Polygonal Approximation of Point Sets

Transcription

1 Poygona Approximation of Point Sets Longin Jan Latecki 1, Rof Lakaemper 1, and Marc Sobe 2 1 CIS Dept., Tempe University, Phiadephia, PA 19122, USA, atecki@tempe.edu, akamper@tempe.edu 2 Statistics Dept., Tempe University, Phiadephia, PA 19122, USA, marc.sobe@tempe.edu Abstract. Our domain of interest is poygona (and poyhedra) approximation of point sets. Neither the order of data points nor the number of needed ine segments (surface patches) are known. In particuar, point sets can be obtained by aser range scanner mounted on a moving robot or given as edge pixes/voxes in digita images. Poygona approximation of edge pixes can aso be interpreted as grouping of edge pixes to parts of object contours. The presented approach is described in the statistica framework of Expectation Maximization (EM) and in cognitivey motivated geometric framework. We use oca support estimation motivated by human visua perception to evauate support in data points of EM components after each EM step. Consequenty, we are abe to recognize a ocay optima soution that is not gobay optima, and modify the number of mode components and their parameters. We wi show experimentay that the proposed approach has much stronger goba convergence properties than the EM approach. In particuar, the proposed approach is abe to converge to a gobay optima soution independent of the initia number of mode components and their initia parameters. 1 Introduction Expectation Maximization (EM) is a very popuar and powerfu method that aows simutaneous estimation of mode parameters and assignment of data points to components of the mode. However, EM produces an optima soution ony if the number of mode components is we estimated and the initia vaues of mode parameters are cose to the goba optimum. If this is not the case EM is ony guaranteed to produce a ocay optima soution. This is iustrated in Fig. 1, where (a) shows data points and the initia configuration of two straight ine segments. The number of mode components (2 ine segments) is correcty initiaized, but their position is not sufficienty cose to the goba optimum. Fig. 1(b) shows the fina, ocay optima, resut obtained by the cassica EM agorithm. Fig. 1(c) shows the gobay optima approximation obtained by the proposed method on the same input. Due to the oca optimum probem, a correct estimation of the number of components and the initia parameters of a statistica mode is crucia in a EM appications, and therefore, beongs to one of the most chaenging probems in statistica reasoning. The proposed approach provides a soution to the probem

2 (a) Ange 9.; Standard Deviation. segments: 2 subsegeng.1 (b) (c) Fig. 1. (a) shows the data points and the initia position of mode ines. (b) shows the optima approximation of the data points obtained by EM. (c) shows the optima approximation resut obtained by the proposed method. of oca optimum in EM that is based on cognitivey motivated oca support evauation of EM components. The exampe shown in Fig. 2 motivates the proposed approach. It is obvious to humans that the approximation in (c) of the underying data points is significanty better then the approximation in (a). Observe the ack of oca support in the data points of the midde part of the ine in (a). This observation is the key argument for the proposed extension of EM. We wi evauate the support in data points of each EM component, and remove parts of components with insufficient support. The existing approaches to determine the optima number of EM components, of which AIC and BIC (Bayesian Information Criterion), which is equivaent to MDL (Minimum Description Length) [1], are most known, do not base their decision on the oca support in data points of the mode components. They assume ony a fix cost per each mode parameter. In particuar, this means that a mode component with high data support (i.e., positioned in a data region with high point density) costs the same as a component with ow data support (i.e., positioned in a regions with ow density of data points) athough it is intuitivey cear that a component with ow data support is far ess reevant than a component with arge data support. AIC, BIC, and MDL require separate EM runs unti convergence for a possibe number of mode components, each run composed of severa EM iterations,

