Direction-Driven Shape-Based Interpolation of Volume Data

Transcription

1 Diretion-Driven Shape-Based Interpolation of Volume Data Jiří Hladůvka and Eduard Gröller Vienna University of Tehnology, Institute of Computer Graphis and Algorithms Favoritenstraße 9 11, E186, A-1040 ien, Austria hladuvka Abstrat e present a novel approah to shape-based interpolation of gray-level volume data In ontrast to the segmentation-based tehniques our method diretly proesses the salar volume requiring no user interation The key idea is to perform the interpolation in the diretions given by analysis of the eigensystem of the struture tensor Our method proesses a slie within a ouple of seonds yielding satisfatory results e give a quantitative and a visual omparison to the linear interslie interpolation Analysis of the results lead us to the onlusion that our tehnique has a strong potential to ompete with well-established shapebased interpolation algorithms 1 Introdution Interpolation is required in imaging, in general, whenever the aquired image data are not at the same level of disretization as the level that is desired [1] In volume visualization by ray asting, for instane, it is neessary to aquire samples in regular distanes on a ray The interpolation mehanism has to provide the ray aster with values at any position between grid points Algorithms for volume analysis are often based on filtering tehniques whih presume isotropy The input data has to be resampled in most ases to an isotropi disretization Medial imaging systems usually aquire the data in slie-by-slie order resulting to different sampling ratios in,, and diretions For an appropriate medial treatment it is neessary that the data at requested loations are reonstruted as preisely as possible taking into aount not only the harateristis of a D signal onveying the data but also the topologial properties of the eplored strutures 2 Related work Interpolation algorithms an broadly be divided into two ategories sene-based (image-based) and objet-based (shape-based) Although sene-based filters reeived a lot of attention in the visualization ommunity in reent years [9, 7, 15, 14], there is repeated evidene in the literature of the superior performane of objet-based over sene-based interpolation tehniques [2] From the beginning shape-based algorithms required a segmented volume, ie, a desription of objets to be interpolated The first approahes are based on ontour interpolation Raya and Udupa [12] proposed an interpolation of distane fields omputed from binary slies Higgins et al [5] also fous on the problem of interpolating binary objets rather then high-resolution gray-sale images The presented method etends the approah of Raya and Udupa [12] employing the original density information to avoid inonsistenies in transition of objets ross setions and their entroids Turk and O Brien [16] benefit from both the ontour desription and distane fields enoded as impliit funtions providing a powerful framework for interpolation of analytially given objets Moshfegi [10] proposes a tehnique aiming for removing stairase effets in a manner that is onsistent with MI The diretion of interpolation is aligned adaptively with the aes of the vessels The template mathing proposed here onsiders a pair of san-lines iteratively looking for the best math in referene windows due to mean square error and a orrelation oeffiient The interpolation method is, however, applied after the projetion providing thus a solution only for 2D senes The first purely gray-level-based approah yielding reasonable results seems to be the three-pass algorithm proposed by Grevera and Udupa [1] 1

