Vanshng Hull Jnhu Hu Suya You Ulrch Neumann Unversty of Southern Calforna {jnhuhusuyay uneumann}@graphcs.usc.edu Abstract Vanshng ponts are valuable n many vson tasks such as orentaton estmaton pose recovery and 3 reconstructon from a sngle mage. Many methods have been proposed to address the problem however a consstent framework to quanttatvely analyze the stablty and accuracy of vanshng pont estmaton s stll absent. Ths paper proposes a new concept vanshng hull whch solves the problem. Gven an edge error model the range of a true edge can be modeled usng a fan regon. The ntersecton of all these fan regons s a convex hull whch s called vanshng hull. A vanshng hull gves the regon of a true vanshng pont and ts dstrbuton determnes the probablty of the vanshng pont. The expectaton of the vanshng hull s the optmal soluton of the vanshng pont ts varance defnes the accuracy of the estmaton and ts shape determnes the stablty of the vanshng pont. Hence we can quanttatvely analyze the stablty and accuracy of the vanshng pont estmaton usng vanshng hull. Smulaton results show that our method s sgnfcantly better than one state-of-the-art technque and real data results are also promsng.. Introducton A vanshng pont s defned as the ntersecton pont of a group of mage lnes that correspond to the projecton of parallel lnes n 3 wth an deal pn-hole camera model. The poston of a vanshng pont n the mage plane s only determned by the camera center and the orentaton of the 3 lnes n the camera system. Vanshng ponts are valuable n many vson tasks ncludng buldngs detecton n aeral mages[5] camera poses recovery [5] robots navgaton [4] and 3 reconstructon from a sngle mage [9]. Many research have been conducted n accurately dentfyng the poston of vanshng ponts. Ths vares from smple lne groupng [8] to more complcated methods usng a statstcal model [6]. The two key problems n dentfyng vanshng ponts are fndng the group of mage lnes that correspond to a true vanshng pont and computng the poston of the vanshng pont wth the presence of mage noses. Most prevous research work [] [] focus on the frst problem whle the computng of vanshng ponts are accomplshed usng the least square technque or searchng for the maxma vote n the Gaussan sphere [4]. The performances are often evaluated emprcally furthermore few work have been conducted on fndng a theory to quanttatvely analyze the stablty and accuracy of vanshng ponts estmaton wth mage noses. The mage lnes that correspond to a vanshng pont can be grouped usng clusterng methods or votng methods. Clusterng methods frst fnd possble clusters usng the ntersecton of all pars of lnes [] or mage gradent orentatons [] then assgn each lne to dfferent clusters usng a dstance or angle crteron. The drawbacks of clusterng methods are the hgh computatonal complexty and that a hard threshold s needed to group lnes nto clusters. Votng methods nclude mage space methods [3] and Gaussan sphere methods [4]. Hough Transform n Gaussan sphere space s a global feature extracton method hence senstve to spurous maxma. The knowledge of prmtve models can be used to reduce spurous maxma [5]. However the accuracy of Gaussan sphere methods s lmted to the dscretzaton of the accumulator space hence t s hard to acheve the precson that an mage can provde. For computng the vanshng ponts wth mage noses the least square method s generally used []. Several mprovements have been made to the least square method. McLena and Kottur [] ntegrate edge detecton and lne clusterng to the process of vanshng ponts detecton and then use a non-lnear method to compute the poston of vanshng ponts wth a statstcal edge error model. Shufelt [5] uses a fan edge error model to mprove the robustness to mage noses n the Gaussan sphere. However the performances of both methods are evaluated emprcally. Lebowtz and Zsserman [] use a Maxmum-Lkelyhood estmator to compute the poston of the vanshng pont however the performance s only vsually analyzed. In general a consstent framework to quanttatvely analyze the computaton of vanshng ponts wth mage noses s stll absent whch s the man mpetus of ths work. Observng that the regon of an edge error model wth mages noses s a fan regon (Fgure ) we ntersect all the