3 69 iteration 2, no spitting 69 iteration 2, no spitting (a) (b) iteration (c) Fig. 2. It is obvious to us that the approximation in (c) of the underying data points is significanty better then the approximation in (a). (a) shows the best possibe approximation of the data points obtained by EM. (b) iustrates the ine spit (LS) based on subsegment remova. The removed subsegments are marked with crosses. (c) shows the fina approximation resut obtained by EM after the spit. which may even be in the order of severa thousands. For AIC, BIC, and MDL to be successfu, it is impicity assumed that EM converges to goba optimum in each run. However, as iustrated in Fig. 1(b) this is not aways the case, since even with a correct number of components EM may get stuck in oca minimum. Therefore, the correct number of two mode components woud not be seected by AIC, BIC, or MDL in our exampe. By ocay evauating the support in the data points of the two ines in Fig. 1(b), we can ceary determine that they form a bad approximation of the data points. By removing most of their parts, and retaining ony sma parts around the data points, we create a better input for the EM agorithm. This finay eads to a gobay optima approximation in Fig. 1(c). The proposed approach provides a soution to the probem of oca optimum in EM by adding two new steps that are we integrated with the standard E and M steps of EM. The two new steps are geometricay motivated and can be interpreted as spit and merge steps in the context of ine fitting. However, the proposed extension of EM is not restricted to any particuar shape of mode components. In the first new step, the spit step, the mode components obtained by a previous EM iteration are examined for support of the data points. The main idea (iustrated by the above exampe) is that higher point density around a mode component (ine segment in our appication) indicates a presence of a inear structure in the data points around the segment. Parts of the segment that do not have sufficient support are removed. This may ead to segment remova

4 but generay eads to a spit of the segment into severa subsegments. The second new step is merging simiar mode components. It prevents generating statistica modes that overfit the data, i.e., fit noise in the data. This step requires a simiarity measure of statistica mode components. The merging step can be interpreted as perceptua grouping that dates back to the first resuts of Gestat psychoogy in the beginning of 2th century [2]. We wi show that integrating the spit and merge operations in the EM framework eads to a gobay optima soution. In our experiments, we were abe to obtain a gobay optima soution after just a few iterations (between and 3). Two exampe appications of our approach are outined in Fig. 3. (a) shows an origina input toy image. (b) shows the edges obtained by Canny edge detector with a substantia amount of added noise, and the initia mode for our agorithm. It consists of ony two ine segments. (c) shows an intermediate step of our agorithm. The fina poygona approximation obtained after 27 iterations is shown in (d). (e) shows an image obtained by samping 3 ground truth segments (1 points) with a substantia amount of noise (2 points). (f) shows the initia mode segments for our agorithm. We present the resuts of our agorithm after 8 in (g) and 19 iterations in (h). An overview of techniques for poygona approximations of curves (when the order of data points is known), which have been studied at east since eary seventies in computer vision, can be found in [3]. To some popuar greed poygona approximation methods in digita images beong [4] and [, 6]. An overview of approaches to obtain poygona maps from aser range data can be found in [7, 8]. We do not make any assumptions about the order of data points and extent of noise. The proposed method avoids the probem of a ocay optima soution and produces stabe approximations not ony to straight but aso to curved ines. Moreover, the fina number of fitted ine segments depends on extent of noise. This means that the number of mode components is adjusted to achieve the best possibe approximation accuracy as the function of noise extent. In order to show that the geometric and cognitivey motivated spit and merge steps can be incorporated into a statistica formaism, we introduce in Section 2 a new target function to be estimated in the EM framework, and reformuate the E and M steps in Section 3. Then we introduce statistica tests for the proposed spit and merge steps in Section 4. Finay in Section, we describe the geometric parts of the spit and merge steps. 2 Optimizing Kuback-Leiber Divergence Our goa is to approximate the ground-truth density q(x) with a member p Θ (x) of a parametric famiy {p Θ (x) : Θ S} of densities. We use Kuback-Leiber divergence (KLD) to measure dissimiarity between the ground-truth and parametric famiy of densities. By definition, the KLD between the ground truth q(x)

