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A Robust Nonparametric Estimation Framework for Implicit Image Models

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A Robust Nonparametric Estimation Framework for Implicit Image Models
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  A Robust Nonparametric Estimation Framework for Implicit Image Models Himanshu Arora Maneesh Singh Narendra AhujaUniversity of Illinois Siemens Corporate Research University of IllinoisUrbana, IL61801, USA Princeton, NJ08540, USA Urbana, IL61801, USA harora1@uiuc.edu msingh@scr.siemens.com n-ahuja@uiuc.edu Abstract  Robust model fitting is important for computer vision tasksdue to the occurrence of multiple model instances, and, un-known nature of noise. The linear errors-in-variables (EIV)model is frequently used in computer vision for model fit-ting tasks. This paper presents a novel formalism to solvethe problem of robust model fitting using the linear EIV  framework. We use Parzen windows to estimate the noisedensity and use a maximum likelihood approach for robust estimation of model parameters. Robustness of the algo-rithm results from the fact that density estimation helps usadmit an a priori unknown multimodal density function and  parameter estimation reduces to estimation of the densitymodes. We also propose a provably convergent iterativealgorithm for this task. The algorithm increases the like-lihood function at each iteration by solving a generalized eigenproblem. The performance of the proposed algorithmis empirically compared with Least Trimmed Squares(LTS) — a state-of-the-art robust estimation technique, and To-tal Least Squares(TLS) — the optimal estimator for addi-tive white Gaussian noise. Results for model fitting on realrange data are also provided. 1. Introduction Robust model fitting is central to many computer visiontasks. Examples include tracking or registration under Eu-clidian, affine, or projective transformations; surface normaland curvature estimation for 3D structure detection; and, fit-tingintensitymodels forobjectrecognitionandobjectregis-tration. Robust estimation implies a framework which toler-ates the presence of outliers — samples not obeying the rel-evant model. Consider the problem of segmenting a rangeimage with planer patches: Here, each plane satisfies a lin-ear parametric model. For estimating parameters of eachplane, samples from all other planes should be consideredas outliers, i.e. they should not contribute to the error in fit.Another scenario is when the noise model for the observedsamples is not known. It is not possible to come up with acost function which is optimal for every kind of (unknown)noise model. Robust estimation seeks to provide reliableestimates in such cases — when data is contaminated withoutliers in form of samples corrupted by unknown noise orwhen multiple structures are present in the data, some or allof which need to be detected.Much work has been done in robust estimation in statis-tics, and more recently in vision. We refer the reader to[13] for a recent review of robust techniques used in com-puter vision. The two major classes of robust methodsproposed in statistics, M-estimators, and least median of squares (LMedS), are regularly used by computer vision re-searchers to develop applications. M-estimators, a general-ization of maximum likelihood estimators and least squaresmethod, werefirstdefinedbyHuber[6]andtheirasymptoticproperties were studied by Yohai et al. [14], and Koenkeret al. [8] in separate works. Least median of squares(LMedS) was proposed by Rousseeuw [10], wherein thesum of squared residuals in traditional least squares is re-placed by median of squared residuals. The Hough Trans-form [7], [9], and RANSAC [4] were independently devel-oped in computer vision community for robust estimation.For Hough Transform, entire parameter space is discretizedand optimal parameters are estimated by a voting schemedue to each data sample. It can be