2 % In order to adopt the previously introdued tehnique [12], the -dimensional grey sene is lifted to a -dimensional binary sene in the first step Then binary shape-based interpolation [12] is applied Finally, the newly interpolated -binary sene is ollapsed bak to dimensions A drawbak of this method is a high time and spae ompleitytwo years later this method and two variants thereof are ompared [2] to the five most referred-to interpolation tehniques The paper emphasizes the superior performane of objet-based over senebased interpolation tehniques, too To see how far this is evident in a speifi appliation, ie, detetion of brain lesions, Grevera and Udupa [] statistially ompare 100 data sets resulting from 10 patients, 2 modalities and 5 interpolation methods In ontrast to the frequently used tehniques we propose a diretion-driven interpolation The interpolation is onduted by the eigenvetors of the struture tensor Struture tensors arry information on the density distribution in simple neighborhoods As they are omputed diretly from salar data our method does not require any segmentation or user interation This makes a reasonable differene between our method and most shape-based interpolation tehniques An introdution to simple neighborhoods and the related tensor analysis as well as the onept of our method are presented in setion Setion 4 disusses implementation issues To see the performane, qualitative and quantitative evaluation of our method is given in setion 5 attern-driven interpolation Usual strategies to study loal neighborhoods are based on analyzing disontinuities in the intensity [8] The human visual system, however, an easily reognize objets that do not differ from a bakground by their mean gray value but only by the orientation or sale of pattern To perform this reognition task with a digital image proessing system, operators whih determine the orientation of patterns are needed [6] 1 Simple Neighborhoods and Struture Tensor For an appropriate representation of the emplaement of an objet or a teture it is neessary to distinguish between diretion given by an angle from the interval and orientation! As we are loally not able to distinguish between patterns that are rotated by 180 degrees, the operators are also required to make no distintion A representation of the orientation by one salar value representing the angle turns out to be not appropriate, beause a measure for ertainty whih desribes the neighborhood independently from the absolute density is not involved This leads to a at least two-omponent vetorial desription The attempt to desribe a neighborhood by the gradient vetor fails beause it does not allow to distinguish between neighborhoods with onstant values and isotropi orientation distribution where gradients anel eah other as they are integrated over the neighborhood This is due to the opposite signs of gradients at inreasing and dereasing edges The following optimization method has been proposed [6] to find an operator whih enodes the angle of orientation, provides a ertainty measure and distinguishes between onstant and isotropi distributions The neighborhood " of the point of interest $# will be desribed by a unit vetor % whih ehibits the least deviation from the orientations of all gradients from " As the squared salar produt &('*),+% - between the gradient vetor '/) and meets riteria for measuring this deviation the following integral will be maimized: 0214 & $#657 89&('*)2& 8 + : % <; (1) where determines a weighting funtion defined on " The optimization problem given by equation (1) an be rewritten as % >AB 0DC E C =+?>% A@ maimum, where & # 5 89&('*)2& 8F'/)2& 8 + ; (2) is a symmetri G matri onsisting of the following elements: >HJIB 0KC E C & L#M5N 8$OQ )2& 8 RH )2& 8 LITS ; () 2

3 q s % \ q % % q % q q q % % % The matri > is referred to as the struture tensor and its onstrution is in more details eplained, eg, in U [4, 6] The solution to % is the prinipal eigenvetor of > In this work we will refer to the eigensystem of > V$ as % X! [Z?\ and V \/] assume V ] thev following ordering of the eigenvalues and orresponding arrangement of the eigenvetors \ % % % e will assume, w/o log, that eigenvetors are of unit length, ie, _% B` 2 Eigenvetors-Aligned Kernel The aim of our approah to interpolation is to preserve to the most possible etent the boundaries of objets, as well as the lines, the edges, and the ridges To ahieve this we propose to ondut the interpolation in the diretions given by these strutures The presumption here is that the hange of the density values in these diretions reahes the minimum Although the struture tensor has originally been proposed to desribe the orientation of a neighborhood by its prinipal eigenvetor \ %, there is more useful information from its analysis For a fied vetor a %, the term a % + >% a gives the square of the density hange in its diretion Under the assumption that > is represented by a regular matri it follows immediately that the minimum of the squared density hange and therefore also the minimum of the density hange are reahed in the diretion of the eigenvetor % Aording to the b!d =e of tensor > the following four ases an be distinguished in a D image (refer also to Table 1): b, =e2&(>f9bg : The salar values do not hange in any diretion The represented neighborhood features onstant gray values Interpolation an simply be done with the nearestneighborhood interpolation method b, =e2&(>f9bh : The salar values signifiantly hange in the diretion of the prinipal eigenvetor %, while in the remaining perpendiular diretions, % and %, remain (nearly) onstant The orresponding neighborhood ontains either a layered teture or a boundary between two objets The interpolation will be onduted by a dis spanned by the eigenvetors % and % b, =e2&(>f9bki : The salar values signifiantly hange in the diretions \ % and % while \ remain (nearly) onstant in % The orresponding neighborhood features either an edge of an objet or an etruded teture The interpolation will be driven by a stik determined by the eigenvetor % b!, :e2&(>fjbk : The salar values hange in all diretions There is either a orner of an objet or a distributed D teture in the neighborhood The interpolation therefore shall onsider all of the eigenvetors The seond row of Figure 2 gives eamples of olor oding of the rank of the struture tensor The areas with b!, =e=&(>fqb 1, 2, and are enoded by green, red, and blue olor, respetively Homogeneous areas are eluded from the rendering To perform a diretional interpolation at a point # we will weight the density ontributions )2&l m of voels from its neighborhood aording to their relative position to an ellipsoid no& $#J The main aes of the ellipsoid will be given by saled eigenvetors % as follows: where qts b!, =el> % 1 p4% \ 2 p4% \ \ % pr% (4) utvp The purpose of saling (4) is to suppress the diretions of a large density hange and to emphasize the diretions of the small density hange e define the weight of a point from the interior of ellipsoid no& $# as: (z w &l m9b`q5 [Zf\4y l 45 # f% 8% 9% z { (5) and set it to zero for the eterior points and the boundary of no& $#J This sets a smooth fade-of of weights from the enter of the ellipsoid to its borders Fig 1 demonstrates how the situation may look like in a plane given by the eigenvetors % and % The gray-level oding of the ontributing grid points orresponds to their weights Finally, we define the interpolated density value as a weighted average: 1f ƒ )2& # 9B~} } 1f ˆƒ w &l L8)2&l m w &l m (6)

