XI International PhD Workshop OWD 2009, October 2009

Transcription

1 XI Internatonal PhD Workshop OWD 009, 17 0 October 009 Vessel Detecton Method Based on Egenvalues of the Hessan Matrx and ts Applcablty to Arway Tree Segmentaton Marcn Rudzk, Slesan Unversty of Technology Abstract Ths paper presents a 3D mage processng method that s based on the analyss of Hessan matrx egenvalues combned wth a multscale mage analyss approach. The method, orgnally developed for blood vessels detecton n medcal mages, can also be used n other areas, where fndng lne-lke structures n the mage s requred. Theoretcal background, advantages and dsadvantages of the method are descrbed. Possble modfcatons requred to allow the method to detect structures of dfferent character (arway tree) are mentoned. An mplementaton of the method was tested on synthetc mages contanng arway-lke structures as well as on real medcal mages from chest CT scan. Results show that the method n general can be used to arway detecton n 3D medcal mages, however t requres mprovements and some adaptaton to ths specfc purpose. 1. Introducton Arway tree segmentaton s an mportant step of medcal mage analyss. It helps the radologst to assess the state of patent s arways, fnd anomales (lke stenoss, nodule, foregn body) and perform surgery plannng, when usng the mnmally nvasve surgery approach. In the case of D mages (e.g. chest radogram) manual mage analyss s usually suffcent n the terms of performng the dagnoss and coarse surgery plannng. In the case of 3D volumetrc data lke CT, MR (Computed Tomograph Magnetc Resonance), manual approach s tedous and tme consumng because many (from 50 to 00) mages have to be analyzed. For the medcal analyss of patent s blood vessels the problem s qute smlar. There were developed methods for automatc detecton or enhancement of vessel-lke structures n medcal mages, manly addressed to blood vessel detecton. Presented n ths paper vessel detecton method s based on the analyss of egenvalues of the mage Hessan matrx [, 3, 4, 6] combned wth a multscale mage analyss approach [1]. The advantage of ths approach s that t can operate n D and/or 3D. Furthermore, basng on the egenvalues not only vessel-lke, but also sheet-lke or blob-lke structures can be detected [3, 6]. Combned wth the multscale mage analyss approach [1] gves a versatle tool for blood vessel enhancement and detecton. However, because the method was developed manly for blood vessels detecton, t cannot be drectly used for arway tree detecton/enhancement, whch n general has the same tubular structure lke blood vessels, but dfferent ntensty cross secton profle. The paper consders the possblty to adapt the method to arway tree detecton The paper s organzed as follows: Secton presents theoretcal background of the method: propertes of the egenvalues of the Hessan matrx and the multscale mage analyss approach. Secton 3 ponts out advantages of the method and drawbacks that need to be overcome f the general method s to be adopted to enhancement and/or segmentaton of the arway tree. In Secton 4 testng and results of an mplementaton of the method are presented. Secton 5 ndcates some deas that should allow the method to be used for the arway tree segmentaton. The paper ends wth a short summary n Secton 6. Throughout the paper followng notons are used: I grayscale nput mage (volumetrc data represented as a 3D array), x, z coordnates of a voxel wthn I, H Hessan matrx, λ egenvalues of the Hessan matrx, G gaussan kernel wth standard devaton.. Theory of operaton.1 Analyss of the egenvalues of the Hessan matrx For a gven voxel of the nput mage a Hessan matrx s composed from the mage nd order partal dervatves (1). I I I y z I I I H = y y yz (1) I I I z zy z 100

