Acosta et al 2022a - Revision history

Rimni: /* 6. Discussion about rod counting method */

2022-07-01T11:18:23Z

‎6. Discussion about rod counting method

Rimni: /* 7. Conclusions */

2022-07-01T11:09:42Z

‎7. Conclusions

Rimni at 11:06, 1 July 2022

2022-07-01T11:06:41Z

Rimni: /* References */

2022-06-30T14:02:45Z

‎References

Rimni: /* References */

2022-06-29T14:38:47Z

‎References

Rimni at 14:15, 29 June 2022

2022-06-29T14:15:35Z

Rimni at 14:10, 29 June 2022

2022-06-29T14:10:59Z

Rimni at 14:10, 29 June 2022

2022-06-29T14:10:29Z

Rimni at 14:08, 29 June 2022

2022-06-29T14:08:59Z

Rimni at 14:08, 29 June 2022

2022-06-29T14:08:39Z

@@ Line 411: / Line 411: @@
 ==6. Discussion about rod counting method==
-In  [[#table-5|Table 5]] results of calculation by rod counting and box counting are shown and compared with the the theoretical calculations reported <span id='citeF-10'></span>[[#cite-10|[10]]]-<span id='citeF-12'></span>[[#cite-12|[12]]], also results obtained by <span id='citeF-9'></span>[[#cite-9|[9]]] are shown in the last column of  [[#table-5|Table 5]] for comparison. The penultimate row of the table [[#table-5|5]] refers to the calculated Koch's snowflake fractal with non-regular sizes.
+In  [[#table-5|Table 5]] results of calculation by rod counting and box counting are shown and compared with the the theoretical calculations reported <span id='citeF-10'></span>[[#cite-10|[10]]]-<span id='citeF-12'></span>[[#cite-12|[12]]], also results obtained by <span id='citeF-9'></span>[[#cite-9|[9]]] are shown in the last column of  [[#table-5|Table 5]] for comparison. The penultimate row of the [[#table-5|Table 5]] refers to the calculated Koch's snowflake fractal with non-regular sizes.
 <div class="center" style="font-size: 75%;">'''Table 5'''. Comparison of measurements made on different fractal figures using different procedures. The last column takes into a count results given by <span id='citeF-9'></span>[[#cite-9|[9]]]</div>

← Older revision		Revision as of 11:09, 1 July 2022
Line 473:		Line 473:

	* Both measurement methods are applied to the Mandelbrot fractal, in a perimeter measurement, and it is established that <math display="inline">D_{Bb} = 1.1965</math> (box count), and that <math display="inline">D_{Br} = 1.15397</math> (rod count) which is considered as the correct one, because it is a linear measurement.		* Both measurement methods are applied to the Mandelbrot fractal, in a perimeter measurement, and it is established that <math display="inline">D_{Bb} = 1.1965</math> (box count), and that <math display="inline">D_{Br} = 1.15397</math> (rod count) which is considered as the correct one, because it is a linear measurement.
		+

	Therefore, it is established that in the linear measurements made on fractals, the counting of rods provides a better approximation to the result of applying the Hausdorff-Besicovitch definition than those obtained with the counting of boxes, even with “non-standardized” rod sizes.		Therefore, it is established that in the linear measurements made on fractals, the counting of rods provides a better approximation to the result of applying the Hausdorff-Besicovitch definition than those obtained with the counting of boxes, even with “non-standardized” rod sizes.

