Estimating Model Parameters and Boundaries By Minimizing a Joint, Robust Objective Function

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1 Proceedings 999 IEEE Conf. on Compter Vision and Pattern Recognition, pp Estimating Model Parameters and Bondaries By Minimiing a Joint, Robst Objective Fnction Charles V. Stewart Kishore Bbna Amitha Perera Department of Compter Science, Rensselaer Polytechnic Institte Troy, New York 8 59 Abstract Many problems in compter vision reqire estimation of both model parameters and bondaries, which limits the seflness of standard estimation techniqes from statistics. Example problems inclde srface reconstrction from range data, estimation of parametric motion models, fitting circlar or elliptic arcs to edgel data, and many others. This paper introdces a new estimation techniqe, called the Domain Bonding M-Estimator, which is a generaliation of ordinary M-estimators combining error measres on model parameters and bondaries in a joint, robst objective fnction. Minimiation of the objective fnction given a rogh initialiation yields simltaneos estimates of parameters and bondaries. The DBM-Estimator has been applied to estimating line segments, srfaces, and the symmetry transformation between two edgel chains. It is naffected by otliers and prevents bondary estimates from crossing even small magnitde discontinities. Introdction Estimation techniqes from the statistics literatre are not well-sited for many compter vision problems. Standard definitions of least-sqares and maximm likelihood estimators give procedres for estimating the parameters of lines, circles, ellipses, planes, qadratic srfaces and many other fnction models from a data set, and these methods may be made robst [8, ] and nbiased [4]. Missing from these estimation techniqes and indeed from the fnction models themselves is any notion of the domain or sbset of the data over which the models apply. This is problematic becase fnction models in compter vision are sally applicable over limited domains, which generally are not known in advance. For example, a srface in the world has a limited spatial extent and therefore a srface model estimated from range data mst cover a limited image region; the same observation holds abot the image description of a moving object. In a very different context, opposing otlines of a symmetric object shold map onto each other sing an estimated symmetry transform [5, 7], bt parts of the object and of the nearby backgrond may not be symmetric and therefore shold not be mapped by the transform. In these and many other cases, accrate estimation of model or transform parameters reqires accrate estimation of the domain, the bondaries, or the sbset of the data over which the model Figre : Example illstrating the DBM-Estimator in fitting a line segment to synthetic D points near a small step discontinity: the pper frame shows a skewed initial fit; the middle frame shows the DBM-Estimator recovering from the skewed fit and growing the segment to the left bt not across the discontinity on the right (in fact it plls away slightly); the lower frame shows the converged fit and bondary estimate. applies. Inclsion of points otside the proper domain can prodce biased parameter estimates, while inclsion of too few points can lead to inaccrate parameter estimates and redndant models. This problem, referred to here as the domain estimation problem, has been addressed nder many different gises in many different areas of compter vision. Existing techniqes, which work with varying degrees of effectiveness [], inclde many different segmentation methods [, 9], snake and balloon models [6, 6], robst estimators [8,, ], and mixtre models [4, ]. The approach introdced in this paper is qite different and more general than any of these. It attacks the limitations of standard estimation techniqes directly by incorporating model parameters and bondaries into a single, robst objective fnction. The objective fnction is a generaliation of standard M-estimator objective fnctions, and depends on two distance measrements for each point: the sal fitting error or residal distance and the distance of the point from the crrent domain. The latter will be called the domain distance. The new techniqe, called the Domain Bonding M- Estimator ( DBM-Estimator ) has several important properties. These can be contrasted against existing techniqes for the domain estimation problem. Bondary estimation is entirely driven by the optimia-

