BUILDING RECONSTRUCTION BASED ON MDL PRINCIPLE FROM 3-D FEATURE POINTS

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1 BUILDING RECONSTRUCTION BASED ON MDL PRINCIPLE FROM 3-D FEATURE POINTS Y. Ishkawa a, I. Myagawa a, K. Wakabayash a, T. Arkawa a a NTT Cyber Space Laboratory, NTT Corporaton, - Hkarnooka Yokosuka-Sh Kanagawa, , Japan (shkawa.yu, myagawa.sao, wakabayash.kaoru, arkawa.tomohko@lab.ntt.co.p Commsson III, WG III/7 KEY WORDS: Reconstructon, Urban, Three-dmensonal, Method, Buldng ABSTRACT: For 3-D buldng reconstructons of urban areas, we present a fully automatc shape recovery method that uses 3-D ponts acqured from aeral mage sequences. Ths paper focuses on shape recovery of flat rooftops that are parallel to the ground. We recover each rooftop from a set of 3-D ponts located at nearly the same heght. Such 3-D pont sets are made by mergng pont sets under the MDL (Mnmum Descrpton Length prncple, whch fnds sutably concse pont sets for 3-D buldng models. Often, only parts of rooftop shapes can be recovered because of the 3-D poston errors beng generated n the ponts. To refne the recovered shapes, we merge the parts under a heurstc condton n whch shapes wll have a par of orthogonally orented edges. To optmze parameters and estmate the vablty of our method, we defned a success rate, called the cover rato, as the area n whch the recovered shape and a correct shape (gven as reference data overlap to the combned area of the recovered and correct shapes. Expermental results showed that our method acheved a cover rato of 75.5%, and through mproved cover rato we also confrmed effectveness of shape refnement. We also found that even f only one-nnth of the reference data could be used n the optmzaton of parameters, the cover rato was 70.96%. The expermental results we obtaned showed that our pont-based method was effectve n enablng the recovery of buldngs n urban areas.. INTRODUCTION Three-dmensonal buldng models n urban area,.e. 3-D cty models, are used wdely to numercally smulate events such as floods, fre spreadng, wave propagaton and heat convecton. Computer graphc models for car navgaton have also been rapdly shftng from -D to 3-D systems. Accordngly, 3-D cty model reconstructon technologes are becomng ncreasngly n demand. There are manly two ways to categorze recovery methods: measurement (aeral or ground systems and sensng devces (camera or laser proflers. The arborne approach s sutable for reconstructon over a wde area. Especally over the past few decades, a consderable number of studes have been made on recovery methods n whch lne segments were extracted from aeral mages and 3-D nformaton was obtaned by stereoscopc vson (Herman, 986; Henrcsson, 998; Paparodts, 998. Not lmted to the recovery of 3-D buldng shapes, lne segments have been one of the mportant prmtves n shape reasonng. However, two knds of lne segments,.e. edge and pattern, frequently mngle together n mages. Whle edge lnes are very useful for contour detecton, lne patterns on flat surfaces have nothng to do wth the contours. Ths makes t dffcult to recover shapes on the bass of lne segment connectons. On the other hand, snce ponts are more prmtve than lne segments, they have several advantages over lne segments. For one thng, they make t easer to acqure 3-D postons and to recover 3-D shapes. For example, 3-D shapes can be recovered smply by generatng a trangle mesh over 3-D ponts. Actually, trangle meshes are appled wdely n recoverng curved surfaces (Terzopoulos, 99. Applcaton users, however, requre a 3-D cty model that s not a mnute mesh model but a CAD model,.e., one whch s a set of concse polyhedrons. Recently, t has become easer to acqure hgh-resoluton, hghframe-rate mage sequences, and therefore t s becomng feasble to recover buldng shapes wth Shape from Moton (Myagawa, 000. In ths paper, we present a fully automatc method of recoverng shapes from 3-D ponts acqured from aeral mage sequences to enable 3-D reconstructon of buldngs n urban areas. The paper focuses on shape recovery of flat rooftops beng parallel wth the ground. Thus we recover each rooftop from a set of 3-D ponts located at nearly the same heght. Such 3-D pont sets are made by mergng pont sets under the MDL (Mnmum Descrpton Length prncple (Rssanen, 984, whch fnds sutably concse pont sets for 3- D buldng models. Often, only parts of rooftop shapes can be recovered because of 3-D poston errors n the ponts. To refne the recovered shapes we merge the parts under a heurstc condton under whch the shapes wll have a par of orthogonally orented edges after mergence. In Secton we show a recovery method based on MDL from 3- D ponts. In Secton 3 we explan refnement of recovered rooftop shapes consderng 3-D poston errors and recovered shapes. We show expermental results n Secton 4. Fnally, we summarze the man ponts of ths paper n Secton 5.. RECOVERY METHOD FROM 3-D POINT SET Fgure depcts an outlne of our recovery method. Here, the fgure shows the (subsectons where each step n the recovery process s explaned.