@@ Line 519: / Line 519: @@
 <div id="cite-13"></div>
-[13]  Husain A., J. Reddy, D. Bisht and M. Sajid Fractal dimension of coastline of Australia, Nature Scientific Report,  2021,  doi.org/10.1038/s41598-021-85405-0
+[13]  Husain A., Reddy J., Bisht D.,  Sajid M. Fractal dimension of coastline of Australia. Nature Scientific Report,  11:6304 2021.  doi.org/10.1038/s41598-021-85405-0
 <div id="cite-14"></div>
-[14]  C. Schneider,  W.  Rasband and K. Eliceiri NIH Image to ImageJ: 25 years of image analysis,  Nature Methods 9, 2012, pp.  671&#8211;675.
+[14]  Schneider C.,  Rasband W., Eliceiri K. NIH image to ImageJ: 25 years of image analysis.  Nature Methods, 9:671-675, 2012.
 <div id="cite-15"></div>
-[15]  P. Shanmugavadivu and V. Sivakumar, Fractal dimension-bound spatiotemporal analysis of digital mammograms,  The European Physics Journal Special Topics, vol. 225, pp. 137-146.
+[15]  Shanmugavadivu P., Sivakumar V.  Fractal dimension-bound spatiotemporal analysis of digital mammograms.  The European Physics Journal Special Topics, 225:137-146, 2016.
 <div id="cite-16"></div>
-[16]  B. Bárány, M. Hochman and A. Rapaport, Hausdorff dimension of planar self-affine measures, Inventiones Mathematicae. vol. 16, 2019, pp. 601&#8211;659.
+[16]  Bárány B.,  Hochman M., Rapaport A. Hausdorff dimension of planar self-affine measures. Inventiones Mathematicae, 16:601-659, 2019.
 <div id="cite-17"></div>
-[17]  H. Christensen, and O. Driver, The fractal nature of clouds in global storm-resolving models,  ArXiv:2108.08565v1,  19 Aug 2021.
+[17]   Christensen H.,  Driver O. The fractal nature of clouds in global storm-resolving models.  ArXiv:2108.08565v1,  Aug 2021.
 <div id="cite-18"></div>
-[[#citeF-18|[18]]]  C. Acosta, O. Carvente and F. Peñuñuri, Fractal dimension measured over areas and perimeters using Box Counting technique applied over a Mandelbrot figures, RIMNI,  Vol. 38(1), 2022, doi.org/10.23967/j.rimni.2022.03.007
+[[#citeF-18|[18]]]  Acosta C., Carvente  O.,   Peñuñuri F. Fractal dimension measured over areas and perimeters using Box Counting technique applied over a Mandelbrot figures. Rev. int. métodos numér. cálc. diseño ing.,  38(1), 11, 2022. doi.org/10.23967/j.rimni.2022.03.007