viewed as a discreteversion of M-estimation. RANSAC [4] uses the number of points with residual below a threshold as the objective func-tion. It has similarities with both M-estimators, and LMedS.Recently, Chen et al. [2] showed that all robust techniquesapplied to computer vision, i.e. those imported from statis-tics, and those developed in computer vision literature, canbe described as specific instances of the general class of M-estimators with auxiliary scale. In a separate work [1],Chen et al. explore the relationship between M-estimatorsand kernel density estimators, and propose a technique forrobust estimation based on kernel density estimators.Many parameter estimation problems in computer visioncan be formulated with the linear errors-in-variable model(EIV) [15], where the observations are assumed to be cor-rupted by additive noise. Further, it is often desirable to useimplicit functional form. For instance, consider the problemof range image segmentation mentioned earlier. We can ex-tract the 3D world coordinates, ( x i ,y i ,z i ) from the rangedata. If the range measurements, r i are noisy, ( x i ,y i ,z i )  will be noisy. The linear EIV model can be used to fit aplane through these noisy observations. Further, we shouldnot use any explicit scheme like z = ax + by + c , since itdoes not support the case when the srcinal plane has theequation ax + by + c = 0 , i.e. a plane perpendicular to z-axis. An implicit scheme is thus essential in this case. Thelinear EIV model has been used for analysis in some com-puter vision papers recently [2], [1].In this work, we assume that the image data may consistof a number of unknown structures, all of which obey thelinear EIV model. We also assume that the observed sam-ples are generated by additively corrupting unknown true samples with i.i.d. noise. However, the noise model is notavailable to us. We present a robust estimation algorithmthat detects these structures irrespective of the number of structures or the noise model. The robustness is achievedas we use a nonparametric (kernel) estimator to estimate thenoise density rather than assuming it to be known a priori.We also prove the convergence of the algorithm under mildconditions on the estimating kernels.In Section 2, we prove that the parameter estimationproblem for the linear EIV model amounts to solving a gen-eralized eigenproblem. We then show in Section 3, that arobust estimation framework can be developed by model-ing the pdf of the additive noise using nonparametric ker-nel density estimators. We then propose an iterative al-gorithm as a solution to the parameter estimation problemusing the ML (maximum likelihood) framework and provethe convergence of this algorithm. In Section 4, we empir-ically compare the proposed approach with Least TrimmedSquares (LTS), a state-of-the-art robust estimation tech-nique, and Total Least Squares (TLS), which is optimal forgaussian noise. We also present results of model fitting onreal data extracted from range images. 2. Linear Errors-in-Variables The linear errors-in-variables (EIV) approach assumes thatthe observed samples are generated from the true data sam-ples by additively corrupting them by independent, identi-cally distributed (i.i.d.) noise. The true samples obey somelinear, functional constraints that capture the a-priori physi-cal nature of the problem. Thus, we define, Definition 2.1 (Linear EIV model). Let  S  ox . = { x io } ni =1 bea data sample set of size n satisfying the constraints, f  ( x io ) = x ioT  θ − α = 0 i = 1 ,...,n (1) The observed data sample set  S  x . = { x i } ni =1 is related to S  ox by i.i.d. samples from an unknown, additive noise process  such that  x i = x io +  i . The ambiguity in parameters θ and  α is resolved by imposing the constraint   θ  = 1 . Consider the case when the noise samples are i.i.d. Gaus-sian i.e.  i ∼ N  (0 ,σ 2 I   p ) . It is well known that the max-imum likelihood estimate of the parameters and noise freesamples