2 Z Y An aeral mage sequence X 3-D ponts (. Layers (.,.3 Clusters Refned clusters ( 3 Buldng models Fgure. Outlne of buldng recovery method. Acquston of 3-D Pont Set A 3-D pont set s acqured automatcally as follows: detect feature ponts, track them, acqure 3-D postons, remove outlers, and transform the camera coordnate system nto a ground coordnate system (X,Y,Z based on nformaton of camera poston and orentaton. For the robust acquston of 3- D postons we adopt an teratve perspectve factorzaton method (Chrsty, 996. In the removal step, 3-D ponts are proected nto all mages. For each pont we then measure the dfference between -D postons found by the trackng and those obtaned by the proecton for all mages. Snce these dfferences reflect errors n the trackng process, ponts whch dffer sgnfcantly are removed from the 3-D pont set. Through ths process, only hghly precse ponts reman the 3-D pont set.. Layer and Cluster Generaton In ths subsecton we show our method of recoverng shapes from the 3-D pont set. Establshng a Z coordnate for each pont heght, the method starts by slcng the pont set nto layers, whch are sets of ponts at almost the same heght. After generatng the layers, each layer s dvded nto clusters, whch are sets of ponts gathered together on an X-Y plane. A -D convex polygon coverng each cluster s then formed as a rooftop -D shape. Our method recovers each rooftop as a flat roof buldng, the heght of whch s the average of the heghts of ponts n each cluster. The recovery process s completed by addng vertcal walls to the rooftops. In contrast to the results obtaned usng a laser profler, wth our method horzontal dstrbuton of the 3-D ponts acqured from mages s not unform. Ths makes t dffcult to generate layers and clusters based on a heght hstogram (Ledur, 998. The procedures our method follows are gven below. Layer generaton. Each ntal layer s an ndvdual 3-D pont. Let ntal layers be {L,..., L n } n order of the heght of the ponts. For each (,,n-, the standard devaton σ of the heghts of ponts n L and L + s calculated. Mnmal standard devaton σ mn s found from n- σ. Then L mn merges wth L mn+ and n n-. For each merge step MDL (defned n Subsecton.3 s computed. Fnally {L,..., L k } whose MDL s mnmal s output as the optmal layer set. Cluster generaton. The followng process s executed for each layer L. Each pont n L becomes an ntal cluster. Frst of all, dstances for all pont pars (p,q are calculated. Then, n order of the dstance lengths of the pars, the closest two clusters (ncludng the respectve ponts p and q are checked and are merged f they pass the check. The check begns wth creatng the smallest convex polygon ncludng all ponts of the two clusters on the X-Y plane. If the polygon also ncludes a pont belongng to a layer lower than L, the two clusters are not merged. If there are no such lower ponts, the two clusters are merged. Ths check enables us to form rooftop regons that are consstent wth the upper and lower relatons of layers..3 MDL-based Determnaton of the Number of Layers To prevent layers from beng dvded too much n the layer generaton process, our proposed method determnes the number of layers based on the MDL prncple as follows: Suppose that the 3-D pont set s dvded nto k layers {L,,L k } and m( clusters {C,,,C,m( } are generated from each layer L (,...,k. As a result, the 3-D pont set s dvded nto sets of clusters { {C,,, C,m( },...,k }. Ths means that the 3-D pont set s expressed by rooftop models recovered from the cluster sets. Gven the pont set and the parameter k, the cluster set {{C,,,C,m( },,k } can be determned. For ths reason we regard k as the degree of freedom of the rooftop models. k should be determned by balancng the smplcty of the