@@ Line 492: / Line 492: @@
 <div id="cite-4"></div>
-[[#citeF-4|[4]]]  Santos E., Souza F. and Ambrosío P. Fractal dimension for characterization of focal breast lesion, Conference WCCM XI, July 20 - 25, 2014, Barcelona, Spain. DOI: 10.13140/2.1.4186.9441
+[[#citeF-4|[4]]]  Santos E., Souza F., Ambrosío P. Fractal dimension for characterization of focal breast lesion. Conference WCCM XI, July 20-25, Barcelona, Spain, 2014. DOI: 10.13140/2.1.4186.9441
 <div id="cite-5"></div>
-[[#citeF-5|[5]]]  Maipas S, Nonni A, Politi E, et al. (November 24, 2018) The Goodness-of-fit of the Fractal Dimension as a Diagnostic Factor in Breast Cancer. Cureus 10(11): e3630. doi:10.7759/cureus.3630
+[[#citeF-5|[5]]]  Maipas S., Nonni A., Politi E., Sarlanis H., Kavantzas N.G. The goodness-of-fit of the fractal dimension as a diagnostic factor in breast cancer. Cureus, 10(11):e3630, 2018. doi:10.7759/cureus.3630
 <div id="cite-6"></div>
-[[#citeF-6|[6]]]  Martinez B. de Oliveira Plotze R. Falvo M. and de Castro M. Fractal dimension applied to plant identification, Information Sciences, Elsevier, vol 178(12), 2008, pp 2722-2733.
+[[#citeF-6|[6]]]  Martinez B., de Oliveira Plotze R., Falvo M., de Castro M. Fractal dimension applied to plant identification. Information Sciences, 178(12):2722-2733, 2008.
 <div id="cite-7"></div>
@@ Line 504: / Line 504: @@
 <div id="cite-8"></div>
-[[#citeF-8|[8]]]  Geogebra, GeoGebra to teach and learn Mathematics www.geogebra.org
+[[#citeF-8|[8]]]  Geogebra, GeoGebra to teach and learn Mathematics https://www.geogebra.org
 <div id="cite-9"></div>
-[[#citeF-9|[9]]]  Gun-Baek So, Hye-Rim So, Gang-Gyoo Jin, Enhancement of the Box-Counting Algorithm for fractal dimension estimation, Pattern Recognition Letters, Volume 98, 2017, Pages 53-58, http://dx.doi.org/10.1016/j.patrec.2017.08.022
+[[#citeF-9|[9]]]  Gun-Baek So, Hye-Rim So, Gang-Gyoo Jin. Enhancement of the box-counting algorithm for fractal dimension estimation. Pattern Recognition Letters, 98:53-58, 2017. http://dx.doi.org/10.1016/j.patrec.2017.08.022
 <div id="cite-10"></div>
-[[#citeF-10|[10]]]  H. Strogatz, ``Nonlinear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering, Westview Press, Boulder CO. USA, 2nd edition (2015).
+[[#citeF-10|[10]]]  Strogatz H. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry and engineering. Westview Press, Boulder CO. USA, 2nd edition, 2015.
 <div id="cite-11"></div>
-[[#citeF-11|[11]]]  R. Devaney A first course in chaotic dynamical systems theory and experiment CRC Press, Florida USA, 2nd edition (2020).
+[[#citeF-11|[11]]]  Devaney R. A first course in chaotic dynamical systems theory and experiment. CRC Press, Florida USA, 2nd edition, 2020.
 <div id="cite-12"></div>
-[[#citeF-12|[12]]]  D. Gatzouras and S. Lalley, Hausdorff and box dimensions of certain self-affine fractals,  Indiana University Mathematics Journal, vol. 41(2), 1992, pp. 533-568.
+[[#citeF-12|[12]]]  Gatzouras D.,  Lalley  S. Hausdorff and box dimensions of certain self-affine fractals.  Indiana University Mathematics Journal, 41(2):533-568, 1992.
 <div id="cite-13"></div>
-[13]  A. Husain, J. Reddy, D. Bisht and M. Sajid Fractal dimension of coastline of Australia, Nature Scientific Report,  2021,  doi.org/10.1038/s41598-021-85405-0
+[13]  Husain A., J. Reddy, D. Bisht and M. Sajid Fractal dimension of coastline of Australia, Nature Scientific Report,  2021,  doi.org/10.1038/s41598-021-85405-0
 <div id="cite-14"></div>

