The feature extraction and analysis of flaw detection and classification in BGA gold-plating areas

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1 Available online at Expert Systems with Applications Expert Systems with Applications 35 (2008) The feature extraction and analysis of flaw detection and classification in BGA gold-plating areas Yih-Chih Chiou a, Chern-Sheng Lin b, *, Bor-Cheng Chiou a a Department of Mechanical Engineering, Chung Hua University, Hsin Chu, Taiwan b Department of Automatic Control Engineering, Feng Chia University, Taichung, Taiwan Abstract In this paper, the measurements along with color image segmentation to detect all possible defects in BGA (ball grid array) type PCB (printed circuit boards) were presented. We use feature extraction and analysis as well as BPN (back-propagation neural) network classification to classify the detected defects. There are variable defects to be detected and classified including stain, scratch, solder-mask, and pinhole. The experimental results show that the proposed algorithm is successful in detecting and classifying the defects on gold-plating regions. The recognition speed becomes faster and the system becomes more flexible in comparison to the previous system. The proposed method, using unsophisticated and economical equipment, is also verified in providing highly accurate results with a low error rate. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Color image segmentation; BGA; Neural network; Flaw detection/classification 1. Introduction A color change on a bond finger or a ball pad usually indicates an anomaly that occurs in the bond finger or the ball pad. For example, a scratch on the bond finger may expose nickel underneath causing a change in color. Accordingly, the research of flaw detection on the bond finger can be conducted based on color machine vision. The items of BGA board normally to be inspected include subtract surfaces, gold-plating regions, metal traces, via holes, burrs, chip out, and discoloration. The gold-plating regions include bond regions and ball pads. Furthermore, the bond regions consist of ground rings, power rings, and bond fingers. However, we focus on the inspection of bond fingers and ball pads within the goldplating regions. Moreover, the detected defects will be classified as stain, scratch, pinhole, or residual solder-mask. Fig. 1 shows the images containing defects that occur in the bond finger and ball pads. Color machine vision is becoming more useful for online industrial applications as it makes color measurements and color recognition more feasible than grayscale images. Referring to Fig. 2, in contrast to grayscale images, color 1 images provide more detailed information for anomaly detection, classification, and verification. As shown in Fig. 2a, the color of a metal trace is green, whereas the color of a bond pad is golden. Accordingly, it is easy to distinguish one from the other. However, the image shown in Fig. 2a is transformed into a grayscale image, in which the metal traces and the bond pads are difficult to be distinguished because their colors look extremely close. Thus, color information is much more appropriate than grayscale information for the detection and classification of flaws. In view of this problem, this research applied color machine vision to inspect BGA substrates. Although there are many techniques involved in flaw inspection, we mainly focus on the color image segmentation and the flaw classification. * Corresponding author. Tel.: ; fax: address: lincs@fcu.edu.tw (C.-S. Lin). 1 For interpretation of color in Figs. 2, 5 7 and 10 the reader is referred to the web version of this article /$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi: /j.eswa

2 1772 Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) Fig. 1. Defective BGA images: (a) scratch; (b) pinhole; (c) residual solder-mask; (d) stain. Fig. 2. Comparison of color and grayscale PCB images: (a) color image; (b) grayscale image. As to the color image segmentation, a few color-based image segmentation techniques were developed to subdivide an image into regions with specific colors. These techniques include neural network color segmentation (SOM (self-organizing map), ART (adaptive resonance theory), BPN) (Darwish & Jain, 1988; Francisco, 1998; Jiang & Zhou, 2004; Lin, Lay, Huan, Chang, & Hwang, 2003; Lin & Lue, 2001), watershed color segmentation (Lezoray, Elmoataz, & Cardot, 2003; Lin, 1997), histogram thresholding (1-D, 2-D, 3-D) (Schuster & Ahmad, 1999), and Bayesian classifier. Jiang and Zhou (2004) adopted SOM neural network to segment an image into several regions. The five-dimensional feature vector inputting to the SOM network consists of x and y positions and