models wth the conformty of the ponts to the models. The MDL prncple (Rssanen, 984 s used n many research areas such as pattern recognton (Leclerc, 990. Gven a model and data, suppose that code length M s the sum of the two code lengths M m and M d requred for descrpton of the model and the data. Based on the MDL prncple, the model that mnmzes M s the optmal model for descrbng data. For our problem, we defne M as Equaton (: M M m + M d ln n C k + + σ m ln n k m( ( z( p z( C, p C, ( where n s the number of 3-D ponts, m s the number of clusters, σ s the standard devaton of the heght of ponts n one cluster (we assume σ s constant over all clusters, Z(p s the heght of a pont p, Z( s the average of the heghts of ponts n a cluster C. The dervaton of Equaton ( s shown n Appendx A. We employ the parameter k that mnmzes M as the number of layers.

3 3. CLUSTER REFINEMENT BASED ON CLUSTER SHAPE AND POSITION ERROR 3. Relaxaton of Exclusve Constrant As mentoned n Subsecton., clusters are merged under a constrant such that a convex polygon of any cluster n a layer L may not cover any pont ncluded by any layer lower than L. In ths paper we call the constrant the exclusve constrant. As ponted out n Subsecton., ponts that have large proecton errors are removed from the 3-D pont set. After the removal, the 3-D poston of the pont set stll contans some errors. Unless the exclusve constrant s relaxed, therefore, clusters are nsuffcently merged and only fragments of rooftop shapes are obtaned. The optmzaton method Regularzaton s adopted n prevous works n whch the 3-D surface s recovered from 3-D ponts (e.g. Grmson, 983. Suppose that D s a gap estmate functon between 3-D ponts and a surface and S s a shape estmate functon of the surface. Under these assumptons, the optmal surface s found by mnmzng E D + λs, where λ s a parameter. Mnmzaton of E s often solved usng numercal analyss methods for dfferental equatons or usng search methods, especally Dynamc Programmng. For our method, however, clusters cannot be expressed n numercal form and cannot be ndependent of the generaton process. Therefore nether the use of numercal analyss nor that of search methods s effectve for our method. We do not apply the Regularzaton framework rgdly to our method. Based on estmate functons smlar to D and S, we decde whether or not to relax the exclusve constrant and merge clusters. Parameters n the estmate functons are determned by optmzaton for reference data. To acheve the optmzaton, t s mportant to reduce the number of parameters n the estmate functons. We do not employ the parameter λ and we hold the number of parameters to a mnmum, that s, we ntroduce only one parameter nto the D and S functons. 3. Penalty D for Breakng Constrant It s assumed that on the X-Y plane a convex polygon Pc ncludes all ponts n a cluster C n a layer L and a pont q n a layer lower than L. As a 3-D model the cluster C expresses a prsm whose shape on the X-Y plane s Pc and whose heght s Z(, and then q s located nsde the prsm (see Fgure. Y Z mn(d e q Pc X Fgure. A pont q breakng exclusve constrant For each edge e of the Pc, horzontal dstance d to q s computed, and mn(d s the mnmal value of d. Then the depth of q for C s defned as smaller than horzontal depth mn(d or vertcal depth Z(-Z(q (see Equaton (. Z Z( Z(q Y q X d mn Z( Z( q, mn( d e Pc ( Usng d, Equaton (3 defnes probablty P(C, q that the prsm expressed by C represents as an actual buldng, where L s the perpheral length of Pc and α s a parameter. d P( C, q exp( αl (3 Suppose that Pc ncludes n ponts {q,..., q n }. Assumng that P(C, q s are ndependent of each other for all q, probablty P( of the rooftop shape beng represented by Pc s the followng: (4 Lettngα be a