2 The partal dervatves are calculated as voxel ntensty dfferences n the neghborhood of the voxel. The Hessan matrx descrbes the nd order local mage ntensty varatons around the selected voxel []. For the obtaned Hessan matrx ts egenvalues λ and egenvectors are calculated. Egenvector decomposton extracts an orthonormal coordnate system that s algned wth the second order structure of the mage [3]. Havng the egenvalues and knowng the (assumed) model of the structure to be detected and the resultng theoretcal behavor of the egenvalues, the decson can be made f the analyzed voxel belongs to the structure beng searched. In the lterature several models were analyzed to fnd the relaton between the egenvalues and vesselness of a voxel (meanng the lkelhood that the voxel belongs to a blood vessel) [4]. What all proposed models have n common s that the ntensty wthn the vessel exhbts a gaussan dstrbuton. For example a smple cylndrcal model () s shown n Fg.1 center and Fg.1 top left. x + y I( x, = G ( x, = const e () Fg.1. Cylndrcal vessel model (center vessel crosssecton, top left ntensty dstrbuton, top rght, bottom left, bottom rght analytcal egenvalues). For the proposed vessel model the analytcal expressons of λ are calculated and analyzed how they behave n the center of the vessel model (Fg.1 top rght, bottom). Krssan et al. [4] shown that the egenvector correspondng to the egenvalue of the smallest magntude determnes the drecton along the vessel (drecton of smallest ntensty varatons). However, at the vessel contours the method fals because two of the egenvalues become zero [5]. Analytcal expressons of the egenvalues and egenvectors for several vessel models can be found n [4]. Other presented models, despte ther ncreasng complext also share the same ntensty profle across the vessel. Ths s because of the appearance of blood vessels n medcal mages (CT, MR) as flled curvlnear cylnders. Followng [3] the egenvalues are sorted so that λ 1 λ λ 3. Tab.1 summarzes the relatons between λ and orentaton of a structure n the mage. Tab.1. Egenvalues of the Hessan matrx and mage structure orentaton (L low, H+ hgh postve, H- hgh negatve) λ 1 λ λ 3 structure orentaton L L L nose (no preferred structure) L L H brght sheet-lke structure L L H+ dark sheet-lke structure L H H brght tubular structure L H+ H+ dark tubular structure H H H brght blob-lke structure H+ H+ H+ dark blob-lke structure Several formulas were proposed to calculate the vesselness of a voxel basng on the values of λ [, 3, 4, 6]. For the aforementoned example one of the egenvalues s always zero and two other exhbt a large negatve value n the center of the vessel. Ths ndcates the character of the structure wthn the mage (see Tab.1) and s used as a crteron n the functon that calculates voxel s vesselness. Addtonally the vesselness functon should ncorporate a term mnmzng the nfluence of mage nose [3]. The output mage s created voxelwse usng the calculated vesselness values. The method s usually combned wth the multscale mage analyss theory [1] that allows usng the same method for fndng small and large objects provded that the object s smlar n the terms of ts model, but ts sze (length, dameter) vares.. Multscale mage analyss The dea of multscale mage analyss s to add a new dmenson to the analyss mage scale. The mage s transformed nto a set of derved mages, each representng the orgnal, but at a dfferent scale. Wth ncreasng the scale the mage gets less detaled. The obtaned set s called the scale-space representaton of the mage. The scale-space theory ntroduced by Lndeberg uses for the purpose of detal removal a convoluton wth a gaussan kernel [1]. For an n-d mage I ts scale-space representaton L(t) at scale t s the mage I convolved wth an n-d gaussan kernel G where t= : L = = ( t) I( x1,..., ) G ( x1,..., ) t (3) Spatal dervatves of the scale-space mage representaton L(t) can be calculated as a convoluton of the mage wth the dervatve of the gaussan kernel at scale t (4): L( t) = G ( x1,..., ) I ( x1,..., ) (4) = t 101

