A Method for Identifying Irregular Lattices of Hexagonal Tiles in Real-time

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1 S. E. Ashley, R. Green, A Method for Identifying Irregular Lattices of Hexagonal Tiles in Real-Time, Proceedings of Image and Vision Computing New Zealand 2007, pp , Hamilton, New Zealand, December A Method for Identifying Irregular Lattices of Hexagonal Tiles in Real-time Steven E. Ashley and Richard Green steven.e.ashley@gmail.com Abstract In this paper I propose a real-time method for the detection of hexagonal tiles arranged in an irregular lattice. The method exploits the geometric properties of hexagons to aid detection. The recognition algorithm works by detecting objects on the boundary and peeling them away possibly exposing further internal objects. The algorithm terminates when no further objects can be detected. Keywords: object recognition, hexagonal lattice, contour. 1 Introduction This paper proposes a computational strategy for recognizing irregular lattices of hexagonal tiles in real-time. One possible application of this technology is to provide real-time help with board games that use hexagonal tiles such as Tantrix. Other techniques that work but are particularly slow include Hough-transformations and per-pixel image matching. Haar classifiers seem a viable alternative however they require extensive training data. 2 Proposed Solution The proposed solution is based on partial shape recognition and is broken into several steps. 2.1 Board Identification To identify the location of the board I apply several levels of thresholding until an appropriately sized square can be found. Squares are detecting by applying Suzuki s algorithm[1] for connectedcomponent detection, applying the Douglas- Peucker algorithm[2] and finally selecting only polygons with four vertices and angles of 90 ± 10 degrees. Figure 1: Tantrix pieces on a square sheet of paper. The goal of Tantrix is to connect the tiles in such a way that the colors of all connecting links must match. There are many other rules that place further restrictions on the placement of tiles. When tiles are touching (as in Figure 1) many simple techniques fail as they are unable to detect the boundary between pieces. These techniques include simple image-segmentation, blob-tracking and contour based methods that rely on the contour being an accurate representation of the outline of an object. 2.2 Board Projection The orientation of the board is determined by calculating the distances from each point in the square to each corner of the bitmap. (1) Equation (1) maps any point (x,y) on the destination image, to a pixel (u,v) on the source image. The projection matrix of the board is calculated by solving equation (1) for. is given as 1. A 271

2 square bitmap is created with a width and height equal to the maximum distance between any two of the corners of the board. Each pixel on the destination image is projected onto the source image (using the projection matrix calculated above) to determine its pixel value. The resulting contours are shown in Figure 4. Figure 4: Extracted contours (outlined in red). 2.4 Corner Detection Figure 2: The projected board in Figure Contour Extraction The projected board is now prepared by passing it through a grayscale threshold function to (> 170). The resulting contour is approximated as a polygon using Douglas-Peucker[2] algorithm with approximation accuracy 5. As the order of vertices is clockwise (the polygon is a hole) we can assume that any vertex with a clockwise angle of 120 degrees (10 degrees tolerance) is the corner of a tile. Sequences of one of more corners (chains) are kept for processing. Figure 3: Figure 2 after thresholding. The outline shapes in the above image are found by applying Suzuki s algorithm [1]. Any contours that occupy less than %0.1 of the image are removed, along with any contours that are not holes. Figure 5: Vertices are colored green if they conform to the criteria for being the corner of a tile or red otherwise. 272

