RANSAC for (Quasi-)Degenerate data (QDEGSAC)

Transcription

1 RANSAC fo (Quasi-)Degeneate data (QDEGSAC) Jan-Michael Fahm and Mac Pollefeys Depatment of Compute Science, Univesity of Noth Caolina at Chapel Hill, Chapel Hill, NC {jmf, Abstact The computation of elations fom a numbe of potential matches is a majo task in compute vision. Often RANSAC is employed fo the obust computation of elations such as the fundamental matix. Fo (quasi-)degeneate data howeve, it often fails to compute the coect elation. The computed elation is always consistent with the data but RANSAC does not veify that it is unique. The pape poposes a famewok that estimates the coect elation with the same obustness as RANSAC even fo (quasi-)degeneate data. The appoach is based on a hieachical RANSAC ove the numbe of constaints povided by the data. In contast to all peviously pesented algoithms fo (quasi-)degeneate data ou technique does not equie poblem specific tests o models to deal with degeneate configuations. Accodingly it can be applied fo the estimation of any elation on any data and is not limited to a special type of elation as pevious appoaches. The esults ae equivalent to the esults achieved by state of the at appoaches that employ knowledge about degeneacies. 1 Intoduction The computation of a elation given sets of potential matches is necessay fo many compute vision applications. It is applied fo example computing fundamental matices out of 2D-2D coespondences, pojection matices fom 2D-3D coespondences, and 3D homogaphies fom 3D-3D coespondences. The pesented famewok is not limited to these applications and can also be applied to the estimation of geometic entities like quadics fom 3D points and conics fom 2D points. In such estimation poblems it is not possible to povide a set of pefect coespondences due to noise in the image data and mismatches caused by ambiguities in the featue desciptions. Accodingly the computation of the elation always has to deal with petubed data and mismatches called outlies. The most common technique to deal with outlies in Figue 1. Two views of the tay scene whee most matches ae on the plane (tay). Only a few matches ae on the candlestick (light gay lines). the matches is to employ the RANSAC algoithm [2, 7]. It solves the two poblems of computing a elation that best fits the data and classifying the data as inlies (coect matches) and outlies. The classification is done by employing a cost function togethe with a theshold. The elation is selected as the one with the highest numbe of inlies o the lagest obust likelihood [2, 9]. If only degeneate data ae given it is not possible to compute the coect elation. Degeneacy means that the data do not povide enough constaints to compute the elation uniquely, but up to a family of elations that all explain the data. Fo example, fo the computation of a fundamental matix, matches that ae on a plane ae degeneate data [3] since they only povide constaints to compute a homogaphy. Futhemoe the data ae often quasi-degeneate which means most data do not povide sufficient constaints to compute the elation uniquely (degeneate data) and only a small faction of the data povides the emaining constaints. In ode to compute the coect elation this small potion of the data has to be incopoated. Hence fo quasidegeneate data the elation can always be uniquely defined, but as explained in Section 3 the RANSAC algoithm has a low pobability to compute the coect elation fo quasidegeneate data. Quasi-degeneate data fo the fundamental matix computation can be seen in Figue 1. The next section eviews the existing techniques fo the estimation of elations fom (quasi-)degeneate data. Aftewads, Section 3 intoduces and eviews the popeties of RANSAC in moe detail along with the notation used

2 thoughout the pape. In Section 4 we popose a famewok to obustly detect degeneate o quasi-degeneate data fully automatically. It selects the appopiate model fo the given data without equiing any knowledge about the degeneacies. Fo quasi-degeneate data it computes the coect elation by using the small potion of the data that povide the emaining constaints in addition to the constaints povided by the degeneate faction of the data. Accodingly it can be applied to a wide vaiety of estimation poblems. We apply the famewok to vaious estimation poblems in compute vision, namely fundamental matix estimation, pojection matix estimation, 3D homogaphy estimation, and quadic fitting in Section 5. Since the famewok is not limited to compute vision it can be employed fo all othe linea estimation poblems. 