threshold of P(, we can transform a condton P( α nto d / L α log( α. For penalty that C ncludes the ponts {q,..., q n } n the lower layer, an estmate functon D( s defned as Equaton (5, (5 Fnally, the exclusve constrant s relaxed by allowng D( to reach a thresholdα(α log(α. 3.3 Shape Estmaton Functon S P( P( C, q exp d αl D( L d Most rooftop shapes n urban areas have rght angles. Ths feature s often exploted for extractng rooftop shapes. Thus for cluster C we desgn the shape estmaton functon S( so that S( takes a larger value as an angle θ of an edge par (e, e n Pc becomes closer to a rght angle. Note that (e, e s taken from not only a par of adacent edges but also every edge par n Pc. Even though Pc does not have a rght angle, the postonng of the edge par s evaluated hghly n S( f Pc has orthogonally orented edge pars. S( s defned as Equaton (6, whch s an average of an exponental functon wthθ weghted by l l, where l s the length of e and β s a parameter. S( l l (,, θ π / exp( β l l (,, (6

4 From Equaton (6, f Pc has orthogonally orented pars of long edges, S( becomes hgh, and f Pc s an excessvely slender rectangle, S( becomes low. Thus we consder S( as an effectve functon n selectng shapes for rooftops. 3.4 Constrant Relaxaton Based on Cluster Shapes We relax the exclusve constrant based on S(. It s better for the cluster shape to be estmated at fnal step n the generaton process, but there are too many combnatons of clusters for mergence to enumerate all clusters at the fnal step. Therefore, clusters are ntally generated wth a rgd exclusve constrant,.e., wthout relaxaton. After ths step only fragments of a rooftop shape can be obtaned. In ths step every cluster par (C,C for whch C and C have cancelled mergence by the exclusve constrant s lsted n the order n whch the cancellaton happens. Secondly, suppose that C s a cluster ncludng all ponts of C and C on the X-Y plane. Then S( s evaluated and t s decded whether or not we should relax the exclusve constrant and merge the two clusters. These evaluatons and decsons are conducted n the order n whch they appear on the cancellaton lst. We call the second process cluster refnement. The mergence s accepted f Condton (7 s satsfed, where max( expresses a functon whch fnds a maxmum: D( C α and max( S( C, S( C S( C (7 reference polygon P, let maxmal E(P;Pc be cover rato E(P for P. E(P can be computed automatcally wth ths rule. Reconstructed 3-D cty models are often appled for vsualzaton or numercal smulaton. In many cases shape errors of larger buldngs have more negatve nfluence on the applcatons. For ths reason we employ an area weghted average of E(P as cover rato E({P } for a set of reference polygons {P }. Assumng that S P s the area of P, E({P } s gven by ΣS P E(P /ΣS P. In the followng secton, we regard the cover rato as the recovery rato D Pont Set and Reference Data 4.. Acquston of 3-D Pont Set: Each 3-D pont set was acqured from an aeral mage sequence that conssts of 80 frames ( pxels. In selectng nne urban area scenes, feature ponts were detected n the center area of the mddle frame n each sequence. The center area was pxels, whch was roughly equvalent to 0 60 meters on the ground. The feature ponts were tracked over all 80 frames by the Lucas- Kanade technque (Bradsk, D ponts were proected nto all 80 frames, and then ponts that had a proecton error of more than three pxels were removed as outler. In the end we obtan ponts on average per sequence. Fgure 3a s a frame example for the feature pont detecton, n whch the whte-lned rectangle expresses the detecton area. We also selected eght other dense urban area scenes as shown n Fg. 3a. Fgure 3b shows a 3-D pont set acqured from a sequence for whch the mddle frame s Fg. 3a. In Fg. 3b the more black there s, the hgher the pont s. Usng ths condton, f a mergence brngs a rooftop shape ncludng orthogonally orented long edge pars, the exclusve constrant s relaxed and the mergence s executed. 