2 tion process, placing it on par with parameter estimation and making heristic growth techniqes nnecessary. The optimiation process may case the srface bondary to grow in some directions and shrink in others, with growth rates determined by the objective fnction s gradient. For D domains, this growth is similar to that of a balloon model [6] with no need to simlate an artificial force! The DBM-Estimator inherits local robstness properties of standard M-estimators while avoiding their ssceptibility to leverage points, which cases their poor global robstness (breakdown point). Domain growth is determined by the cmlative effect of points in small regions near the domain bondary. This means the objective fnction does not need to be tned to the tail of the error distribtion to ensre proper domain growth [9], allowing the DBM- Estimator to distingish between noise and small scale discontinities. An example illstrating the behavior of the DBM-Estimator is shown in Fig.. Despite its advantages, the DBM-Estimator does have limitations. First, since it ses a gradient search techniqe, its stopping point is a local minimm, making initialiation important. Usally, however, it is sfficient to avoid seed regions that sbstantially overlap a bondary. Second, thogh working better than previos techniqes, the DBM- Estimator is not capable of localiing extremely small magnitde discontinities (e.g. step heights of. standard deviations). The DBM-Estimator is illstrated here on three different problems: line segment fitting for D data (Fig. ); estimating the parameters and bondaries of a planar srface in a range image; and estimating the planar projective transform characteriing the symmetric relation between (parts of) two edgel chains. Application of the DBM-Estimator to other problems shold be relatively straightforward. M-Estimators This section briefly reviews standard M-estimators [8] to set the context for derivation of the DBM-Estimator. Consider N data points x i, model parameters a, which are to be estimated, and a fnction r x i ;a mapping a point and a parameter vector to a residal. In the simple case of data points x i x i i T and linear regression, r x i ;a i x i T a. The scale normalied residal is i r x i ;a σ i, where σ i is the scale or noise standard deviation which may need to be estimated. The M-estimate of the parameters is â argmin a ρ i () i where ρ is the robst loss fnction. Among the many possible forms of ρ, only those reaching an asymptotic Rho Wgt Figre : M-estimator ρ fnction (left) and weight fnction (right). I(D) Figre : Illstration of a domain over which a srface model might apply in a range image. Both the interior I D and bondary regions Ω D are shown. vale for large are at all effective near discontinities []. One of these, the Tkey biweight [] (Fig. ), is adapted below to the DBM-Estimator. Its eqation is ρ T B 6 if B B B 6 if B. Most often, eqation is solved by iteratively reweighted least sqares (IRLS) [] sing a weight fnction entirely determined by ρ: w ψ ρ. Figre shows the weight fnction compted from (). General Formlation The DBM-Estimator is formlated by defining the domain and domain distance, by defining the objective fnction, and by specifying an iterative search techniqe. These are considered in trn below.. The Domain and the Domain Distance The domain D consists of an interior region I D and a bondary region Ω D. These will be image regions for the srface fitting application (Fig. ), sections of arc length for the crve symmetry application, and jst an interval of the x-axis for line segment fitting. The next step is to define the distance d D x i x j in D between any two points, x i and x j. This may be a Eclidean distance in the image plane or the difference in parameter vales for an arc length parameteriation of a crve. The domain distance of a point x i is then defined as d i () min d x x j I D i x j! () Points in I D, interior points, have d i. Ω D is then defined as points i sch that " d i " d max, for some d max. Now that both a domain distance and a residal distance (Section ) are defined, two different types of otliers mst

3 # ><< / be defined. Domain otliers are points with large d i. Residal otliers are points with large magnitde i. Some points will be both. Neither type shold ndly inflence the estimated model parameters or bondaries.. The Objective Fnction Estimate The DBM-Estimator objective fnction is a D ρ i d i (4) i and the DBM-Estimate of model parameters and bondaries is â ˆD argmin a$ D i ρ i d i (5) where ρ d is the generaliation of the robst loss fnction for ordinary M-estimators. (No specific representation of the domain is assmed here, so the domain is jst written as D, with ˆD as its estimate. Also, the dependence of i and d i on a and D is left ot of these eqations to simplify notation.) The smmation is taken over all data points, althogh, as will be clear later, evalating this smmation will only reqire points in I D and Ω D. Defining the generalied loss fnction ρ d is clearly the heart of the isse.. Designing ρ% & d' Intitively, ρ d shold be designed to grow the domain to inclde residal inliers, stop growth or shrink away from discontinities, and be naffected by residal otliers properties evident in the example in Fig.. Hand specified data illstrating the reqirements for line segment fitting are shown in Fig. 4; the intitions are the same for the other applications. Plots (a) and (b) show small scale depth and orientation discontinities where the domain shold not expand frther to the right despite points in Ω D having fit residals less than σ. (This illstrates a limitation of region growing techniqes which typically inclde as inliers all neighboring points falling within ( σ of an extrapolated model.) In (c), the strctre to the right, well otside the domain, shold neither bias the line segment parameters nor force expansion of the domain. In (d), residal otliers falling in Ω D shold not prevent domain growth from crossing occlded regions