3D PART EGMENTATION UING IMULATED ELECTRICAL CHARGE DITRIBUTION Kenong Wu and Matin D. Levine Cente fo Intelligent Machines & Dept. of Electical Engineeing McGill Univesity, Monteal, Quebec, Canada, H3A A Abstact A novel appoach to 3D pat segmentation is pesented. Beginning with ange data of a 3D object, we simulate the chage density distibution ove an object's suface which has been tessellated by a tiangula mesh. We then locate the object pat bounday at deep suface concavities by tacing local chage density minima. Finally, we decompose the object into pats at the pat bounday points. Intoduction The poblem of segmenting a 3D object into pats has attacted much attention in compute vision. The esolution of this poblem is impotant fo computing pat-based desciptions and ecient object ecognition [, ]. It has been customay to detemine object pats by analysing geometical popeties of objects, such as suface cuvatue o volumetic shape. In this pape, we popose a novel appoach to object segmentation into pats which is motivated by physics: An object to be segmented is viewed as a chaged conducto. Thus a coespondence between the chage density distibution ove the object's suface and the actual pats can be established. The object is then boken into pats based on the simulated chage density distibution on the object. Pat segmentation algoithms can be categoised as being shape- o bounday-based. hape-based appoaches [,, 9, ] decompose objects into pats by measuing the shape similaity between image data and an aangement of pedened pat models. In contast, bounday-based methods segment objects into pats by seeking pat boundaies instead of shape. ince this type of appoach concentates on pat boundaies, it can segment an object without incopoating pat shape infomation. Fo example, Koendeink and Van Doon [] have poposed paabolic lines as pat boundaies. Rom and Medioni [3] have pefomed pat decomposition based on this theoy using ange data as input. Inspied by a paticula egulaity in natue - tansvesality, Homan and Richads [] have poposed segmenting objects into pats at deep suface concavities. A few algoithms based on this concept have been developed to segment objects in ange data [,, ]. The wok epoted in this pape is also boundaybased but employs a new suface popety. tating with single-view ange data, we appoximate an object suface by a tiangula mesh and then simulate the electical chage density distibution. We constuct a Diect Connection Gaph based on the tiangula mesh, which povides a convenient coodinate system on the object's suface. Using this gaph, we locate object pat boundaies at deep suface concavities whee the chage density achieves a signicant local minimum. We then decompose the object into pats at the boundaies. Thee ae cetain advantages to ou appoach. Othe bounday-based segmentation techniques ae dominated by the need to analyse suface cuvatue. Howeve, because of inheent input data noise and the necessity of computing deivatives, cuvatue computation has poven uneliable []. An assumption about suface smoothness is mandatoy when using a cuvatue computation. In compaison, ou appoach involves the solution of a set of integal equations and theefoe does not equie such an assumption. ince the algoithm uses all data on the object's suface to compute the chage density at each location, the in- uence of noise is geatly educed. Physics When a chaged conducto with an abitay shape is in electostatic equilibium, all chage esides unevenly on the oute suface of the conducto [3]. The chage density is vey high at shap convex edges and cones. Convesely, almost no chage accumulates at shap concavities. Theefoe deep suface concavities, which have been shown to delineate pat bound-
O O T i, f i= T, f = k k Figue : The obsevation point and the chage souce point on the suface of an ellipsoid. O is the oigin of the coodinate system. aies [], can be detected by isolating signicant chage density minima. The physical model we have used is the chage density distibution on a pefect conducto in fee space, whee thee is no othe conducto o chage. Let be a vecto position of an obsevation point and be a position of a souce point chage q on the suface, as shown in Figue. uppose chage is continuously distibuted ove the object suface (see Figue ). Then the electical potential at contibuted by all of the chage on can be expessed as follows []: () = Z ( ) j? j d () Hee ( ) is the chage density at, epesents a unit aea on, and is a constant, known as the pemitivity of fee space. Accoding to physics [3], all points on a chaged conducto in electic equilibium ae at the same electic potential. ince we estict in Equation () to the conducto suface, () is constant. Accodingly, () may be ewitten as follows: V = Z ( ) j? j d () Hee V = () is a constant. ince in Equation () is an abitay suface, it is impossible to solve the equation analytically. Howeve, we can obtain an appoximate solution to Equation () by using nite element methods [], as descibed in next section. 