4. EXPRERIMENTAL RESULTS 4. Defnton of Recovery Rato In our method parameters σ, α and β are used n the calculaton of MDL and n udgng whether to relax the exclusve constrant. We determne the parameters by maxmzng the reference data recovery rato. In ths case the recovery rato defnton s nseparably connected to the determnaton of the parameters. Thus we here explan the defnton of recovery rato that we have used n our experments. Recovery rato s often defned based on the number of recovered buldngs. Judgment whether each buldng s recovered or not, however, cannot avod beng subectve to some degree. In partcular, udgng shapes recovered by our method s often dffcult, because polygons of the pont-based method are more lkely to have many vertces than n the lnesegment based method. Therefore we evaluate the recovered shapes obectvely and quanttatvely by comparng them wth reference data. For each target obect on the ground a rooftop polygon P s obtaned manually on mages as reference data. On the other hand, a rooftop polygon Pc s supposed to be recovered from a cluster C that expresses the rooftop of the obect. Cover rato E(P;Pc of P for Pc s defned as 00*S P C / (S P +S C -S P C, where S P, S C and S P C are respectvely the areas of P, Pc, and the overlap of P and Pc. If two or more Pc lap over one Fgure 3: An example of an mage frame (3a:upper and 3-D ponts (3b:lower 4.. Collecton of Reference Data: From the mddle frames of each of the nne sequences we manually obtaned reference data, whch was a -D rooftop shape of man-made obects on the ground satsfyng the followng condtons: a every shape was a convex polygon, b area was over 000 pxels ( 8 m, c heght was over 0 meters. In the end we obtaned 96 polygons as reference data. There were 0.7 polygons on average per scene, and the number of polygons n one scene s 8 at the maxmum and sx at the mnmum. 4.3 Evaluaton Results of Recovery Method We evaluated our method n both the cases where the exclusve constrant was relaxed and where t was not. We call the former

5 case the Relax method and the latter case the Rgd method. To optmze the parameters n the estmate functons we utlzed the Powell method, whch s one type of drecton set method (Press, Optmzng Wth All Reference Data: The Rgd method uses only one parameter σ contaned n Equaton (. On the other hand, the Relax method has two addtonal parameters α and β. We optmzed the parameters wth all reference data and generated clusters usng both the Rgd and Relax methods. The cover rato wth the clusters are shown n Table 4, σ[m] α β E [%] Rgd Relax Dfference Table 4. Optmzaton and estmaton results. The Rgd and Relax methods acheved cover ratos of 68.8% and 75.5% respectvely. Consderng that the recovery was executed usng only nformaton about the 3-D pont poston, these results show the effectveness of the 3-D pont-based method. In partcular, the Relax method rased the cover rato to more than 7% that of the Rgd method, whch confrms the effectveness of the cluster refnement descrbed n Subsecton 3.4. We consder values of σ n Table 4 as estmated values of the standard devaton of the heght of ponts ncluded n one cluster. Consderng that the heght of Japanese buldngs per floor s about 4m accordng to statstcs, the values seem vald. And ths result shows that layer dvson based on the MDL prncple worked well. Layers and clusters are generated so that M n Equaton ( s mnmal. If σ s small, M d has more nfluence on M than M m does. Ths means that heght error Z(p-Z( s