.

B. Regularization of the Flux Maximizing Flow

Vasilevskiy and Siddiqi in [22] proposed the flux maximizinggeometric flow (FLUX) for low contrast blood vessel segmen-

tation. This is an edge integration method which finds the loca-tions where is maximized. The flux functional to beminimized (maximized in norm) can be witten as

(17)

This integration is an estimation of the inward flux, and, similarto the outlined proof in the Appendix B, the EL equation can beeasily shown to be

(18)

The relevent point to our paper is that, using FLUX in order toobtain a smooth segmenting surface, the data should be initiallysmoothed. This filtering damages the available edge informa-tion and introduces size complexity. Introduction of geometricalregularization term can prevent leakage from noisy edges and isalso essential for obtaining a smooth surface without much al-tering the original data. Hence, in this paper, the FLUX energyfunctional in (17) is integrated with the proposed geometric reg-ularization functional. For comparison purposes, we suggest tominimize a more general from

(19)

Replacing the mean curvature with the minimum principal cur-vature, this corresponds to the follwing minimizer equation:

(20)

where and control the smoothness and anisotropyconstraints respectively.

As in the original version of the FLUX, to handle the varioussizes of the vessels, we compute the Laplacian operator using amutliscale method, i.e., the inward flux is computed on the pe-riphery of 2-D discs or 3-D spheres with various scales, then theresult is divided by the number of the points on the periphery[22] and the largest magnitude over the scales is chosen. De-pending on the application, the scales are linearly varied fromthe minimum (1 pixel or voxel) to the maximum radius of theavailable vessels (up to 6 pixels or voxels).

V. RESULTS AND EXPERIMENTS

In this section, the effect of anisotropy constraint on the seg-mentations of synthetic image volumes as well as actual 2-Dretinal angiogram and 3-D MRA data sets is evaluated. We ap-proximate: , where

and are the eigenvalues and their corre-spondent eigenvectors of , so that the propagation is onlyapplied in the main orientation of the local surface structure.

A. Selection of Parameters

The size of the neighborhood, i.e., can be decided basedon the scale of the target vessels, as it is the case with our 2-Dexperiments. Nevertheless, since usually thiner vessels are more


Fig. 5. (a) Helix target volume with decreasing thickness and increasing cur-vature; (b)–(d) MIPs of corrupted data sets by Gaussian noise distributions withstandard deviations of , respectively.

challenging, in our 3-D experiments, we set . This suf-fices to produce reliable orientations. Also, since our volumetricMRA data sets had different dynamic ranges and setting an ap-propriate value for was difficult, the data sets were normalizedto the range of [0, 65535] prior to segmentation. Upon this nor-malization, better results were obtained usingfor the regularization of CURVES, and forFLUX. Nevertheless, these settings can be further refined by theuser.

B. Synthetic Image Volumes

In this study, a synthetic image as shown in Fig. 5 with theresolution of 101 101 101 voxels is used to evaluate the per-formance of the proposed combinational segmentation schemes.The helix tube is chosen to resemble an actual vessel patternby decreasing in thickness from 4 voxels in the bottom up tothe pixelation level in the top as shown in panel (a). The curva-ture is increasing in the same bottom-to-top direction as it canbe seen from the MIP images. The original binary volume wasconvolved with a low pass averaging filter. We observed thatmaximum intensity value in the original data set was 1280. Thiswas corrupted by different Gaussian noise levels having zeromeans and standard deviations of 50, 100, 200, 300 as shownin Fig. 5(b)–(d), respectively. Fig. 6 shows the middle slice ofthe sample noisy volumes, indicating the relative strength of therandom noise in the corresponding slice. As shown, the highernoise levels effectively destroy the thinner parts.

In these experiments, in order to have the maximum gain onthe elongations, all curvature smoothing terms are ignored, i.e.,we set in (20) and ignore the constant in (16). Thefront evolution starts from seeding in the thicker parts and ap-proaches the thiner and curly parts. This process ends upon the

Fig. 6. (a)–(d) 51 slices of the corrupted data sets by Gaussian noise distribu-tions with standard deviations of , respectively.