that are at most d max wide. Here are for reqirements on ρ d imposed by the reqired behavior.. For points in I D, ρ shold behave as a standard M- estimator. Ths, ρ ρ T.. ρ d shold be constant for d d max, independent of, preventing domain otliers from inflencing model parameters (Figre 4c). Ths, as d increases from to d max, ρ d shold evolve from ρ T to a constant fnction (for fixed d). Points that are initially domain This has the added advantage of redcing ρ) (and therefore the IRLS weight fnction) with increasing d, so that points in Ω* D+ have redced inflence on estimated fit parameters. otliers may eventally be drawn into the domain, bt only throgh its contined expansion.. For small, ρ d shold decrease monotonically as d decreases, driving the domain toward the point. For larger, sch as for points on the tail of the residal distribtion, ρ d shold decrease monotonically as d increases, casing the domain to back away from the point. The transition vale of between these two effects, a vale denoted as and shown in Fig. 5(a) and (b), is important since the cmlative inflence of the points in a small section of Ω I will determine whether D expands or shrinks locally. The exact choice of is based on statistical analysis smmaried in Sec Residal otliers within Ω D shold not, by themselves, prevent contined domain growth. This reqirement implies that for all d, lim, ρ d ρ. The first three reqirements are met by a simple fnction (see Figre 5a): ρ d.-/ " " 5 ρ T f d ρ T 4 f d ρ if d if d d max ρ ρ T if d d max, (6) This joint loss fnction combines a standard M-estimator loss fnction for points in I D (the first reqirement) with a constant loss fnction for points otside Ω D (the second reqirement). The fnction f d controls the transition from one to the other as a point moves throgh Ω D, and is piecewise qadratic over 6 d max7 with f, f d max, and f f d max. All three reqirements are clearly met by ρ d. Unfortnately, ρ d from (6) does not satisfy the forth reqirement and therefore does not realie the desired behavior pictred in Figre 4(d). As ρ d is crrently specified, residal otliers in Ω D, which may be cased by partial occlsions or clstered otliers, can prevent growth of D or even case it to shrink. The problem is that the first three reqirements imply ρ d has different asymptotic vales (for large ) for different d; the forth implies these asymptotes mst be the same. The soltion is to relax the first reqirement, altering ρ T to redescend to an asymptote of ρ : B B ρ M 98;:<< C D if E EGF BH A B ρ? CLNM MO J CL BO C DQP ρ if B RSE ETF CH ρ if E EGU C. Sbstitting ρ M for ρ T in (6) yields the crrent ρ d. Plots of ρ d and the associated weight fnction to be sed in IRLS parameter estimation are shown in Fig. 5(b) (7)

4 X Ω (D) I(D) x Ω (D) I(D) x I(D) x (a) (b) (c) (d) Figre 4: Data illstrating the desired behavior of domain growth. The plots show a cross-section of an image, with the x axis as the image coordinate and the axis as the depth. In each case, a linear srface is shown, with dashed lines above and below showing approximate σ ncertainty bands. Ω D and I D are indicated by segments of the x axis..5 d max < d < d max d=.5 d=. d=. d=.5 d=. d=.. I(D) d=. d=. d=.5 d=. d=. x.8 ρ(, d).5 ρ(, d).5 wgt (a) (b) (c) Figre 5: Plots illstrating ρ H d (sing d max 8 ); (a) shows a version meeting the first for reqirements; (b) shows the final version; and (c) shows its associated weight fnction. Only the positive axis is shown. In stdying these plots, the reader shold note that the expected vale of E E nder a Gassian distribtion V W 79. This means a sbstantial majority of inlier fit residals will have E ETR. and (c). It is important to note that the negative weights for larger seen in (c) have only a minor effect on estimation, which in fact is advantageos: near discontinities they psh the estimated model (very) slightly away from data corresponding to the other strctre..4 Minimiation Conceptally, solving (5) is straightforward. The minimiation mixes IRLS steps to refine the model parameters and differential search steps on the domain. The IRLS steps are exactly as in ordinary M-estimators. The differential search steps are applied to each section of the discretied bondary Ω D. For example, in line segment fitting, the bondary segments are the two intervals 6 x l d max x l and x r x r d max7 (see Fig. 4), which are determined by x l and x r. Hence, the differential search steps compte changes to x l and x r. Considering jst x r, the partial derivative of the objective fnction is # x r iy x i x r $ x rz d max[ \ ρ i d i d! T Only partial derivatives for points in the bondary segment are non-ero, and the is jst the derivative of d i with respect to x r. This partial derivative is normalied by its expected vale calclated assming all bondary segment residals are normally distribted. The step change in x r, which may be positive or negative depending on ] x r, is then jst proportional to this normalied derivative. Extension to two-dimensional domains is straight-forward for polygonal representations of the domain bondary. In this case, the step compted is along the segment normal (see Sec. 5) d max d domain d domain Figre 6: The minimm reqired for different d max sch