2 Pevious wok Estimating elations on data that contain matches as well as mismatches is a poblem in most applications in compute vision. The most popula technique to deal with it is RANSAC intoduced by Fishle and Bolles [2]. RANSAC is eviewed in moe detail in Section 3. The poblem of degeneate data fo the fundamental matix computation was addessed by Kanatani [4]. To intoduced a obust extension in [8, 9]. He poposed to employ model selection to ovecome the limitations of RANSAC with degeneate data. The goal is to compute the model that explains the given data best by employing knowledge about the degeneacies in the pesence of outlies. To addessed the case of 2D-2D point coespondences. If these matches ae coplana only a homogaphy can be detemined uniquely. The technique poposed by To computed a homogaphy and a fundamental matix fo the data. Fo each of the elations a cost function is applied that measues the obust likelihood of the mapping fo the coespondences. Additional penalties ae applied fo a lowe numbe of constaints povided by a coespondence. Fo quasi-degeneate data the model selection typically votes fo the homogaphy since the cost fo it is usually lowe than the cost fo the fundamental matix as a esult of the small numbe of off-plane inlies. These non-degeneate inlies ae outlies to the homogaphy but thei small atio in the data only slightly inceases the cost fo the homogaphy. Hence due to the bette cost of the homogaphy and the fact that each element detemines two degees of feedom it is still cheape fo quasi-degeneate data than the coect fundamental matix that has highe penalty fo the lowe numbe of constaints detemined by each match. Additionally the RANSAC fo the fundamental matix has a low pobability to compute the coect epipola geomety [1] as explained in the next section. The model selection equies two RANSAC s and two nonlinea optimizations of the obust likelihood one fo each elation. The fist RANSAC appoach fo computing the epipola geomety that can deal with quasi-plana data was ecently intoduced by Chum et al. [1]. They employ a citeion to detect degeneate samples duing the RANSAC fo the fundamental matix computation by exploiting the knowledge that coplana points ae a degeneate configuation. It was done by examining samples that lead to an epipola geomety with a highe numbe of inlies than all pevious samples. Such a elation is expected to be a bette elation than all pevious ones. Then the sample is tested fo degeneacy. If it contains too many degeneate inlies, it is extended to contain a sufficient numbe of non-degeneate inlies if those ae available. Fo non-degeneate data the appoach [1] computes the coect epipola geomety and fo degeneate data it chooses the appopiate model. Fo the fundamental matix computation this appoach epesents the state of the at. Note that it equies an explicit test fo dealing with the plana degeneacy. Late we will show that ou technique povides equivalent esults without equiing a specific test fo the degeneacy which makes ou appoach geneally applicable to linea model fitting poblems with unknown degeneacies. Tang et al. [6] poposed a tenso voting based appoach that poses the poblem of estimating the fundamental matix as one of finding the most salient hypeplane in an eightdimensional space. It detects (quasi-)degeneate data without pio knowledge but does not poduce a fundamental matix fo these cases. To summaize, the diffeence between the existing techniques and the poposed algoithm is that existing techniques except [6] equie explicit models fo the degeneate cases to avoid the computation of ambiguous elations. In contast to those appoaches, ou technique does not equie any specific knowledge of the degeneate and quasidegeneate data. It detects cases of degeneate data automatically and chooses the appopiate model. All pevious techniques can only be applied to explicitly modeled (quasi-)degeneacies wheeas the poposed technique is geneally applicable. The poposed technique will be explained in detail in the next section. 3 RANSAC In the following we biefly summaize RANSAC and explain its behavio fo (quasi-)degeneate data as this is impotant to show the popeties of ou poposed technique. RANSAC is employed to estimate an n-paametic elation T on the data {p}. Simultaneously it classifies the data {p} into inlies {in} and outlies {out}. It selects m andom elements fom {p} and computes a candidate elation T c fom this andom sample. The minimal numbe m = n of elements equied to compute the elation depends on the