Fig. 7. Segmentation of the noisy data sets corresponding to the second columnof the Table I. (a) CURVES, (b) regularized CURVES ,

(c) FLUX, (d) regularized FLUX .

convergence of the segmentation algorithm.1 This is repeatedfor three times for each of levels. The segmentation errors arecalculated using the Dice similarity coefficient [37]

(21)

where and are the target and obtained segmentation sets.The regularized CURVES and FLUX schemes are obtained bysetting and in (16) and (20) respectively.Fig. 7 shows the sample segmentation results obtained fromnoisy data sets with . By comparing the CURVES

1Through our experiments, we observed that convergence for the higher noiselevel of in FLUX segmentation was not possible since thesegmentation was adapting itself toward the background. So, in that case, weused a fixed number of iterations to evaluate the efficiency.


TABLE ISEGMENTATION ERRORS OF CURVES AND FLUX METHODS AND THEIR REGULARIZED SCHEMES FOR DIFFERENT

GAUSSIAN NOISE STANDARD DEVIATION LEVELS APPLIED ON THE SYNTHETIC DATA SET

TABLE IISEGMENTATION ERRORS OF ORIGINAL AND REGULARIZED CURVES SCHEMESAPPLIED ON THE SYNTHETIC DATA SETS FOR DIFFERENT VALUES OF FOR

TWO NOISE LEVELS OF AND

TABLE IIISEGMENTATION ERRORS OF ORIGINAL AND REGULARIZED FLUX SCHEMESAPPLIED ON THE SYNTHETIC DATA SETS FOR DIFFERENT VALUES OF FOR

TWO NOISE LEVELS OF AND

segmentations in panels (a) and (b), a significant elongationimprovement using our method is observable. This correspondsto the improvement of the segmentation errors from 68.9% to43.41% as indicated in the column of the Table I.Also by comparing the segmenations of FLUX in Fig. 7(c)and (d), we can notice the start of some leakgage in (c) whichis not present in (d). It shows that while preserving a bettertubular shape, elongation improvement in the main orientationis achievable using our regularization method. The last columnof the Table I indicates the elapsed CPU time for differentsegmentation methods. These are for a C++ code running on a3.0 GHz PC under Linux. As shown, the inclusion of our reg-ularization terms increases the execution time by a maximumfactor of two, but improvements of segmentation is noticeablein most cases.

We further segmented our model volume using values ofand for CURVES and FLUX re-

spectively. As shown in Table II, the performance of regularizedCURVES method degrades beyond an optimal value of . Thereason is that by too much enforcing the anisotropic constraint,the segmented tubes do not follow the actual paths in the highercurvature parts. On the other hand, as shown in the Table III, seg-mentation error of the anisotropic regularized FLUX, does notsignificantly change by varying . The reason can be explainedas follows: for lower noise levels, the FLUX propagates throughlower contrast regions without significant leakage. Therefore, inthat case, anisotropic constraint is not much effective. The effi-ciency of the FLUX is more improved in higher noise levels,where the leakage happens using the original method. We alsonote that over enforcing the shape constraint by higher values of

deteriorates the segmentation performance.

C. Two-Dimensional Examples

Fig. 8 shows example retinal angiogram image segmentationsusing the regularized and relax FLUX schemes. The multiscaleoutward flux of the image gradient field is obtained as indicatedin Fig. 8(b). This is in fact a bi-directional speed image that con-trols the front expansion and contraction. Initial seeds are indi-cated in Fig. 8(c). Fig. 8(d) is the segmentation from the relaxflux maximizing scheme obtained by and in (20).Note that significant adaption to background has occurred, be-cause the flux image is expansive (negative) at those regions.Fig. 8(e) is the segmentation obtained from applying our de-fined anisotropic constraint by and . The neigh-borhood radius is selected for this experiment only byvisual assessment of the approximate vessels width. Note thatthe leakage has been controlled and the main structure of thedim vessel is extracted successfully.

Fig. 9 is another selected region from the same retinal an-giogram. The data was initially smoothed by application of aGaussian filter and the multiscale-flux image was computed.2

Fig. 9(d) is the segmentation achieved by the original FLUX.Note that, although the main vessels are captured, leakage hasoccurred into the background region. Inclusion of curvature inevolution by setting and , has resulted a betterregularized segmentation in Fig. 9(e), but at the same time ithas been a constraint to allow enough elongations. Fig. 9(f) isthe segmentation obtained from applying both smoothness andanisotropy constraints by and . The neighborhoodradius is selected for this experiment based on approxi-mate visual assessment of the vessel widths. Note that while themain vascular region has been extracted, leakage to backgroundhas been significantly limited compared to Fig. 9(f).