3 Finite Element olution To compute the chage density distibution based on Equation (), the integation ove the complete suface is conveted to a summation of integations ove a numbe of plana tiangles. These tiangles fom a tiangula mesh tessellated on the object's suface. The mesh has N plana tiangles, T k ; k = ; :::; N. Each Figue : Tiangula mesh on the suface of an ellipsoid. tiangle is assumed to possess a constant chage density, k, as shown in Figue. A set of basis functions f k ; k = ; :::; N is dened on this tiangula mesh as follows: f k ( ) = if T k othewise (3) The basis function, f k, is nonzeo only when (taken as the centoid of the tiangula patch) is on the tiangle T k, as shown in Figue. Theefoe, the chage density ( ) can be appoximated by a piecewise constant chage density function as follows: NX ( ) k f k ( ) () k= ubstituting () into Equation (), we have NX Z V = k () k= Tk j? j d ince the chage density is assumed to be constant on each T k, we may take i as the obsevation point on each T i and ewite Equation () as: NX Z V = k i = ; ::::::; N: () k= Tk j i? j d Because of chage consevation, the sum of the chages on each tiangle equals the total chage on the suface of the conducto. Let Q be the total chage on the conducto and k be the aea of T k. Then we have NX Q = k k () k= Assuming Q is known, and using () and (), we obtain a set of linea equations with N + unknowns, ; :::; N and V. ince the integal in () can be evaluated analytically [], the chage density distibution k and the constant V can be obtained by solving a set of linea equations. This is accomplished using a conjugate gadient squaed method [].
3 (b) 3 (b) (c) Figue 3: ingle-view ange data of an object. Fontal view. (b) ide view. Due to self-occlusion, suface infomation on the othe side cannot be seen by the lase angende. Tiangula Mesh Constuction The chage density computation equies a closed tiangula mesh to be tessellated on the complete suface of the object. Howeve, the ange data obtained fom a paticula view only eect the visible sufaces, as shown in Figue 3. Thus it is impossible to pefom the mesh tessellation based on the actual shape of the invisible pats of the object. In pactice, we aticially constuct a mesh on the invisible side in ode to make up a closed tiangula mesh. This pemits us to compute the chage density. We note that the actual shape of the invisible suface only aects the absolute value of the chage density on the visible sufaces. The position of the extema of the chage density distibution emains almost the same and thus it makes sense to constuct an aticial mesh on the invisible suface. This agument is justied in []. The closed tiangula mesh fo the object is composed of thee patches of tiangula meshes, as shown in Figue. The st, called the top patch, is obtained by tiangulating the ange data on the visible suface. The second, called the bottom patch, is plana, and is actually the (spatial) pojection of the top patch onto an abitay plane pependicula to the Z axis. These two patches ae illustated in Figue. The thid one, called the side patch, lls the gap between the top and the bottom patches, as shown in (b). The complete closed tiangula mesh in Figue (c) is obtained by meging the patches in and (b). The details petaining to the suface tiangulation algoithm ae given in []. (b) (c) Figue : Tiangulation of the ange data in Figue 3. shows the top and the bottom patches of the closed tiangula mesh. (b) shows the side patch. (c) gives the closed tiangula mesh obtained by meging patches in and (b). Figue : Diect Connection Gaph (DCG). A tiangula mesh. (b) DCG of the tiangula mesh in. (c) ubgaphs of (b) afte bounday node deletion. Hee tiangula patches,, 3 and ae assumed to be located on the pat bounday. Object Decomposition The object is decomposed into pats afte obtaining the simulated chage density distibution. The method is based on a Diect Connection Gaph(DCG) dened on the tiangula mesh, as shown in Figue. Hee nodes epesent tiangula patches in the mesh and banches epesent the connections