5 iteration 1, after (merge) & EM iteration 2, after (merge) & EM (a) (b) (c) iteration 27, after spit, before EM 2 iteration 1, after (merge) & EM (d) (e) (f) iteration 4, after spit & EM iteration 19, after spit & EM (g) (h) Fig. 3. (a) An origina input image. (b) The edges obtained by Canny edge detector, and two initia ine segments. (c) We see the poygona approximation of the edge pixes obtained after (d) The fina poygona approximation obtained after 27 iterations is shown in (e)-(g) Iustrate our approach on simuated data generated by 3 ground truth segments with ony 1 signa and 2 noise points. and the density, p Θ (x) is: D(q(x) p Θ (x)) = og q(x) p Θ (x) q(x)dx = og q(x)q(x)dx og p Θ (x)q(x)dx (1) Observe that KLD is abe to determine the optima number of mode components of p Θ. This is due to the fact that KLD D(q p Θ ), viewed as a functiona on the space { p Θ } of Gaussian mixtures, is convex and hence has a unique minimum. It can be easiy derived that the parameters Θ minimizing (1) are given by Θ = argmax Θ { og p Θ (x)q(x)dx } (2)

6 We obtain the cassica maximum ikeihood estimator by appying the MC (Monte Caro) integra estimator to (2) under the assumption that the observations x 1,..., x n are i.i.d. (independenty and identicay distributed) sampe points seected from the distribution q(x). Θ = argmax Θ og p Θ (x i ) (3) However, as we derive beow (equation (6)), if some proportion of the observations x 1,..., x n are noisy, a more accurate estimator of Θ in (2) is given by: Θ = argmax θ og p θ (x i )sdd(x i ), (4) i where sdd is caed the smoothed data density and is defined in (7) beow. Equation (4) is the basis of the proposed approach. We demonstrate theoreticay and experimentay that maximization in (4) yieds substantiay better resuts than the cassica EM maximization in (3). To demonstrate the significance of (4), we consider the probem of estimating the optima number of mode components by minimizing the KLD D(q(x) p Θ (x)) in Θ. The parametric famiy { p Θ (x) }, being a famiy of Gaussian mixture distributions, is convex. It foows that there is a unique member (x) of the Gaussian mixture famiy p Θ with minimum KLD from q. This minimizing mixture must have the correct number of mode components. However, it is we known that (3) cannot be used to estimate the correct number of mode components, since (3) increases when the number of mode components increases. In contrast, we are abe to determine the correct number of mode components when using (4) to estimate the KLD, D(q(x) p Θ (x)). Thus, the modified EM agorithm that maximizes (4) is not ony abe to estimate mode parameters but aso the right number of mode components. One of the key steps in the derivation of (4) is the Monte Caro (MC) estimate of the integra given by the right hand side of equation (1). Let x 1,..., x n be i.i.d. sampe points drown from the probabiity density function (pdf) q(x). Then we can approximate the integra of a continuous function f by its MC estimate: f(x)q(x)dx 1 f(x i ) () n In the usua approach to inference, it is a commony accepted assumption that sampe data points x 1,..., x n are distributed according to the (estimated) density q(x). This assumption is the key to insuring that maximum ikeihood estimators are appropriate for purposes of estimating parameters of interest. However, in a rea appications, the sampe data points are corrupted by a certain amount of noise. Usuay the proportion of noisy points does not decrease when the number of sampe points is increased. Due to the noise, the foowing equation provides a substantiay better estimate f(x)q(x)dx f(x i )sdd(x i ). (6) i i i