D. Three-Dimensional Examples

Volumetric segmentations were achieved using ten phasecontrast (PC) and time-of-flight (TOF) MRA data sets obtainedfrom different scanners of variable magnetic strengths. Sometypical examples are given here for illustration purposes.Qualitative comparisons are mainly presented since obtainingthe ground-truth segmentation of these low-contrast complexvascular structures was difficult. Maximum intensity projection(MIP) images are included for visual evaluation, though webelieve that MIP partially indicates the low-contrast vessels.

Fig. 10 is the first example using a 3T TOF-MRA data set.One important objective is to illustrate the intrinsic difference

2The initial Gaussian filtering contributes to enlarge the original size of ves-sels, as the segmented vessels may slightly appear wider. This filtering, however,is necessary to make the data well smooth [22].


Fig. 8. Retinal angiogram with size of pixels used for regularization of FLUX segmentation. (a) Original image, (b) multi scale outward flux, (c) initialseeds, (d) segmentation using original FLUX method , (e) regularized flow with anisotropy constraint .

Fig. 9. (a) Part of another retinal angiogram image with size of pixels, (b) mutliscale flux image, (c) initial seeds, (d) segmentation using originalFLUX method , (e) FLUX without anisotropy constraint , (f) Regularized FLUX .

between our proposed “anisotropic constraint regularization”and “minimum surface curvature” methods. The latter has beenemployed in CURVES for smoothing. To that end, anisotrop-ically regularized GAC was compared to the CURVES. For afair comparison image driven terms were kept the same, thiswas necessary since without the heuristic term of ofthe CURVES, segmentation was not possible. We can observethat because of minimum curvature smoothing in Fig. 10(b), thinvessels in the CURVES segmentation appear to be contractedand obscured. While using anisotropic constraint regularizationin panel (d), the surface is equivalently smooth and the shape in-formation has introduced significant extensions of thinner ves-sels. Anisotropic constraint regularization was also applied onFLUX segmentation. The mutliscale FLUX image was com-puted for the discrete scales ranging from 1 to 6 voxels usingthe principle described in [22]. As shown the extracted ves-sels in the original FLUX segmentation appear to be irregularlyshaped, while a few low-contrast segments have been missedFig. 10(c). Irregularity was even increasing by further iterationsmaking the visualization difficult. Whereas, using our regular-ization method, the extracted vessels appear more tubular andsmooth Fig. 10(e). Most importantly, this smoothness has notcompromised thin vessels segmentation. In fact, those structuresshow better segmentation (indicated by arrows).

Fig. 11 is our second example using another 3T TOF-MRAdata set. By comparing the results of CURVES and anisotropic

constraint regularized GAC in panels (b) and (d), respectively,we similarly find out that minimum surface curvature can ob-scure the extension of the segmentation toward thinner vessels,but using our method obtaining both smooth and extended struc-tures is possible. Also by comparing the results of the mut-liscale FLUX and anisotropic constraint regularized FLUX inpanels (c) and (e), respectively, we realize enhanced tubularityand smoothness of the surface in (e), as well as extension of afew low-contrast vessels, that is achieved by introduction of theanisotropic constraint.

Since the minimum curvature smoothing term discouragesexpansion, a question may arise is that if the smoothing termis excluded from CURVES, how would it compare to ouranisotropic regularization method? Fig. 12 shows the result ofsuch study. The surface curvature term is ignored in CURVES,and, therefore, Fig. 12(b) is the maximum gain out of unregu-larized GAC segmentation. We observe that even though thesmoothing term is excluded from evolution there are still somemissing faint vessels, which are segmented using anisotropicconstraint as shown by arrows in (d). Also, by comparingthe results of the mutliscale FLUX and anisotropic constraintregularized FLUX in panels (c) and (e), respectively, we findout that although the difference is not so significant (partlybecause of smaller compared to Figs. 10 and 11),there are still a few gaps between some vessel segments whichare merged in (e). It should be noted that the FLUX can detect


Fig. 10. Portion of a 416 512 112, 3T TOF-MRA data used for illustration of regularization of FLUX and CURVES. (a) MIP images from different directions.Segmented vessels using (b) minimum curvature regularized GAC (CURVES), inclusion of in CURVES results in contracted structures, (c) original FLUX, (d)anisotropically regularized GAC , better elongations of the thin vessels are obtained, (e) anisotropically regularized FLUX , segmentedstructures are more tubular and some thin vessel extensions are achieved (indicated by arrows).

the missing gaps by further iterations, however, vessels losetheir tubular shape.

