The Comparison of Region Growing Algorithms with Using EMST for Point Clouds Zeynep Hasirci and Mehmet Ozturk Abstract— Close region interference is a problem in point cloud processing at the selection of local region step. Thus, growing local region with discarding the unwanted points is a crucial stage before post processing. Euclidean minimum spanning tree (EMST) is widely used to overcome this problem. This study aims to investigate the utilization of EMST for three commonly used region growing algorithms k-nearest neighbor (k-NN), circular region growing (CRG) and hybrid connection table (HCT). The experiments are conducted on two different data sets that one represents close region interference while the other has varying thickness problems. The results are presented quantitatively by measuring similarity of the reconstructed curve with the original one and the algorithm runtime. Visual results are also presented. EMST usage is improving the accuracy of local region at close region interference while increasing algorithm run time. Consequently, HCT with or without EMST is the most preferable method. Keywords—Close region interference, connection EMST, region growing, point cloud processing.
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I. INTRODUCTION
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urve reconstruction has an important place in pattern recognition, computer aided design, image processing and reverse engineering areas. Thus, several related studies have been conducted and applied in different fields. Point clouds can be simply defined as unordered point clusters that are sampled from a curve or surface. The sampling here is to get coordinates of a point on a physical object. For example, a dinosaur model is sampled with a laser scanner to obtain its virtual model in computer to be used in entertainment industry. The sampling points can be evenly spaced or this spacing mostly can be different due to noise caused by scanner hardware or amount of curvature of sampling objects. The aim of studies in this field is to model the sampled objects most accurately in computer environment. The main reason of the problems is unordered point sets. The studies about curve reconstruction from point clouds can be divided into four groups according to the properties of curves [1]. The first one is grouping studies according to sampling intervals as mentioned before. The second group
Manuscript received February 1, 2014. This work was supported in part by Karadeniz Technical University OSA Student Chapter. Z. Hasirci is with Karadeniz Technical University, Dep. of Electrical and Electronics Engineering, 61080 Trabzon, Turkey (corresponding author phone: +90-462-377 2992; e-mail:
[email protected]). M. Ozturk is with Karadeniz Technical University, Dep. of Electrical and Electronics Engineering, , 61080 Trabzon, Turkey
includes open or closed curves. The high curvature curves which has sharp corner are in the third group. The last group is made of branching or self-intersecting curves (simple or non-simple). All the curves mentioned previously could be noisy or noiseless. In this study, noisy point clouds are examined. There are limited studies in this field. Cheng at al. [2] reconstruct a curve with the assumption of additive noise along the normal direction of uniformly sampled curve. The local region growing method used in this study is based on a circular region with an increasing radius step by step (circular region growing-CRG). The stopping condition for region growing is samples inside the disk fit in a strip whose width is small relative to the radius of the circular region. Lee [3] proposed a two steps algorithm for curve reconstruction which includes moving least squares method-MLS enhanced with Euclidean minimum spanning tree-EMST. Additionally, the necessity of dynamic bandwidth selection is mentioned. To be able to determine thickness of the point cloud and adjust bandwidth accordingly the author used correlation coefficient. Alexa at al. [4] are studied on surface reconstruction with a similar approach and also emphasized the importance of the dynamically adjustment of bandwidth. Taubin and Ronfard [5] defined curves and surfaces with linear functions and applied this method to solve 2D curve reconstruction problems. Lin at al. [6] divided point samples from curve into rectangular shaped sub-regions with a new clustering technique and reconstructed curves via B-spline method. Liu at al. [7] proposed a method based on growing a B-spline curve along its local tangent directions calculated on a rectangular local region. Wang at al. [8] used a CRG technique with commonly used MLS and suggested a theoretical method to weight points inside circular region. Unnikrishnan at al. [9] also recommended statistical approach to determine the local region size. Ozturk and Hasirci [1] utilized a modified EMST connection table (hybrid connection table-HCT) to grow a circular local region and proposed an eigenvalue based bandwidth selection method. The effect of using EMST with region growing strategies on the performance of curve reconstruction is not dealt much in the literature according to the best of authors’ knowledge. The purpose of this paper is to show the performance of using EMST with region growing methods on curve fitting. In this context, possible region growing strategies with or without EMST such as k-nearest neighbor (k-NN), CRG and HCT are compared in terms of speed and accuracy.
