249:
54:, object, image, etc. into smaller and smaller pieces, typically "box"-shaped, and analyzing the pieces at each smaller scale. The essence of the process has been compared to zooming in or out using optical or computer based methods to examine how observations of detail change with scale. In box counting, however, rather than changing the magnification or resolution of a lens, the investigator changes the
24:
541:, including the minimum and maximum sizes to use and the method of incrementing between sizes. Many such details reflect practical matters such as the size of a digital image but also technical issues related to the specific analysis that will be performed on the data. Another issue that has received considerable attention is how to approximate the so-called "optimal covering" for determining
353:
343:
102:
where choosing boxes of the right relative sizes readily shows how the pattern repeats itself at smaller scales. In fractal analysis, however, the scaling factor is not always known ahead of time, so box counting algorithms attempt to find an optimized way of cutting a pattern up that will reveal the
482:
Box counting may also be used to determine local variation as opposed to global measures describing an entire pattern. Local variation can be assessed after the data have been gathered and analyzed (e.g., some software colour codes areas according to the fractal dimension for each subsample), but a
310:
Every box counting algorithm has a scanning plan that describes how the data will be gathered, in essence, how the box will be moved over the space containing the pattern. A variety of scanning strategies has been used in box counting algorithms, where a few basic approaches have been modified in
301:
is counted). For other types of analysis, the data sought may be the number of pixels that fall within the measuring box, the range or average values of colours or intensities, the spatial arrangement amongst pixels within each box, or properties such as average speed (e.g., from particle flow).
263:
The relevant features gathered during box counting depend on the subject being investigated and the type of analysis being done. Two well-studied subjects of box counting, for instance, are binary (meaning having only two colours, usually black and white) and gray-scale
1938:
580:
To address various methodological considerations, some software is written so users can specify many such details, and some includes methods such as smoothing the data after the fact to be more amenable to the type of analysis being done.
356:
Figure 4. It takes 12 green but 14 yellow boxes to completely cover the black pixels in these identical images. The difference is attributable to the position of the grid, illustrating the importance of grid placement in box
272:
from such still images in which case the raw information recorded is typically based on features of pixels such as a predetermined colour value or range of colours or intensities. When box counting is done to determine a
281:, the information recorded is usually either yes or no as to whether or not the box contained any pixels of the predetermined colour or range (i.e., the number of boxes containing relevant pixels at each
557:
One known issue in this respect is deciding what constitutes the edge of the useful information in a digital image, as the limits employed in the box counting strategy can affect the data gathered.
577:
illustrates, the overall positioning of the boxes also influences the results of a box count. One approach in this respect is to scan from multiple orientations and use averaged or optimized data.
370:
shows the typical pattern used in software that calculates box counting dimensions from patterns extracted into binary digital images of contours such as the fractal contour illustrated in
446:
illustrates the basic pattern of scanning using a sliding box. The fixed grid approach can be seen as a sliding box algorithm with the increments horizontally and vertically equal to
346:
Figure 3. Retinal vasculature revealed through box counting analysis; colour-coded local connected fractal dimension analysis done with FracLac freeware for biological image analysis.
62:). Computer based box counting algorithms have been applied to patterns in 1-, 2-, and 3-dimensional spaces. The technique is usually implemented in software for use on patterns
1065:
Plotnick, R. E.; Gardner, R. H.; Hargrove, W. W.; Prestegaard, K.; Perlmutter, M. (1996). "Lacunarity analysis: A general technique for the analysis of spatial patterns".
539:
209:
127:
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464:
420:
396:
299:
229:
187:
167:
147:
565:
The algorithm has to specify the type of increment to use between box sizes (e.g., linear vs exponential), which can have a profound effect on the results of a scan.
1016:
King, R. D.; George, A. T.; Jeon, T.; Hynan, L. S.; Youn, T. S.; Kennedy, D. N.; Dickerson, B.; the
Alzheimer’s Disease Neuroimaging Initiative (2009).