Finally, similar results have been obtained using low-TeslaMRA data sets. These experiments can be regarded as thebenchmark studies with lower SNR. Figs. 13 and 14 aresegmentations of sample intraoperative 0.4T PC-MRA and0.3T TOF-MRA data sets by CURVES and FLUX alterna-tives. In these implementation the curvature term is includedin CURVES to obtain a smooth surface. This has partlycontributed to occlude the segmenting front in some thinervessels as shown in (b), whereas anisotropic regularization ofCURVES has resulted a better segmentation in (d). Note thathow several distinct segments have merged together and overall the continuity seems much better. Similarly, the smoothnessand tubularity of the segmentation surface of anisotropicallyregularized FLUX in Fig. 14(e) compared to the original FLUXin Fig. 14(c) is noticeable.

A quantitative validation experiment is shown in Fig. 15. Forthis case we had the possibility to scan a volunteer with two dif-

ferent magnetic strengths; the lower field was used for segmen-tation and higher field for validation. The first data, a TOF-MRA256 256 153 matrix used for validation purpose only, wasobtained from a higher 3 Tesla MR scanner. A noise clean re-gion of interest was separated as shown in panel (a). The seconddata was obtained from a 1.5 Tesla scanner and registered to thefirst volume. The corresponding ROI is shown in (b). Clarity ofthe vessels in high Tesla field, allowed us to make a referencesegmentation volume using an appropriate thresholding value asshown in panel (c). Fig. 16 shows the segmentation results. Nocurvature was employed in this segmentation to obtain the max-imum gain from CURVES. By visual comparison of (a) to (b)and (c) to (d), we see that most of extensions achieved in (b) and(d) are in accordance with the reference volume in Fig. 15(c),showsing the advantageous of our method to improve the elon-gations of low contrast vessels. Moreover, by comparing thepanels (c) and (d), we may observe that using our regularizationmethod, extracted structures appear to preserve better tubularshapes. The segmentation errors were obtained using (21) and


Fig. 11. Portion of 3T TOF-MRA data set with the size of 200 256 73. (a) MIP image. Segmented vessels using (b) CURVES (minimum surface curvatureregularization), (c) FLUX, (d) anisotropic regularization of GAC , thin vessels show better segmentation, (e) anisotropic regularization of FLUX

. Compared to (c), enhanced tubularity of the segmented structures and elongation of a few low-contrast vessels (indicated by arrows) can be observed.

indicated in Table IV. Note the lower segmentation errors of theregularized methods versus original methods, i.e., 49.55% and43.89% versus 52.88% and 46.64%, respectively.

VI. DISCUSSION AND CONCLUSION

A new geometric regularization method is proposed forsegmenting human blood vessels. This regularization enforcesstructural anisotropy and has the ability to detect the major frontorientation to enforce the elongation in that orientation. Themethod takes the advantage of shape induced information, and,therefore, the anisotropic expansion at the ending of narrowvessels is independent from the image content. The advantage isthat these straight extensions encourage thin vessel extensionsif no image gradient data is available, so a few closely apartvessel segments can merge without losing the tubular shape.However, as it has been notified by one of our anonymousreviewers, leakage in the form of over-extension may arise,particularly when the anisotropic constraint or the value ofis rather high. Unfortunately given a 2-D/3-D image, settingan appropriate value for is not straightforward in general,but for a typical PC and TOF MRA data set, as discussed inthe Section V-A upon normalizing the image dynamic range,an optimum value can be selected from a conservative rangeof . Within this range, improvement of segmentationfrom visual and quantitative aspects was achievable withoutsignificant leakage. One research direction to control the oversegmentation at the vessel endings, is to select a differentdefinition , e.g., so that it cantolerate higher values of isotropy in the vessel endings.

The size of neighborhood used for computation of localsurface structures, is a feature which should be assigned to thescale of target vessels. Small neighborhood size fails to pro-vide robust orientation estimation and large sizes will includeother surrounding structures and consequently may produce er-roneous results. Fortunately, for our 3-D MRA data sets, inwhich the most challenging part is to detect thin vessels, se-lecting suffices. We emphasize that this selection doesnot affect the final segmented vessel thickness, but is a feature toencourage segmenting vessels thinner or equal to that size. Thiscan be realized if we note that with a same in both CURVESand FLUX algorithms, segmented vessels using CURVES ap-pear thinner than actual MIP images.