3 numbe of constaints povided by each element and the numbe n of paametes of the elation. To achieve a bette pefomance RANSAC often employs linea estimation of the elation T. Fo a linea estimation we have an linea inlie function f in given by f in (t, w) = A w t with w {p} (1) whee the unit vecto t epesents the elation T and A w is the matix containing the data fom the match w. The matix A w has linealy independent ows because each match w povides linealy independent constaints. An inlie to elation T is a data point w fo which f in (t, w) < c t. In geneal a linea equation system is defined fom the inlie function f in and data {p} by A w }{{} A t = 0 with w {p}, (2) whee the data matix A IR {p} (n+1) consists of all sub matices A w IR (n+1) induced by the matches w {p}. Hence the ows of the data matix A contain the linea inlie test given by the inlie function f in. The elation t is detemined as the nullspace N of the data matix A. Hence the data matix should have a ank A of at most n to obtain a non tivial solution of (2). Accodingly the dimension n of the nullspace (codimension) is at least one. Fo noise fee degeneate data the ank A is educed to d < n as a smalle numbe of independent constaints is povided by the data and the computed candidate t c becomes ambiguous. Then the candidate elation T c is a membe of an n dimensional subspace N which can be epesented by the matix N IR (n+1) n containing the ighthand singula vectos which ae a base of the nullspace N. Afte the computation of the candidate elation T c, RANSAC applies it to all given data {p} and classifies by thesholding (1) the data in inlies {in c } and outlies {out c }. The andom sampling is epeated until a sufficient numbe of samples has been evaluated. The numbe of equied samples S is adaptively detemined by exploiting the of inlies in the data of the best cuently known elation and the desied pobability η that a good candidate elation has aleady been computed. The standad appoach consists of stopping the RANSAC when the numbe of samples S is at least faction ɛ = {inc} {p} S = log(1 η) log(1 ɛ m ), (3) whee m is the numbe of elements in the sample. Aftewads the best candidate elation, the one with the most inlies is deliveed as elation T RANSAC which best fits the given data. Futhemoe it gives the classification of the potential matches {p} as inlies {in} and outlies {out}. In the following fo (quasi-)degeneate data we will distinguish between two disjoint sets of inlies. Degeneate inlies ae those inlies that ae in degeneate configuation. These degeneate inlies do not detemine all degees of feedom of the elation uniquely. Hence the ank of the povided data matix A deg is always less than n. The set of inlies that ae not in degeneate configuation is denoted as non-degeneate inlies. By adding the non-degeneate inlies the elation is uniquely detemined. Fo the fundamental matix degeneate inlies ae all matches that ae coplana and non-degeneate inlies ae the off-plane inlies. ɛ = 0.9 ɛ = 0.8 ɛ d = 0.85 ɛ d = 0.88 ɛ d = 0.75 ɛ d = 0.78 P nd P s P nd P s P nd P s P nd P s F 3% 36% 0.5% 5% 1.4% 47% 0.3% 8% P 2.1% 19% 0.4% 3% 1.3% 26% 0.2% 4% H 1.6% 54% 0.7% 9% 1.4% 66% 0.3% 13% Q 3.6% 12% 0.3% 2% 1.13% 18% 0.2% 3% Table 1. Pobability fo a non-degeneate sample P nd and a successful estimation P s fo a RANSAC fo quasi-degeneate data (F = fundamental matix (8-point), P = pojection matix, H=3D homogaphy, Q = quadic). is the numbe of elements that can be Samples containing only degeneate inlies give a high numbe of inlies fo (quasi-)degeneate data without poviding a sufficient numbe of constaints to compute all degees of feedom of the candidate elation T c. The emaining constaints ae detemined by the outlies in the sample o the noise in the degeneate inlies. So the obust nullspace 1 N has dimension n > 1. Accodingly all these samples contain all degeneate inlies and up to n 1 outlies, whee n 1 abitaily chosen to fix the emaining n 1 constaints, e.g. two fo the fundamental matix. The pobability P d,s of a sample not poviding a sufficient numbe of constaints is n 1 P d,s = 1 P nd ( ), (4) whee P nd ( n 1 ) is the pobability to choose n 1 nondegeneate inlies. It is given by: m n 1 n 1 P nd ( ) = j=0 ( ) m ɛ j d j (ɛ ɛ d) m j, (5) whee ɛ is the inlie faction in {p} and ɛ d is the faction of degeneate inlies in {p}. Accodingly the pobability of 1 Fo a sample the data matix A in is the data matix induced by only the inlies in the sample. Then the obust nullspace is the space that has the nullvectos belonging to singula values σ i < ɛ with i = 1,..., d of the data matix A in as base.