fan edges to form a convex polygon called the Vanshng Hull. Ths paper shows that the vanshng hull gves the regon of the true vanshng pont and ts dstrbuton gves the probablty of the vanshng pont. The expectaton of the vanshng hull the centrod for a unform dstrbuton gves the optmal soluton of the vanshng pont under statstcal meanngs ts varance defnes the accuracy of the vanshng pont and ts shape determnes the stablty of the vanshng pont. Hence the vanshng hull concept provdes a theoretcal framework to quanttatvely analyze the regon optmal soluton stablty and accuracy of vanshng ponts. Besdes a framework for analyzng vanshng ponts wth mage noses we also present a novel edge groupng method based on heurstc flters wthout any hard thresholds. Snce the dea of vanshng hull s derved from the ntersecton of edge regons we frst present a smple edge error model (Secton.) and the correspondng edge groupng method (Secton.). The vanshng hull concept s frst ntroduced based on the smple edge error model (Secton.3) and then extended to general edge error models (Secton.4). The performance of our method s quanttatvely compared wth one state-of-the-art technque [](Secton 3) and we conclude the paper n Secton 4.. Vanshng hull.. Edge error model fferent edge error models have been presented. McLean and Kottur [] use a statstc model to present both the error of the lne centrod and orentaton. Shufelt [5] presents a smple but effectve edge error model. Inspred by the dea of a fan edge regon [5] we derve the concept of vanshng hull by the ntersecton of all these regons. We frst adapt ths smple edge error model to our vanshng hull framework and then extend the concept to general edge error models. Consder the representaton of a lne segment usng two endponts and assume the two end ponts have one pxel precson then two fan regons wth a rectangular regon n the mddle can be formed by movng the two end ponts freely n the pxel squares (Fgure ). Ths regon s a concave regon so we cannot guarantee that the ntersecton of such regons wll be convex. Fortunately a true vanshng pont cannot le n the mage lne segment (Secton.) so the rectangular regon has no effect on the shape of the ntersecton of edge regons. We smply take the mddle pont of the edge and form two fan regons wth the two end ponts. Furthermore a true vanshng pont Fgure. Edge error model. can only le n one drecton of the edge so we can just take one of the fan regon whch s a convex regon (Fgure )... Lne clusterng Snce we are nterested n fndng the ntersecton regons of all the edges t s necessary to frst dentfy the mage lnes that can form a possble vanshng pont.e. we need to group lnes nto dfferent clusters. We opt to use the clusterng method n mage space rather than Gaussan sphere because of several reasons. Frst Gaussan sphere method s a global feature extracton method whch s senstve to spurous maxma. Second the accuracy s lmted to the dscretzng accuracy hence hard to acheve the precson that an mage can offer. The last and most mportant reason the ntersecton of the fan regons of the edges that belong to the maxmum cell n the Gaussan sphere may be empty whch makes the vanshng hull meanngless. Fndng clusters The Canny edge detector s used to fnd sub-pxel edges then Hough Transform s used to fnd possble lnes and nearby lnes are connected usng some user defned threshold. The lnes are represented usng two end ponts. The ntersectons of all pars of lne segments are computed to fnd all possble clusters. The computatonal complexty s O( n ) where n s the number of lnes. Groupng lnes nto dfferent clusters takes O(n) tme so the overall tme complexty s O( n 3 ) whch s expensve for a large number of lnes. We wll reduce the complexty usng a flterng step and the RANSAC algorthm later. Groupng After fndng the clusters we need a crteron to assgn lnes to dfferent clusters. A dstance crteron gves prorty to close vanshng ponts whle an angle crteron gves prorty to far vanshng ponts. A reasonable threshold s to use a tuple of both dstance and angle or use normalzed angle error. However all these methods need a hard threshold whch may be nconsstent wth the edge error model. We use a method that s consstent wth the edge error model wthout any hard threshold. For each cluster of two lnes we can fnd the ntersecton regon A of the edges and a test edge s assgned to ths cluster when ts edge regon overlaps wth regon A. Furthermore we use a strong constrant for clusterng (Fgure ). An edge s assgned to a cluster only f ts edge regon covers the ntersecton pont of the cluster. Ths guarantees that the ntersecton regon of the edge regons n each Fgure. Groupng wth an edge error model.