3 In order to be able to compare the dervatves across multple scales one can normalze the free x varables: x ˆ =. Then the dervatves of the gaussan become normalzed by ts standard devaton : G = G = G, leadng ˆ ˆ to: L( t) = G ( x1,..., ) I( x1,..., ) (5) = t what allows the responses across scales to compared. Smlarl the nd order normalzed dervatve of L(t) s calculated usng: j L( t) = G (...) I (...) (6) = t Scale-space representaton smplfes the contents of the mage dependng on the chosen scale. Ths allows to search for objects of smlar dmensons as the chosen scale, or to analyze the mage across wde range of scales to see f any object of unknown sze but known model can be found..3 Multscale vessel detecton The multscale vessel detecton s performed for scales between t mn and t max (correspondng to mn and max). For each mn ; the Hessan max matrx entres are calculated usng (6). Then the egenvalue analyss s performed as descrbed n Secton.1 and the result for a gven scale s obtaned. The fnal result of the multscale analyss s the voxel-wse maxmum of obtaned results over all analyzed scales. 3. Propertes In general, the method based on the Hessan egenvalues analyss s capable of detectng not only tubular structures, but also blob-lke and sheet-lke structures wthn the mage [3, 6]. Ths only requres fndng proper formulas for blobness and sheetness as functons of λ. Also dark vessel detecton s not a problem [3] as t only requres change of condtons mposed on values of λ durng calculaton of voxel s vesselness. However, the ntensty dstrbuton wthn the vessel stll has to be more-or-less of gaussan shape to ensure maxmal values of the two egenvalues at the center of the vessel. The man advantage of the method s that there s no dscretzaton of the vessel orentaton by egenvector decomposton the prncpal drectons of the nd order mage structure are found. Ths approach s less computatonally expensve than performng multple flterng n multple dscrete j orentatons [3]. Due to the fact that the analyss of the theoretcal egenvalues s performed at the center of the vessel, the method by tself extracts the centerlnes of the vessels, what can be consdered as another advantage. Also because the output s calculated for each voxel separately the method may successfully detect vessels wth hgh degree of stenoss (or even dsconnected segments), whereas methods based on voxel connectvty fal to segment the part of the vessel after the obstructon and requre specal detecton of such cases. However, the lack of connectvty and neghborhood analyss s also a drawback when the vessels exhbt other than gaussan ntensty profle (although the multscale approach partally takes care of ths ssue see Fg. and Fg.3) and the behavor of the egenvalues farther from the vessel center should also be consdered. Addtonall the formulas used to calculate the vesselness provde good response to tubular objects and good nose and other structure (blobs, sheets) suppresson, however at vessel bfurcatons the method s response s weak. Ths s due to the fact that the vessel bfurcaton does not exhbt a tubular structure, but rather s blob-lke. Another ssue s connected wth small vessels (few voxels n dameter), where due to fnte mage resoluton and partal volume effect, the vessel center s generally not n the center of a voxel. In such case the egenvalues are not calculated at the vessel center, thus the response s weak [4]. For those reasons the method s usually used as a preprocessng step or as a support n a more complex vessel segmentaton methods [4, 5]. 4. Testng and results Because the arways exhbt a tubular structure the method was tested to assess ts applcablty to arway detecton from CT data. Arways cross-secton profles are smlar (but not suffcently) to the dark vessel cylndrcal model. Fg. shows three exemplary cross-secton ntensty profles from real CT dataset: top trachea, mddle man bronchus, bottom bronchole. As can be seen the trachea (Fg. top) s smlar to a dark vessel, because tssues of hgher densty surround t. Although, the profle has a flat bottom comparng to the cylndrcal vessel model, the multscale approach allows that model to be used (Fg.3). The farther from the trachea the arway s, the more ts shape becomes a ppe a hollow tube wth thn walls surrounded by tssues of almost the same densty as the nsde of the tube (bronchole, Fg. bottom). However the ntensty profle s then more-less of gaussan shape. Fg.3 shows the frst two cross-sectons from Fg., but convolved wth gaussan kernel wth smlar to the radus of the arway. Ths llustrates what happens durng the multscale mage analyss. It 10

4 can be seen that although the trachea orgnally does not have a gaussan profle (and the all egenvalues at the center would be close to zero), at a larger scale the profle becomes of gaussan shape, thus the Hessan egenvalue analyss can gve correct results. An mplementaton of the Frang s vesselness flter [3] was tested to assess ts usefulness to detecton of the arways n 3D medcal CT mages. The testng was performed on synthetc 3D mage data (cross secton shown n Fg.4) contanng ppes wth brght walls and dark vessels wth gaussan ntensty dstrbuton and rad from 1 to 8 pxels. The performance of the method n the presence of nose was also tested (Fg.5). Fg.4. Synthetc mage contanng arways and dark vessels cross sectons (rad from 1 to 8 pxels). Fg.5. Synthetc mage contanng arways and dark vessels cross sectons wth supermposed gaussan nose. The multscale mage analyss was performed for from 1 to 8 that approxmately correspond to the dameters of the objects n the nput mages. The results can be seen n Fg.6 and Fg.7 respectvely. From the results one can notce that although the flter response s much hgher to the dark vessels, the method s stll able to detect the arways correctly. Fg.. Arway cross-secton profles: top trachea, mddle man bronchus, bottom bronchole. Vertcal scale n the profles s n Hounsfeld Unts Fg.6. Testng results of the Frang s method on noseless mage. Fg.3. Arway cross-secton profles after convoluton wth a gaussan kernel: top trachea, =10; bottom man bronchus, =5. Vertcal scale n the profles s n Hounsfeld Unts. Fg.7. Testng results of the Frang s method on nosy mage. Fnall the mplemented method was used to detect arways from real medcal mages. The orgnal CT data was preprocessed n order to extract only the lungs volume. Obtaned lung volume was subject to further analyss. Fg.8 presents volume rendered result (wthout any postprocessng) of the arway detecton from two exemplary CT datasets. As can be seen the trachea and two man bronch were found. The bfurcatons followng the man bronch are mssng and hgher order broncholes are not detected. Addtonally some other anatomcal structures are present n the results. 103