4 RANSAC to succeed is P s = 1 (1 P nd ) S with S fom (3). Examples of the pobabilities P nd and P s ae shown in Table 1 fo common estimation poblems. Table 1 illustates the need fo an estimation that is obust with espect to (quasi-)degeneate data, since fo quasi-degeneate data the pobability P s of RANSAC to compute a coect solution is small. The elation T c computed fom these samples is an abitay element of a family of possible solutions fo the degeneate data and fits the degeneate inlies pefectly esulting in a high numbe of inlies without poviding the coect elation T c. Hence the adaptive temination fom (3) stops the sampling too ealy to have a pobability η fo the computation of a coect candidate elation T c. 4 RANSAC fo (quasi-)degeneate data (QDEGSAC) The pevious section showed that fo (quasi-)degeneate data the standad RANSAC appoach is highly unlikely to compute the coect elation. The model selection technique poposed by To [8, 9] exploited explicit knowledge about the degeneacies of the fundamental matix computation without ovecoming the poblem caused by quasidegeneate data. The degeneacy test poposed by Chum et al. [1] elies on the availability of a special test fo degeneacy to handle quasi-degeneate data coectly. As discussed in Section 2 the ank of the data matix A can be used to detect degeneate data if they ae noise fee. The computation of the ank is inaccuate since it is sensitive to noise in the data. The distubance esults in small singula values. Hence it is still possible to estimate the ank by using an appopiate theshold on the singula values. If a sample contains degeneate inlies and outlies the latte incease the ank of the data matix. That means the ank of the data matix appeas to be equal to the expected ank fo the nondegeneate case. So the ambiguity can not be detected by analyzing the singula values of the data matix. In this section we popose a new famewok, QDEGSAC that employs RANSAC to compute the coect solution including model selection fo the given data. The advantage of the novel famewok is that it does not equie any specific knowledge about the degeneacy, in paticula no test is equied to detect degeneate samples. Accodingly it can be applied to a wide vaiety of estimation poblems not only to those whee the degeneacies ae known. Ou famewok can be intepeted as a obust measuement of the ank A of the data matix A. It detects automatically and obustly the highe codimension fo (quasi- )degeneate data. Fo degeneate data it selects the ight model to epesent the data. If the data ae quasi-degeneate the novel technique efficiently seaches fo additional inlies among the initial set of outlies to povide the highest possible ank A of the data matix A. Figue 2. Oveview of the QDEGSAC. The poposed famewok consists of thee phases as shown in Figue 2. The fist RANSAC estimates the full model assuming that the data ae not degeneate. The classification of the data into inlies {in n } and outlies {out n } of this pocess is used fo the next two phases. A test fo the numbe of constaints povided by the inlies {in n } is pefomed. This step is denoted as model selection in Figue 2. The model selection is the obust ank detection in the famewok. Aftewads the model completion tests the outlies of the pevious phases fo non-degeneate inlies to compute the coect elation. In the following we will discuss the details of each phase. RANSAC A geneal RANSAC(n) is pefomed to compute the desied elation T with n-degees of feedom. Depending on the numbe n of degees of feedom of the elation T it will take at least m = n elements to compute the n degees of feedom. The equied numbe of elements m depends on the numbe of constaints that ae povided by each element. Fom those samples the data matix A s is constucted and used to compute the elation. The RANSAC(n) delives a elation T RANSAC,n employing n constaints and a classification of the set of potential matches {p} in inlies {in n } and outlies {out n }. These sets ae tested in the following two phases. Fo only degeneate data in the sample the codimension n is geate than one. This is a esult of the insufficient numbe of constaints povided by the data. In case of quasi-degeneate data in the sample the codimension n is one. Fo degeneate data and outlies in the sample the codimension is also one as the outlies povide the emaining constaints. All samples that contain only degeneate data o degeneate data and outlies commit a high numbe of inlies as explained ealie. Accodingly the elation T RANSAC,n deliveed by the RANSAC(n) does not have pobability η to be the coect elation. So often not all degees of feedom of the elation T RANSAC,n ae fixed