4 ) ) ) z Figure 1: The ellipsoid given by the saled eigenvetors and the weighting of the ontributing grid points The similarity in weighting of the three light-gray points is due to equation (5) Tensor ropagation The omputation of the struture tensor due to equation () applies only to grid points To alulate the diretional information also at off-grid points a mehanism to ompute the struture tensor there is needed A straightforward solution would be to generalize equation () involving derivative filters for omputing the partial derivatives between grid points [9, 14] An indiret approah an benefit from the already omputed information at the surrounding grid points employing an interpolation of the orresponding eigensystems Using quaternions for this task is the first hoie for a high quality interpolation On the other hand, sine there is a strong oherene both in orientation and the magnitude of the struture tensor, it is not neessary to sample the tensor field densely [6] Therefore nearest-neighbor interpolation provides a stable and a muh simpler solution 4 Implementation As the interpolation usually is to be performed for a larger amount of voels the implementation an be divided into two steps In preproessing, the eigenvalue analysis of the struture tensor is performed for a set of voels whih surround the positions to be interpolated 1 omputation of the struture tensor J: Identifying the onvolution in equation () with a smoothing of the produt of partial derivatives, the elements >!HJI of the struture tensor > an be omputed in terms of two-pass onvolution and a salar produt In the first step the differential operators are applied resulting to the partial 2 derivatives Then the salar produts H! I are omputed In the final step the salar produts are smoothed with a filter orresponding to a weighting funtion e have used an optimized Sobel filter [6] for the differentiation step and the Gauss filter defined on a 777 neighborhood for the averaging step To redue the omputational overhead we eploited the separability of both filters 2 eigenvalue analysis of J: Sine the struture tensor > is represented by a symmetri, real-valued matri, the fast-onverging Jaobi method an be used for eigen-analysis, as reommended by ress et al [11] The tuples of eigenvalues and V:\ eigenvetors ] V ] V are then rearranged to satisfy After the preproessing step for eah requested position L# the interpolation involves the following steps I1 inheriting the tensor information: In aordane to setion we have used the nearest neighbor interpolation for the tensor transfer I2 interpolation: For all voels from the neighborhood q of # determined by the saling fator weights w &l are omputed aording to the equation (4) and (5) and the weighted average (6) is assigned q to the result of interpolation The values of and p have been set empirially after an error analysis of several eperiments to pšbk, q and Bkd i 5 Evaluation The purpose of this setion is to provide a threefold analysis of the performane of the proposed method Firstly, in setion 51 we give a quantitative error analysis and a omparison to the linear filtering in diretion Seondly, a visual evaluation of error volumes is presented in setion 52 Thirdly, the time and spae ompleity of our method is disussed in setion 5 For the following analysis and omparison we have used 11 data sets overing a broad spetrum 4