R, G, and B components of each pixel. An ensemble-based SOM neural network was used to segment an image into several different regions. Francisco (1998) proposed a neural network architecture, which integrated ART, COS (color opponent system), CSS (chromatic segmentation system); and BCS/ FCS (boundary and feature contour system), for the segmentation and classification of colored and textured images. Color segmentation techniques are also used for the segmentation of facial regions. A simple method can coarsely segment the input image into regions and the skin color is learned by using a back-propagation learning algorithm. At the final stage, they use a Gaussian-based Bayesian classifier to distinguish skin from non-skin. Lezoray et al. (2003) first use a gradient-based color watershed segmentation to segment the cell from an image. Then, distinguishable features used for the subsequent classification of cells are extracted. Finally, a multiple ordinate neural network architecture (MONNA) is used to classify the cell into different kinds. The method can roughly locate the facial regions even if with the occurrence of over-segmentation. Schuster and Ahmad(1999) partitioned an RGB space using a 3-D RGB histogram. They tried three mathematical models, i.e., ellipsoid model, cylinder model, and mixed density model, to describe a 3-D color histogram. The experimental results show that the mixed density model is the most suitable for approximating the 3-D histogram of a color image. Other techniques are also used to segment an image into regions (Breen, 1994; Capson & Tsang, 1990; Geren & Lo, 1997; Hara, Doi, Karasaki, & Iida, 1988; Moganti & Ercal, 1995; Xiang, 1997). Breen (1994) employed regression methods to automated segment images. Xiang (1997) segmented a color image by minimizing the maximum-discrepancy between original pixel colors and the corresponding quantized colors. There are several techniques of colorbased classification including neural networks, statistical methods (nearest neighbors, Bayesian classifier, scatter measured by variances and standard deviation), decision trees, and normalized correlations (Lin, Ho, Wu, Miau, & Lin, 2004; Tao, 1996; Tragesser, 1998). The fruit sorter invented by Tao (1996) used color space transformation to transform input color images from RGB model to HSI model. Then, the H component is used to grade the fruits. They adopted different formulas for calculating the Hue

3 Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) values of red and golden apples to enhance the red color on the red apples and the golden color on the yellow-green apples. To provide an objective method and procedure for evaluating the seed quality, Tragesser (1998) first used the means of the individual R, G, and B components to calculate H, S, and I values. Then, the kernel area and the percentage of the hard endosperm, HE%, were measured. Finally, seeds were classified according to the obtained features including H, S, I, kernel area, and HE% of the seeds. In comparison to the above-mentioned techniques, we describe a useful method for automated flaw detection and classification by using cascade ANN networks. The proposed method of the feature extraction and analysis neither relies on time-consuming watershed color segmentation technique to identify gold-plating regions, nor requires establishing complicated mathematical 3-D model for describing a 3-D histogram. It can be fast and precise enough to detect the small flaw of BGA board manufacturing process. 2. Flaw detection scheme By inspection, normal bond fingers and ball pads possess particular colors. However, once scratches, stains, pinholes, or residual solder-masks occur on their surfaces, the colors on the defected regions will be changed. For example, a scratch on the gold-plating regions that expose the copper or nickel underneath will change its color from golden to copper or silver. Furthermore, a stain or oxidation on the gold-plating regions will also change the color on the defected area. Accordingly, our research takes the advantage of color variation to extract the gold-plating regions from a BGA image, and classifies the defects into different types. Using color variations among gold, nickel, copper, and solder-masks, we can successfully detect any defect on the surface of the gold-plating regions. We first use color image segmentation to extract goldplating regions from a color BGA image. Then, we use blob analysis, morphological operations, and size-filtering techniques to locate the defects in the gold-plating regions. After that, feature extraction and analysis techniques are applied to extract distinguishable features from each defect. Finally, we use the extracted features as an input