that there is a.99 probability of domain growth across a large interval for -d (pper, dotted crve) and -d (lower, solid crve) domains..5 Parameter Tning and Scale The DBM-Estimator is controlled by for main parameters, B, C, and d max and is inflenced by estimates of scale (noise standard deviation), σ. The first two parameters can be easily fixed and ignored. B, inherited from the standard Tkey biweight, is set at 4. [], and C B. The parameters and d max are interdependent. They mst be set together to determine the sensitivity of the DBM- Estimator to small magnitde discontinities, while ensring contined growth across a fit s correct domain. The relationship between and d max can be established nmerically by finding for each d max the minimm sch that there is a.99 probability of (correct) domain growth across a large (e.g. pixel) interval. These vales are shown in Fig. 6 for one-dimensional domains and for two-dimensional domains, where a polygonal bondary segment is assmed to be 5 pixels wide. The choice of d max therefore dictates. Rather than sing a fixed d max, however, the DBM- Recall that is defined in terms of scale normalied residals (Sec. ).

5 Estimator changes it in a coarse-to-fine manner. Using large d max initially prevents the DBM-Estimator from crossing small magnitde discontinities, and allows it cross narrow occlded regions (gaps in the data). Using a small final d max allows more precise bondary localiation. We se the initial d max and the reslts shown in Fig. 6 to fix. Three comments abot scale, σ, are important. First, robst estimation of scale within the DBM-Estimator is straightforward sing standard techniqes [8], althogh scale shold be fixed once the domain is sfficiently large. Second, since both noise and modeling error contribte to the vale of σ, fixing σ from otside the DBM-Estimator controls tolerance of modeling error in domain estimation. See Sec. 6 for an example of the effect of this. Third, σ, whether estimated or fixed externally, shold not be sed directly in calclating the scale normalied residal i for points in Ω D. The calclation of i mst instead se prediction interval techniqes [9] to factor in the ncertainty of estimated model parameters. This is especially important to allow recovery from poor initial fits, as shown in Fig...6 Instantiating the DBM-Estimator Applying the DBM-Estimator to a specific problem reqires for isses to be addressed. () A mechanism is needed to provide an initial domain, and perhaps initial model parameter estimates. () The domain representation and domain distance metric mst be specified. () A method for weighted least-sqares model parameter estimation is needed. (4) A techniqe for calclating fit residals and scale mst be provided. All bt the second are reqired for properly formlated estimation problems [5]. 4 Line Segment Estimation This and the next two sections briefly describe application of the DBM-Estimator to the problems of line segment estimation, srface fitting, and symmetry estimation. The simple example of line segment fitting has been sed to describe the DBM-Estimator throghot this paper. Experiments are presented here to characterie DBM- Estimator performance in accrate bondary location given a small starting region. Reslts are shown in Fig. 7 for step and crease discontinities and for two different starting regions. Step discontinities are characteried by scale normalied step height h σ, while crease discontinities are characteried by angle α, sch that the two lines forming the discontinity make an interior angle of π α. Fig. gives an example near a step edge with h σ 5. The fit of interest is always the left half of the discontinity. Starting regions are either far from the discontinity or close to it. Fig. 7 shows that the starting region makes a difference for small magnitde discontinities: starting far away allows the fit to stabilie before reaching the discontinity. Still, however, discontinities are localied extremely accrately by the DBM-Estimator for all bt the smallest discontin- av. bias in discont. position(in pixels) h/sigma av. bias in discont. position (in pixels) alpha in degrees Figre 7: The average bias in the position of x r, the right domain bondary of the line segment, for step (left) and crease (right) discontinities. The left and right halves of the discontinities were each pixels wide. Datasets with σ 8 W noise and % otliers were generated from the discontinity models. In each plot the dotted line shows the reslts of starting from near the discontinity while the solid line shows the reslts of starting far away. ities. This performance is far better than that of ordinary robst estimators [], which sally fail at step heights below 6σ. 5 Bonded srface estimation A bonded srface estimated from range data is represented by a low-order polynomial eqation x y and a twodimensional region I D. This region is characteried by an enclosing polygon and one or more non-intersecting interior polygons corresponding to holes in the srface. Initialiation reqires only a small initial region, which may be specified manally or by finding the centers of connected regions having similar robst estimates of local srface normals. The domain polygon consists of short