is then given by [ˆ θ, ˆ α, ˆ x io ] = argmin θ,α,x io 1 n n  i =1  x i − x io  2 (2)subject to the constraints on θ , α , and x io as specified inDefinition 2.1. Clearly, in minimization of (2), for fixed val-ues of  θ and α , the estimates for noise free samples x io aregiven by the orthogonal projection of the observed samples x i onto the hyperplane given by (1), with min x io  x i − x io  =  x i − ˆ x io  = | x iT  θ − α | (3)This indicates that the minimization can be reduced to justminimizing the sum of squared projections,  ni =1 | x iT  θ − α | 2 , with respect to parameters θ and α . The theorem belowshows that this is indeed true. Theorem 2.2. Define [˜ θ, ˜ α ] as the total least squares(TLS)solution as below, [˜ θ, ˜ α ] . = argmin θ,α 1 n n  i =1 | x iT  θ − α | 2 ,  θ  = 1 (4) Then, [˜ θ, ˜ α ] = [ˆ θ, ˆ α ] where [ˆ θ, ˆ α ] are as defined in (2) withthe constraints as specified in Definition 2.1.Proof. For the optimization problem specified by (4), thesolution [˜ θ, ˜ α ] should satisfy the following equations ob-tained using the Lagrange multiplier method. These equa-tions are obtained by setting the derivative with respect to θ and α equal to zero. θ :  ni =1 ( x iT  ˜ θ − ˜ α ) x i + λ ˜ θ = 0 (5) α :  ni =1 ( x iT  ˜ θ − ˜ α ) = 0 (6)Similarly, the solution, [ˆ θ, ˆ α ] , to the problem specified by(2) subject to the constraints in Definition 2.1, should satisfythe following equations: x io : x i − ˆ x io = λ i ˆ θ i = 1 ,...,n (7) θ :  ni =1 λ i ˆ x io + γ  ˆ θ = 0 (8) α :  ni =1 λ i = 0 (9)subject to the constraints in Definition 2.1. Now, from (7),we get ˆ x io = x i − λ i ˆ θ . Also, taking the transpose of thisequation and post-multiplying ˆ θ gives us, λ i = x T i ˆ θ − ˆ α .These two equations can be used in (8) to get, n  i =1 ( x iT  ˆ θ − ˆ α ) x i + ( γ  − n  i =1 λ i 2 )ˆ θ = 0 (10)Substituting the values of  λ i in (9) also gives, n  i =1 ( x iT  ˆ θ − ˆ α ) = 0 (11)  Further, (5) and (6) can be used to show that λ =  ni =1 ( x T i ˜ θ − ˜ α ) 2 . Similarly, (7)-(9) imply γ  −  ni =1 λ 2 i =  ni =1 ( x T i ˆ θ − ˆ α ) 2 . Thus, solutions to (5)-(6) and (10)-(11)are the same. Hence both problems are equivalent.Now, let us examine the problem in (4) more closely.Defining β  = ( θ,α ) T  , A =  ni =1  x i x iT  x i x iT  1  and B =  I   p 00 0  , where I   p is the p ×  p identity matrix, wecan rewrite (4) as ˆ β  = argmin β β  T  A β  ; β  T  B β  = 1 (12)Solving for β  leads to A β  = λ B β  , where λ is mini-mum eigenvalue for the generalized eigenproblem. Thus,the solution is the generalized (minimum) eigenvector of  A with respect to B . Consequently, The Maximum likelihoodestimation of the linear EIV model parameters in case of Gaussian noise reduces to a generalized eigenproblem.The assumption, made above, of Gaussian noise is notalways desirable. The model of noise is often unknown, andthe estimator proposed above may not be optimal in gen-eral. In particular, for heavy tailed distributions (e.g. log-normal distribution as will be discussed in section 4), theapproach above might have a really bad performance. Also,the structure that we need to detect (in this case the modelthat we need to fit) might only be valid locally. For instance,while detecting multiple planer segments in a range image,the model parameters are valid only on (local) segments of data. Since the segmentation is not a priori available, robustestimation becomes important for the discovery of any lo-cal models. It tolerates the presence of data samples that donot obey the model that is to be estimated. In the next Sec-tion, we propose a principled approach to carry out robustestimation. 