5 though constaints povided by inlies, since those constaints esult fom noise in the degeneate inlies o fom outlies. Please note fo non-degeneate data RANSAC(n) will delive the coect elation T as the codimension n is one with a pobability depending on the confidence theshold η. The idea of the poposed famewok is to detect obustly the case whee the codimension n is lage than one. Model selection To detemine the codimension a seies of RANSACs is pefomed to estimate elations T RANSAC,dim with a smalle numbe of constaints dim < n. It stats with a RANSAC(n 1) that lowes the numbe of constaints n employed fo computation by one. That means it detemines the elation T RANSAC,n 1 by employing the closest ank n 1 appoximation fo the data matix A s of each sample. Accodingly RANSAC(n 1) also uses a smalle numbe of elements m = n 1 m in each sample than RANSAC(n) does. The input set fo RANSAC(n 1) is the set of inlies {in n } of the fist RANSAC(n). It detemines fo those inlies {in n } if they can be epesented by all elations in the nullspace of the esulting equations. The evaluation fo each potential match in {in n } employs the inlie citeion of RANSAC(n) to check if it is an inlie to all elations in the nullspace. The following theoem gives a necessay and sufficient condition to test this. Theoem 1 (Inlie to all elations in nullspace) Given a data matix A its n -dimensional obust nullspace defines a set of elations N. If an inlie w is an inlie to all elations N j with j = 1,..., n in an othogonal base of N with cost A w N i = c i then its cost is bound by n i=1 c2 i fo all elations t N. poof: Each elation t in the obust nullspace of A can be witten as a linea combination of the base vectos N j with j = 1,..., n. The tansfomation t is defined by t = Nn whee N is the matix containing the base vectos N j and n is the unit vecto that contains the weights fo the base vectos. Then the maximal possible cost A w t is the maximum eigenvalue of A w because t is one. The cost c w of the inlie w to a elation t N is n c 2 w = n T N T A T wa w Nn c 2 i, i=1 because N T A T wa w N is positive semi-definite its maximal eigenvalue is positive and bound by its tace. The tace is given by n i=1 N i T AT wa w N i = n i=1 c2 i. Theoem 1 shows that the squaed algebaic eo is bound by the sum of squaed algebaic eos obseved fo the elations in the base of the nullspace N. This means we can decide fo each match w if it is an inlie to all elations t N by only computing n eos. In pactice the c i ae often simila, i.e. c 2 w is pactically bound by max i c 2 i and not by n i=1 c2 i. The geometic eo behaves simila to the algebaic eo fo nomalized data. So it can also be used fo the decision. Fo the expeiments in Section 5 the esults of the geometic and the algebaic eo wee equivalent. RANSAC(n 1) tests fo all inlies {in n } of the elation T RANSAC,n if they ae also inlies to all elations in the obust nullspace of the data matix A. So if a elation T RANSAC,n 1 with fewe paametes eceives sufficient suppot, indicated by the numbe of inlies {in n 1 } divided by the numbe of inlies {in n } of the oiginally computed elation T RANSAC,n, then it follows that the data did not povide a sufficient numbe of constaints to detemine the n degees of feedom of T RANSAC,n. If the elation does not have sufficient suppot in the inlies {in n }, the data {in n } povided n constaints. The pocess of educing the numbe of constaints exploited to compute the elation T RANSAC,n 1 is continued until the elation T RANSAC,n i does not have a sufficient suppot in the inlies {in n }. Sufficient is detemined by t ed as atio of the inlies {in n i } of RANSAC(n i) to the inlies {in n } of the fist RANSAC(n). As we will see late, the technique is not sensitive to the atio t ed and it can be safely chosen in a ange of 50% 80% without significant impact. If a elation that cannot epesent the inlies {in n } is found the appopiate model fo the inlies {in n } has been computed in the pevious step i 1. It has n i + 1 degees of feedom and a codimension of i 1. The successive eduction of the numbe of constaints used to compute the elation detemines how many constaints