@@ Line 479: / Line 479: @@
 ==References==
 <div id="cite-1"></div>
-'''[[#citeF-1|[1]]]'''  B. Mandelbrot, “The fractal geometry of nature”, W. H. Freeman 1982. <div id="cite-2"></div>
+[[#citeF-1|[1]]]  Mandelbrot B. The fractal geometry of nature. W.H. Freeman and Company, pp. 460,  1982.
-'''[[#citeF-2|[2]]]'''  B. Mandelbrot, “Self-affine and Fractal dimension”, Physica Scripta, vol.  32(4),  1985, pp. 257-260. <div id="cite-3"></div>
-'''[[#citeF-3|[3]]]'''  B. Mandelbrot,  “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension”, Science, 156(3775), pp. 636-638, DOI: 10.1126/science.156.3775.636 <div id="cite-4"></div>
+<div id="cite-2"></div>
-'''[[#citeF-4|[4]]]'''  Santos E., Souza F. and Ambrosío P. “Fractal dimension for characterization of focal breast lesion”, Conference WCCM XI, July 20 - 25, 2014, Barcelona, Spain. DOI: 10.13140/2.1.4186.9441 <div id="cite-5"></div>
+[[#citeF-2|[2]]]  Mandelbrot B. Self-affine and Fractal dimension. Physica Scripta, 32(4):257-260, 1985.
-'''[[#citeF-5|[5]]]'''  Maipas S, Nonni A, Politi E, et al. (November 24, 2018) The Goodness-of-fit of the Fractal Dimension as a Diagnostic Factor in Breast Cancer. Cureus 10(11): e3630. doi:10.7759/cureus.3630 <div id="cite-6"></div>
-'''[[#citeF-6|[6]]]'''  Martinez B. de Oliveira Plotze R. Falvo M. and de Castro M. “Fractal dimension applied to plant identification”, Information Sciences, Elsevier, vol 178(12), 2008, pp 2722-2733. <div id="cite-7"></div>
+<div id="cite-3"></div>
-'''[[#citeF-7|[7]]]'''  ImageJ “Processing and Analysis in Java” https://imagej.nih.gov <div id="cite-8"></div>
+[[#citeF-3|[3]]]  Mandelbrot B.  How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156(3775):636-638, 1967. DOI: 10.1126/science.156.3775.636
-'''[[#citeF-8|[8]]]'''  Geogebra, “GeoGebra to teach and learn Mathematics” www.geogebra.org <div id="cite-9"></div>
-'''[[#citeF-9|[9]]]'''  Gun-Baek So, Hye-Rim So, Gang-Gyoo Jin, “Enhancement of the Box-Counting Algorithm for fractal dimension estimation”, Pattern Recognition Letters, Volume 98, 2017, Pages 53-58, http://dx.doi.org/10.1016/j.patrec.2017.08.022 <div id="cite-10"></div>
+<div id="cite-4"></div>
-'''[[#citeF-10|[10]]]'''  H. Strogatz, ``Nonlinear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering, Westview Press, Boulder CO. USA, 2nd edition (2015). <div id="cite-11"></div>
+[[#citeF-4|[4]]]  Santos E., Souza F. and Ambrosío P. Fractal dimension for characterization of focal breast lesion, Conference WCCM XI, July 20 - 25, 2014, Barcelona, Spain. DOI: 10.13140/2.1.4186.9441
-'''[[#citeF-11|[11]]]'''  R. Devaney “A first course in chaotic dynamical systems theory and experiment” CRC Press, Florida USA, 2nd edition (2020). <div id="cite-12"></div>
-'''[[#citeF-12|[12]]]'''  D. Gatzouras and S. Lalley, “Hausdorff and box dimensions of certain self-affine fractals”,  Indiana University Mathematics Journal, vol. 41(2), 1992, pp. 533-568. <div id="cite-13"></div>
+<div id="cite-5"></div>
-'''[13]'''  A. Husain, J. Reddy, D. Bisht and M. Sajid “Fractal dimension of coastline of Australia”, Nature Scientific Report,  2021,  doi.org/10.1038/s41598-021-85405-0 <div id="cite-14"></div>
+[[#citeF-5|[5]]]  Maipas S, Nonni A, Politi E, et al. (November 24, 2018) The Goodness-of-fit of the Fractal Dimension as a Diagnostic Factor in Breast Cancer. Cureus 10(11): e3630. doi:10.7759/cureus.3630
-'''[14]'''  C. Schneider,  W.  Rasband and K. Eliceiri “NIH Image to ImageJ: 25 years of image analysis”,  Nature Methods 9, 2012, pp.  671&#8211;675. <div id="cite-15"></div>
-'''[15]'''  P. Shanmugavadivu and V. Sivakumar, “Fractal dimension-bound spatiotemporal analysis of digital mammograms”,  The European Physics Journal Special Topics, vol. 225, pp. 137-146. <div id="cite-16"></div>
+<div id="cite-6"></div>
-'''[16]'''  B. Bárány, M. Hochman and A. Rapaport, “Hausdorff dimension of planar self-affine measures”, Inventiones Mathematicae. vol. 16, 2019, pp. 601&#8211;659. <div id="cite-17"></div>
+[[#citeF-6|[6]]]  Martinez B. de Oliveira Plotze R. Falvo M. and de Castro M. Fractal dimension applied to plant identification, Information Sciences, Elsevier, vol 178(12), 2008, pp 2722-2733.
-'''[17]'''  H. Christensen, and O. Driver, “The fractal nature of clouds in global storm-resolving models”,  ArXiv:2108.08565v1,  19 Aug 2021. <div id="cite-18"></div>
-'''[[#citeF-18|[18]]]'''  C. Acosta, O. Carvente and F. Peñuñuri, “Fractal dimension measured over areas and perimeters using Box Counting technique applied over a Mandelbrot figures”, RIMNI,  Vol. 38(1), 2022, doi.org/10.23967/j.rimni.2022.03.007
+<div id="cite-7"></div>