cluster s not empty. The normalzed length of each edge s accumulated n ts assgned cluster and the maxmum clusters are chosen to compute potental vanshng ponts. Flterng spurous vanshng ponts Most of our testng mages are outdoor buldng mages wth heavy occluson by trees (Fgure 5) whch causes many spurous vanshng ponts. Knowledge of the mage and vanshng ponts are used to flter spurous vanshng ponts. Frst we roughly classfy the extracted lnes nto x and y groups accordng to the lne orentaton whch dramatcally reduce the sze of lne number. Then vanshng ponts are fltered usng the followng three flters. ) Iteratve lne length. Accordng to the edge error model longer lnes are more relable however we would lke also to keep shorter lnes. So we frst flter the lnes usng large length threshold then estmate the possble vanshng ponts and these ponts are used to fnd more lne supporters accordng to the groupng method. ) Coverng area. Another observaton of the mage s that edges of trees only cover a small part of the mage regon so the rato of the coverng area aganst the mage area s also used to flter spurous vanshng ponts. 3) Vald vanshng pont. Vanshng ponts are the ntersecton of mage lnes that correspond to parallel lnes n 3. So by defnton a vald vanshng pont wll not le on the mage segment n the mage space whch s very effectve n reducng spurous clusters. RANSAC Even though we classfy lnes nto two drectons to reduce the lne number and use flters to reject spurous clusters the number of clusters may stll be large. Snce we are nterested n fnd vanshng ponts that correspond to domnant drectons the RANSAC algorthm s used to fnd the maxmum cluster of lnes for x-y drecton. The vanshng pont of z drecton s estmated usng the orthogonal property of the three drectons and ts supportng lnes are found usng our groupng method to refne the poston usng vanshng hull..3. Vanshng hull efnton and property A vanshng hull s defned as the ntersecton of the fan-shape edge regons wth a gven edge error model. Fgure 5. (d) shows a vanshng hull of a real mage. A vanshng hull has the followng propertes. Property I. A vanshng hull s not empty. Proof: accordng to the groupng method an edge s assgned to a cluster only f ts edge regon covers the ntersecton pont of the cluster so the vanshng hull contans at least the ntersecton pont. Property II. A vanshng hull s convex. Proof: the ntersecton of convex regons (edge regons are fan shape) s convex. Property III. A true vanshng pont les nsde the regon of ts vanshng hull wth the assumpton of correct edge error model (the true edge les nsde the edge fan). Proof: By defnton the true vanshng pont must les nsde the unon of all the edge regons. Now assumng the vanshng pont VP les outsde of the vanshng hull but nsde the unon of the edge regon. Then there must exst some edge say L whose edge regon does not cover VP. Hence the edge error model of edge L s wrong whch s contradctory wth our assumpton. Hence VP must be nsde the vanshng hull. Ths property s mportant t tells us where to fnd the true vanshng pont. Property IV. The centrod of a vanshng hull s the optmal estmaton of the vanshng pont wth a unform dstrbuton model. Proof: the optmal estmaton of the vanshng pont s the expectaton of the probablty dstrbuton of the vanshng ponts nsde the vanshng hull under statstc meanng. Wth a unform dstrbuton the expectaton of a vanshng hull s ts centrod. Property V. The varance of a vanshng hull determnes the accuracy of the estmated vanshng pont. Proof: ths s drectly from probablty theory. Property VI. The shape of a vanshng hull determnes the stablty of the estmaton of the vanshng pont. A vanshng hull can be open t can also be a closed non-trval convex polygon a lne segment or a pont. When the mage lnes are parallel the vanshng hull s an open convex hull; the centrod s undetermned whch means the estmaton of the vanshng pont s unstable. Ths s reasonable because edges have nose any non-zero nose wll be enlarged to nfnty when the vanshng pont s