ISSN 0253-2778

CN 34-1054/N

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An improved box-counting method for calculating image fractal dimension

  • Department of Fuel, Army Logistics University, Chongqing 401311, China

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https://doi.org/10.3969/j.issn.0253-2778.2018.06.009
  • Received Date: 01 November 2017
  • Accepted Date: 10 April 2018
  • Rev Recd Date: 10 April 2018
  • Publish Date: 30 June 2018
    Abstract

  • Abstract

    A fractal dimension is a useful feature parameter for texture analysis, segmentation and classification in many fields. The differential box-counting method is frequently used to estimate image fractal dimension because of its simplicity. However this method is flawed with lack of accuracy and stability. A new box-counting method is presented. First, more nodes are into the discrete intensity surface of a digital image to make it relatively more approximate to a continuous surface. This step makes it possible to distinguish different images at the smallest scale. Then, the fractal dimension of the digital image is estimated directly according to the box number at the smallest scale without the fitting step. Experimental results show that this method is more accurate and stable compared with some typical methods. For some special test images, such as pulse images, the proposed method outperormed unreasonable estimates. In addition, because there is no need to calculate the box numbers at other scales, the computational complexity of our method is lower.

    Abstract

    A fractal dimension is a useful feature parameter for texture analysis, segmentation and classification in many fields. The differential box-counting method is frequently used to estimate image fractal dimension because of its simplicity. However this method is flawed with lack of accuracy and stability. A new box-counting method is presented. First, more nodes are into the discrete intensity surface of a digital image to make it relatively more approximate to a continuous surface. This step makes it possible to distinguish different images at the smallest scale. Then, the fractal dimension of the digital image is estimated directly according to the box number at the smallest scale without the fitting step. Experimental results show that this method is more accurate and stable compared with some typical methods. For some special test images, such as pulse images, the proposed method outperormed unreasonable estimates. In addition, because there is no need to calculate the box numbers at other scales, the computational complexity of our method is lower.
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    • References(20)
  • [1]
    MANDELBROT B B. The Fractal Geometry of Nature [M]. San Francisco: Freeman, 1982.
    [2]
    PENTLAN A. Fractal-based description of nature scenes [J]. IEEE Transactions on Pattern Analysis & Machine Intelligence， 1984, 6(6): 661-647.
    [3]
    VOSS R. Random Fractals: Characterization and Measurement [M]. New York: Plenum, 1986.
    [4]
    AI T, ZHANG R, ZHOU H, et al. Box-counting methods to directly estimate the fractal dimension fo a rock surface [J]. Applied Surface Science, 2014, 314: 610-621.
    [5]
    XU J, JIAN Z, LIAN X. An application of box counting method for measuring phase fraction [J]. Measurement, 2017, 100: 297-300.
    [6]
    KARPERIEN A L, JELINEK H F. Box-counting fractal analysis: A primer for the clinician[M]// The Fractal Geometry of the Brain. New York: Springer, 2016:13-43.
    [7]
    SAMAJDAR T, PATTNAIK P K. Experimental study of multi-fractal geometry on electronic medical images using differential box counting[M]//Computational Intelligence in Data Mining. Singapore: Springer, 2017: 363-370.
    [8]
    RISTANOVIC D, STEFANOVIC B, PUSKAS N. Fractal analysis of dendrite morphology using modified box-counting method [J]. Neuroscience Research, 2014, 84: 64-67.
    [9]
    慕永云, 王荣本, 赵一兵, 等. 基于多特征融合的前方车辆检测方法研究[J]. 计算机应用研究, 2011, 28(9): 3572-3575.
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    MU Yongyun, WANG Rongben, ZHAO Yibing, et al. Study on detecting method of preceding vehicle based on multi-feature fusion [J]. Application Research of Computers, 2011, 28(9): 3572-3575.
    [11]
    张建明, 张玲增, 刘志强. 一种结合多特征的前方车辆检测与跟踪方法 [J]. 计算机工程与应用, 2011, 47(5): 220-223.
    [12]
    ZHANG Jianming, ZHANG Lingzeng, LIU Zhiqiang. Approach to front vehicle detection and tracking based on multiple feature [J]. Computer Engineering and Applications, 2011, 47(5): 220-223.
    [13]
    LI J, DU Q, SUN C. An improved box-counting method for image fractal dimension estimation [J]. Pattern Recogn, 2009, 42(11): 2460-2469.
