Hough变换关键问题研究及其在航迹起始方法中的应用
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摘要
通过大量数据的非相干积累，利用Hough变换可在低信噪比下提取信号特征，这在低信噪比自动航迹起始以及低信噪比图像处理中有重要的应用价值。本文在“十五”预研项目关于低信噪比目标跟踪研究的资助下，研究了标准Hough变换的参数空间量化以及参数提取问题。主要工作如下：
1．以基于Hough变换的航迹起始为具体研究对象，比较深入地分析了标准Hough变换在应用中存在的三个关键问题：参数量化、积累方式和参数提取。
2．推导了参数积累单元投票的概率分布，并根据所得的分布指出了因变量参数量化时应注意的问题，为航迹可分辨性的参数设计提供了依据。
3．提出了一种参数积累单元的权值积累方法，改善了峰值单元与其附近单元的积累值的可区分性，从而为参数提取创造了有利条件。仿真表明该方法能够有效起始航迹。
4．给出了一种峰值提取算法，并在航迹起始中，将采用该算法与不采用该算法的航迹起始结果进行了比较，验证了该算法的可行性和有效性。
Through the noncoherent integration of great numbers of data, and by using Hough transform the features of signals in low SNR (signal-to-noise-ratio) can be extracted. Therefore, the Hough transform may play an important role in the automatic track initiation and the processing of images with low SNR environment. Supported by the key research project related to target tracking in low SNR and electronic warfare's environment, a part of the tenth Five-year plan our research focuses on the problems of the parameter quantization and parameter extraction. The main works are as follows.
1. Three key problems of Hough transform in track initialization field are presented: parameter quantization , accumulate method and parameter extraction .
2. Firstly, the vote distribution of parameter accumulate cells is proposed. Hence several important aspects of the dependent variable are proposed and which gives a reference for how to quantize the parameters.
3. Secondly, a method of weight integration of 'votes' is proposed. Simulation results show that the method can significantly improve the separability between the local peak cell and its neighbors compared with the standard binary integration method.
4. Finally, a peak extraction method is proposed for further improving the separability between the local peak cell and its neighbors in multiple target tracking. The simulation of multiple track initiation shows its feasibility and effectiveness.
1．以基于Hough变换的航迹起始为具体研究对象，比较深入地分析了标准Hough变换在应用中存在的三个关键问题：参数量化、积累方式和参数提取。
2．推导了参数积累单元投票的概率分布，并根据所得的分布指出了因变量参数量化时应注意的问题，为航迹可分辨性的参数设计提供了依据。
3．提出了一种参数积累单元的权值积累方法，改善了峰值单元与其附近单元的积累值的可区分性，从而为参数提取创造了有利条件。仿真表明该方法能够有效起始航迹。
4．给出了一种峰值提取算法，并在航迹起始中，将采用该算法与不采用该算法的航迹起始结果进行了比较，验证了该算法的可行性和有效性。
Through the noncoherent integration of great numbers of data, and by using Hough transform the features of signals in low SNR (signal-to-noise-ratio) can be extracted. Therefore, the Hough transform may play an important role in the automatic track initiation and the processing of images with low SNR environment. Supported by the key research project related to target tracking in low SNR and electronic warfare's environment, a part of the tenth Five-year plan our research focuses on the problems of the parameter quantization and parameter extraction. The main works are as follows.
1. Three key problems of Hough transform in track initialization field are presented: parameter quantization , accumulate method and parameter extraction .
2. Firstly, the vote distribution of parameter accumulate cells is proposed. Hence several important aspects of the dependent variable are proposed and which gives a reference for how to quantize the parameters.
3. Secondly, a method of weight integration of 'votes' is proposed. Simulation results show that the method can significantly improve the separability between the local peak cell and its neighbors compared with the standard binary integration method.
4. Finally, a peak extraction method is proposed for further improving the separability between the local peak cell and its neighbors in multiple target tracking. The simulation of multiple track initiation shows its feasibility and effectiveness.
引文
[1] llingworth,J, and Kittler,J. "A survey of the Hough transform", Computer Vision, Graphic,and Image Processing, 44, 1988,pp. 87-116.
[2] D.S. Le, G. R. Thoma and H. Wechsler, "Automated page orientation and skew angle detection for binary document images", Pattern Recognition, Vol. 27, Issue 10, Oct. 1994, pp. 1325-1344.