3. Robust EIV Estimation UsingParzen Windows If we know the noise model for the linear EIV problem, thenwe can use the maximum likelihood approach to estimatethe EIV model parameters. However, quite often, we do nothave access to such a model. In that case, one can take re-course to estimating the noise density and then applying themaximum likelihood framework. In this section, we presentsuch an approach. We first formulate the problem in termsof a noise density estimate using Parzen windows and sub-sequently, we propose a solution to the said problem.Parzen windows or kernel density estimators are a pop-ular non parametric density estimation technique in patternrecognition and computer vision [3]. The Parzen windowestimate of the pdf from a given set of data samples can bedefined as follows: Definition 3.1 (Kernel Density Estimator). Let the ob-served samples y i ∈ R  p  , i = 1 ,...,n be generated indepen-dently from an underlying probability distribution function f  ( x )  , f  : R  p → R + . Then the kernel density estimate for  f  is defined as, ˆ f  ( y ) . =1 n det( H  ) n  i =1 K  ( H  − 1 ( y − y i )) (13) where, H  is a nonsingular bandwidth matrix, and  K  : R  p → R + is the kernel function with zero mean, unit area,and identity covariance matrix. The kernel function K  ( · ) used above is often assumedto be rotationally symmetric. We find it convenient to de-fine the profile of this rotationally symmetric kernel as aunivariate kernel function κ : R → R + , where K  ( y ) = c k κ ( − y  2 ) , c k being a normalization constant.The kernel density estimate of an arbitrary set of datasamples can be computed as shown above. However, theabove density estimate does not factor in any prior knowl-edge that one may have of the data. For example, the datamight be generated using a parametric model. For sucha case, we proposed a zero bias-in-mean kernel estimatorin our earlier work [12]. We used this estimator for ro-bust (parameter) estimation where the image is specified us-ing an explicit parametric formulation. In this paper, weadapt the aforementioned approach to define a robust ker-nel maximum likelihood estimation framework for the EIVmodel. We draw the reader’s attention to the fact that theEIV model is an implicit function formulation unlike ourprevious work.Now we explain our approach in terms of noise densityestimation: Let us assume that the noise free values x io andthe parameters [ θ,α ] are known such that the constraintsin Definition 2.1 are satisfied. Then, the noise can be es-timated as  i = x i − x io , with x T io θ = α , i = 1 ,...,n , and  θ  = 1 . The noise is nothing but the deviation of the ob-servation from the model described by the parameters. Thekey question is to decide the metric that is to be chosen onthese deviations to estimate the model parameters. If thenoise density was known, one could easily formulate such ametric using the maximum likelihood framework. However,since the noise density is not known, we take the next bestapproach — we use Parzen windows to estimate the noisedensity using Definition 3.1.For a set of observed data samples { x i } ni =1 , the ker-nel density estimate of noise given the noise free samples { x io } ni =1 and parameters [ θ,α ] can be written as ˆ f  (  | θ,α,x io ) =1 n n  i =1 K  ( H  − 1 (  − ( x i − x io ))) (14)under the constraint x T io θ = α ,  θ  = 1 . Let us define space S  . = { Θ . = [ θ,α, { x io } ni =1 ] |  θ  = 1 , x T io θ = α ∀ i =  1 ,..,n } . Then the model parameters Θ ∈ S  , and assumingthe noise to be zero-mean, the maximum likelihoodestimateof model parameters is given by Θ ML = argmax Θ ∈S ˆ f  (0 | Θ) (15)Note that the above definition is not restrictive, since anyshift in the  -space can be accounted for by a shift in x io and α . In absence of a disambiguating prior, we assume azero-mean noise process.In general, there might be multiple structures in the data,all of which we might need to discover (akin to the Houghtransform). Thus, the estimated density function ˆ f  (0 | Θ) might be multimodal. In such a case, one seeks all localminima of the density function. Consequently, we definethe parameter estimates as follows, Θ kml = argLmax θ ∈S ˆ f  (  = 0 | Θ) (16)where argLmax denotes a local maximum. It can be shownthat the estimator above is a redescending