5 Fg.9. Detected blood vessels nsde lung volume (no post-processng appled). 5. Possble mprovements future work Frst ssue to be consdered s the poor performance of the method n detectng small arways. Ths could be seen n the results of synthetc mage and real mage processng. An arway cross secton can be modeled by a Laplacean of a Gaussan (7) functon (smpler to analyze, but not exactly fttng nto the real arway cross secton) or a gaussan rng wth a relatvely large radus R comparng to, for example by (8): x + y + x y I ( x, = const e + e (7) y Fg.8. Detected arway trees (no post-processng appled). For a comparson, Fg.9 shows the result of blood vessels detecton nsde the lung volume usng the orgnal Frang s vesselness flter. + R I ( x, = const e (8) x y For those models the egenvalue analyss has to be done not only at the center of the vessel (especally the n the second case where due to flat profle all λ are close to zero), but also n the closest neghborhood, where the rng s present. Thus the condtons for arwayness should nclude smlar condtons as for vesselness and addtonally those mposed by the presence of the rng around the center of the vessel. In ths way the drawback concernng the lack of neghborhood analyss could be fxed. Another possblty s to combne the vessel detecton wth wall detecton. Arway walls are brght and of planar nature. By detectng planar objects around the vessels, the arways could be detected. However, there s rsk of too hgh false postve detectons as n real CT data there exst many 104

6 anatomcal structures whose boundares could be detected. Mentoned above deas are to be tested durng further research on the presented methodology. 6. Summary The general dea behnd the method based on the Hessan egenvalues analyss seems promsng, but accordng to several authors [4, 5] by tself may be nsuffcent as a standalone vessel segmentaton tool. However, obtaned results show that the method s able to perform the arway tree detecton, but t requres further development and parameter tunng to be fully adapted to ths specfc purpose. Thus the research s stll beng performed on the adaptaton of the method to arway tree segmentaton from real lfe medcal mages. [6] Yang Yu and Hong Zhao: Enhancement Flter for Computer-Aded Detecton of Pulmonary Nodules on Thoracc CT Images, n proc. 6 th Internatonal Conference on Intellgent Systems Desgn and Applcatons, 006 Author: MSc. Rudzk Marcn Slesan Unversty of Technology ul. Akademcka Glwce tel. (03) fax (03) 37 5 emal: marcn.rudzk@polsl.pl Bblography [1] Lndeberg Tony: Scale-space: A framework for handlng mage structures at multple scales, n proc. CERN School of Computng, Egmond aan Zee, The Netherlands, Sept. 1996, avalable onlne: ftp://ftp.nada.kth.se/cvap/reports/ ln-cern-summ-school-96.pdf [] Sato Yoshnobu, et al.: 3D Mult-Scale Lne Flter for Segmentaton and Vsualzaton of Curvlnear Structures n Medcal Images, n proc. of the Frst Jont Conference on Computer Vson, Vrtual Realty and Robotcs n Medcne and Medal Robotcs and Computer-Asssted Surger volume 105 of Lecture Notes n Computer Scence, pages 13, Mar. 1997, avalable onlne: spl-pre007/pages/papers/yosh/cr.html [3] Frang Alejandro F., et al.: Multscale Vessel enhancement Flterng, n W. M. Wells, A. Colchester, and S. Delp, eds, Medcal Image Computng and Computer Asssted Interventon (MICCAI), volume 1496 of Lecture Notes n Computer Scence, pages , Oct. 1998, avalable onlne: mcca1998.pdf [4] Krssan Karl, et al.: Model Based Detecton of Tubular Structures n 3D Images, Computer Vson and Image Understandng, 80:, pages , Nov. 000, avalable onlne: /Publ/cvu.pdf [5] Krssan Karl, et al.: Multscale Segmentaton of the Aorta n 3D Ultrasound Images, n proc. 5 th Annual Int. Conf. of the IEEE Engneerng n Medcne and Bology Socety EMBS, Cancun Mexco, pages , Sep. 003, avalable onlne: HomePage/Publ/KrssanEMBS03.pdf 105