at nfnty whch makes the estmaton unrelable. An open vanshng hull ndcates a vanshng pont at nfnty whch corresponds to a pont at the great crcle parallel to the mage plane n the Gaussan sphere. We handle ths case by settng the vanshng pont to nfnty. When the vanshng hull s a non-trval convex polygon the vanshng pont can be relably estmated usng the centrod wth the varance of the dstrbuton as the estmaton accuracy. When the vanshng hull shrnks to a lne segment the uncertanty s along just one drecton and the vanshng pont can be precsely computed when the vanshng hull degenerates to a pont whch corresponds to an error-free edge model. etermnng vanshng hull Now let s consder how to fnd the vanshng hull. A fan shape edge regon can be consdered as the ntersecton of two half planes (Fgure ) so the problem of ntersecton of the edge regons can be cast as the problem of half-planes ntersecton. A naïve way to solve the problem s to add one half-plane bound lne a tme and compute the ntersecton regon whch takes O( n ) tme. However a more elegant algorthm wth O( n lg n) tme can be presented by
study the property of dual space [3]. There exsts an nterestng property called dualty between lnes and ponts [3]. Gven a non-vertcal lne L t can be expressed usng two parameters (k b). We can defne a pont wth k as the x coordnate and b as the y coordnate. Ths pont s called the dual pont (L* ) of lne L. The space of the lnes s called the prme space whle the space of the ponts s called the dual space. The half-plane ntersecton problem n the prme space can be mapped as the problem of fndng the convex hull of the ponts n the dual space. We frst dvde the bound lnes of half-planes nto two sets: an upper set (half-planes le above the bound lnes) and a lower set (half-planes le below the bound lnes). Let s consder the upper set frst. For each lne L n the upper set we can fnd a dual pont L*(k b). The ntersecton of all the half-planes n the upper set can be mapped as fndng the lower convex hull (the edges of the lower boundary of the convex hull) of the dual ponts. The proof s out of the scope of ths paper readers may refer to [3]. Smlarly we can fnd the ntersecton of all the half-planes n the lower set by determnng the upper convex hull of the dual ponts. Notce that accordng to our dual mappng the upper convex hull and the lower convex hull wll not ntersect although the upper and lower half-planes do ntersect and that s the reason we splt the whole half planes nto two sets. Fnally the two regons are merged to fnd the vanshng hull. The algorthm s summarzed as followng:. Splt the half plane nto two sets an upper set and a lower set.. Map each set nto a dual space and fnd the correspondng upper and lower convex hull (notce the two convex hulls are dfferent). 3. Map the two half convex hulls back to prme space and merge them to fnd the vanshng hull. Fndng a convex hull of a pont sets s a well-defned problem [3] whch takes O( n lg n) wth n as the number of ponts. The mappng and mergng takes lnear tme so the overall tme complexty s O( n lg n). Expectaton and varance of vanshng hull Wth a unform probablty dstrbuton the expectaton and varance of the poston of a true vanshng pont can be computed usng Equaton () where s the regon of the vanshng hull and A s ts area. We only show x coordnate due to lack of space y coordnate can be computed n a smlar way. Gven a lst of vertces of the vanshng hull t s easy to show that the mean can be computed usng the coordnates of these vertces (Equaton 3). µ ( x) = xda var( x) = ( x µ ( x)) da () A A = n ( x y+ x+ y ) = A () n ( x) = ( x + x+ )( x y+ x+ y ) 6A = µ (3) The varance of the vanshng hull can also be represented as a smple expresson of the coordnates of the vertces. Accordng to Green s calculus theorem [6] an ntegral n the regon can be converted to an ntegral on the boundary: g f f ( x dx + g( x dy = ( ) dxdy (4) x y 3 Let f = g = ( x µ ( x)) we have 3 g f = ( x µ ( x)) (5) x y So: g f var( x ) = ( x µ ( x)) da = ( ) da A A x y = f ( x dx + g( x dy (6) A Let s consder the ntegral on lne ( x y ) to ( x + y+ ) we can parameterze the pont on the lne usng t such that: x x + t x x y = y + t y + y t [] (7) = ( + ) ( ) Let a = ( x µ ( x)) b = ( x + x ) c = ( y + y ) t s easy to show that the varance of x coordnate can be computed as: n 3 3 b var( x) = c( a + a b + ab + ) 3 = 4 (8) Smlarly we can compute the varance of y coordnate..4. Vanshng hull for general edge error model The vanshng hull concept can be easly extended to general statstcal edge error models. Frst we show an augmented vanshng hull consderng the full edge regon and then we show that a general vanshng hull can be derved n a smlar way. In Secton. we gnored the rectangular regon and use one of the edge fan to derve the vanshng hull concept. We clam that the shape of the vanshng hull wll not change wth ths approxmaton whch s true. However the probablty dstrbuton n the vanshng hull s not a unform dstrbuton. For a full edge span regon an angle (called extreme angleθ ) s formed by the vanshng pont VP and two extreme ponts P and P (Fgure 3). Accordng to the edge error model the two end ponts have equal probablty nsde the one-pxel-sze square. The probablty of a true edge passng through the vanshng pont VP s determned by the overlappng area of the extreme angle and the pxel squares. Let p ( l VP) be the probablty of a true edge l pass- 3
ng through VP and e be the two end ponts of l and e S and S be the overlappng regon of the extreme angle wth the two pxel squares then: p l VP) = p( e S & e ) (9) ( S Assumng the two end ponts are ndependent wth probablty densty functon (PF) f ( x and g( x y ) respectvely then the jont PF s f ( x * g( x y ) whch s a 4 unform dstrbuton. Then p ( l VP) s the ntegral of the jont PF over the regon S S. p ( l VP) = f ( x * g( x y ) dsds e S & e S We can use a lne-sweepng method to compute the ntegral n a regon. Now consder a sweep lne L that passes VP and ntersects the two squares wth lne segment and l (Notce L s a lne nsde the extreme angle l whle l s a true edge nsde the whole edge regon). Lne L sweeps the whole overlappng regon when ts angle vares from to θ the ntegral of the jont PF over the two lne segments s: p( e l & e l ) = f ( x * g( x y ) dl dl Vanshng hull Extreme angle θ Full edge regon e l & e l enote the end ponts of lne segment ( x ( x y ) and l as ( x y ) ( x y ) they form a lne segment n a 4 space (because we have three lnear constrants: l l and they have the same slope). The ntegral of a unform dstrbuton over a lne segment n 4 s just the length of the lne segment hence: p( e l & e l ) = [( x + ( x x ) + ( y y ) P L Fgure 3.Compute probablty dstrbuton for a full edge regon. x ] ) + ( y Now we can ntegrate over the angle to get the regon ntegral: P VP e l e l l () () as () p ( l VP) = p( e l( ϑ) & e l ( ϑ)) dϑ (3) ϑ The full expresson of the analytcal PF of the vanshng hull for a full edge regon s complcated and the dstrbuton functon may not be contnuous over the entre regon. Even when the dstrbuton for a sngle edge s contnuous the overall PF s very hgh order due to the large number of edges. An analytcal soluton to the ntegral of such a hgh order non-lnear PF s very complcated and may not exst. We use a dscretzng method to solve ths problem. The dscretzaton process can acheve hgh precson (one pxel) because the vanshng hull regon s bounded. Consder the center pont VP( x of a cell wth one pxel sze we can fnd the extreme angle relatve to edge l and then compute the probablty p ( l VP( x ) accordng to Equaton 3. Then the probablty of the pont VP( x over all the edges s computed and normalzed (Equaton 4). The expectaton and varance can be easly computed n the dscretzed vanshng hull. n = * P( x P ( x p ( l VP( x ) P ( x = (4) = P( x Such a vanshng hull consderng the full edge span regon s called augmented vanshng hull. In practce we found that the vanshng hull often conssts of only a few vertces (less than vertces for lnes) whch means