M-estimator [12]. Θ kml is a solution to a constrained nonlinear program.The local maximum of  ˆ f  ( · ) can be sought in general by gra-dient ascent. We now propose an iterative algorithm to seek the modes of distribution ˆ f  ( · ) given a starting point. Underthe constraint that the profile of the kernel, κ ( · ) , is a convexbounded function, the algorithm is guaranteed to increasethe objective function at each iteration and converges to alocal maximum.Since, Θ kml is constrained to lie in the space S  , we candefine the objective function q : S → R + as q (Θ) =1 n n  i =1 κ ( − H  − 1 ( x i − x io )  2 ) (17)Now, let the derivative of the profile κ be κ  = g . Assumingthat the initial estimate of  Θ is Θ (0) , and using the convexityof the profile, we see that q (Θ) − q (Θ (0) ) ≥ 1 n n  i =1 g ( − H  − 1 ( x i − x (0) io )  2 )(  H  − 1 ( x i − x (0) io )  2 − H  − 1 ( x i − x io )  2 ) (18)Defining the weights w i = g ( − H  − 1 ( x i − x (0) io )  2 ) , weseek the next iterate Θ (1) as the maximizer of right handside of (18), i.e., Θ (1) = argmin Θ ∈S n  i =1 w i (  H  − 1 ( x i − x io )  2 ) (19)The problem in (19) is similar to the ML estimation for i.i.dGaussian noise samples with identity covariance matrix, asdiscussed in Section 2. It can be reduced to the followingminimization on the space ˜ S  = { [˜ θ, ˜ α, { ˜ x io } ni =1 ] | ˜ x T io ˜ θ =˜ α, ˜ θ T  H  − 2 ˜ θ = 1 } ˜Θ (1) = argmin Θ ∈ ˜ S n  i =1 w i  H  − 1 x i − ˜ x io  2 (20)where θ = H  − 1 ˜ θ α = ˜ α x io = H  ˜ x io (21)where ˜Θ (1) = [˜ θ (1) , ˜ α (1) , { ˜ x (1) io } ni =1 ] . By Theorem 2.2, wecan write [˜ θ (1) , ˜ α (1) ] = argmin ˜ θ T  H  − 2 ˜ θ =1 n  i =1 w i (( H  − 1 x i ) T  θ − α ) 2 (22)with ˜ x io being equal to the perpendicular projectionof  H  − 1 x i onto the plane defined by [˜ θ, ˜ α ] . Defining A =  ni =1 w i  H  − 1 x i x iT  H  − 1 − H  − 1 x i − x iT  H  − 1 1  and B =  H  − 2 00 0  , from the discussion in Section 2, we get [˜ θ T  ˜ α ] as the generalized eigenvector of  A with respect to B corresponding to the minimum eigenvalue. Θ (1) can thusbe estimated using (20).The above mentioned process is repeated iteratively withnew estimates to yield a sequence { Θ ( n ) } ∞ i =1 of parame-ter estimates, and a sequence { q (Θ ( n ) ) } ∞ i =1 of function val-ues. Clearly, for Θ = Θ (1) , the right hand side of (18) ispositive, implying that q (Θ (1) ) ≥ q (Θ (0) ) . The sequence { q (Θ ( n ) ) } ∞ i =1 is thus increasing and bounded above(since κ is bounded), implying that it is convergent. 4. Experiments and Results First, we empirically compare the performance of the algo-rithm proposed in Section 3 with (a) the total least squares(TLS) solution, and (b) Least Trimmed Squares(LTS) [11].TLS, as discussed in Section 2, is the optimal estimator foradditive, white Gaussian noise (AWGN) and comparisonwith TLS shows the comparable performance of our algo-rithm to the optimal solution in case of often-used AWGNmodel. To test the robustness, we compare our algorithmwith Least Trimmed Squares(LTS) which is a state-of-the-art method for robust regression.We use line fitting in 2D space as the testbed for our ex-periment. The true samples ( x io ,y io ) satisfy ay io + bx io + c = 0 where b = c = 1 , a = − 1 . The true values are setas x io = i 50 − 1 , and y io = x io + 1 , i = 0 ,.., 100 . Thedata samples ( x i ,y i ) are generated by adding uncorrelatednoise samples to ( x io ,y io ) . The noise samples are generatedfrom the Gaussian distribution (a standard noise model) andtwo-sided log-normal distribution (to simulate outliers) withseveral different variance values.Figure 1 shows two sample realizations. At each value of variance, we generated 1000 realizations and computed the  −0.5 0 0.5 1 1.5 2 2.5−1.5−1−0.500.511.5   −0.5 0 0.5 1 1.5 2 2.5 3−2−1.5−1−0.500.511.522.53 (a) (b)Figure 1: Points generated according to the model ( x i ,y i ) =( x io ,y io ) +  i and {  i } ∈ R 2 (a)  i is gaussian with mean 0 andvariance 0 . 