ae deliveed by the inlies {in n }. At this point the model selection is finished and the model is the elation family that is othogonal to the constaints povided by the inlies {in n }. The computational cost of each of the RANSACs except fom the last one is smalle than the cost of the fist RANSAC(n). It esults fom the fact that the set of potential matches is the set of inlies {in n } of the fist RANSAC and the inlie faction is compaably high fo degeneate data. It follows fom (3) that the numbe of steps as well as the numbe of evaluations needed to decide between outlies and inlies is lowe than fo the fist RANSAC(n). So the computation is less than i-times the cost of the fist RANSAC(n). The computationally most expensive RANSAC is the RANSAC(n i) that can not find a elation that epesents the inlies of RANSAC(n). It equies S n i iteations accoding to (3), with log(1 η) n i S n i = log(1 t q ed ) with q =. (6) The theshold t ed influences the untime of RANSAC(n i) as it is the minimum equied inlie atio. Accoding to (6) the RANSAC(n i) needs a significant numbe of tials to pove that the inlies ae not suppoted by a elation

6 computed by employing n i constaints. Model completion Afte finishing the model selection fo quasi-degeneate data it is still possible to find the coect n-paametic elation employing data that is not coveed by the selected model. These data ae usually only a small faction of the data {p}. Accodingly the low pobability to select those non-degeneate inlies often leads to an ambiguous elation T RANSAC,n duing the RANSAC(n). The model selection povides a estictive model which classifies the data {p} into degeneate inlies {in s } and outlies {out s }. The degeneate inlies {in s } povide a data matix A deg. The model completion uses the closest ank n i + 1 appoximation of A deg and extends that with the A w matices coming fom a RANSAC with samples of size fom the outlies {out s}. To summaize, the novel famewok fist pefoms a RANSAC to find a tansfomation that explains the data. Aftewads the codimension of the data matix fom the inlies is estimated obustly. It detemines a lage codimension fo degeneate data even if outlies detemine fee constaints. Finally the famewok inspects the data fo inlies that povide the emaining constaints to compute the n-degees of feedom of the elation T. i 1 Figue 3. Coect epipola geomety computed by QDEGSAC valid fo all points. Epipola geomety computed by RANSAC only valid fo points on the tay (plane). dent constaints. Figue 3 shows one example of a wong epipola geomety computed by RANSAC(8) due to degeneate data. The epipola geomety computed by QDEGSAC is shown in Figue 3. 5 Expeimental esults In this section we apply ou novel famewok to vaious estimation poblems in compute vision. Fist the estimation of the epipola geomety is shown fo (quasi-)degeneate data. Aftewads, the estimation of the camea pojection matix, the estimation of 3D homogaphies as well as the estimation of quadics fo (quasi-)degeneate data ae discussed. The QDEGSAC technique was tested on thee images of the tay scene, two of which ae shown in Figue 1. The 365 potential point tiplets wee established with the wide baseline SIFT-featue matching technique of [5]. It contains 337 matches on the dominant plane (tay). Only 11 matches ae off-plane on the candlestick. The latte matches have to be employed to compute the coect elations. Fundamental matix estimation QDEGSAC uses a linea eight point algoithm fo the estimation of the fundamental matix [3]. Fo all expeiments we assume an algebaic eo compaable to an aveage distance of 1.5 pixel to the epipola line in the image fo inlies. The theshold t ed was set to 70% and η was set to 99%. Fist we tested QDEGSAC on the tay scene shown in Figue 1, fo which the pedicted pobability of computing the coect epipola geomety with RANSAC is P nd = 1.7% and the pobability P s of a successful RANSAC is P s = 12%. Fo the eight point algoithm the codimension is thee fo coplana matches [3, page 281]. instead of one. Hence the computed elation has six linealy indepen- Figue 4. Numbe of inlies (y-axis) fo diffeent numbes of employed constaints (x-axis), with t ed = 0. Fo six constaints the additional inlies found by QDEGSAC ae stacked. Numbe of classifications as inlie fo off plane inlies fo 100 QDEGSAC executions. The QDEGSAC algoithm was executed one hunded times on the potential matches. Fo the 11 