that the probablty of a vanshng pont close to the edge regon s boundary s very low. Snce vanshng ponts close to the mddle of the edge regon has smlar overlappng area wth the pxel squares t s reasonable to assume a unform dstrbuton for the vanshng hull formed of full edge regons. We can extend the vanshng hull concept to general edge error model n a way smlar to augmented vanshng hull. A general edge error model often models the error of the edge centrod and orentaton [] or the two end ponts [5] usng a Gaussan dstrbuton. The edge span regon s stll a fan shape so the ntersecton of the edge regons s a convex hull. Assumng the PF of the lne l passng a vanshng pont (x s f ( x then the PF of the vanshng hull over all the lnes s f ( x. Agan ths functon s a hgh order non-lnear functon we dscretze the vanshng hull and compute the mean and varance of the vanshng hull. 3. Analyss and result 3.. Smulaton data We frst analyze the theory usng synthetc data. The goal of the smulaton s to show that a vanshng hull gves the regon of the true vanshng pont ts expectaton s the
optmal soluton ts shape determnes the stablty and ts varance determnes the accuracy of the estmaton. The smulaton s desgned as followng. A group of 3 parallel lnes are projected by an deal pn-hole camera to an mage plane then random noses wth specfed magntude are added to the end ponts. The vanshng pont s estmated usng the centrod of the vanshng hull assumng a unform Table. Parameter settngs. Parameter Range Other parameter settngs.lne orentaton θ [. 4] fov = 4 l = 5 angle (degree) ε =.5 n =.camera feld of fov [ 8] θ = l = 5 vew (degree) ε =.5 n = 3.mage lne l [ ] θ = fov = 4 length (pxel) ε =.5 n = 4.mage nose ε [.5.5] θ = fov = 4 magntude (pxel) l = 5 n = 5.number of mage n [ ] θ = 5 fov = 4 lnes l = 5 ε =. 5 dstrbuton. We extensvely analyze our Vanshng Hull () algorthm wth dfferent parameter settngs (Table ). For each of the fve groups of parameter settngs we sample the space wth evenly dstrbuted ntervals the other four parameters are set as constant when one parameter vares n ts range to test the performance relatve to each parameter. Vanshng hull s the true vanshng pont regon The smulaton shows that all the true vanshng ponts le nsde the vanshng hull. Ths s logcal because the maxmum nose magntude s specfed so the edge error model exactly predcts the regon of the true edge regon hence the true vanshng ponts le nsde the vanshng hull. Expectaton s the optmal soluton The result of method s compared wth other two methods Least Square (LS) method and Maxmum Lkelyhood () method [] to show that the expectaton s the optmal soluton. The comparson crteron s the recovered orentaton angle error relatve to the ground truth. LS method uses least square to fnd the vanshng pont closest to all lnes and method uses a non-lnear method to mnmze the dstance of the lne that passes the vanshng pont and the md-pont to the two end ponts. Accordng to our mplementaton the dfference of the and LS method s often several pxels so the angle dfference s very small. Ths s because method uses a non-lnear optmzaton method whch often gves a local mnmum close to the result of LS method. We just show the result of and method to make the fgure clear (Fgure 4). The parameter θ and fov are related to the perspectve effect of mages. The result (Fgure 4. (a) (b)) shows that the method gves large errors for weak perspectve mages (up to 4 degrees when the orentaton angle s less than. degrees) whle method performs relably wth maxmum angle error less than.5 degrees and average less than. degrees. When the perspectve effect s strong (orentaton angle s larger than degrees) the vanshng hull shrnks to several dozens of pxels both methods perform well. The parameter l and ε are related to the qualty of edges. The smulaton (Fgure 4. (c) (d))shows that the method gves large orentaton errors for poor qualty edges ( l < 3 or ε >. ) and the maxmum angle error s more than 4 degrees. The method performs sgnfcantly