between diect neighbous. Two tiangles which shae two vetices ae consideed to be diect neighbous. Fo example in Figue, tiangles and ae diect neighbous while and 3 ae not. Thus the DCG povides a convenient coodinate system on the object suface and indicates the spatial elationship between a tiangle and its neighbous. Object decomposition is pefomed by gouping tiangles in the top patch of the closed mesh, which epesents the visible suface. The stategy is to locate these tiangles situated at pat boundaies and then delete them fom the gaph. This divides the oiginal DCG into a set of disconnected subgaphs, as shown in Figue (c). Each object pat epesented by a subgaph can then be obtained by applying a component labelling algoithm to the DCG. The details egading pat bounday localisation can be found in []. Expeimental Results Befoe demonstating pat segmentation, we compae the noise sensitivities fo the chage density and cuvatue computations. The latte has been taditionally used in bounday-based pat segmentation [,, 3]. In ode to clealy illustate the issues, we examine a simple D polygon, as shown in Figue. Due to the image sampling pocess, the bounday of this polygon is contaminated by high fequency noise. Figue shows the computed chage density (the left column) and cuvatue (the ight column). We use incemental cuvatue [] to appoximate the cuvatue of the jagged polygonal contou (incement = ). In the st ow, although no smoothing has been applied to the polygon contou, the chage density clealy indicates both the concavities and convexities (see Fig-
Figue : A polygonal contou. Because of image quantisation, the bounday of the polygon in an image is jagged. (b) (c) (d) chage density incemental cuvatue... Figue : A single view of a stone owl. haded ange data of the owl. (b) The tiangulatessellation on the visible sufaces. (c) The chage density distibution. (d) The segmented pats.. 3 3 3 3. (b) chage density 3 3 (c) incemental cuvatue..3.... 3 3 (d) Figue : Compaison between chage density and cuvatue computations. and (b) show the computed chage density and cuvatue distibutions, espectively, when no smoothing is applied to the contou shown in Figue. (c) and (d) illustate the chage density and cuvatue distibutions, espectively, when the contou is smoothed by a lowpass lte. These esults indicate that the chage density computation is moe obust than the cuvatue computation. ue ), while concave cones ae vey pooly indicated by the cuvatue distibution (see Figue (b)). In the second ow, we illustate the expeimental esults poduced by a smoothed polygonal contou. Hee % of the enegy of the highest Fouie fequency components of the jagged contou have been emoved. It can be seen that the chage density distibution (see Figue (c)) is much smoothe than the cuvatue distibution (see Figue (d)). These expeiments clealy illustate that the chage density computation is moe obust with espect to high fequency noise than the cuvatue computation. The st example of segmenting an object into pats involves the ange data of a caved stone owl. Figue shows the shaded ange data and (b) illustates the tiangula mesh fo the visible sufaces. Hee 3 tiangula facets ae used to epesent the suface. The closed tiangula mesh is composed of 9 tiangles. Figue (c) illustates the computed chage density Figue 9: (b) (c) (d) An alam clock. The shaded ange data. (b) The tiangula tessellation fo the visible sufaces. (c) The chage density distibution. (d) The segmented pats. distibution on the suface. It can be clealy seen that the lowest chage densities ae located at suface concavities. Convesely, the chage density at convexities exhibits a local maximum. As shown in Figue (d), this object is segmented into thee pats, namely, the head, the toso and the feet. In the second example, an alam clock with two inges on top is used. Figue 9 and (b) show the shaded ange data and the tiangula mesh, espectively. Thee ae 9 tiangles tessellated on the visible suface and, tiangles on the closed mesh. The computed chage density distibution is illustated in Figue 9 (c) and the segmented pats ae given in (d). Hee the clock is decomposed into thee pats. These esults ae consistent with ou intuition of object pats. We note that although only patial shape infomation of the complete objects ae available in these expeiments, and the constuction of the closed tiangula meshes is athe abitay, ou algoithm can still poduce the desied esults fo the visible sufaces. Also, the elative sizes of the tiangles is not cucial to the chage density computation. Duing ou many expeiments, we obseved that even if the atio of maximum to minimum tiangle aea was, ou algoithm would still poduce satisfactoy esults. The complexity of the chage density computation is govened by the constuction of the coecient ma-