3 2.4 Tile Identification Chains of corners are processed according to the number of vertices in the chain and whether it loops or not. Looping chains with exactly 6 vertices are mapped directly to hexagonal tiles. Looping chains of other lengths are (perhaps incorrectly) discarded. 2.5 Removal of Identified Tiles Finally, the identified tiles are removed by painting over them with the background color. The image is also eroded slightly to prevent the holes inside each tile from leaking out. The algorithm is repeated from the Tile Identification step until no additional tiles are detected. Known Next Angular Bisection Previous Opposing Figure 6: Calculating remaining hexagon vertices given a single vertex. Non-looping chains with 1 to 5 vertices are mapped to hexagons. Anything else is (perhaps incorrectly) discarded. The hexagon calculations assume the pieces have a constant size with respect to the image size. The identification results are shown in figure 7 below. Figure 8: Remaining tiles once identified tiles are removed. 3 Results The final result is shown in Figure 9. Each frame takes milliseconds to process. Test machine is a Pentium 4, 3.06 GHz Dell Inspiron 5150 Laptop running Windows XP Professional (140MB ram available). Contour extraction fails in areas with poor lighting. I believe this to be because an arbitrary thresholding value was used for contour extraction. Figure 7: Identified tiles are outlined. If the board is removed, changed color, or altered in size, the algorithm completely fails. In several parts of the algorithm I assume the shape and color of the board to be white / square and of a certain size. As long as the camera can see the entire board, it copes well with camera rotations and minor lighting changes. 273

4 Figure 10 shows a situation where the algorithm fails by producing an extra hexagon. and Image Understanding, vol. 89, issue. 1, January [2] D. H. Douglas, and T. K. Peucker, Algorithms for the reduction of the number of points required to represent a digitized line or its caricature, Cartographica: The International Journal for Geographic Information and Geovisualization, vol. 10, issue. 2, pp , [3] N. Ansari, and E.J. Delp, Partial shape recognition: a landmark-based approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, issue 5, Figure 9: All tiles are identified. 4 Conclusion Figure 9 above shows the algorithm working in a single situation. There are a number of situations (such as Figure 10) where the algorithm fails but I believe these can be resolved. Further work includes changing the algorithm to be more robust to changes in the environment by removing assumptions such as the existence of the playing board. General accuracy could also be improved. Further work is also required to compare this algorithm with other algorithms such as Hough transformations and Haar classifiers. [4] J. Revaud, G. Lavoué, Y. Ariki, and A. Baskurt, Fast and cheap object recognition by linear combination of views, Proceedings of the 6 th ACM international conference on Image and Video Retrieval, pp , [5] P. Suetens, P. Fua, and A.J. Hanson, Computational strategies for object recognition, ACM Computing Surveys, vol. 24, issue. 1, pp. 5-62, [6] J. Gausemeier, J. Fruend, C. Matysczok, B. Bruederlin, and D. Beier, Development of a real time image based object recognition method for mobile AR-devices, Proceedings of the 2nd international conference on Computer Graphics, Virtual Reality, Visualisation and Interaction, pp , [7] M.W. Spratling, Learning Image Components for Object Recognition, The Journal of Machine Learning Research, pp , Figure 10: The center hexagon is detected twice, once for each pair of edges. 5 References [1] K. Suzuki, I. Horiba, and N. Sugie, Lineartime connected-component labeling based on sequential local operations, Computer Vision [8] H. Noor, S.H. Mirza, Y. Sheikh, A. Jain and M. Shah, Model generation for video-based object recognition, Proceedings of the 14th annual ACM international conference on Multimedia, pp , [9] Qiaohui Zhang, Kentaro Go, Atsumi Imamiya, and Xiaoyang Mao, Robust objectidentification from inaccurate recognitionbased inputs, Proceedings of the working 274

5 conference on Advanced Visual Interfaces, pp , [10] S. Belongie, J. Malik, and J. Puzicha, Shape matching and object recognition using shape contexts, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 24. issue 4, pp , April [11] L. De Floriani, A graph based approach to object feature recognition, Proceedings of the third annual symposium on Computational geometry, pp , 1987, [12] L. Stark, and K. Bowyer, An aspect graphbased control strategy for 3-D object recognition, Proceedings of the 1st international conference on Industrial and engineering applications of artificial intelligence and expert systems, vol. 2, pp , [13] OpenCV Square Detector program, ncvlibrary/opencv/samples/c/squares.c?revisi on=1.4&view=markup, rev. 1.4, viewed. 1 July