7 Finay equation (4) ceary foows from (6) and (2). The smoothed data density sdd is defined as sdd(x) K( d(x, x i) ) = 1 h nh G(d(x, x i ),, h), (7) where proportionaity refers to the fact that sdd(x i ) = 1, d(x, y) is the Eucidean distance, and G(d(x, y),, h) is a Gaussian with mean zero and the standard deviation (std) h. An intuitive motivation for sdd is as foows: If a given data point x j is samped from the true distribution q(x), then x j ies in a dense region of the observed sampe points and consequenty sdd(x j ) is arge. If a given data point x j is samped from the noise distribution, then x j is ikey to ie in a sparse region of the sampe space, and consequenty sdd(x j ) is sma. To estimate the bandwidth parameter h, we can draw from a arge iterature on nonparametric density estimation [9, 1]. As we show in the presented experimenta resuts, an accurate bandwidth estimation in not crucia in our approach. 3 E and M Steps We introduce atent variabes z 1,..., z n which serve to propery abe the respective data points x 1,..., x n. It is assumed that the pairs (x i, z i ) for i = 1,..., n are i.i.d. with common (unknown) joint (ground truth) density, q(x, z) = q(x)q(z x); q(x) is the margina x-density and q(z x) is the conditiona density of the abe z given x. In this new framework, the KLD between the joint density q(x, z) and a parametric counterpart density p Θ (x, z) is D(q(x, z) p Θ (x, z)) = D(q(x)q(z x) p Θ (x)p Θ (z x)) { [ ] [ ]} q(x) q(z x) = og + og q(x)q(z x)dzdx x z p Θ (x) p Θ (z x) [ ] [ ] q(x) q(z x) = og q(x)dx + q(x) og q(z x)dz (8) p Θ (x) p Θ (z x) x x We are now ready to introduce the expectation (E) and maximization (M) steps. Both steps aim at minimizing the same target function (8) in our framework. The expectation step yieds the standard EM formua; considerations discussed above ead to a different soution for the maximization step. Expectation Step: For a fixed set of parameters Θ, we want to find a conditiona density q(z x) that minimizes D(q(x, z) p Θ (x, z)). Since KLD is aways nonnegative, and the second summand in (8) is minimized for q(z x) = p Θ (z x) (in which case it is equa to zero), we obtain from (8) that q(z x) = p Θ (z x) minimizes D(q(x, z) p Θ (x, z)). z

8 In particuar, for given sampe points x 1,..., x n, we obtain q(z i = x i ) = p Θ (z i = x i ) = p(z i = x i, Θ) (9) = p(x i z i =, Θ)p(z i = Θ) p(x i Θ) (1) p(x i z i =, Θ)p(z i = Θ) = k j=1 p(x i z i = j, Θ)p(z i = j Θ) = p(x i z i =, Θ)π k j=1 p(x, i z i = j, Θ)π j (11) where π = p(z i = Θ) and π j = p(z i = j Θ) are the prior probabiities of component abes and j correspondingy. Maximization Step: For the fixed margina distribution q(z x) = p Θ (z x), we want to find a set of parameters Θ that maximizes (8). Substituting q(z x) = p Θ (z x) in (8), we obtain D(q(x, z) p Θ (x, z)) = og( q(x) p Θ (x) )q(x)dx = D(q(x) p Θ(x)) (12) Thus, minimizing D(q(x, z) p Θ (x, z)) in Θ is equivaent to minimizing D(q(x) p Θ (x)) in Θ. Using the estimate derived in equation (4), minimizing (12) in Θ is equivaent (in the MC setting discussed above) to maximizing the weighted margina density W M(Θ) = sdd(x i ) og p Θ (x i ) = sdd(x i ) og p(x i Θ) = = k sdd(x i ) og[ p(x i z i =, Θ)p(z i = Θ)] =1 k sdd(x i ) og[ p(x i z i =, Θ)π ] (13) =1 where π = p(z i = Θ) are the prior probabiities of component abes = 1,..., k. Now we expicity use the incrementa update steps of the EM framework. Using the prior probabiities of component abes π (t) = p(z i = Θ (t) ) obtained at stage t for = 1,..., k, we obtain from (13) that an update of W M(Θ) is estimated by maximizing W M(Θ; Θ (t) ) = k sdd(x i ) og[ p(x i z i =, Θ)π (t) ] (14) in Θ with Θ (t) denoting the vaue of Θ computed at stage t of the agorithm. The crucia difference between this and the standard EM update is that our target function is weighted with terms sdd(x i ). We note that the known convergence proofs for the EM agorithm appy in our framework, since adding the weights sdd(x i ) in (14) does not infuence the convergence. =1