tix fo the set of linea equations and the conjugate gadient squaed method to solve the equation. The complexity of both is of the ode of O(N ), whee N is the numbe of tiangula facets. On a GI R wokstation, the actual computing time fo the chage distibution fo the owl is 9 seconds, with about two seconds fo suface tiangulation, and one second fo pat decomposition. Conclusions This pape pesents a new physics-based appoach to 3D pat segmentation. Unlike most pevious appoaches, which pefomed the suface cuvatue computation, we solve a set of integal equations ove the whole object suface. Because of this, ou algoithm does not equie an assumption on suface smoothness and is obust to noise. Once segmented pats have been obtained, pat-based desciptions can be computed [9] and utilised fo ecient object ecognition. Acknowledgements We wish to thank Pofesso Jonathan Webb and Gilbet oucy fo thei kind help. MDL would like to thank the Canadian Institute fo Advanced Reseach and PRECARN fo its suppot. This wok was patially suppoted by a Natual ciences and Engineeing Reseach Council of Canada tategic Gant and an FCAR Gant fom the Povince of Quebec. Refeences [] R. Baett, M. Bey, T. F. Chan, et al. Templates fo the olution of Linea ystems: Building Blocks fo Iteative Methods. IAM, Philadelphia, 99. [] I. Biedeman. Human image undestanding: Recent eseach and a theoy. Compute Vision, Gaphics, and Image Pocessing, 3:9{3, 9. [3] F. J. Bueche. Intoduction to Physics fo cientists and Enginees. McGaw-Hill Book Company, New Yok, 3d edition, 9. [] F. P. Feie, J. Lagade, and P. Whaite. Daboux fames, snakes and supequadics: Geomety fom the bottom up. IEEE Tansactions on Patten Analysis and Machine Intelligence, ():{, August 993. [] F. P. Feie and M. D. Levine. Deiving coase 3D models of objects. In Poceedings of IEEE Confeence on Compute Vision and Patten Recognition, pages 3{33, Ann Abo, Michigan, June 9. [] H. Feeman. hape desciptions via the use of citical points. Patten Recognition, (3):9{, 9. [] A. Gupta and R. Bajcsy. Volumetic segmentation of ange images of 3D objects using supequadic models. CVGIP: Image Undestanding, (3):3{3, Novembe 993. [] D. Homan and W. Richads. Pats of ecognition. Cognition, :{9, 9. [9] T. Hoikoshi and. uzuki. 3D pats decomposition fom spase ange data using infomation citeion. In Poceedings of 993 IEEE Compute ociety Coneence on Compute Vision and Patten Recognition, pages {3, New Yok City, NY, June 993. IEEE Compute ociety Pess. [] J. J. Koendeink and A. J. Van Doon. The shape of smooth objects and the way contous end. Peception, :9{3, 9. [] A. Lejeune and F. Feie. Pationing ange images using cuvatue and scale. In Poceedings of 993 IEEE Compute ociety Coonfeence on Compute Vision and Patten Recognition, pages {, New Yok City, NY, June 993. IEEE Compute ociety Pess. [] A. P. Pentland. Recognition by pats. In Poceedings of the Fist Intenational Confeence on Compute Vision, pages {, London, June 9. [3] H. Rom and G. Medioni. Pat decomposition and desciption of 3D shapes. In Poceedings of the th Intenational Confeence on Patten Recognition, volume I, pages 9{3, Jeusalem, Isael, Octobe 99. IEEE compute ociety, IEEE Compute ociety Pess. [] P. P. ilveste and R. L. Feai. Finite Elements fo Electical Engineeing. Cambidge Univesity Pess, Cambidge, nd edition, 99. [] F. olina, A. Leonadis, and A. Macel. A diect patlevel segmentation of ange images using volumetic models. In Poceedings of 99 IEEE Intenational Confeence on Robotics and Automation, pages { 9, an Diego, CA, May 99. IEEE Robotics and Automation ociety, IEEE Compute ociety Pess. [] E. Tucco and R. B. Fishe. Expeiments in cuvatuebased segmentation of ange data. IEEE Tansactions on Patten Analysis and Machine Intelligence, ():{, Febuay 99. [] D. Wilton,. M. Rao, A. W. Glisson, et al. Potential integals fo unifom and linea souce distibutions on polygonal and polyhedal domains. IEEE Tansactions on Antennas and Popagation, AP-3(3):{, 9. [] K. Wu. Computing Paametic Geon Desciptions of 3D Multi-Pat Objects. PhD thesis, McGill Univesity, Monteal, Canada, Apil 99. [9] K. Wu and M. D. Levine. Recoveing paametic geons fom multiview ange data. In Poceedings of IEEE Confeence on Compute Vision & Patten Recognition, pages 9{, eattle, June 99.