[3] P. Bosdogianni, M. Petrou, and J.Kittler, "The use of Hough transform to mixed pixel classification", Image Processing for Remote Sensing, IEE Colloquium on, Feb. 1996, pp. 1/1-1/6.
[4] F. Vajda, "Implementation issues of the Hough Transform, Journal of Systems Architecture", Vol.43, Issues 1-5, March 1997, Pages 163-165.
[5] G.H. Watson and K. Gilholm, "Signal and image feature extraction from local maxima of generalised correlation", Pattern Recognition, Vol, 31, Issue 11, Nov. 1998, pp. 1733-1745.
[6] Athanasios Nikolaidis and Ioannis Pitas, "Facial feature extraction and pose determination", Pattern Recognition, Vol. 33, Issue 11, Nov. 2000, pp. 1783-1791.
[7] L. Iocchi and D. Nardi, "Hough Localization for mobile robots in polygonal environments", Robotics and Autonomous Systems, Vol. 40, Issue 1, July 2002, pp.43-58.
[8] 王绍霖，付永生“Hough变换边缘参数提取算法” 同济大学学报(自然科学版)，Vol．4，1996．
[9] 胡正平，王成儒，于莉娜 “基于改进随机Hough变换的虹膜定位算法”，仪器仪表学报，2003年第5期．
[10] 孙亦南，刘伟军，王越超，孙亦蓬 “一种用于圆检测的改进Hough变换方法”，计算机工程与应用，2003年第20期．
[11] 董志荣“论航迹起始方法”情报指挥控制系统与仿真技术，1999．2．
[12] F.R. Castella, "Sliding window detection probabilities", IEEE Trans.on AES, Vol.12, Nov.1976, pp.815-819.
[13] Leung H, Hu Z, Blanchette M, "Evaluation of multiple target track initiation techniques in real radar tracking environments" [J],IEE Proceedings on Radar, Sonar Vol. 143,No.4,1996 pp.246-254.
[14] J.E. Holmes "Development of algoritms for the formation and updating of tracks", Proceedings of IEE International Radar Conference, London: 1997, 81-85.
[15] X.R.Li, Ning Li and V.P.Jilkov "SPRT-Based Track Confirmation and Rejection",Proceedings of the Fifth International Conference on Information Fusion, Vol.2 8-11, July 2002, pp.951-958.
[16] 朱洪艳，韩崇昭，韩红．“基于期望极大化算法的航迹起始方法研究”，计算机工程与应用 2003年14期 pp．66-69．
[17] P.G. Casner, R.J. Prengaman "Intergration and automation of multiple collocated radars", Proceedings of the IEE 1977 International Radar Conference, London: 1977,pp. 145-149.
[18] W.G.Bath, "False alarm control in automated radar surveillance systems", Proceedings of the IEE 1982 International Radar Conference,London: 1982 71-75.
[19] Jing Zhongliang, Zhou Hongren and Wang Peide "Tracking initiation and termination of multiple maneuvering target in a dense multi-return environment" Proceedings of the 29th Conference on Decision and Control Honolulu,Hawaii Dec. 1990.
[20] C.B.Chang, J.A.Tabaczynski "Application of state estimation to target tracking" IEEE Transactions on Automatic Control, 1984,29(2), pp.98-109
[21] G.V. Trunk, J.D. Wilson. "Track initiation in a dense detection environment", AD-A060744,1987.
[22] G.V. Trunk, J.D. Wilson "Track initiation of occasionally unresolved radar targets" IEEE Trans. on AES, Vol. 17, No.1, Jan. 1981, pp. 122-130
[23] J.J. Stem, S.S. Blackman. "Generalized correlation of multitarget track data", IEEE Trans.on AES, Vol.11, No.6, 1975, pp.1207-1217.
[24] 周宏仁、敬忠良、王培德 “机动目标跟踪” 国防工业出版社 1991．8
[25] S.S. Blackman "Multiple-target tracking with radar application" Dedham: Artech House, Inc., 1986.
[26] D.B. Reid "An Algorithm for Tracking Multiple Targets", IEEE Trans. On AC., Vol.24, No.6, Dec. 1979, pp.843-854.