9 4 Spit and Merge The proposed spit and merge steps adjust the number of mode components by performing compnent spit and merge steps ony if they increase the vaue of our target function (14). Our framework is very genera in that it aows many possibe seections of the candidate components for the spit and merge steps. We present specific seection methods of the candidate components in Section. They are based on a Maximum A Posteriori principe. In the foowing formuas, we assume that the candidate components are given. Spit: Assume that we are given two candidate mode components 1, 2 ; we consider repacing the mode component with components 1, 2. Since our goa is maximizing QM(Θ; Θ (t) ) in formua (14), we simpy need to check whether repacing with 1, 2 increases W M, where j {1,..., k}: < W M(Θ; Θ (t) ) = sdd(x i ) og[ j sdd(x i ) og[ j + p(x i z i = 1, Θ)π (t) 1 p(x i z i =, Θ)π (t) p(x i z i = j, Θ)π (t) j ] + p(x i z i = 2, Θ)π (t) 2 ] (1) We ony need to perform oca computation to perform this test, i.e., we ony need to compute the corresponding probabiities for the candidate components 1, 2, subject to the condition that π (t) = π (t) 1 + π (t) 2. The parameters are estimated foowing the sparse EM step in Nea and Hinton [11], (see equation (1)). In accordance with the resuts of [11] this oca computation guarantees that the target function increases after each iteration (if (1) hods). Convergence is aso guaranteed in this way. Merge: Given a candidate component, we merge two existing mode components 1, 2 to if for j {1,..., k} > W M(Θ; Θ (t) ) = sdd(x i ) og[ j sdd(x i ) og[ j + p(x i z i = 1, Θ)π (t) 1 p(x i z i =, Θ)π (t) p(x i z i = j, Θ)π (t) j ] + p(x i z i = 2, Θ)π (t) 2 ] (16) Again we ony need to perform oca computations to perform this test. For merge, we ony need to compute the corresponding probabiities for the candidate component, subject to the same constraint π (t) = π (t) 1 + π (t) 2. If (16) hods and we repace 1, 2 with, the convergence of our agorithm foows from the resuts of [11]. We note that the proposed spit and merge steps do not work in the cassica EM framework. To see this, consider sdd(x i ) = 1 for a the data points

10 (i = 1,..., n). The merge inequaity (16) is not satisfied even if the ground truth mode is assumed to be a singe component, since mutipe components can better fit the data, and consequenty have a arger og ikeihood vaue. Anaogousy, if the spit inequaity (1) hods for a reasonabe seection of candidate component modes, the cassica EM framework incorrecty spits ground truth components. Thus, a mixture mode of arger number of components is aways prefered in the cassica EM framework. In the proposed framework, sdd represents an estimated density of the data points. Consequenty, in the proposed spit and merge steps, the divergence of parametric components, 1, 2 from the ground truth is evauated with respect to this nonparametric density. Line Segments as Components We present specific detais concerning our use of ine segments as EM mode components in the appications presented beow. We stress that this section appies aso to hyper panes in any dimensions, but the presentation is given in terms of ine segments for purposes of simpification. The proposed approach requires a minor extension of EM ine fitting to work with ine segments, which we wi ca Expectation Maximization Segment Fitting (EMSF). The difference between EMSF and EM ine fitting is that our mode components are ine segments (rather than ines). The input, for our mode, is a set of ine segments and a set of data points. As with EM the proposed EMSF is composed of two steps: (1) E-step The EM probabiities are computed based on the distances of points to ine segments instead of the distances of points to ines. (2) M-step Given the probabiities computed in the E-step, the new positions of the ines are computed by minimizing squared regression error weighted with these probabiities. As in the case of EM ine fitting, the output of the M-step is a new set of ines (not ine segments). Since we need ine segments as input to the E-step, we trim ines to ine segment based on their support in the sampe data. This is done by the spit process described in Section.2. Now we describe the specific detais reated to ine segments for steps (1) and (2). In order to derive the soution of (14) for EM mode components being ine segments, we introduce so caed EM weights. In the cassica EM, the weight = p(z i = x i, Θ (t) ) represents the probabiity that x i corresponds to segment s for = 1,..., k. We use the notation θ for the parameters of the ine segment s itsef. In our framework w (t) i w (t) i sdd (t) (x i ) p(z i = x i, Θ (t) ), (17) and the weights are normaized so that k =1 w(t) i = 1 for each i. After the E- step associated with the t th iteration is accompished, we obtain a new matrix ). Intuitivey, each row i = 1,..., n of this matrix corresponds to weighted (w (t) i