@@ Line 116: / Line 116: @@
 <div class="center" style="font-size: 75%;">'''Table 2'''. Richardson's method used to calculate the statistical self-similar fractal dimension of (a) Hilbrts's curve, (b) Koch 85's curve and (c) Cantor middle-thirds set</div>
-<div id='table-1'></div>
+<div id='table-2'></div>
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
@@ Line 279: / Line 279: @@
 <div class="center" style="font-size: 75%;">'''Table 3'''. The <math>P_i</math> points are obtained with rod lengths different from those used in the construction of the Koch curve</div>
-<div id='tab-1'></div>
+<div id='table-3'></div>
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
@@ Line 349: / Line 349: @@
 <div class="center" style="font-size: 75%;">'''Table 4'''. Results of rod counting measurements on a Mandelbrot fractal</div>
-<div id='tab-1'></div>
+<div id='table-4'></div>
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
@@ Line 418: / Line 418: @@
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-| style="border-left: 2px solid;border-right: 2px solid;" |  Geometry
+!  Geometry !!  Theoretical !!  Rod C. !!  B. Count. !!  E. B. Count.
-| style="border-left: 2px solid;border-right: 2px solid;" | Theoretical
+|- style="text-align:center"
-| style="border-left: 2px solid;border-right: 2px solid;" | Rod C.
+|  Koch Snowflake
-| style="border-left: 2px solid;border-right: 2px solid;" | B. Count.
+|  1.2620
-| style="border-left: 2px solid;border-right: 2px solid;" | E. B. Count.
+|  1.26186
-|- style="border-top: 2px solid;"
+|  1.2848
-| style="border-left: 2px solid;border-right: 2px solid;" |  Koch Snowflake
+|  1.267
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.2620
+|- style="text-align:center"
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.26186
+|   Hilbert´s curve
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.2848
+|  2.0000
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.267
+|  2.04097
-|- style="border-top: 2px solid;"
+|  1.8984
-| style="border-left: 2px solid;border-right: 2px solid;" |  Hilbert´s curve
+|  1.974
-| style="border-left: 2px solid;border-right: 2px solid;" | 2.0000
+|- style="text-align:center"
-| style="border-left: 2px solid;border-right: 2px solid;" | 2.04097
+|   Koch 85' curve
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.8984
+|  1.7850
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.974
+|  1.78505
-|- style="border-top: 2px solid;"
+|  1.7032
-| style="border-left: 2px solid;border-right: 2px solid;" |  Koch 85' curve
+|  1.788
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.7850
+|- style="text-align:center"
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.78505
+|  Cantor Set
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.7032
+|  0.6310
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.788
+|  0.63093
-|- style="border-top: 2px solid;"
+|  0.8082
-| style="border-left: 2px solid;border-right: 2px solid;" |  Cantor Set
+| - -
-| style="border-left: 2px solid;border-right: 2px solid;" | 0.6310
+|- style="text-align:center"
-| style="border-left: 2px solid;border-right: 2px solid;" | 0.63093
+|   Koch NIS
-| style="border-left: 2px solid;border-right: 2px solid;" | 0.8082
+|  1.2620
-| style="border-left: 2px solid;border-right: 2px solid;" | - -
+|  1.25966
-|- style="border-top: 2px solid;"
+|  1.2848
-| style="border-left: 2px solid;border-right: 2px solid;" |  Koch NIS
+|  1.267
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.2620