line segments, with differential search steps calclated in two parts: (a) the change in position along the normal to each segment is fond as described in Sec..4, and then (b) the change in each vertex is fond by combining motion vectors from adjacent segments. Segments growing too large are split; those shrinking too small are merged. Changes in topology are detected and corrected along the polygons [7]. The overall effect is similar to that of a snake or balloon model [6, 6], withot any simlation of artificial forces. Crrently there is no smoothing of the DBM-Estimator bondary polygon, so it appears jagged. Example reslts are shown in Fig Symmetry Transform Estimation Given two pairs of edgel seqences, S _^ p i i n` and S a^ pi i m`, the problem is to find a correspondence map C : ^ n`cb ^ m`, a sbseqence ^ p imin p imax` of S, and a plane projective transform d mapping each p i in the sbseqence onto pc i (with small error). Symmetry is enforced by reqiring d to map pc i onto p i as well, making d at least approximately an involtion. An example may be seen in Fig. 9 where the left and right halves of the otline appear symmetric over a large interval. The vageness of this problem description, characteristic of many domain estimation problems, may be eliminated by casting it into the DBM-Estimator formlation sing the steps otlined in Sec..6: () An initial, small sbseqence

6 o o o Figre 8: Domains of bonded planar srfaces estimated by the DBM-Estimator sing range data taken from the Soth Florida dataset []. The top left and right show intermediate stages of growth on a gens srface, before and after changes in topology. The middle shows grond trth segmentation reslts. The bottom shows DBM-Estimator srface interiors, each of which is estimated independently based on atomatically extracted seed regions. Observe that crrently the reslts show doble bondaries at discontinities becase adjacent domain estimates are nrelated. Of special note here, the three center srfaces (5,, and ) on the foregrond object are particlarly difficlt to estimate and most pblished techniqes merge them [, 9]. and associated correspondence map is assmed to be provided by an algorithm sch as described in [5, 7]. () The domain and domain distance are simply represented by an arclength reparametriation, s i, of the seqences. This makes the domain representation exactly as in line segment estimation with the x coordinate being replaced by arc-length. From this we can explicitly write the problem formlation as argmin e $ fg $ fih n ij ρ GkGdml nno ρ GkGdml n p nl p o kq σ sr ft n9o l kq σ sr ft (8) Clearly d depends on C and vice-versa, so they are estimated by alternating matching based on estimated d and refining d based on new matches, as in []. () Weighted least-sqares parameter estimation is a straightforward extension of Hartley s [] centering and normaliation algorithm. (4) As discssed above, scale in this application represents modeling error inexactness of the symmetry more than noise, and therefore scale is assmed to be set externally. Example reslts illstrating se of the DBM-Estimator in estimating d and the symmetry bondaries, with emphasis on the role of scale are shown in Fig Discssion The DBM-Estimator has been proposed to address the general problem of simltaneosly and robstly estimating model parameters and model bondaries. This model may be a crve, a srface, a transformation, or an image motion, depending on the problem instance. Formlation of the DBM-Estimator is intitive, controlled by only a few parameters, and easily adapted to nmeros problems. The major isse in each problem instance is defining the domain and the domain bondary representation. Example implementations and reslts on three problem instances have been shown. These illstrate the breadth of possible applications of the DBM-Estimator, different possible domain bondary representations, and the ability of the DBM-Estimator to tolerate otliers and localie small scale discontinities. The polygonal bondary representation for D domains has the added ability to adapt to changes in the domain bondary topology dring minimiation of the DBM-Estimator objective fnction. Several isses are still nder investigation. First we have begn experimentation with incorporating the DBM- Estimator into a mixtre model formlation [4, ]. Promising reslts, in terms of recogniing and localiing extremely small-scale bondaries, have been obtained in the line segment estimation problem. Second, we are working on improvements to the smoothness of the polygonal bondary representation for srface fitting. Third, we are developing a generaliation that ses piecewise polynomial (spline) crves and srfaces instead of a single model for the entire domain. Together, these improvements will add to the tility of what we believe is already a widely applicable new techniqe. Acknowledgements The athors grateflly acknowledge the spport of the National Science Fondation nder award IRI References [] F. Arman and J. Aggarwal. Model-based object recognition in dense range images. Compting Srveys, 5():5 4, 99. [] A. E. Beaton and J. W. Tkey. The fitting of power series, meaning polynomials, illstrated on band-spectroscopic data. Technometrics, 6:47 85, 974.