5 \ s in terms of modality, resolution, noise harateristis and objet detail The first group onsists of the small-resolution, noise-free artifiial data sets generated by the vt library [1] The seond group omprises the data aquired from CT and MRI sanners All data set were quantized to 256 levels 51 Quantitative Evaluation The approah to the omparison is to pretend there is a slie missing in the volume, to estimate this slie and ompare it to the original For the omparison, we reuse the framework for error analysis by Grevera and Udupa [2] The authors introdued three figures-of-merit (FOM) In the following epression of FOMs, Œ denotes an interpolation method (linear in and diretional), is the data set being interpolated, )2& X represents the original density value in the voel given by oordinates X, and )Ž*& X represents the density estimation due to the method Œ at the voel given by oordinates X Interpolation runs over all slies of, where slie 4 is defined by its z-oordinate e 1 Mean-squared differene: FOM Ž &( 99B R f & )2& X -5u) Ž & X (7) where is the total number of voels involved in the omparison 2 Number of sites of disagreement FOM Ž &( 99B where & 8?B Largest differene FOMŽ &( j9b šœž Ÿ! : if, & )2& X :5u) Ž & X otherwise (8) (9) )2& X f5u) Ž & X (10) The measurements due to the equations (7) (10) for the linear interpolation and the proposed diretional interpolation are summarized in Table 2 e make the following three observations regarding the performane of our tehnique 1 Our tehnique yields in average better estimates of the original density values than the linear interpolation This is espeially remarkable for the vt data As the Engine data set features many z-ais aligned objets the diretional interpolation does not outperform the linear interpolation so dramatially 2 An eat reonstrution of the original density values happens at more sites in the ase of diretional interpolation Again, it is more obvious in the ase of the vt data and gets almost negligible for the sanned data sets In our opinion, however, these measurements are of a lower importane than the mean-squared error The maimal error is generally smaller in the ase of diretional interpolation 52 Visual Evaluation To evaluate the performane of both interpolation methods visuals we reate two error volumes onsisting of voels with densities set to ž) Ž & X?5u)2& X 6, where Œ denotes the interpolation method The error volumes are then displayed with splatting The rd and the 4th rows of Figure 2 bring eamples on rendering of error introdued by the linear and the diretional interpolation, respetively Sine the diretional interpolation yields very small differenes in the ase of the vt Faet Otahedron dataset, almost every piel gets rendered to the bakground olor in the orresponding image In ase of the CT head with markers the improvement by our interpolation is mostly apparent at the positions featuring objets, eg, the wires in the markers As the error measurements for the Engine data set do not differ signifiantly a visual evaluation is almost impossible in this ase 5 Time and Spae Compleity Referring to the Table, the performane of our method is learly dominated by the smoothing step (setion 4, 1) The total timing therefore strongly depends on the size of the support of the weighting funtion Our implementation proesses in average voels per soond on a 400 MHz entium II This orrespond approimately to 6 seonds for reonstrution of one slie To store the eigenvetors of the struture tensor, our implementation requires a spae of & : where is the size of the data to be interpolated This an, however, be optimized to an on-the-fly omputation and ahing of the struture tensors 5

6 whih gives the same spae ompleity as the one of linear interplation, ie, &Fž 6 Conluding Remarks e present a new method for interpolating the greylevel volume data taking into aount the topologial properties of the strutures Unlike the methods whih are based on an a priori information from segmentation, our method proesses the density data diretly requiring no user interation In order to preserve the boundaries of objets, lines, and the ridges to the most possible etent we proposed to ondut the interpolation by the diretions of these strutures The information on the diretion of interpolation is inherited from the tetural desription of the objets by the struture tensors e ompared the performane of our method to the most used linear interpolation in the diretion of the ais for 11 data sets onluding that our method outperforms the linear interpolation in the mean error, maimal error and in the number of sites with eat reonstrution There are two main issues not addressed in this work Firstly, the size of the oriented filter is given by two onstatnts whih have been set empirially after many eperiments In our opinion this is not appropriate if the method is required to perform robustly for all kind of data Instead of using fied onstants we think of using parameters whih reflet the magnitude of the density hange separately in eah neighborhood A good hoie for this purpose seem to be the eigenvalues of the struture tensor Seondly, we find it neessary to ompare our method to other tehniques, espeially to the stateof-the-art in shape-based interpolation introdued by Grevera and Udupa [1] e would like to onlude with remarks on possible results of this omparison Intuitively, our method requires far less spae and time than the interpolation in [1] To interpolate one slie of voels, the algorithm by Grevera and Udupa requires spae of &( : where denotes the quantization level and tens of minutes (as measured on a Spar 2) In ontrast, our method interpolates one slie in a ouple of seonds requiring &! - spae Finaly, sine we have used the same error measurements and a statistial omparison to the same interplotation method as in [1] we were able to indiretly ompare the error performane of both methods Even though this preliminary omparison sounds promising for our method, further work is required to be done e are onvined that suh a omparison has to be done diretly for the same data sets and address it to the future work Aknowledgments The work presented in this publiation has been funded by the V is M ed projet 1 V is M ed is supported by Tiani Medgraph, 2 Vienna, and the Forshungsförderungsfonds für die gewerblihe irtshaft, Austria A part of the work has also been supported by the BandViz projet 4 whih is supported by FF under projet number Referenes [1] G Grevera and J Udupa Shape-based interpolation of multidimensional grey level images IEEE Transations on Medial Imaging, 15(6): , De 1996 [2] G Grevera and J Udupa An objetive omparison of -D image interpolation methods IEEE Transations on Medial Imaging, 17(4): , Aug 1998 [] G Grevera, J Udupa, and Miki A taskspeifi evaluation of three-dimensional image interpolation tehniques IEEE Transations on Medial Imaging, 18(2):17 14, Feb 1999 [4] H Haußeker and B Jähne A tensor approah for loal struture analysis in multidimensional images In roeedings D Image Analysis and Synthesis 96, Universität Erlangen-Nürnberg, 1996 [5] Higgins, C Morie, and E Ritman Shapebased interpolation of tree-like strutures in three-dimensional images IEEE Transations on Medial Imaging, 12():49 450, 199 [6] B Jähne Digital Image roessing, hapter 11 First-Order Tensor Representation, pages Springer Verlag Berlin Heidelberg, 4 edition,