899:
Landini, G.; Murray, P. I.; Misson, G. P. (1995). "Local connected fractal dimensions and lacunarity analyses of 60 degrees fluorescein angiograms".
1956:
789:
757:
442:
Another approach that has been used is a sliding box algorithm, in which each box is slid over the image overlapping the previous placement.
378:. The strategy simulates repeatedly laying a square box as though it were part of a grid overlaid on the image, such that the box for each
252:
Figure 2. The sequence above shows basic steps in extracting a binary contour pattern from an original colour digital image of a neuron.
1402:
845:
Karperien, Audrey; Jelinek, Herbert F.; Leandro, Jorge de Jesus Gomes; Soares, JoĂŁo V. B.; Cesar Jr, Roberto M.; Luckie, Alan (2008).
1435:
600:
323:
317:
519:
The implementation of any box counting algorithm has to specify certain details such as how to determine the actual values in
1962:
331:
1818:
1775:
98:
scaling, but from a practical perspective this would require that the scaling be known ahead of time. This can be seen in
1896:
1978:
1462:
211:, a measuring element that is typically a 2-dimensional square or 3-dimensional box with side length corresponding to
483:
third approach to box counting is to move the box according to some feature related to the pixels of interest. In
2016:
1628:
969:"Signal attenuation and box-counting fractal analysis of optical coherence tomography images of arterial tissue"
1484:
1108:
Plotnick, R. E.; Gardner, R. H.; O'Neill, R. V. (1993). "Lacunarity indices as measures of landscape texture".
484:
1370:
1921:
1143:
McIntyre, N. E.; Wiens, J. A. (2000). "A novel use of the lacunarity index to discern landscape function".
1970:
1529:
1395:
810:
Li, J.; Du, Q.; Sun, C. (2009). "An improved box-counting method for image fractal dimension estimation".
542:
427:
375:
278:
366:
The traditional approach is to scan in a non-overlapping regular grid or lattice pattern. To illustrate,
1755:
1447:
546:
471:
1330:"The Use of Fractal Analysis and Photometry to Estimate the Accuracy of Bulbar Redness Grading Scales"
1916:
1911:
1701:
1633:
1236:
1074:
819:
643:
1674:
1651:
1586:
1534:
1519:
1452:
1227:
Chhabra, A.; Jensen, R. V. (1989). "Direct determination of the f( alpha ) singularity spectrum".
522:
192:
110:
1901:
1881:
1845:
1840:
1603:
1295:
1160:
1125:
949:
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374:
or the classic example of the coastline of
Britain often used to explain the method of finding a
1018:"Characterization of Atrophic Changes in the Cerebral Cortex Using Fractal Dimensional Analysis"
694:"Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals"
490:
449:
405:
381:
284:
214:
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74:
can be used to investigate some patterns physically. The technique arose out of and is used in
2011:
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is used to scan a pattern or data set (e.g., an image or object) according to a predetermined
103:
scaling factor. The fundamental method for doing this starts with a set of measuring elements—
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47:
32:
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Fernández, E.; Bolea, J. A.; Ortega, G.; Louis, E. (1999). "Are neurons multifractals?".
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Figure 2c. Boxes laid over an image concentrically focused on each pixel of interest.
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945:
632:"Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging"
611:
467:
402:). This is done until the entire area of interest has been scanned using each
79:
1086:
1760:
1706:
1618:
1469:
1355:
1291:
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1213:
1180:"Error estimation of the fractal dimension measurements of cranial sutures"
1051:
1002:
882:
673:
1094:
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717:
169:-sized boxes are applied to the pattern and counted. To do this, for each
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here for convenience, of sizes or calibres, which we will call the set of
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1346:
1329:
984:
51:
342:
1691:
1608:
1411:
1121:
863:
847:"Automated detection of proliferative retinopathy in clinical practice"
322:
316:
268:(i.e., jpegs, tiffs, etc.). Box counting is generally done on patterns
95:
28:
967:
Popescu, D. P.; Flueraru, C.; Mao, Y.; Chang, S.; Sowa, M. G. (2010).