09 , (b)  i is log-normal with M  = − 4 and S  = 1 . 5 . Table 1: TLS, LTS, and KML estimates of  ( b,c ) for ( x i ,y i ) =( x io ,y io ) +  i and {  i } ∈ R 2 are i.i.d. Gaussian with mean 0 and variance σ 2  I  2 . Ground-truth values are ( b,c ) = (1 , 1) . Meanand deviation of the estimated values for 1000 experiments arepresented in the top and bottom four rows, respectively. MeanTLS LTS KML σ b c b c b c 0.03 1.000 1.000 0.999 0.999 1.000 1.0000.06 1.002 1.000 0.991 1.000 1.003 1.0000.09 1.001 0.998 0.979 1.000 1.001 0.9980.12 1.001 1.000 0.963 0.999 1.003 1.001Standard Deviation0.03 0.007 0.004 0.008 0.004 0.007 0.0040.06 0.015 0.007 0.015 0.009 0.015 0.0070.09 0.020 0.013 0.023 0.013 0.020 0.0130.12 0.027 0.016 0.032 0.020 0.029 0.016means and variances of the estimated parameters. Table 1shows the results for Gaussian noise. The estimated param-eters here are normalized with respect to a since the LTSalgorithm is implemented only for explicit function model.The upper and lower halves of the table shows means andvariances of the estimated parameters respectively. TLS,LTS, and KML denote Total Least Squares, Least TrimmedSquares, and Kernel Maximum likelihood (proposed algo-rithm). As we can see, the TLS is the best for this case, i.e.has means closest to 1 and lowest variances, but the perfor-mance of our algorithm is comparable. LTS has a bias inestimation and the variances are higher as well. This showsthat the proposed algorithm is comparable to LTS, whichis the optimal estimator for this case. Table 2 shows theperformance for log-normal noise. The table exposes thenon robustness of TLS. Its variance blows up as the noisevariance increases. Both KML and LTS perform well, withKML being better for higher noise variances. We also notethat our algorithm is simpler than LTS and is faster by al-most one order of magnitude.We nextdemonstrate the ability of the algorithm to detectmultiple structures in data: The algorithm was used to esti-Table 2: TLS, LTS, and KML estimates of  ( b,c ) for ( x i ,y i ) =( x io ,y io ) +  i and {  i } ∈ R 2 are i.i.d. log-normal with pa-rameter mu = − 4 and S  2 = σ 2  I  2 . Ground-truth values are ( b,c ) = (1 , 1) . Mean and deviation of the estimated values for 1000 experiments are presented in the top and bottom four rows,respectively. MeanTLS LTS KML σ b c b c b c 0.5 1.000 1.000 0.990 1.000 1.000 1.0001.0 1.002 1.000 0.976 1.000 1.003 1.0001.5 1.149 0.996 0.953 1.000 1.001 0.9982.0 5.314 1.497 0.919 0.996 1.003 1.001Standard Deviation0.5 0.016 0.009 0.019 0.011 0.016 0.0091.0 0.038 0.019 0.025 0.014 0.025 0.0151.5 2.087 0.084 0.038 0.019 0.038 0.0202.0 186.2 24.79 0.065 0.032 0.044 0.024   −50050250300350400450−50050xz       y (a) (b)Figure 2: (a) Intensity image from perceptron Ladar USF RangeDatabase (b) Cartesian coordinates extracted from the range datacorresponding to (a). mate plane parameters from 3D data extracted from rangeimages with planer patches. We used the perceptron ladarrange images from the USF Range Database [5]. Carte-sian coordinates ( x i ,y i ,z i ) corresponding to points r i in therange image are first extracted. Estimation of (all) plane pa-rameters is formulated as a robust EIV  model parameter es-timation problem. The algorithm has following steps: (1)Estimate TLS estimate Θ  p for the parameters for each datapoint due to points within δ neighborhood of  p to providean initial guess. (2) Arrange the points and correspondingparameter values in decreasing order of likelihood q (Θ  p ) and put on a stack  S  . (3) Choose the value of parametersfrom top of the stack, and apply the iterations according to(19) till convergence. Append this value in the estimated pa-rameter list. (4) Remove all points from stack  S  which arewithin a perpendicular distance τ  from the estimated plane.(5) Repeat steps (3),(4) till all points are exhausted.The above steps were applied to the data depicted in Fig-
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