The proposed regularization method has various improve-ments over original FLUX and CURVES methods. Elongationof thin vessels is significantly improved when our anisotropicconstraint is applied on CURVES. On the other hand, althoughthe FLUX is capable to detect the low contrast vessels, themethod is basically weak in preventing leakage and by furtheriterations the segmented structures may lose their tubularpatterns. This is shown in most of our experiments as well as ina recent publication [38]. Anisotropic constraint when appliedon the FLUX improves the tubularity of the extracted vessels,inhibits the leakage, and enhances the elongations in the mainorientations.

Compared to the minimal surface curvature smoothingmethod, the proposed method is computationally expensive butit can provide smoothness and anisotropic elongation which isparticularly important for thin structure segmentation. Sinceour regularization method enforces an implicit tubular shape


Fig. 12. Portion of a 256 320 128 3T TOF-MRA data set: (a) MIP images from different orientations. Segmented vessels using: (b) CURVES without curva-ture smoothing term, (c) FLUX, (d) anisotropically regularized CURVES , (e) anisotropically regularized FLUX . Arrows indicate somevessels missing in the corresponding unregularized versions.

prior, vessel bifurcation is inhibited because such structurescontribute to orientation ambiguity (or isotropy) at bifurca-tions. One possible interesting direction for research is thedevelopment of a mechanism to suppress our regularizationmodel at vessel branchings. Whether this can be achieved usinggeometrical features or image content features remains as ourfuture research activities. Finally, it should be noted that thoughwe applied our method for segmentation of vessels in medicalimages, the proposed method is quite general and it can beused for segmenting other elongated patterns in nonmedicalapplications, e.g., road detection in synthetic aperture radarimages.

APPENDIX AFRÉCHET DERIVATIVE AND ITS PROPERTIES

The following review of basic derivation concepts and sup-plementary propositions is adapted from [33].

Definition A1 Fréchet Derivative: Let and be linearnormed vector spaces, and be an open subset of function

is called Fréchet differentiable at if thereexists a bounded linear operator such that

(A1)

where denotes the space metric. We write andcall it the Fréchet derivative of at . is called the Fréchetdifferential of at .

The basic differentiation properties for Fréchet derivative areas follows.

Proposition A1 (Linearity): If and are two maps fromwhich are differentiable at , and and are scalars

(two real or complex numbers), then is differentiable atwith .Proposition A2 (Product Rule): Let and be

normed spaces. If andare differentiable at , and is a bounded


Fig. 13. Portion of a 256 256 60 0.4T phase contrast MRA data set. (a) MIP images. Segmentation obtained from (b) CURVES (minimum curvatureregularized GAC), (c) original FLUX, (d) anisotropic constraint regularization of GAC with significant improvement of thin vessel segmenta-tion, (e) anisotropic constraint regularization of FLUX . Arrows indicate a few low-contrast vessel obtained using our regularization method.

bilinear form, then the operator isdifferentiable at , and for

(A2)

Proposition A3 (Chain Rule): If is differentiableat in , and is differentiable at , thenthe composition is differentiable in and the derivative isthe composition of the derivatives:

.We are interested in minimizing our shape-dependent func-

tional defined in (6); therefore, the Fréchet derivative should beapplied for , where is the set of continuouslevel-set functions defined over the region and is the setof non-negative real numbers.

The necessary condition for to have an extrema oncan be stated using Fréchet derivative as

for all (A3)

To keep the consistency with notations on [32] and [27], wedenote: as the Fréchet differential of

at in direction.