II. METHOD Most of the curve reconstruction algorithms for noisy point cloud require a method to select local linear regions due to achieve the best ordering for the proper implementation of polynomial fitting. In the literature, local region growing strategies are usually based on growing of a circular region. However, this kind of selection is not adequate for point clouds which include sharp corners or branches. The involving of two distinct parts of the point cloud is the most common problem. Additionally, the nonhomogeneous distribution of noise or varying thickness of point clouds is the other difficulties. Thus, local region selection method plays an important role on the result of the curve reconstruction algorithms. For the automatic adjustment of bandwidth, Ozturk and Hasirci [1] used a normalized eigenvalue analysis based method. Cheng at al. [2] proposed a method based on the ratio of the radius and width of the disk which fit in a strip. The usage of correlation coefficient for this purpose is suggested by Lee [3]. In this study, normalized eigenvalue analysis based method is preferred. For local region selection, circular region growing is an essential aim to determine local linearity in a most accurate way. However, using this method only can cause inclusion of unwanted points into local region (close region interference). k-NN and HCT without EMST also produce similar results with CRG. This problem is solved by including EMST into the common region growing algorithms. Fig. 1 shows a curve reconstruction problem by using CRG without EMST.
without EMST includes the following steps: 1. Calculate the EMST table. 2. Set k=2. 3. Select initial point index, i=1. 4. Select n=k+1 points by setting k = k+ 1. 5. IF EMST is used, check EMST connectivity* of selected points, 5.1. Calculate normalized eigenvalue (eig). 5.1.1. IF eig≤ɛ (threshold), set k = k + 1. 5.1.2. ELSE do polynomial fitting (pol fit). 6. ELSE go to step 5.1. 7. Set i=i+1, set k=2 and repeat step 4 until all the points are processed.
EMST connectivity* 1. Do Breath First Search (BFS) for ith point and get number of levels, l. 2. Discard the points whose level ls ≥ round(l/2). B. Circular Region Growing Method (CRG) CRG is the most fundamental and simple method to grow local region. In this method, the radius of circle which is centered at initial point is increased with a certain amount of step size and the points inside the circle are selected. If EMST is used, the connectivity of selected points is checked as mentioned in Section A. Pseudo code with or without EMST is given below: 1. Determine step size, δ, for increasing radius, r. 2. Select initial point index, i=1. 3. Set r = r + δ and select the points inside the circle (n). 3.1. IF n≤2, repeat step 3. 3.2. ELSE IF EMST is used, check EMST connectivity* of selected points. 3.2.1. Calculate normalized eigenvalue (eig). 3.2.1.1. IF eig≤ɛ (threshold), set r = r + δ. 3.2.1.2. ELSE do polynomial fitting (pol fit). 3.3. ELSE go to step 3.2.1. 4. Set i=i+1, set r=0 and repeat step 3 until all the points are processed.
Fig. 1. An example for curve reconstruction by using CRG without EMST.
A. k-Nearest Neighbor Method (k-NN) k-NN method grows local region by selecting k nearest neighbors according to the Euclidean distances between points. If EMST is used, the algorithm additionally checks the connectivity between neighbors and the initial point over EMST table. If there is no an available path from initial point to any neighbor, it is discarded from selection. The pseudo code of k-NN region growing algorithm with or
C. Hybrid Region Growing Method (HCT) HCT uses graph theory for growing. In this method, the connections between the points according to some criteria are used to increase local region size. The growing procedure includes levels that come from breadth first search (BFS). The search tree is built up with the help of Euclidean distances between points. Two points are connected if they are close to each other less than a threshold value, ɛ. This threshold is calculated from local mean and standard deviation of MST connections. Pseudo code for HCT with or without EMST is shown below: 1. Calculate the EMST table. 2. Calculate mean and standart deviation of connection
distances, λort, λstd , respectively. 3. Create a new connection table (HCT) according to λ≤λort+3*λstd.. 4. Set initial point index, i=1. 5. Set BFS level as s=0. 6. Set s=s+1, select the points ns which connected to ith point with sth level. 6.1. IF ns≤2 , repeat step 6. 6.2. ELSE IF EMST is used, check EMST connectivity* of selected points. 6.2.1. Calculate normalized eigenvalue (eig). 6.2.1.1. IF eig≤ɛ (threshold), go to step 6. 6.2.1.2. ELSE do polynomial fitting (pol fit). 7. Set i=i+1, set s=0 and repeat step 6 until all the points are processed.
show the performance of EMST on close region interference problem. On the other hand, beside the close region interference, DS2 is used to test the effect of EMST utilization for sharp corner. feature. The DS2 has uniformly distributed noise along the normal directions of original curve with a mean of 0 and standard deviation σ=2.5.