11 probabiities that the data point x i is associated with the corresponding ine segments; each coumn = 1,..., k can be viewed as weights representing the infuence of each point on the computation of new ine positions in the M-step. Beow, we use the notation x i = (x ix, x iy ) with (i = 1,..., n) for the coordinates of the observed data points, and ( x, ȳ) for the coordinate averages. The ine L, constructed beow, is constructed to go through the point ( x, ȳ). To obtain the soution of (14), we perform an orthogona regression weighted with the matrix (w i ). The soution is given as the norma vector to ine L, which is the vector corresponding to the smaest eigenvaue of the matrix M defined as [ n w i(x ix x) 2 n w ] i(x ix x)(x iy ȳ) n w i(x ix x)(x iy ȳ) n w i(x iy ȳ) 2 (18) Finay the parameters θ (t+1) are given as parameters of the ine segment s (t+1) obtained by trimming the ine L to the data points. We are now ready to introduce particuar reaization of spit and merge for EM mode components being ine segments. The proposed spit and merge EM segment fitting (SMEMSF) agorithm iterates the foowing three steps (1) EMSF (2) Spit (3) Merge Spit step is presented in detai in Section.2 whie Merge step is described in Section.1. Spit evauates the support in the data points of ines obtained by EMSF and removes the parts that are weaky supported. Since we have a finite set of data points, this has the effect of trimming the ines to ine segments. Finay the merge step merges simiar ine segments. Thus, spit and merge steps adjust the number of mode components to better fit the data..1 Merging If inequaity (16) hods, we merge two mode components represented by parameters 1, 2 into one mode componet given by parameter. Whie components 1, 2 are present at step t (they are ine segments s 1, s 2 ), we did not yet specify hot to compute the candidate component. Now we describe a particuar method to generate a candidate component in the particuar case in which the mode components are ine segments. We stress that other methods are possibe and that inequaity (16) appies to them too. A support set S(s j ) for a given ine segment s j (mode component ) is defined as set of points whose probabiity of supporting segment s j is the argest, i.e., S(s j ) = {x i : w ij = max(w i1,..., w ik )}. This maps each data point to a unique segment using the Maximum A Posteriori principe. Given two ine segments s 1, s 2, the merged segment s is obtained by trimming the straight ine obtained by regression on data points in S(s 1 ) S(s 2 ). Trimming is performed by ine spit described in Section.2.

12 .2 Line spit (LS) A cassica case of EM oca optimum probem is iustrated in Fig. 2(a), where the ine segment is in a ocay optima position. Ceary, the probem here is that we have a mode consisting of one ine ony, whie two ine segments are needed. Fig. 2(b) iustrates a spit operation described in this section. It is based on remova of subsegments that do not have sufficient support in the data points. As the resut we obtain two ine segments. Finay, Fig. 2(c) shows the gobay optima approximation of the data points obtained by EM appied to the two segments. The main idea is that higher point density aong a segment indicates a presence of a inear structure in the data points around the segment. Each ine or ine segment is examined on having sufficient support in data points measured as point density around it. Ony parts of segments that have sufficient support of the data points remain. This eads to spit of existing ines or segments aowing us to adjust the number of the ine segments (i.e., the number of EM mode components) to better fit the input data points. In [12] the number of points in discretey enumerated rectanguar strips (e.g., neighborhoods of a possibe segments with endpoints in some finite set) is counted. Then signa strips are seected based on the ratio of the number of points to the area. Then strips are grouped to poyines based on proximity of their endpoints and their anges. The main difference of our approach to the approach in [12] is that we do not seect the signa segments, but evauate the existing structures seected by EM. This makes our computation more efficient, since we do not need to numerate a possibe strips, and more accurate, since the ine segments are optimay fitted to the data points in our approach. Line Spit (LS) is composed of the foowing steps: (2.1) Subsegment support computation. (2.2) Remova of subsegments with insufficient support that satisfy inequaity (1). The input to LS are segments s 1,..., s k obtained by cipping the ines 1,..., k created in EMSF to the image rectange. We divide each segment s j {s 1,..., s k } into subsegments of a predefined ength 2r, i.e., s j = I j 1... Ij, so that two consecutive subsegments overap in their common endpoint, where is the number of subsegments. (For simpicity we assume that the ength of s j is exacty mutipe of 2r.) For each subsegment I j k, we define its support as the number of data points in the square S(I j k ) whose two sides are parae to subsegment I j k and whose center is contained in I j k, i.e., support(i j k ) = #({x i} S(I j k )). A few such squares are iustrated in Fig. 2(b). In each iteration a support threshod C is computed from the statistics of support(i j k ) vaues over a subsegments of a ine segments. Finay subsegments I j k with support(i j k ) C are removed. The subsegments to be removed are marked with crosses in Fig. 2(b). New segments are created as connected