not coplana matches the numbe of detections as inlie was counted and ae shown in Figue 4 as well as the numbe of inlies fo the models with diffeent numbe of employed constaints. The evaluation shows that the QDEGSAC always found the coect solution in contast to the taditional RANSAC that had success in 17% of the uns. Futhemoe it always detected the six linea constaints povided by the degeneate inlies. Figue 4 shows that the famewok is not sensitive to t ed which was set to zeo fo the chat. The ange of t ed could be between 20% and 80% without influencing the esult. QDEGSAC needs fo this scene about 4-7 tials in the fist RANSAC(8). The RANSAC(7) and RANSAC(6) need togethe 4 tials accoding to the high inlie pobability in {in 8 }. The unsuccessful RANSAC(5) needs 26 samples accoding to (3) and t ed = 70%. The RANSAC fo model

7 completion needs up to 28 samples. Fo non-degeneate data the equied ovehead would have been 54 RANSAC samples fo an unsuccessful RANSAC(7). Figue 5. Example fom [1] of a scene that mostly contains featues on one plane (gound plane). Off plane inlies used by QDEGSAC to compute the coect epipola geomety shown in. The poblem of the estimation of the fundamental matix fom (quasi-)degeneate data has been addessed by Chum et al. [1] on the data shown in Figue 5. They detemined the ate of detection fo the off-plane inlies in one hunded uns. The detection ate of thei algoithm is shown in Figue 6. The poposed QDEGSAC algoithm was tested on the same set of matches and the detection ate is also shown in Figue 6. The compaison shows that QDEGSAC achieves the same esults as the technique fom [1] without incopoating any knowledge about the degeneacy. The computed epipola geomety fo the box scene is shown in Figue 5-. Figue 6. Numbe of inlies (y-axis) fo diffeent numbes of employed constaints (xaxis). Fo six constaints the additional inlies of QDEGSAC ae stacked. Compaison fo detection of the off plane inlies. The model selection of To [8, 9] tests fo coplana matches fo which a homogaphy is computed. Othewise the fundamental matix is computed. Fo the decision the GRIC citeion [8] is employed. We computed the GRIC fo both tansfomations on the tay scene. The GRIC GRIC F fo the fundamental matix was computed fo the fundamental matix computed by the fist RANSAC(8) in the famewok and was GRIC F = fo the potential matches {p}. The GRIC F is only slightly deceased fo the coect epipola geomety as expected. The GRIC fo the homogaphy was GRIC H = Hence the model selection would always decide on a homogaphy fo the tay scene wheeas ou famewok is able to compute the epipola geomety without knowledge about the potential degeneacies. Theefoe, while [8] woks well fo degeneate data, it can not deal with quasi-degeneate data. Pojection matix estimation The linea estimation of the pojection matix also suffes unde 3D points lying on a 3D wold plane. The techniques poposed in [1, 9] can be extended to detect coplana points which is one of the known degeneate cases. In this section we apply the novel famewok fo the linea estimation of the pojection matix. The expeiments used the tay scene fom Figue 1. Figue 7. Numbe of inlie (y-axis) as a function of the numbe of employed constaints (x-axis) fo the computation of P. Compaison of detection ates (y-axis) fo QDEGSAC and RANSAC fo the off plane inlies on the tay scene The pojection matices of the fist two cameas whee deduced fom the fundamental matix estimated employing ou novel fame wok. The 3D points ae tiangulated fom the fist two images of the scene. These 3D points whee used to estimate the pojection matix of the thid view. To evaluate the pefomance of the novel technique we pefom 100 estimations on the 2D-3D matches and count the numbe of detections fo the off plane inlies (on the candlestick). The detection ates ae shown in Figue 7. It can be seen that the estimation employing the new famewok ovecomes the poblem of the taditional RANSAC. The off-plane inlies ae always detected as inlies and the estimated pojection matix is always coect. The epojection eos fo QDEGSAC and taditional RANSAC ae shown in Figue 8. The new poposed famewok always detects the geate codimension of the data matix of fou instead of one. The detection is obustly pefomed egadless of the pojection matix extacted in the fist RANSAC. The tests with the