[27] S. Mori, C. Y. Chong, R. P. Wishner and E. Tse, "Multi-Target Multi-Sensor Tracking Problems: a General Approach", in Proc. 1983 American Control Conf. San Francisco, CA, June 1983.
[28] S. Mori, C. Y. Chong, E. Tse and R. P. Wishner, "Tracking and Classifying Targets without A Priori Identification", IEEE Trans. On AC., Vol.31, No.5, 1986, pp.401-409.
[29] M.C. Smith and E.M. Winter "Feature space transform for multitarget detection" in Proc. IEEE Conf. On Decision and Control, Albuquerque, NM (1980), pp.835-836
[30] B.D. Carlson, E.D. Evans and S.L. Wilson, "Search radar detection and track with the Hough transform, Part Ⅰ: System Concept" IEEE Trans. on AES Vol.30, No.1, Jan. 1994, pp.102-108
[31] S.L. Wilson, B.D. Carlson and E.D. Evans, "Search radar detection and track with the Hough transform, Part Ⅱ: Detection statistics" IEEE Trans. on AES Vol.30, No.1, Jan. 1994, pp.109-115
[32] E.D. Evans, S.L. Wilson and B.D. Carlson "Search radar detection and track with the Hough transform, Part Ⅲ: Detection performance with binary integration" IEEE Trans. on AES Vol.30, No. 1, Jan. 1994, pp.116-125
[33] Kiril M. Alexiev, Ljudmil V. Bojilov, "A Hough Transform Track Initiation Algorithm for Multiple Passive Sensors", Proc. of the International Conf. On Multisource - Multisensor Information Fusion, FUSION'2000, pp. TuB2-11-TUB2-16.
[34] Kiril M. Alexiev, "Implementation of Hough Transform as Track Detector" Proc. of the International Conf. On Multisource - Multisensor Information Fusion, FUSION'2000, pp.ThC4-11-THC4-16.
[35] S.W. Yankowich and M.Farooq "Hough transform based multisensor,
multitarget, track initiation technique", Opt. Eng Vol.37, No.7, July 1998,pp.2064-2077.
[36] 程洪玮、孙仲康 “利用Hough变换实现目标检测与航迹启动” 国防科技大学学报 Vol．20，No．4，1998，pp．53-58．
[37] 王怀理、王德生、田立生 “Hough变换在航迹起始中的应用” 清华大学学报(自然科学版)Vol．42，No．7，2002，pp．909-912．
[38] 王国宏、苏峰、毛士艺、何友 “杂波环境下基于Hough变换和逻辑的快速航迹起始算法” 系统仿真学报Vol．14，No．7，2002，pp．873-875
[39] P.V.C. Hough, "A method and means for recognizing complex patterns", U.S. Patent 3,069,654.
[40] D.H. Ballard "Generalizing the Hough transform to detect arbitrary shape", Pattern Recognition, Vol. 13, No.2, 1981,pp. 111-122.
[41] R.O. Duda and P.E. Hart "Use of the Hough transformation to detect lines and curves in pictures" Communication of the ACM, Vol. 15, No.1, Jan. 1972,pp.11-15.
[42] John Immerker "Some remarks on the straight line Hough transform", Pattern Recognition Letters, 19,1998, pp. 1133-1135.
[43] H. Li, M.A. Lavin and R.J. LeMaster, "Fast Hough transform", Computer Vision, Graphics and Image Processing, 36, 1986, pp. 139-161.
[44] J. Illingworth and J. Kittler, "Adaptive Hough transform", IEEE Trans. On Pattern Analysis and Machine Intelligence, PAMI-9(5), Sept. 1987, pp. 690-698.
[45] J. Princen, J. Illingworth and J. Kittler, "A hierarchical approach to line extraction", Proc. IEEE Computer Vision and Patt. Recogn. Conf., San Diego, CA(June 1989), pp.92-97.
[46] D. Ben-Tzvi, and M.B. Sandier, "A combinatoryial Hough transform", Patt. Recogn. Lett., Vol,11,No.3, 1990, pp.167-174.
[47] Xu, L, Oja, E and Kultanen, P "A new curve detection method: Randomized Hough Transform(RHT)", Patt. Recogn. Lett., Vol. 11, No.5, 1990, pp. 331-338.
[48] Xu, L and Oja, E "Randomized Hough Transform(RHT): Basic mechanisms, algorithms, and computational complexities", GVGIP: Image Understanding,
Vol.57, No.2, 1993, pp.131-316.