APPENDIX BPROOF OF THEOREM 2

In order to derive the EL equation of (7), we follow the vari-ational framework outlined in [32]. For abbreviation purposes,we denote , where the matrix has


Fig. 14. Another example using a 256 256 90 0.3T TOF-MRA data set. (a) MIP images. Segmentation obtained from (b) CURVES (minimum curvatureregularized GAC), (c) original FLUX, (d) anisotropic constraint regularization of GAC with significant improvement of thin vessel segmentation, (e)anisotropic constraint regularization of FLUX . Compared to (c), segmentation surface obtained in (e) is smoother and it shows better low contrastvessel detection (indicated by arrows).

been defined in (1). According to definition in (A1), derivativecan be written as

(B1)

where and , using the product rule (A2) and setting, i.e., normal scalar product, (B1) can be

written as

(B2)

(B3)

The term in (B2) is in fact the scalardifferential and with implied dependency on can be written as

(B4)

where as . After replacing (B4) in (B2),can be obtained as

(B5)


Fig. 15. TOF-MRA 256 256 153 data sets with two different magneticstrengths used for our validation experiment. (a) MIP from a portion of a1.5 Tesla data, used for segmentation. (b) MIP of a the 3 Tesla data of the samevolunteer used for validation. (c) The reference volume obtained from 3T data.

By using Green (divergence) theorem, (B5) is equal to

(B6)

Simplification of in (B3) is also straightforward but requiresfurther manipulation. Using the chain rule introduced in propo-sition (A3) and according to derivative of matrix trace and in-verse operator properties

(B7)

where the dependency of and on is implied. The termusing definition of in (1) can be expressed as

(B8)

Fig. 16. Segmentations of the 1.5T TOF data set. (a) Original CURVES, (b)regularized CURVES with anisotropy constraint, (c) original FLUX, (d) FLUXwith anisotropy constraint, arrows indicate a few merging vessels using ourmethod. Note that the most of extensions and merging obtained in (b) and (d)are in accordance with the reference segmentation in Fig. 15(c).

TABLE IVSEGMENTATION ERRORS OF ORIGINAL AND REGULARIZED FLUX AND

CURVES SCHEMES ON A 1.5T TOF-MRA DATA SET

Replacing (B8) in (B7), and changing the order of trace andintegral operators, i.e., , we get

(B9)

where

(B10)

(B11)

(B12)


We note that is a positive symmetric matrix; hence, furthersimplifications can be achieved by noticing that

(B13)

Other terms can be written in the same way and therefore

(B14)

(B15)

(B14) and (B15) can be replaced back in (B9) to obtain . Afterchanging the order of integrals in (B3) and defining as

(B16)

can be obtained as

(B17)

Similar to (B6), after applying divergence theorem, and ex-panding the divergence on , (B17) can be written as

(B18)

Replacing with in (B18) and addition with in (B6),finally, we obtain

(B19)

According to (A3), this should vanish for every ; therefore, byusing gradient decent method, we get

(B20)

Equation (B20) is the level-set EL equation for unconstrainedminimization of . The constrained optimization in (7) canachieved by replacing the regional update values with valuesextended from zero level set [32]. We note that

is a nonsymmetric regional “source” term thatclearly violates our constraint. This must be replaced by valuesfrom border. However, since we are dealing with sharp cornersin which finding the closeset points on the border is inaccurate,the second methodology is to ignore this regional source termand use reinitialization. This has been shown to be equal to theextension method in the steady state solution [32]. Therefore,the EL equation of constraint minimization problem introducedin (7) can be written as

Starting from an initial signed distance function, every after afew iterations is explicitly reinitialized to the signed distancefunction.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers thathelped us to improve the quality of this paper significantly. Theywould also like to thank M. Goto for provding the sample MRAdata sets from Radiological Center, The University of TokyoHospital.

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Ali Gooya (S’07) received the B.S. degree in elec-tronic engineering from the the Iran University ofScience and Technology in 1997, the M.S. degreein bioelectric engineering from Tehran University,Iran, in 2000, and the Ph.D. degree in the fieldof bioinformatics from The University of Tokyo,Tokyo, Japan, in 2007.

He joined TAM Iran Khodro and developedseveral computer vision applications. In 2004, hereceived a Monbusho scholarship from the JapanGovernment. He was awarded a postdoctoral fel-

lowship at The University of Tokyo by the Japan Society for Promotion ofScience. His research interest includes medical image analysis, variationalimage processing and segmentation, pattern recognition, and computer vision.

Hongen Liao (M’04) received the B.S. degree inmechanics and engineering sciences from PekingUniversity, Beijing, China, in 1996, and the M.E. andPh.D. degrees in precision machinery engineeringfrom The University of Tokyo, Tokyo, Japan, in2000 and 2003, respectively.