Fig. 2 shows the first three steps of the HCT based region growing procedure.
Fig. 2. The first three iteration of the HCT.
III. EXPERIMENTAL RESULTS In this study, the methods are tested over two different point sets to show the effect of EMST for different conditions such as variable thickness and close region interference. The similarity between processed points and original curve which the point cloud created is measured via Modified Housdorff Distance (MHD) [10] beside the algorithm runtime (τ (s)). The simulation studies are performed on a notebook computer, Intel® Core™2 Duo CPU P8700 @2.53GHz with 4GB RAM in MATLAB environment. The first data set (DS1) shown in Fig. 3 is an artificially generated integral symbol and includes N=1000 points. The aim of this experiment is to show the performance of EMST on varying thickness problem. The DS1 has also uniformly distributed noise along the normal directions of original curve with a mean of 0. To be able to generate a varying thickness along the medial axis of the curve, the standard deviation of additive noise is changed between 0 and 2.5. The second data set (DS2) shown in
Fig. 4 is artificially hand generated ohm symbol shaped and includes N=1000 points. The purpose of this experiment is to
Fig. 3. The comparison of region growing algorithms with or without EMST for DS2 (a)-(b) k-NN, (c)-(d) CRG, (e)-(f) HCT
All region growing algorithms are incorporated with normalized eigenvalue method and threshold is selected as 0.9. The detailed explanation about the selection of the threshold value and normalized eigenvalue method can be found in [1]. In Table II, performance values (algorithm runtime τ (S) and Modified Housdorff Distance (MHD)) of DS1 are given for algorithms k-NN, CRG and HCT.
In Table I, algorithm runtime τ (S) and Modified Housdorff Distance (MHD) of DS2 are listed for the same algorithms which are mentioned above.
22.89
HCT
τ (s) k-NN CRG
25.84 18.43
MHD 0.546 0.441
With EMST
τ (s) 54.84 22.39
MHD 0.545 0.439
23.48
0.233
TABLE II PERFORMANCE ANALYSIS FOR DS2
TABLE I PERFORMANCE ANALYSIS FOR DS1 Without EMST
0.245
Without EMST
τ (s) k-NN CRG HCT
18.79 15.61 18.29
MHD 0.827 0.769 0.970
With EMST
τ (s) 40.60 19.67 19.60
MHD 0.758 0.667 0.575
Fig. 4. The comparison of region growing algorithms with or without EMST for DS1 (a)-(b) k-NN, (c)-(d) CRG, (e)-(f) HCT
IV. CONCLUSION This study intends to show the effect of EMST utilization on two different data sets for three different region growing algorithms. The achievements of the tested algorithms are measured over the similarity of original curve with the reconstructed one by MHD. Besides, algorithm runtime τ (s) is considered as a performance criterion and is given. The main problem of the DS1 is the non-homogeneous distributed points and variable thickness as a result (Fig. 3). The purpose of this experiment is to show the effect of EMST usage on a different problem data set except close region interference. Theoretically, EMST should not change the MHD significantly. As it is investigated, Table I supports this guess. In terms of algorithm runtime, τ (s), the EMST causes an increase. An overall evaluation on Table I show that HCT is a preferable method. DS2 includes a close region interference problem unlike the DS1. For DS2, MHD values which are obtained by using EMST are lower than MHD values of the tested algorithms without EMST as shown in Table II. Additionally, this improvement can be seen in Fig. 4. It is an expected result for data sets which include close region interference and sharp corner problems. The fundamental problem is the selection of non-local data into local region specifically at the close regions. Non-local data selection into local region causes trouble for curve fitting step. EMST algorithm prevents inclusion of unwanted data into local region. Thus, the result of polynomial fitting will represent the local data in a more accurate way. On the other hand, usage of EMST increases algorithm runtime, τ (s), for all of the methods. If Table II is summarized over all, it can be said that HCT with EMST is the most effective algorithm again.
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