[49] H. K(?)lvi(?)inen, L. Xu and E. Oja, "Recent versions of the Hough transform and the randomized Hough transform: Overview and comparisons", Reserach Report No 37, Department of the information Technology, Lappeenranta University of Technology, Finland(1993).
[50] R.S. Stephens, "Probabilistic approach to the Hough transform", Image & Vision Comput., Vol.9, No, 1, Feb. 1991, pp.66-71.
[51] W. Niblack and D. Petkovic, "On Improving the Accuracy of the Hough Transform: Theory, Simulations, and Experiments" on Proceedings of CVPR.'88, June 1988, pp.574-579.
[52] Ming Zhang, "On the Discretization of Parameter Domain in Hough Transformation", Proceedings of the 13th International Conference on Pattern Recognition, Vol.2, Aug. 1996, pp.527-531.
[53] H.Goto and H.Aso, "The Efficient Sampling Interval of the Scanning Parameter in the Hough Transform", Systems and Computers in Japan, Vol. 29, No. 11, pp. 9-19, 1998, Scripta Technica, Inc. (in English)
[54] H.Goto and H.Aso, "Designing Efficient Hough Transform by Noise-Level Shaping", IEICE Trans. INF.& SYST., Vol. E83-D, No. 2, Feb. 2000, pp.242-250.
[55] Y. Furukawa and Y. Shinagawa, "Accurate and robust line segment extraction by analyzing distribution around peaks in Hough space", Computer Vision and Image Understanding, Vol. 92, Issue. 1, October 2003, pp. 1-25.
[56] Q. Ji and R.M. Haralick, "Error propagation for the Hough transform", Pattern Recognition Letters, Vol. 22, Issues 6-7, May 2001, pp.813-823.
[57] M.K. Ibrahim, E.C.L.Ngau, and M.F.Daemi "Weigthed Hough Transform", SPIE Vol.1607 Intelligent Robots and Computer Vision X:Algorithms and Techniques (1991), pp.239-241.
[2] D.S. Le, G. R. Thoma and H. Wechsler, "Automated page orientation and skew angle detection for binary document images", Pattern Recognition, Vol. 27, Issue 10, Oct. 1994, pp. 1325-1344.
[3] P. Bosdogianni, M. Petrou, and J.Kittler, "The use of Hough transform to mixed pixel classification", Image Processing for Remote Sensing, IEE Colloquium on, Feb. 1996, pp. 1/1-1/6.
[4] F. Vajda, "Implementation issues of the Hough Transform, Journal of Systems Architecture", Vol.43, Issues 1-5, March 1997, Pages 163-165.
[5] G.H. Watson and K. Gilholm, "Signal and image feature extraction from local maxima of generalised correlation", Pattern Recognition, Vol, 31, Issue 11, Nov. 1998, pp. 1733-1745.
[6] Athanasios Nikolaidis and Ioannis Pitas, "Facial feature extraction and pose determination", Pattern Recognition, Vol. 33, Issue 11, Nov. 2000, pp. 1783-1791.
[7] L. Iocchi and D. Nardi, "Hough Localization for mobile robots in polygonal environments", Robotics and Autonomous Systems, Vol. 40, Issue 1, July 2002, pp.43-58.
[8] 王绍霖，付永生“Hough变换边缘参数提取算法” 同济大学学报(自然科学版)，Vol．4，1996．
[9] 胡正平，王成儒，于莉娜 “基于改进随机Hough变换的虹膜定位算法”，仪器仪表学报，2003年第5期．
[10] 孙亦南，刘伟军，王越超，孙亦蓬 “一种用于圆检测的改进Hough变换方法”，计算机工程与应用，2003年第20期．
[11] 董志荣“论航迹起始方法”情报指挥控制系统与仿真技术，1999．2．
[12] F.R. Castella, "Sliding window detection probabilities", IEEE Trans.on AES, Vol.12, Nov.1976, pp.815-819.
[13] Leung H, Hu Z, Blanchette M, "Evaluation of multiple target track initiation techniques in real radar tracking environments" [J],IEE Proceedings on Radar, Sonar Vol. 143,No.4,1996 pp.246-254.