better than the method and very relable over the whole range (maxmum angle error.3 degrees and average angle error less than. degrees). The last parameter (Fgure 4. (e)) compares the performance aganst the number of mage lnes. The smulaton shows that the number of lnes has no strong effects on the performance of method however t s a crtcal parameter for the performance of the method. When the number s more than 5 method performs very relably when the number drops to t stll performs sgnfcantly better than the method. However when the number of lnes drops below 5 the result s mxed. There are several reasons for ths. Frst the regon of a vanshng hull shrnks wth the ncreasng number of lnes so t s more relable for more lnes. Second the expectaton of the vanshng hull s the optmal soluton for a vanshng pont under statstcal meanng. However when the sample (number of lnes) s small the true value of the vanshng pont may devate from the statstcal value. The last reason s that we use a unform dstrbuton whch s an approxmaton as we showed n secton.4. In general the method performs sgnfcantly better than the other two methods especally for weak perspectve mages and the performance of the method s very relable wth several hundreds of lnes a reasonable number for hgh-resoluton mages of buldngs and aeral mages. Ths shows that the expectaton of the vanshng hull s the optmal soluton of vanshng ponts estmaton. Stablty and accuracy We can also predct unstable vanshng ponts usng the method. For most of the cases the vanshng hull s closed whch ndcates that the vanshng pont s stable. When the vanshng hull s open t ndcates that the vanshng pont s at nfnty. We vsualze unstable vanshng ponts wth blue color n Fgure 4.(f) whereθ =.. Notce that the method acheves an average error less than.5 degrees even for such an ll-condtoned case. The error of the method s also wthn the magntude of the
Error of orentaton (degree) Error of orentaton (degree) 4 35 3 5 5 5 6 5 4 3 3 4 Orentaton angle (degree) Error of orentaton (degree) 3 5 5 5 3 4 5 6 7 8 Camera FOV (degree) Error of orentaton (degree) 6 4 8 6 4 4 6 8 Lne length (pxel) a b c Error of orentaton (degree).4..8.6.4. Error of orentaton (degree) 8 6 4...3.4.5 Nose magntude (pxel) varance for all fve groups of parameters whch mples that the varance determnes the accuracy of the estmaton. We dd not show the graph of the varance due to lmted space. 4 6 8 Number of lnes 3.. Real data The vanshng hull theory s also tested usng real data. Snce the concept of the vanshng hull s derved from the edge error model ts property (optmal estmaton stablty and accurac depends on the edge error model whch makes t a valuable tool to analyze the performance of dfferent edge error models. We tested the algorthm wth two mages. The camera s calbrated and lens dstorton s corrected. The nose magntude of the end ponts of edges s modeled as: ε = c l where l s the length of the edge and c s set to 3.5 pxels. Ths model shows a better result than settng the nose magntude as a constant value for all lnes. The frst mage s a strong perspectve mage occluded by heavy trees. Fgure 5. (a) (b) compares the result before and after flterng vanshng ponts spurous vanshng ponts caused by trees and small-scale textures are fltered usng our method whch shows the effectveness of our flter. The vanshng hull (Fgure 5. (d)) s found usng dual-space algorthm the vanshng pont s computed as the centrod assumng a unform dstrbuton and the mage s rectfed (Fgure 5. (c)) []. It s hard to verfy the property of the vanshng hull for a real mage due to lack of ground truth so we opt to use a manual verfcaton by 4 6 8 Number of lnes d e f Fgure 4. Performance comparson of and method on synthetc data. measurng the horzontal and vertcal lne angle error. Careful manual user verfcaton shows that the horzontal angle error s less than. degrees (the standard devaton of the vanshng hull s σ =. degrees) and vertcal angle less than.5 degrees ( σ =. 