vector to the neural networks for the defect classification. Subsequently, we use the developed flaw detection and classification algorithm to extract the defects from goldplating regions and to classify them into variable types of flaws. The inspection is conducted according to the procedures depicted in the flow diagram shown in Fig. 3. Some of the primarily processing steps in the flowchart will be discussed in detail as follows Gold-plating area segmentation Prior to flaw detection, we need to separate gold-plating pixels from non-gold-plating pixels such that subsequent search of defects can be confined in the gold-plating Start Gold-plating Areas Segmentation Color Space Transformation regions. Unlike statistical methods, neural network approaches do not require exact knowledge of the statistical distribution of input items. In general, provided that the data contained in the training set are the representatives of the objects to be recognized and the training process is successful, the classification results will be satisfactory. Hence, we employ BPN networks to distinguish gold-plating and non-gold-plating regions. As shown in Fig. 4, a pixel-separating ANN (artificial neural networks) network consists of one input layer, one hidden layer, and one output layer. The input layer includes three input features, i.e., R, G, and B components of each pixel. The hidden layer has 5 units. The output layer has 2 output units, i.e., gold-plating pixel and non-gold-plating pixel. The reason for using such a simple ANN network is due to the RGB features of the gold-plating and the non-gold-plating pixels as shown in Fig. 5 that are well separated. Let the mean square error between the actual output and the desired output of a neural network be expressed as E min ¼ 1 2 X ðoi y i Þ 2 ; Image Pre-processing Flaw Detection Edge Detection Feature Extraction Flaw Classification using BPN Stop Fig. 3. Flow diagram of the proposed flaw detection processes. where y i is the desired output and O i is the actual output. The back-propagation training algorithm uses iterative gradient algorithm to minimize the mean square error between the actual output and the desired output, Dw ji ¼ ad pj O pj ; where a is known as the learning rate, d pj refers to the error signal at each node j in the hidden layer, and O pj refers to the error signal at each node in the output layer. Repeat all test patterns until the output layer error is within the specified tolerance for each pattern and for each neuron. To train the networks, the training patterns are added to the training sets by enclosing a region in an image representing gold-plating or non-gold-plating region using a mouse selection. Each pixel enclosed in the rectangle area selected by the mouse then becomes a sample representing a gold-plating region or non-gold-plating region. And the R, G, and B components of a pixel are the 3 units of the input vector. As long as the training using back-propagation algorithm is successful, i.e., the error norm converges before it reaches the pre-specified maximum iteration, the network will be able to segment a BGA image into goldplating or non-gold-plating regions. At the recognition stage, the R, G, and B components of each pixel of the image are fed to the networks. And the

4 1774 Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) hidden layer (5) R input layer (3) output layer (2) gold-plating pixel G B non-gold-plating pixel Adjust Weights error Feature Extraction Actual Output Desired Output Flaw Detection and Classification input layer hidden layer output layer Fig. 4. Cascade ANN Network structure for total system. Fig. 6. Results of color image segmentation: (a) (c) the original images; (d) (f) the resulting images containing only non-gold-plating regions. Fig. 5. The three-dimensional RGB color distributions of gold-plating and non-gold-plating pixels contained in the training set. pixel will then be classified as a gold-plating or non-goldplating pixel according to the output vector of the networks. In the meantime, the gold-plating pixels are set to black. As shown in Fig. 6, the proposed BPN-based color segmentation technique can successfully distinguish goldplating and non-gold-plating regions. It is obvious that the defects in the gold-plating areas have been isolated after segmentation. One of the major problems occurred in color machine vision is that the illumination change seriously affects the color appearances of a gold-plating region during camera operation. This makes it difficult to obtain stable color features for color recognition in the real world. In order to make the recognition result invariant to the illumination change, we need to normalize R, G, and B components of each pixel.