    [14]
    GE M, LIN Q. Realizing the box-counting method for calculating fractal dimension of urban form based on remote sensing image [J]. Geo-Spatial Imformation Science, 2009, 12(4): 265-270.
    [15]
    BARNSLEY M, DEVANEY R, MANDELBROT B B. The Science of Fractal Image [M]. Berlin: Springer-Verlag, 1988.
    [16]
    LONG M, PENG F. A box-counting method with adaptable box height for measuring the fractal feature of images [J]. Radio Engineering, 2013, 22(1): 208-213.
    [17]
    CHEN W, YUAN S, HSIEH C. Two algorithms to estimate fractal dimension of gray-level images [J]. Optical Engineering, 2003, 42(8): 2452-2464.
    [18]
    KAEWARAMSRI Y, WORARATPANYA K. Improved triangle box-counting method for fractal dimension estimation[M]// Recent Advance in Information and Communication Technology 2015. Cham: Springer, 2015:53-61.
    [19]
    LIU Y, CHEN L, JIANG L. An improved differential box-counting method to estimate fractal dimensions of gray- level images [J]. Journal of Visual Communication and Image Representation, 2014, 25(5): 1102-1111.
    [20]
    SARKAR N, CHAUDHURI B B. An efficient differential box-counting approach to computed fractal dimension of image [J]. IEEE Transactions on systems, man, and cybernetics, 1994, 24(1): 115-120.)
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    [1]
    MANDELBROT B B. The Fractal Geometry of Nature [M]. San Francisco: Freeman, 1982.
    [2]
    PENTLAN A. Fractal-based description of nature scenes [J]. IEEE Transactions on Pattern Analysis & Machine Intelligence， 1984, 6(6): 661-647.
    [3]
    VOSS R. Random Fractals: Characterization and Measurement [M]. New York: Plenum, 1986.
    [4]
    AI T, ZHANG R, ZHOU H, et al. Box-counting methods to directly estimate the fractal dimension fo a rock surface [J]. Applied Surface Science, 2014, 314: 610-621.
    [5]
    XU J, JIAN Z, LIAN X. An application of box counting method for measuring phase fraction [J]. Measurement, 2017, 100: 297-300.
    [6]
    KARPERIEN A L, JELINEK H F. Box-counting fractal analysis: A primer for the clinician[M]// The Fractal Geometry of the Brain. New York: Springer, 2016:13-43.
    [7]
    SAMAJDAR T, PATTNAIK P K. Experimental study of multi-fractal geometry on electronic medical images using differential box counting[M]//Computational Intelligence in Data Mining. Singapore: Springer, 2017: 363-370.
    [8]
    RISTANOVIC D, STEFANOVIC B, PUSKAS N. Fractal analysis of dendrite morphology using modified box-counting method [J]. Neuroscience Research, 2014, 84: 64-67.
    [9]
    慕永云, 王荣本, 赵一兵, 等. 基于多特征融合的前方车辆检测方法研究[J]. 计算机应用研究, 2011, 28(9): 3572-3575.
    [10]
    MU Yongyun, WANG Rongben, ZHAO Yibing, et al. Study on detecting method of preceding vehicle based on multi-feature fusion [J]. Application Research of Computers, 2011, 28(9): 3572-3575.
    [11]
    张建明, 张玲增, 刘志强. 一种结合多特征的前方车辆检测与跟踪方法 [J]. 计算机工程与应用, 2011, 47(5): 220-223.
    [12]
    ZHANG Jianming, ZHANG Lingzeng, LIU Zhiqiang. Approach to front vehicle detection and tracking based on multiple feature [J]. Computer Engineering and Applications, 2011, 47(5): 220-223.
    [13]
    LI J, DU Q, SUN C. An improved box-counting method for image fractal dimension estimation [J]. Pattern Recogn, 2009, 42(11): 2460-2469.
    [14]
    GE M, LIN Q. Realizing the box-counting method for calculating fractal dimension of urban form based on remote sensing image [J]. Geo-Spatial Imformation Science, 2009, 12(4): 265-270.
    [15]
    BARNSLEY M, DEVANEY R, MANDELBROT B B. The Science of Fractal Image [M]. Berlin: Springer-Verlag, 1988.
    [16]
    LONG M, PENG F. A box-counting method with adaptable box height for measuring the fractal feature of images [J]. Radio Engineering, 2013, 22(1): 208-213.
    [17]
    CHEN W, YUAN S, HSIEH C. Two algorithms to estimate fractal dimension of gray-level images [J]. Optical Engineering, 2003, 42(8): 2452-2464.
    [18]
    KAEWARAMSRI Y, WORARATPANYA K. Improved triangle box-counting method for fractal dimension estimation[M]// Recent Advance in Information and Communication Technology 2015. Cham: Springer, 2015:53-61.
    [19]
    LIU Y, CHEN L, JIANG L. An improved differential box-counting method to estimate fractal dimensions of gray- level images [J]. Journal of Visual Communication and Image Representation, 2014, 25(5): 1102-1111.
    [20]
    SARKAR N, CHAUDHURI B B. An efficient differential box-counting approach to computed fractal dimension of image [J]. IEEE Transactions on systems, man, and cybernetics, 1994, 24(1): 115-120.)
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