8 wheeas QDEGSAC always computed the coect quadic. QDEGSAC always detected the educed ank of six instead of nine of the data matix. 6 Conclusion Figue 8. epojection eo fo the pojection matix estimated with QDEGSAC epojection eo of a RANSAC solution. pojection matix show that ou poposed famewok detects the case of degeneate data obustly and always computes the coect pojection matix in contast to the taditional RANSAC which only computes it in 27% of the uns. 3D homogaphy estimation The poposed famewok is also applied to the linea estimation of a 3D homogaphy that maps 3D points to 3D points. We apply it again to the tay scene fom Figue 1 to establish two pojectively skewed scenes. The fist scene is tiangulated fom view one and view two, and the second scene was established fom the second and the thid view. The coplana points ae also a degeneate configuation fo the computation of the 3D homogaphy. The poposed famewok showed a simila behavio fo detection of off-plane inlies as fo the estimation of the fundamental matix and the pojection matix fo one hunded uns of QDEGSAC. The gaph fo the measuements is not included due to the lack of space. The famewok detects the eleven degees of feedom povided by the degeneate inlies on the plane. Aftewads the fou missing constaints povided by the non-degeneate inlies ae used to compute the coect 3D homogaphy. In one hunded uns the taditional RANSAC was successful in about 13% of the uns wheeas QDEGSAC always computed the coect 3D homogaphy. Quadic estimation The pevious paagaphs evaluated the novel famewok on the estimation of elations fequently employed in compute vision applications. The linea estimation of quadics fom 3D points is also done with the famewok. It is done fo a quadic defined by two planes. Most potential matches ae on one plane and only 2% of the potential matches {p} ae on the second plane. The latte have to be employed to detemine the quadic uniquely. The given 1000 potential matches contain 88% degeneate inlies and 10% outlies. The degeneate and non-degeneate inlies ae distubed with Gaussian noise with a standad deviation 2% of the bounding box. We obseve a simila behavio fo the detection of nondegeneate inlies in one hunded QDEGSAC uns as fo all pevious expeiments. The taditional RANSAC only computed the coect quadic once in one hunded uns We have intoduced a new famewok fo the obust computation of a elation fom (quasi-)degeneate data. The famewok simultaneously classifies the data into inlies and outlies with egad to the elation. The novel technique evaluates the computed model in contast to taditional RANSAC. Fo degeneate data an appopiate model is chosen. Fo quasi-degeneate data the small faction of the data that povides the necessay additional constaints is identified. Accodingly the computation does not suffe fom ambiguities. We also show the wide field of applications of the famewok. Finally we compae ou technique with the existing techniques fo the special case of the fundamental matix computation. This compaison showed that ou appoach pefomed as well as the state of the at while being moe geneally applicable. Acknowledgements We thank Ondej Chum fo poviding the data fo the compaison with [1] and Philippos Modohai fo his help with the poof of Theoem 1. This wok was patially suppoted by NSF Caee awad IIS Refeences [1] O. Chum, T. Wene, and J. Matas. Two-view geomety estimation unaffected by a dominant plane. In CVPR (1), pages , [2] M. A. Fishle and R. C. Bolles. Random sampling Consensus: A paadigm fo model fitting with application to image analysis and automated catogaphy. Communications of the ACM, 24(6): , June [3] R. Hatley and A. Zisseman. Multiple View Geomety in Compute Vision. Cambidge Univesity Pess, [4] K. Kanatani. Statistical optimization fo geometic computation: theoy and pactice. Elsevie, [5] D. G. Lowe. Distinctive image featues fom scale-invaiant keypoints. Int. J. of Compute Vision, 60:91 110, [6] C. K. Tang, G. Medioni, and M. S. Lee. N-dimensional tenso voting and application to epipola geomety estimation. IEEE PAMI, 23(8): , August [7] P. H. To and A. Zisseman. Robust computation and paameteization of multiple view elations. In Poceedings IEEE Int. Confeence on Compute Vision, pages , [8] P. H. S. To. An Assessment of Infomation Citeia fo Motion Model Selection. In Poc. CVPR 97, page 47. [9] P. H. S. To. Geometic motion segmentation and model selection. Philosophical Tansactions of the Royal Society, pages , 1998.