[14] J.E. Holmes "Development of algoritms for the formation and updating of tracks", Proceedings of IEE International Radar Conference, London: 1997, 81-85.
[15] X.R.Li, Ning Li and V.P.Jilkov "SPRT-Based Track Confirmation and Rejection",Proceedings of the Fifth International Conference on Information Fusion, Vol.2 8-11, July 2002, pp.951-958.
[16] 朱洪艳，韩崇昭，韩红．“基于期望极大化算法的航迹起始方法研究”，计算机工程与应用 2003年14期 pp．66-69．
[17] P.G. Casner, R.J. Prengaman "Intergration and automation of multiple collocated radars", Proceedings of the IEE 1977 International Radar Conference, London: 1977,pp. 145-149.
[18] W.G.Bath, "False alarm control in automated radar surveillance systems", Proceedings of the IEE 1982 International Radar Conference,London: 1982 71-75.
[19] Jing Zhongliang, Zhou Hongren and Wang Peide "Tracking initiation and termination of multiple maneuvering target in a dense multi-return environment" Proceedings of the 29th Conference on Decision and Control Honolulu,Hawaii Dec. 1990.
[20] C.B.Chang, J.A.Tabaczynski "Application of state estimation to target tracking" IEEE Transactions on Automatic Control, 1984,29(2), pp.98-109
[21] G.V. Trunk, J.D. Wilson. "Track initiation in a dense detection environment", AD-A060744,1987.
[22] G.V. Trunk, J.D. Wilson "Track initiation of occasionally unresolved radar targets" IEEE Trans. on AES, Vol. 17, No.1, Jan. 1981, pp. 122-130
[23] J.J. Stem, S.S. Blackman. "Generalized correlation of multitarget track data", IEEE Trans.on AES, Vol.11, No.6, 1975, pp.1207-1217.
[24] 周宏仁、敬忠良、王培德 “机动目标跟踪” 国防工业出版社 1991．8
[25] S.S. Blackman "Multiple-target tracking with radar application" Dedham: Artech House, Inc., 1986.
[26] D.B. Reid "An Algorithm for Tracking Multiple Targets", IEEE Trans. On AC., Vol.24, No.6, Dec. 1979, pp.843-854.
[27] S. Mori, C. Y. Chong, R. P. Wishner and E. Tse, "Multi-Target Multi-Sensor Tracking Problems: a General Approach", in Proc. 1983 American Control Conf. San Francisco, CA, June 1983.
[28] S. Mori, C. Y. Chong, E. Tse and R. P. Wishner, "Tracking and Classifying Targets without A Priori Identification", IEEE Trans. On AC., Vol.31, No.5, 1986, pp.401-409.
[29] M.C. Smith and E.M. Winter "Feature space transform for multitarget detection" in Proc. IEEE Conf. On Decision and Control, Albuquerque, NM (1980), pp.835-836
[30] B.D. Carlson, E.D. Evans and S.L. Wilson, "Search radar detection and track with the Hough transform, Part Ⅰ: System Concept" IEEE Trans. on AES Vol.30, No.1, Jan. 1994, pp.102-108
[31] S.L. Wilson, B.D. Carlson and E.D. Evans, "Search radar detection and track with the Hough transform, Part Ⅱ: Detection statistics" IEEE Trans. on AES Vol.30, No.1, Jan. 1994, pp.109-115
[32] E.D. Evans, S.L. Wilson and B.D. Carlson "Search radar detection and track with the Hough transform, Part Ⅲ: Detection performance with binary integration" IEEE Trans. on AES Vol.30, No. 1, Jan. 1994, pp.116-125
[33] Kiril M. Alexiev, Ljudmil V. Bojilov, "A Hough Transform Track Initiation Algorithm for Multiple Passive Sensors", Proc. of the International Conf. On Multisource - Multisensor Information Fusion, FUSION'2000, pp. TuB2-11-TUB2-16.
[34] Kiril M. Alexiev, "Implementation of Hough Transform as Track Detector" Proc. of the International Conf. On Multisource - Multisensor Information Fusion, FUSION'2000, pp.ThC4-11-THC4-16.
[35] S.W. Yankowich and M.Farooq "Hough transform based multisensor,
multitarget, track initiation technique", Opt. Eng Vol.37, No.7, July 1998,pp.2064-2077.