5 2.2. Color space transformation Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) The usefulness of color information not only depends on the analysis technique, but also on the format of the color data. Mathematical transformation of the image data from one color space to another may further enhance its usefulness for a particular application. There are many color models beside RGB. Experimental results show that HIS color model is the most suitable for the current application. Accordingly, the original RGB model is transformed into HSI model. Then, the image plane containing the S component is used for the subsequent processes to facilitate the flaw detection. The color space transformation effects are shown in Fig. 7. It is clear that the defects in the gold-plating regions become more distinguishable than it was before Image pre-processing Because of noise and textures presented on the surface of the test object, most images still contain some black points as shown in Fig. 8a after color segmentation and color space transformation. Those black points can be eliminated using low-pass filtering and morphological closing operation. Low-pass filtering has the effect of blurring image such that disconnected regions will be reconnected again. Closing operation has the effects of filling notches on the contour, removing small holes and cracks, and reconnecting the broken narrow necks. Fig. 8b shows the result of the post-processing treatment, in which small black spots in the resulting image have been removed Flaw detection The purpose of flaw detection is to search for defects within the gold-plating regions such that the number of defects and the locations of each defect can be realized. The object can be achieved by using labeling techniques, Fig. 8. The results of post-processing: (a) the image after color image segmentation; (b) the resulting image after post-processing. such as recursive method, sequential method, boundary method, and iterative method. Upon the completion of labeling, each defect will be assigned a unique sequential number. Accordingly, the last assigned number represents the number of defects. Each detected defect is commonly called a blob. Although some noises have been removed from the images after morphological operations, not all the closedboundary regions left in the images denote defects. Size-filtering is a commonly used method to exclude regions that are not defects. As the name suggests, size-filter determines whether a region represents a defect by its size. Indeed, size-filter will exclude the regions out of the specified ranges. Referring to Fig. 9, although gold-plating regions contain five blobs, only three are labeled after size-filtering. This is because the sizes of the two arrow-pointed blobs are not within the specified ranges, i.e., their areas are too small to be identified as defects. Theoretically, noises and objects that are not identified as defects will be removed from the image after the morphological and size-filtering operations Edge detection The purpose of edge detection is to acquire edge points and draw the contour of each defect on the image. Here Freeman s direction chain code (Lin et al., 2003) is used to gather the edge points of each defect. Having obtained the edge points of each defect, the centroid of defects can be obtained by averaging coordinate values of all edge points. Moreover, the distance between centroid and each Fig. 7. Results of color model transformation: (a) (c) are the original RGB images; Their S components are shown in (d) (f), respectively. Fig. 9. These two arrow-pointed blobs will not be regarded as defects because of their small sizes.

6 1776 Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) edge point can be calculated. Finally the longest distance, dist_max, and the shortest distance, dist_min, can be obtained. Fig. 10a shows the results of the edge detection. For the concern of clarity, the defect shown in Fig. 10b has been enlarged. The red closed-curve represents the contour of the defect labeled with numerical number Feature extraction Most defects occur in the gold-plating regions have irregular shapes, and as a result, a set of distinguishing features will be needed such that the object of defect classification can be achieved satisfactorily. As the name implies, the principal concern here is to extract features that are beneficial to the subsequent flaw classification. Commonly used features include position, geometry shape, color, contrast, composition, and texture. After testing with various combination of features, the classification results show that R; G; B, circularity, roughness, and L S factor are effective in distinguishing one defect category from another. These six features can easily be extracted from each defect using edge detection and blob analysis techniques. Use the image shown in Fig. 11a as an example, the extracted features of each of the seven defects have been tabulated and shown in Fig. 11b. Definitions of the six above-mentioned features are briefly described in the following Circularity Circularity, also called compactness, is related to perimeter l p and area A. It is defined as Circularity ¼ l2 p 4pA : ð1þ A circle s circularity is equal to one. For other closedboundary geometry objects, their circularities are larger than one Roughness Roughness is related to perimeter l P and convex perimeter l C. It is defined as Roughness ¼ l P ð2þ l C In general, for a planer object having a rougher contour, its convex perimeter will be shorter than its perimeter, i.e., its Fig. 10. Results of flaw detection and edge detection: (a) the resulting image of flaw detection and edge detection; (b) the enlarged defect image. Fig. 11. The results of feature extraction: (a) the image with seven labeled defects; (b) the tabulated features of the seven labeled defects. roughness value is larger than one. On the other hand, if the planer object has a smoother contour, then its convex perimeter is close to its perimeter, i.e., its roughness value approaches to one L S factor After edge detection, dist_min and dist_max, which are the shortest and the longest distances between a defect s centroid and its boundary, respectively, are available. The L S factor for a planar object is defined as dist max L S factor ¼ dist min : ð3þ In general, a circular planar object has the smallest L S factor, i.e., unity. The more slender the planar object is, the larger its L S factor R; G; B For an RGB color image, each pixel consists of three color components, i.e., R, G, and B. In this research, the R, G, and B values of all the pixels enclosed by the contour of the planar object were summed up. Then the mean values of R, G, and B components were calculated and denoted by R; G and B, respectively.