regarded as more mportant than the number of clusters m. Consderng that σ of the Relax method s smaller than that of the Rgd method, the Relax method acqured fragments of accurate shapes before relaxaton. Then, by relaxng the exclusve constrant, the fragments were merged. Fnally, more accurate shapes were recovered wth the Relax method than wth the Rgd method. Fgure 5 shows the dfference between the cover ratos acheved by the Rgd and Relax methods for all 96 reference data tems, where the horzontal axs s the cover rato of the Rgd method and the vertcal axs s that of the Relax method. Cover rato E [%] (Relax method dfference of over 5% dfference of under 5% Cover rato E [%] (Rgd method Fgure 5. Cover rato: Rgd vs. Relax In Fgure 5, the ponts on the dagonal lne show cases where there was no dfference between the two methods, whle the ponts n the upper-left area show cases where the Relax method acheved better results. The dfference was less than 5% n exactly half of the cases (48 cases, whle n many other cases the cover rato was hghly mproved. In other words, the Relax method tended to mantan the cover rato of the Rgd method or to greatly mprove t n cases where mprovement s possble. Condton (7 means two clusters are merged only f shape estmaton s defntely mproved. Thus, t can be sad that the Relax method tends to relax the exclusve constrant n a conservatve manner. And ths tendency can be seen n evaluatng t Optmzng Wth Partal Reference Data: In actual applcatons, only a small reference data set can be used to optmze parameters. Therefore, wdespread use s made of the Cross Valdaton estmaton method (Shahraray, 989. Ths method as follows. Frst, reference data s dvded nto n sets {A,...,A n }, and then parameters are optmzed wth A. Next, clusters are generated wth the optmzed parameters. Fnally, the cover rato of reference data wthout A s computed. In ths process, we call parameter optmzaton learnng and computaton of cover rato test. We performed Cross Valdaton for,...,n and calculated the average of n test results. For n the sze of every A was 48 (96/, and for n4 the sze of every A was 4±. When n was 9, the sze of A was not unform; t vared between 6 and 8 because each A was assumed to be a set of reference data housed wthn each of nne scenes. Fgure 6 shows learnng and test expermental results, where dashed and sold lnes respectvely express learnng and test cover rato. Cover rato E [%] The number of dvsons n Fgure 6. Cross Valdaton Results As the fgure shows, as n becomes hgher the learnng cover rato mproves, n contrast to the test cover rato, whch decreases due to the reduced amount of learnng data. Even for an n value of 9, however, the Relax method acheved a cover rato of 70.96%, whch was not much worse than 75.5% result obtaned when the parameters are optmzed wth all reference data. 5. CONCLUSIONS Learnng (Relax Test (Relax Learnng (Rgd Test (Rgd In ths paper we proposed a fully automatc method of recoverng shapes from a 3-D pont set to reconstruct a of 3-D cty model of buldngs n urban areas. The features of the proposed method are t determnes the number of layers

6 based on the MDL prncple and t refnes clusters based on the cluster shapes. We estmated the valdty of our method wth 3-D pont sets and reference data acqured from nne aeral mage sequences n a dense urban area. When parameters were optmzed wth all reference data, the Relax method acheved a cover rato of 75.5%, confrmng the effectveness of cluster refnement n comparson to the Rgd method. Furthermore, a cover rato of 70.96% was acheved even f only one-nnth of the avalable reference data could be used for optmzaton. Expermental results showed that our pont-based method recovers the shapes of buldngs