He was a research fellow of Japan Society for thePromotion of Science (JSPS). He has been a facultymember at the graduate school of engineering, TheUniversity of Tokyo, since 2004. He is currently anAssociate Professor with the Department of Bioengi-

neering, Graduate School of Engineering, The University of Tokyo. He is theauthor and coauthor of more than 80 peer-reviewed articles and proceedings,as well as over 100 abstracts and numerous invited lectures. His research inter-ests include image-guided surgery, medical robotics, computer-assisted surgery,and fusion of these techniques for minimally invasive precision diagnosis andtherapy. He has also been involved in long viewing distance autostereoscopicdisplay and 3-D visualization.

Dr. Liao was distinguished by receiving the government award [The Com-mendation for Science and Technology by the Minister of Education, Culture,Sports, Science and Technology (MEXT), Japan]. He is also the recipient ofmore than ten awards, including the OGINO Award (2007), the ERICSSONYoung Scientist Award (2006), the IFMBE Young Investigators Awards (2005and 2006), and several Best Paper Awards from different academic societies.His research is well funded by MEXT, the Ministry of Internal Affairs and Com-munications, the New Energy and Industrial Technology Development Organ-ization, and the JSPS. He is an Associate Editor of IEEE EMBC Conference,Organizing Chair of MIAR 2008, and Program Chair of ACCAS 2008.

Kiyoshi Matsumiya received the B.S. degree fromWaseda University, Japan, in 1999, and the M.S. andPh.D. degrees in engineering from The University ofTokyo, Tokyo, Japan, in 2001 and 2004, respectively.

He was a Research Associate with The Universityof Tokyo from 2004–2007. He is currently aninstructor with the Okayama University of Sci-ence, Okayama, Japan. His research interest coversdifferent biomedical engineering applications andsurgical robotics for minimally invasive surgeries.


Ken Masamune was born in Tokyo, Japan, in 1970.He received the Ph.D. degree in precision machineryengineering from The University of Tokyo in 1999.

From 1995 to 1999, he was a Research Associatein the Department of Precision Machinery Engi-neering, The University of Tokyo. From 2000 to2004, he was an Assistant Professor in the Depart-ment of Biotechnology, Tokyo Denki University,Saitama, Japan, and between 2004 and 2005, he wasan Associate Professor in the Department of Intel-ligent and Mechanical Engineering. Since August

2005, he has been an Associate Professor with the Department of Mechano-In-formatics, Graduate School of Information Science and Technology, TheUniversity of Tokyo. His main interest includes computer-aided surgery and, inparticular, medical robotics, image guided surgery, and visualization systemsfor assisting surgical procedures to achieve minimally invasive surgery usingMRI and other image information.

Yoshitaka Masutani (M’02) received the B.E.,M.E., and Ph.D. degrees for his biomedical imagingresearch on vascular structure analysis in precisionmachinery engineering from The University ofTokyo, Tokyo, Japan.

Since 2001, he has been an Assistant Pro-fessor with the Department of Radiology, TheUniversity of Tokyo Hospital, and the Divisionof Biomedical Engineering Graduate School ofMedicine, The University of Tokyo, where he directsthe Image Computing and Analysis Laboratory

(UTRAD/ICAL). His research interests cover broad areas of computer vision,graphics, and image processing for biomedical imaging, especially for diffusionMRI analysis, computerized detection, and computer-assisted intervention.

Takeyoshi Dohi received the B.S., M.S., and Ph.D.degrees in precision machinery engineering from TheUniversity of Tokyo, Tokyo, Japan in 1972, 1974, and1977, respectively.

After a brief research fellowship with the Instituteof Medical Science, The University of Tokyo, hejoined Tokyo Denki University in 1979 as a Lecturerand then became Associate Professor in 1981. From1981 to 1988, he was an Associate Professor withThe University of Tokyo in precision machineryengineering. Since 1988, he has been a Full Professor

with The University of Tokyo, where he presently is a Professor of informationscience and technology. His research interests include computer-aided surgery,rehabilitation robotics, artificial organs, and neuro-informatics.

Dr. Dohi is a board member and president of numerous domestic and interna-tional professional societies, including the International Society for ComputerAided Surgery (ISCAS), The Japanese Society for Medical and Biological En-gineering, and The Japan Society of Computer Aided Surgery (JCAS). He hasserved as a board member of Medical Image Computing and Computer AssistedIntervention (MICCAI).

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 8 ... · IEEE TRANSACTIONS ON IMAGE PROCESSING,...

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