7 Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) circularity roughness L/S ratio R-mean G-mean B-mean scratch pinhole stain residual solder-m input layer (6) hidden layer (7)(5)(7) output layer (4) Fig. 12. The ANN network structure for classifying defects Flaw classification With regard to flaw classification, we also make use of BPN network to classify the defects into four categories: stain, scratch, pinhole, and residual solder-mask. Similarly, prior to classification, the structure of the network must be determined. As shown in Fig. 12, a five-layer network is used for the flaw classification. It is known that the network structure is composed of an input layer, three hidden layers, and one output layer. The input layer consists of 6 units, i.e., the six features extracted from each defect, including compactness, roughness, L S factor, R; G, and B. The output layer consists of 4 units to be classified, i.e., the four types of defects, including scratch, pinhole, stain, and residual solder-mask. As to the three hidden layers, they have 7, 5, and 7 units, respectively. With respect to the setting of the two parameters during network training, the learning rate and allowable maximum sums-ofsquared-errors were set to be 0.25 and 0.001, respectively. Similarly, providing that network training succeeds the classification process can then be proceeded. In the recognition phase, by inputting the six features extracted from the defects of an unknown category, the exact category of each defect can then be determined according to the values of the output vector. 3. Results and discussion 3.1. Experimental results Fig. 13 shows the image measuring result of convex perimeter. From Fig. 14, it is clear that the configuration is the best choice in terms of MSE and convergent speed. The reason behind the transformation of RGB color model to HSI color models is that we need to extract the S components for further thresholding and blob analysis. Nevertheless, we still need R, G, B components as the input to the ANN for flaw classification. To verify the capabilities of the proposed method in detecting and classifying defects, numerous BGA sample images were used. The experimental results demonstrate that the proposed method is successful both in detecting Fig. 13. Convex perimeter. Fig. 14. The plot of convergences vs. layers. defects in the gold-plating regions and in classifying the defects. It takes 1.05 s on average to inspect an image (Table 1). Flaw detection process is shown in Fig. 15. As observed in Fig. 15b, the gold-plating pixels have been successfully removed from the input image (Fig. 15a). Fig. 15c displays the opening operation results of Fig. 15b. Flaw Table 1 Experimental results Item Specification Training set Includes 56 flaw samples: 11, 28, 13, and 4 for scratch, stain, pinhole, and residual solder-mask, respectively Computer Pentium IV 1.2 GHz Test images RGB color 24-bits images Total s inspection time

8 1778 Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) Discussion Fig. 15. Flaw detection process: (a) the original image; (b) the non-goldplating image; (c) the opening result image of (b) image; (d) the flaw detection result. detection result is shown in Fig. 15d. In the present example, a single defect with a rectangular mark has been detected. More flaw detection results are shown in Fig. 16. As shown in Fig. 16a, one defect is detected and classified as a stain. Fig. 16b contains two scratches. Fig. 16c contains one pinhole, and there is one residual solder-mask and one stain in Fig. 16d. Based on the experimental results obtained, the proposed flaw detection and classification scheme is capable of detecting and classifying flaws occurred on the gold-plating regions. It is worth mentioning that although colors of the goldplating regions of the three images shown in Fig. 6a c are inconsistent, the algorithm still succeeds in segmenting gold-plating regions as a result of network retraining. In other words, different combinations of colors representing gold-plating regions and non-gold-plating regions can be added to the training sets, respectively. If the network retraining is successful, the network will be capable of recognizing any newly added gold-plating color. Flaw detection and classification can be done simultaneously. However, to satisfy the speed requirement, a user can choose to detect flaws only. The classification of flaws can then be executed off-line, if necessary. The proposed color image segmentation technique can be further extended to divide the whole image into gold-plating, metal trace, solder-mask, and text regions, such that the subsequent dimension measuring and flaw detection can be facilitated. There are many advantages of using the proposed method. First, the proposed method is more suitable for industrial application because its high speed surpasses all other known methods. Second, it is capable of retaining the system in response to the presence of new BGA boards by adding new combinations of colors representing goldplating regions to the training sets. Despite the aforementioned advantages, the proposed method possesses some limitations. The most obvious example is that the present approach fails to distinguish a defect that is connected to Fig. 16. The automated inspection results: (a) one stain; (b) two scratches; (c) one pinhole; (d) one solder-mask and one scratch.