+|- style="text-align:center"
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.25966
+|   Mandelbrot
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.2848
+|  - -
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.267
+|  1.15397
-|- style="border-top: 2px solid;border-bottom: 2px solid;"
+|  1.1965
-| style="border-left: 2px solid;border-right: 2px solid;" |  Mandelbrot
+| - -
-| style="border-left: 2px solid;border-right: 2px solid;" | - -
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.15397
-| style="border-left: 2px solid;border-right: 2px solid;" | 1.1965
-| style="border-left: 2px solid;border-right: 2px solid;" | - -
-−
 |}
-In column 4 of table [[#table-5|5]] an area method is used and even when this area collapse to a line, this procedure shows a numerical difference with respect to a perimeter method.
+The application of the “box counting” method implies the measurement of the area of an object, however this same procedure, without any change, is used in the measurement of lengths, but there are differences in the results obtained, as shown in  [[#table-5|Table 5]], when compared with the theoretical results.
 Based on these differences, it is proposed to use the “rod counting” for longitudinal measurements, the “box counting” method is used to measure areas, as well as for the measurement of 3D distributions, the “cube counting” is used.
@@ Line 470: / Line 466: @@
 Taking into account the existing software, the ImageJ application gives us the possibility to efficiently measure the fractal dimension of areas, but it does not have a plugin to perform linear measurements. This is why GeoGebra, even though it is not designed for that purpose, has tools that enable the application of the “rod counting” method in an orderly manner.
-==7 Conclusions.==
+==7. Conclusions==
 * It was possible to show that there is a numerically evaluable difference between the counting of boxes and the counting of rods for linear (perimeter) measurements.
@@ Line 482: / Line 478: @@
 Thus, if the objective is to have a better description of the surrounding reality, the method used to describe the fractal geometry that is generated must be in accordance with the measurement made, to measure lines, the counting of rods, to measure areas, the counting of boxes and to measure volumes, the counting of cubes.
-===BIBLIOGRAPHY===
+==References==
 <div id="cite-1"></div>

@@ Line 402: / Line 402: @@
 <div id='img-6'></div>
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
 |style="padding:10px;"| [[Image:Review_986628909362-Fig6-Chaos.png|497px|Results of processing perimeter of Mandelbrot's fractal with the box counting method, in the “FracLac” plugin of “ImageJ” app, D<sub>Bb</sub> = 1.1965]]

@@ Line 404: / Line 404: @@
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
-|style="padding:10px;"| [[Image:Review_986628909362-Fig6-Chaos.png|397px|Results of processing perimeter of Mandelbrot's fractal with the box counting method, in the “FracLac” plugin of “ImageJ” app, D<sub>Bb</sub> = 1.1965]]
+|style="padding:10px;"| [[Image:Review_986628909362-Fig6-Chaos.png|497px|Results of processing perimeter of Mandelbrot's fractal with the box counting method, in the “FracLac” plugin of “ImageJ” app, D<sub>Bb</sub> = 1.1965]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" style="padding:10px;" | '''Figure 6'''. Results of processing perimeter of Mandelbrot's fractal with the box counting method, in the “FracLac” plugin of “ImageJ” app, <math>D_{Bb} = 1.1965</math>