7 ) whle method gves. and.3 degrees angle error respectvely. The second mage s a weak perspectve mage (Fgure 5. (e)) whch s a very challengng example for vanshng pont algorthms. The result of method shows that x 6 drecton vanshng pont s and y drecton s 4 4. Careful manual user verfcaton shows that the horzontal angle error of method s less than.3 degrees ( σ =.35 ) and vertcal angle less than. degrees ( σ =. ) whle method gves.94 and.55 degrees angle error respectvely. Both mages show that the method gves better performance (optmal soluton) and the user verfed orentaton error s wthn the range of varance (accurac. Hence the true vanshng ponts are wthn the regon of the vanshng hull. All the vanshng ponts are stable because the vanshng hulls are all closed convex polygons. 4. Concluson Vanshng ponts are valuable n many vson tasks such as orentaton and pose estmaton. Ths paper defnes the concept of vanshng hull whch s the ntersecton of the edge regons. The vanshng hull gves the regon of the true vanshng pont and ts probablty dstrbuton determnes the property of vanshng ponts. The expectaton
a b c d e Fgure 5. Real mage results. A strong perspectve mage s tested clusterng result before flterng (a) and after flterng (b) are shown and the mage s rectfed (c). Fgure (d) shows ts vanshng hull (yellow edges) and vanshng ponts estmated usng method (yellow pont) and method (red pont). A challengng example of weak perspectve mage s also tested (e). of the vanshng hull s the optmal soluton of the vanshng pont ts varance defnes the accuracy of the estmaton and ts shape determnes the stablty of the vanshng pont. Future work ncludes usng a more accurate edge error model to reduce the requrement of large number of edges and apply ths method to more real mages. 5. Acknowledgment Ths work made use of Integrated Meda Systems Center Shared Facltes supported by the Natonal Scence Foundaton under Cooperatve Agreement No. EEC-9595. 6. References [] A. Almansa A. esolneux and S. Vamech. Vanshng pont detecton wthout any a pror nformaton. PAMI 5(4): 5 57 3. [] M. Antone and S. Teller. Automac recovery of relatve camera rotatons for urban scenes. In CVPR pp.8 89. [3] M. Berg etc. Computatonal geometry algorthms and applcatons. ISBN: 3-54-656- publshed by Sprnger. [4] B. Caprle and V. Torre. Usng vanshng ponts for camera calbraton. Internatonal Journal of Computer Vson pp.7 4 99. [5] R. Cpolla T. rummond and.p. Robertson. Camera calbraton from vanshng ponts n mages of archtectural scenes. In Proceedngs of Brtsh Machne Vson Conference pp.38 39 999. [6] R. T. Collns and R. S. Wess. Vanshng pont calculaton as statstcal nference on the unt sphere. In ICCV pp.4-43 99. [7] L. Grammatkopoulos G.Karra and E. Petsa. Camera calbraton combnng mages wth two vanshng ponts. ISPRS 4. [8] F.A. van den Heuvel. Vanshng pont detecton for archtectural photogrammetry. Internatonal archves of photogrammetry and remote sensng. 3(5):65 659998. [9] E. Gullou. Meneveaux E. Masel. and K. Bouatouch. Usng vanshng ponts for camera Calbraton and coarse 3 reconstructon from a sngle mage. The Vsual Computer pp.396-4 []. Lebowtz and A. Zsserman. Metrc rectfcaton for perspectve mages of planes. In CVPR pp.48 488 998. []. Lebowtz A. Crmns and A. Zsserman. Creatng archtectural models from mages. Computer Graphcs Forum 8(3):39 5 999. [] G. Mclean and. Kottur. Vanshng pont detecton by lne clusterng. PAMI 7():9-95 995. [3] C. Rother. A new approach for vanshng pont detecton n archtectural envronments. In BMVC. [4] R. Schuster N. Ansar and A. Ban-Hashem. Steerng a robot wth vanshng ponts. IEEE Transactons on Robotcs and Automaton 9(4): 49-498 993. [5] J. Shufelt. Performance evaluaton and analyss of vanshng pont detecton technques. PAMI (3):8 88 999. [6] E. Wessten. Green's theorem. From MathWorld http://mathworld.wolfram.com/greenstheor em.htm. [7] S. Barnard. Interpretng perspectve mages Artfcal Intellgence vol. pp.435-46983. [8] C. Burchardt and K. Voss. Robust vanshng pont determnaton n nosy mages ICPR.