n urban areas effectvely. APPENDIX A. DERIVATION OF CODE LENGTH M In ths secton we elaborate on the dervaton of M (M m +M d n Equaton ( wth the same symbols used n Subsecton.3. We consder only Z coordnates of all ponts as nformaton beng encoded and transmtted. Then Z(p for every pont p s dvded nto three knds of nformaton: a. a selecton of cluster C that ncludes p, b. the heght h of the model recovered from C, and c. the dfference between h and Z(p. The code length that s necessary to dvde a pont sequence of n ponts nto k layers s log n- C k-. Once layers are determned, clusters can be obtaned wth X and Y coordnates. It follows that log n- C k- s long enough to encode nformaton a. And accordng to the MDL prncple (Rssanen, 984, (m/ log n s a code length suffcently long for nformaton b, consderng that m s the number of Z( whch are parameters of models. Fnally, we defne code length for model descrpton M m as (ln n-c k- + (m/ ln n / ln. Next we explan data descrpton length M d. Let h, be the heght of the upper surface of a 3-D model recovered from a cluster C, ; and assume that probablty Q(p for the heght error δz(pz(p-h, ( p C, obeys Normal Dstrbuton whose standard devaton s σ. To mnmze code length we should employ Z(C, as h, because Z(C, s the maxmum lkelhood estmator for h,. Accordng to Shannon s nformaton theory, f the occurrence probablty of data s P, code length log P s long enough to encode the data. Assumng that Q(p are ndependent for all pont p, mnmal code length M d (, for cluster C, s the followng: M d (, log Q( p C, p ( Z ( p Z ( C., + const ln σ p C, (8 M d (, s the sum of mnmal code lengths for nformaton c for all ponts n C,. We can obtan M d usng the non-constant part of M d (, as Equaton (9: Fnally, M s obtaned removng ln, the common constant coeffcent of M m and M d. REFERENCE Bradsk, G., R. and Psarevsk, V., 000. Intel s Computer Vson Lbrary: Applcatons n calbraton, stereo, segmentaton, trackng, gesture, face and obect recognton. In: CVPR000, pp Chrsty, S., and Horaud, R., 996. Eucldean Shape and Moton from Multple Perspectve Vews by Affne Iteratons. In: IEEE Trans. on Pattern Analyss and Machne Intellgence, Vol.8, No., pp Grmson, W., E., L An Implementaton of a Computatonal Theory of Vsual Surface Interpolaton. Computer Vson Graphcs and Image Processng, Vol., pp Henrcsson, O., 998. The Role of Color Attrbutes and Smlarty Groupng n 3-D Buldng Reconstructon. In: Computer Vson and Image Understandng, Vol. 7, No., pp Herman, M., 986. Repersentaton and Incremental Constructon of a Three-Dmensonal Scene Model. In: Technques for 3-D Machne Percepton, pp Leclerc, Y., G, 990. Regon Groupng Usng the Mnmum- Descrpton-Length Prncple. In: Image Understandng Workshop, pp Ledur, R., Maresch, M., and Lesher, C., D Reconstructon of Urban Envronments from Dense Dgtal Elevaton Models. In: Image Understandng Workshop, pp Myagawa, I., Naga, S., and Sugyama, K., 000. Shape Recovery from Hybrd Feature Ponts wth Factorzaton Method. ISPRS 000, TC III-0-05, Amsterdam, The Netherlands. Henrcsson, O., 998. The Role of Color Attrbutes and Smlarty Groupng n 3-D Buldng Reconstructon. In: Computer Vson and Image Understandng, Vol. 7, No., pp Paparodts, N., 998. Buldng Detecton and Reconstructon from Md- and Hgh-Resoluton Aeral Imagery. In: Computer Vson and Image Understandng, Vol. 7, No., pp. -4. Rssanen, J., 984. Unversal codng, nformaton, predcton and estmaton. IEEE Trans. Infomaton Theory, 30(4, pp Terzopoulos, D. and Vaslescu, M., 99. Samplng and Reconstructon wth Adaptve Meshes. In: CVPR 99, pp M d ln σ k m( ( Z ( p Z ( C, p C, (9 Press, W., H., Flannery, B., P., Teukolsky, S., A., and Vetterlng, W., T., 99. Numeral Recpes n C. Cambrdge Unversty Press, Cambrdge, pp