13 components of remaining subsegments of segment sj. If inequaity (1) hods, then the origina input segment sj is removed, and the newy created segments are added to the ist of origina segments for the next iteration of EMSF. If a its subsegments are removed, then a given segment is removed. 6 Appications Two exampes of approximations of point sets in digita images are iustrated in Fig. 3. An exampe appication of our approach in robot mapping is outined in Fig. 4. (a) shows an origina data set of aser range scan points aigned with the agorithm presented in [13]. The origina set is composed of 39 scans, each with 361 points. Thus, the origina input map is composed of 142,9 points. We initiaize our agorithm with 192 segments, the grid segments, as mode components. (b) shows the output with 96 segments after the first iteration of our agorithm. The fina poygona map in (c), obtained after 6 iterations, is composed of 86 segments, i.e., of 172 points. Thus, the proposed approach yieds the data compression ratio of 829:1. The mean distance of scan points to the cosest ine segments is 3.cm. We seected this map, since it contains surfaces of curved objects. The obtained poyines in (c) iustrate that the proposed approach is we suited to approximate inear as we as curved surfaces. For more detais on the appication of the proposed method in robot mapping see [14] (a) 1 1 (b) 1 1 (c) Fig. 4. (a) An origina outdoor map is composed of 142,9 scan points obtained during the Rescue Robot Camp in Rome, 24. We begin the approximation process with 192 ine segments that form the grid. (b) shows the output after the first iteration of our agorithm with 96 segments. (c) The fina poygona map obtained after 6 iterations is composed of ony 86 segments. The obtained compression rate is 829:1, and the approximation accuracy is 3.cm. Exampes iustrating fitting panar patches to 3D range data are given in [1]. Here we show ony one exampe in Fig.. Fig. (a) shows a 3D projection of the surface of some industria part composed of 744,4 aser range scan points, obtained from Fig. (b) shows

14 our approximation with 27 panar patches. The mean distance of each point to the cosest surface patch is.49 with the origina object size of (a) (b) Fig.. (a) An input surface with 744,4 sampe points. (b) Approximation with 27 panar patches. 7 Concusions The combination of Expectation Maximization Segment Fitting with aternating Segment Spitting and Merging was proven to be a powerfu too to gain a poyine representation of edge points in digita images, eading to a geometricay higher representation and an exceent data compression rate. The newy introduced, perceptua grouping based merging step baances the number of segments, created by partitioning and spitting, in a visuay natura way and therefore aows for the number of starting segments for the EM step to be imprecise. The extended EM agorithm is proven to yied gobay optima resuts. Acknowedgment This work was supported in part by the Nationa Science Foundation under grant NSF IIS We aso woud ike to acknowedge the support from the Nationa Institute of Standards and Technoogy, award Nr. NIST 7NANBH We thank Giorgio Grisetti (Univ. La Sapienza, Rome) for providing us the scan data. References 1. Bea, M.J., Ghahramani, Z.: The variationa bayesian em agorithm for incompete data. In: BAYESIAN STATISTICS 7. Oxford Univ. Press (23)

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