Fractal analysis of anatomical structures linear contours: modified Caliper method vs Box counting method

Keywords: fractal analysis, morphometry, linear contour, Caliper, Box counting.


Fractal analysis estimates the metric dimension and complexity of the spatial configuration of different anatomical structures. This allows the use of this mathematical method for morphometry in morphology and clinical medicine. Two methods of fractal analysis are most often used for fractal analysis of linear fractal objects: the Box counting method (Grid method) and the Caliper method (Richardson’s method, Perimeter stepping method, Ruler method, Divider dimension, Compass dimension, Yard stick method). The aim of the research is a comparative analysis of two methods of fractal analysis – Box counting method and author's modification of Caliper method for fractal analysis of linear contours of anatomical structures. A fractal analysis of three linear fractals was performed: an artificial fractal – a Koch snowflake and two natural fractals – the outer contours of the pial surface of the human cerebellar vermis cortex and the cortex of the cerebral hemispheres. Fractal analysis was performed using the Box counting method and the author's modification of the Caliper method. The values of the fractal dimension of the artificial linear fractal (Koch snowflakes) obtained by the Caliper method coincide with the true value of the fractal dimension of this fractal, but the values of the fractal dimension obtained by the Box counting method do not match the true value of the fractal dimension. Therefore, fractal analysis of linear fractals using the Caliper method allows you to get more accurate results than the Box counting method. The values of the fractal dimension of artificial and natural fractals, calculated using the Box counting method, decrease with increasing image size and resolution; when using the Caliper method, fractal dimension values do not depend on these image parameters. The values of the fractal dimension of linear fractals, calculated using the Box counting method, increase with increasing width of the linear contour; the values calculated using the Caliper method do not depend on the contour line width. Thus, for the fractal analysis of linear fractals, preference should be given to the Caliper method and its modifications.


Addison, P. S. (1997). Fractals and chaos: an illustrated course. CRC Press.

Chikui, T., Tokumori, K., Yoshiura, K., Oobu, K., Nakamura, S., & Nakamura, K. (2005). Sonographic texture characterization of salivary gland tumors by fractal analyses. Ultrasound in Medicine & Biology, 31(10), 1297-1304. doi: 10.1016/j.ultrasmedbio.2005.05.012

Di Ieva, A., Esteban, F. J., Grizzi, F., Klonowski, W., & Martín-Landrove, M. (2015). Fractals in the neurosciences, part II: clinical applications and future perspectives. The Neuroscientist, 21(1), 30-43. doi: 10.1177/1073858413513928

Di Ieva, A., Grizzi, F., Jelinek, H., Pellionisz, A. J., & Losa, G. A. (2014). Fractals in the neurosciences, part I: general principles and basic neurosciences. The Neuroscientist, 20(4), 403-417. doi: 10.1177/1073858413513927

Fernández, E., & Jelinek, H. F. (2001) Use of fractal theory in neuroscience: methods, advantages, and potential problems. Methods, 24(4), 309-321. doi: 10.1006/meth.2001.1201

Jelinek, H.F., & Fernandez, E. (1998). Neurons and fractals: how reliable and useful are calculations of fractal dimensions?. Journal of Neuroscience Methods, 81(1-2), 9-18. doi: 10.1016/s0165-0270(98)00021-1

King, R.D., Brown, B., Hwang, M., Jeon, T., George, A. T., & Alzheimer’s Disease Neuroimaging Initiative. (2010). Fractal dimension analysis of the cortical ribbon in mild Alzheimer's disease. Neuroimage, 53(2), 471-479. doi: 10.1016/j.neuroimage.2010.06.050

King, R. D., George, A. T., Jeon, T., Hynan, L. S., Youn, T. S., Kennedy, D. N., Dickerson, B. & Alzheimer’s Disease Neuroimaging Initiative. (2009). Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis. Brain Imaging and Behavior, 3(2), 154-166. doi: 10.1007/s11682-008-9057-9

Kiselev, V. G., Hahn, K. R., & Auer, D. P. (2003). Is the brain cortex a fractal? Neuroimage, 20(3), 1765-1774. doi: 10.1016/s1053-8119(03)00380-x

Lauwerier, H. (1991). Fractals: endlessly repeated geometrical figures. ICON Group International.

Lee, K. I., Choi, S. C., Park, T. W., & You, D. S. (1999). Fractal dimension calculated from two types of region of interest. Dento Maxillo Facial Radiology, 28(5), 284-289. doi: 10.1038/sj/dmfr/4600458

Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. N.Y.: W.H.Freeman & Co.

Mandelbrot, B. B. (1977). Fractals – Form, Chance and Dimension. San Francisco: W.H. Freeman & Co.

Mandelbrot, B. B. (1967). How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, New Series, 3775(156), 636-638.

Maryenko, N. I., & Stepanenko, O. Yu. (2021). Fractal analysis as a method of morphometric study of linear anatomical objects: modified Caliper method. Reports of Morphology, 27(4), 28-34. doi: 10.31393/morphology-journal-2021-27(4)-04

Maryenko, N. I., & Stepanenko, O. Yu. (2021). Fractal dimension of external linear contour of human cerebellum (magnetic resonance imaging study). Reports of Morphology, 27(2), 16-22. doi: 10.31393/morphology-journal-2021-27(2)-03

Raguso, G., Ancona, A., Chieppa, L., L'abbate, S., Pepe, M. L., Mangieri, F. ... Rangayyan, R. M. (2010, January). Application of fractal analysis to mammography. In 2010 Annual International Conference of the IEEE Engineering in Medicine and Biology (pp. 3182-3185). IEEE. doi: 10.1109/IEMBS.2010.5627180

Rajković, N., Krstonošić, B., & Milošević, N. (2017). Box-counting method of 2D neuronal image: method modification and quantitative analysis demonstrated on images from the monkey and human brain. Computational and mathematical methods in medicine, 2017, 8967902. doi: 10.1155/2017/8967902

Rangayyan, R. M., & Nguyen, T. M. (2007). Fractal analysis of contours of breast masses in mammograms. Journal of digital imaging, 20(3), 223-237. doi: 10.1007/s10278-006-0860-9

Ristanović, D., Stefanović, B.D., & Puskas, N. (2013). Fractal analysis of dendrites morphology using modified Richardson's and Box counting method. Theoretical biology forum, 106(1-2), 157-168.

Ristanović, D., Stefanović, B. D., & Puškaš, N. (2014). Fractal analysis of dendrite morphology of rotated neuronal pictures: the modified Box counting method. Theoretical biology forum, 107(1-2), 109-121.

Schneider, C. A., Rasband, W. S., & Eliceiri, K. W. (2012). NIH Image to ImageJ: 25 years of image analysis. Nature methods, 9(7), 671-675. doi: 10.1038/nmeth.2089

Shrout, M. K., Potter, B. J., & Hildebolt, C. F. (1997). The effect of image variations on fractal dimension calculations. Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontics, 84(1), 96-100. doi: 10.1016/s1079-2104(97)90303-6

Zaletel, I., Ristanović, D., Stefanović, B. D., & Puškaš, N. (2015). Modified Richardson’s method versus the box-counting method in neuroscience. Journal of Neuroscience Methods, (242), 93-96. doi: 10.1016/j.jneumeth.2015.01.013

How to Cite
MaryenkoN. І., & Stepanenko, O. Y. (2022). Fractal analysis of anatomical structures linear contours: modified Caliper method vs Box counting method. Reports of Morphology, 28(1), 17-26.