[36] 程洪玮、孙仲康 “利用Hough变换实现目标检测与航迹启动” 国防科技大学学报 Vol．20，No．4，1998，pp．53-58．
[37] 王怀理、王德生、田立生 “Hough变换在航迹起始中的应用” 清华大学学报(自然科学版)Vol．42，No．7，2002，pp．909-912．
[38] 王国宏、苏峰、毛士艺、何友 “杂波环境下基于Hough变换和逻辑的快速航迹起始算法” 系统仿真学报Vol．14，No．7，2002，pp．873-875
[39] P.V.C. Hough, "A method and means for recognizing complex patterns", U.S. Patent 3,069,654.
[40] D.H. Ballard "Generalizing the Hough transform to detect arbitrary shape", Pattern Recognition, Vol. 13, No.2, 1981,pp. 111-122.
[41] R.O. Duda and P.E. Hart "Use of the Hough transformation to detect lines and curves in pictures" Communication of the ACM, Vol. 15, No.1, Jan. 1972,pp.11-15.
[42] John Immerker "Some remarks on the straight line Hough transform", Pattern Recognition Letters, 19,1998, pp. 1133-1135.
[43] H. Li, M.A. Lavin and R.J. LeMaster, "Fast Hough transform", Computer Vision, Graphics and Image Processing, 36, 1986, pp. 139-161.
[44] J. Illingworth and J. Kittler, "Adaptive Hough transform", IEEE Trans. On Pattern Analysis and Machine Intelligence, PAMI-9(5), Sept. 1987, pp. 690-698.
[45] J. Princen, J. Illingworth and J. Kittler, "A hierarchical approach to line extraction", Proc. IEEE Computer Vision and Patt. Recogn. Conf., San Diego, CA(June 1989), pp.92-97.
[46] D. Ben-Tzvi, and M.B. Sandier, "A combinatoryial Hough transform", Patt. Recogn. Lett., Vol,11,No.3, 1990, pp.167-174.
[47] Xu, L, Oja, E and Kultanen, P "A new curve detection method: Randomized Hough Transform(RHT)", Patt. Recogn. Lett., Vol. 11, No.5, 1990, pp. 331-338.
[48] Xu, L and Oja, E "Randomized Hough Transform(RHT): Basic mechanisms, algorithms, and computational complexities", GVGIP: Image Understanding,
Vol.57, No.2, 1993, pp.131-316.
[49] H. K(?)lvi(?)inen, L. Xu and E. Oja, "Recent versions of the Hough transform and the randomized Hough transform: Overview and comparisons", Reserach Report No 37, Department of the information Technology, Lappeenranta University of Technology, Finland(1993).
[50] R.S. Stephens, "Probabilistic approach to the Hough transform", Image & Vision Comput., Vol.9, No, 1, Feb. 1991, pp.66-71.
[51] W. Niblack and D. Petkovic, "On Improving the Accuracy of the Hough Transform: Theory, Simulations, and Experiments" on Proceedings of CVPR.'88, June 1988, pp.574-579.
[52] Ming Zhang, "On the Discretization of Parameter Domain in Hough Transformation", Proceedings of the 13th International Conference on Pattern Recognition, Vol.2, Aug. 1996, pp.527-531.
[53] H.Goto and H.Aso, "The Efficient Sampling Interval of the Scanning Parameter in the Hough Transform", Systems and Computers in Japan, Vol. 29, No. 11, pp. 9-19, 1998, Scripta Technica, Inc. (in English)
[54] H.Goto and H.Aso, "Designing Efficient Hough Transform by Noise-Level Shaping", IEICE Trans. INF.& SYST., Vol. E83-D, No. 2, Feb. 2000, pp.242-250.
[55] Y. Furukawa and Y. Shinagawa, "Accurate and robust line segment extraction by analyzing distribution around peaks in Hough space", Computer Vision and Image Understanding, Vol. 92, Issue. 1, October 2003, pp. 1-25.
[56] Q. Ji and R.M. Haralick, "Error propagation for the Hough transform", Pattern Recognition Letters, Vol. 22, Issues 6-7, May 2001, pp.813-823.
[57] M.K. Ibrahim, E.C.L.Ngau, and M.F.Daemi "Weigthed Hough Transform", SPIE Vol.1607 Intelligent Robots and Computer Vision X:Algorithms and Techniques (1991), pp.239-241.