7 [7] T Lehmann, C Gonner, and K Spitzer Survey: interpolation methods in medial image proessing IEEE Transations on Medial Imaging, 18(11): , Nov 1999 [8] E Lindeberg Edge detetion and ridge detetion with automati sale seletion In roeedings of IEEE Computer Vision and attern Reognition, pages , 1996 [9] T Möller, R Mahiraju, K Müller, and R agel Evaluation and Design of Filters Using a Taylor Series Epansion IEEE Transations on Visualization and Computer Graphis, (2): , 1997 [10] M Moshfeghi Diretional interpolation for magneti resonane angiography data IEEE Transations on Medial Imaging, 12(2):66 79, June 199 [11] H ress, S A Teukolsky, T Vetterling, and B Flannery Numerial Reipes in C, hapter 11, pages Cambridge University ress, 2 edition, 1992 [12] S Raya and J Udupa Shape-based interpolation of multidimensional objets IEEE Transations on Medial Imaging, 9(1):2 42, Mar 1990 [1] M Šrámek and A Kaufman vt: A lass library for voelization of geometri objets vislab/vtools/vt/, 2000 [14] T Theußl, H Hauser, and E Gröller Mastering windows: Improving reonstrution In roeedings of IEEE Volume Visualization, pages , 2000 [15] Thevenaz, T Blu, and M Unser Interpolation revisited IEEE Transations on Medial Imaging, 19(7):79 758, July 2000 [16] G Turk and J F O Brien Shape transformation using variational impliit funtions Computer Graphis, (Annual Conferene Series):5 42,

8 s b, =e=&(>- onditions neighborhood desription V:\ KV gv 0 V:\ s V gv onstant neighborhood 1 V:\ KV s V boundary or layered teture 2 V \ KV gv edge or etruded teture orner or isotropy Table 1: The ases of density distribution in a D sene \ FOM FOM FOM data set z-lin dir z-lin dir z-lin dir vt ire Tetrahedron vt ire Otahedron vt ire Dodeahedron vt Faet Tetrahedron vt Faet Otahedron vt Faet Dodeahedron Lobster Engine Blok CT Head with Markers MRI Head Vertebra Table 2: Error measurements due to the Equations (7) (10) for the linear interpolation in z-diretion and the newly proposed diretional interpolation Input Volume eigensystem of > interpo- total Data set Dimensions Sobel 77 Gauss Jaobi lation vt ire Tetrahedron vt ire Otahedron vt ire Dodeahedron vt Faet Tetrahedron vt Faet Otahedron vt Faet Dodeahedron Lobster Engine Blok CT Head with Markers MRI Head Vertebra Table : Time in seonds for the diretional interpolation as measured on a 400 MHz entium II 8

9 vt Faet Otahedron CT Head with Markers Engine Blok Figure 2: Volume rendering of the original data sets (1st row), olor oding of the rank of the struture tensor (2nd row), error volume of the linear interpolation (rd row), and error volume of the diretional interpolation (4th row) 9