7 Figre 9: Example of symmetry transform estimation. The left shows an approximately symmetric (with small perspective effects) orchid with two overlaid edgel seqences shown as the white otline. The center and right show reslts of estimating symmetry starting from small corresponding sbseqences taken from the lower left and right corners of the orchid. Each image displays the mapping of the interior and bondary sbseqences from the right half of the otline onto the left. For the center image, σ was fixed at pixels, so the DBM-Estimator domain cold not grow across intervals of minor asymmetry (even thogh the bondaries enter these intervals). For the right image, σ was fixed at 4 pixels, so the symmetry grew mch farther. The top end of the mapped sbseqence shows that the bondary interval has entered a region of strong asymmetry, so growth has stopped. [] P. Besl and N. McKay. A method for registration of -d shapes. PAMI, 4():9 56, 99. [4] J. Cabrera and P. Meer. Unbiased estimation of ellipses by bootstrapping. PAMI, 8(7):75 756, 996. [5] T. Cham and R. Cipolla. Symmetry detection throgh local skewed symmetries. Image and Vision Compting, (5):49 45, 995. [6] L. Cohen and I. Cohen. Finite-element methods for active contor models and balloons for -d and -d images. PAMI, 5(): 47, 99. [7] R. Crwen, C. Stewart, and J. Mndy. Recognition of plane projective symmetry. In Proc. ICCV, pages 5, 998. [8] F. R. Hampel, P. J. Rosseew, E. Ronchetti, and W. A. Stahel. Robst Statistics: The Approach Based on Inflence Fnctions. John Wiley & Sons, 986. [9] R. Haralick and L. Shapiro. Compter and Robot Vision, volme. Addison-Wesley, 99. [] R. Hartley. In defence of the 8-point algorithm. In Proc. ICCV, pages 64 7, 995. [] P. W. Holland and R. E. Welsch. Robst regression sing iteratively reweighted least-sqares. Commn. Statist.-Theor. Meth., A6:8 87, 977. [] A. Hoover, G. Jean-Baptiste, X. Jiang, P. Flynn, H. Bnke, D. Goldgof, K. Bowyer, D. Eggert, A. Fitgibbon, and R. Fisher. An experimental comparison of range image segmentation algorithms. PAMI, 8(7):67 689, 996. [] P. J. Hber. Robst Statistics. John Wiley & Sons, 98. [4] S. J, M. Black, and A. Jepson. Skin and bones: Mlti-layer, locally affine, optical flow and reglariation with transparency. In Proc. CVPR, pages 7 4, 996. [5] K. Kanatani. Statistical Optimiation for Geometric Comptation: Theory and Practice. Elsevier, 996. [6] M. Kass, A. Witkin, and D. Teropolos. Snakes: Active contor models. IJCV, (4):, 988. [7] T. McInerney and D. Teropolos. Topologically adaptable snakes. In Proc. ICCV, pages , 995. [8] P. Meer, D. Mint, A. Rosenfeld, and D. Y. Kim. Robst regression methods for compter vision: A review. IJCV, 6:59 7, 99. [9] J. V. Miller and C. V. Stewart. Prediction intervals for srface growing range segmentation. In Proc. CVPR, pages 7, 997. [] P. J. Rosseew and A. M. Leroy. Robst Regression and Otlier Detection. John Wiley & Sons, 987. [] C. V. Stewart. Bias in robst estimation cased by discontinities and mltiple strctres. PAMI, 9(8):88 8, 997. [] Y. Weiss. Smoothness in layers: Motion segmentation sing nonparametric mixtre estimation. In Proc. CVPR, pages 5 56, 997.