9 Y.-C. Chiou et al. / Expert Systems with Applications 35 (2008) in the gold-plating regions have been successfully accomplished. Acknowledgement This work was supported in part by the National Science Council under Contract NSC E References Fig. 17. Misclassification example: (a) the defect pointed by arrow will not be detected; (b) the defect is classed as non-gold-plating region during color segmentation, since it is connected to the solder-mask area. the non-gold-plating regions as shown in Fig. 17. It is because the pixels that make up the defect have been classified as non-gold-plating pixels during the course of color segmentation. 4. Conclusions As the color image segmentation is concerned, the goldplating regions can be extracted successfully by means of a BPN classifier with 96% accuracy. We take advantage of the fact that gold-plating regions are different in colors in regions such as solder-masks, metal traces, and substrate surfaces. Moreover, flaws inside the gold-plating regions with distinguishable colors in comparison with golden color can be successfully extracted. Furthermore, by feeding the well-trained BPN network with a discriminating feature vector, defects can be further classified into different categories. In this paper, the pre-processing and recognition methods of an integrated machine vision system for automated production BGA board inspection were presented. Using these algorithms the recognition rate becomes greater and the system becomes more flexible in comparison to previous system. The proposed method provides highly accurate results with a 5.4% total error rate. As mentioned above, there are many items to be inspected. However, in our research, the inspection regions will be restricted to bond pads and golden fingers only. Besides, the detected flaws will be grouped into four types, i.e., stains, scratches, pinholes, and residual solder-mask. In summary, the objects of the research in detecting and classifying defects Breen, E. J. (1994). Regression methods for automated colour image classification and thresholding. Journal of Microscopy, 174(1), Capson, D. W., & Tsang, R. M. C. (1990). Automatic visual measurement of surface-mount device placement. IEEE Transactions on Robotics and Automation, 6(1), Darwish, A. M., & Jain, A. K. (1988). A rule based approach for visual pattern inspection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(1), Francisco, D. P. (1998). A dynamic network model of the color visual pathways for attentive recognition. Neural Processing Letters, 7(1), Geren, N., & Lo, E. K. (1997). Automated removal and replacement of through-hole components in robotic rework. IEEE Transactions on Components, Packaging, and Manufacturing Technology, Part C, 20(3), Hara, Y., Doi, H., Karasaki, K., & Iida, T. (1988). A system for PCB automated inspection using fluorescent light. IEEE Transactions on Pattern Analysis and Machine Intelligence, 10(1), Jiang, Y., & Zhou, Z. H. (2004). SOM ensemble-based image segmentation. Neural Processing Letters, 20(3), Lezoray, O., Elmoataz, A., & Cardot, H. (2003). A color object recognition scheme: application to cellular sorting. Machine Vision and Applications, 14, Lin, C. S. (1997). Evaluation of defects on an optical disc master plate. Optics and Lasers Technology, 29(8), Lin, C. S., Ho, C. W., Wu, C. Y., Miau, L. H., & Lin, L. (2004). Automatic inspection device for HCV antibody rapid test strips for the production line. Journal of Scientific & Industrial Research, 63(3), Lin, C. S., Lay, Y. L., Huan, C. C., Chang, H. C., & Hwang, T. S. (2003). The preprocessing and recognition methods of an integrated automated production lot number inspection system. Journal of Scientific & Industrial Research, 62, Lin, C. S., & Lue, L. W. (2001). An image system for fast positioning and accuracy inspection of BGA boards. Microelectronics Reliability, 41(1), Moganti, M., & Ercal, F. (1995). Automatic PCB inspection systems. IEEE Potentials, 14(3), Schuster, S., & Ahmad, S. (1999). Method for segmentation of digital color images. US Patent no. 5,933,524, Tao, Y. (1998). Method and apparatus for sorting objects by color including stable color transformation. US Patent no. 5,533,628, Tragesser, S. (1998). Use of color image analyzers for quantifying grain quality traits. US Patent no. 5,835,206, Xiang, Z. (1997). Color image quantization by minimizing the maximum intercluster distance. ACM Transactions on Graphics, 16(3),