1679:
1315:
Defining
Microglial Morphology: Form, Function, and Fractal Dimension
693:
31:
viewed through "boxes" of different sizes. The pattern illustrates
466:. Sliding box algorithms are often used for analyzing textures in
351:
341:
329:
321:
315:
247:
22:
311:
order to address issues such as sampling, analysis methods, etc.
1384:
330:
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Figure 2b. Boxes slid over an image in an overlapping pattern.
235:
to cover the relevant part of the data set, recording, i.e.,
1380:
932:(1997). "Multifractal Modeling and Lacunarity Analysis".
94:
Theoretically, the intent of box counting is to quantify
507:
is centred on each pixel of interest, as illustrated in
487:
box counting algorithms, for instance, the box for each
1328:
Schulze, M. M.; Hutchings, N.; Simpson, T. L. (2008).
525:
493:
452:
408:
384:
320:
Figure 2a. Boxes laid over an image as a fixed grid.
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78:. It also has application in related fields such as
1930:
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630:Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003).
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223:
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161:
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121:
692:Smith, T. G.; Lange, G. D.; Marks, W. B. (1996).
398:never overlaps where it has previously been (see
1334:Investigative Ophthalmology & Visual Science
901:Investigative Ophthalmology & Visual Science
1396:
8:
739:
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107:—consisting of an arbitrary number, called
58:used to inspect the object or pattern (see
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426:has been recorded. When used to find a
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63:
1957:List of fractals by Hausdorff dimension
687:
685:
683:
622:
470:analysis and have also been applied to
243:captured within the measuring element.
71:
1317:. Charles Sturt University, Australia.
805:
803:
801:
782:Fractal Geometry in Biological Systems
775:
773:
771:
769:
423:
7:
924:
922:
430:, the method is modified to find an
1178:Gorski, A. Z.; Skrzat, J. (2006).
527:
197:
115:
14:
1939:How Long Is the Coast of Britain?
1196:10.1111/j.1469-7580.2006.00529.x
478:Subsampling and local dimensions
1272:Journal of Neuroscience Methods
698:Journal of Neuroscience Methods
508:
443:
367:
1963:The Fractal Geometry of Nature
748:The Fractal Geometry of Nature
1:
1284:10.1016/s0165-0270(99)00066-7
710:10.1016/S0165-0270(96)00080-5
656:10.1016/S0006-3495(03)74817-6
601:Minkowski–Bouligand dimension
515:Methodological considerations
832:10.1016/j.patcog.2009.03.001
534:{\displaystyle \mathrm {E} }
239:, for each step in the scan
204:{\displaystyle \mathrm {E} }
122:{\displaystyle \mathrm {E} }
55:
1979:Chaos: Making a New Science
1249:10.1103/PhysRevLett.62.1327
780:Iannaccone, Khokha (1996).
2033:
1022:Brain Imaging and Behavior
784:. CRC Press. p. 143.
752:. Henry Holt and Company.
574:
431:
399:
371:
99:
59:
16:Fractal analysis technique
1034:10.1007/s11682-008-9057-9
973:Biomedical Optics Express
500:{\displaystyle \epsilon }
485:local connected dimension
459:{\displaystyle \epsilon }
415:{\displaystyle \epsilon }
391:{\displaystyle \epsilon }
294:{\displaystyle \epsilon }
240:
232:
224:{\displaystyle \epsilon }
182:{\displaystyle \epsilon }
162:{\displaystyle \epsilon }
142:{\displaystyle \epsilon }
43:
42:is a method of gathering
1087:10.1103/physreve.53.5461
1229:Physical Review Letters
1157:10.1023/A:1008148514268
946:10.1023/A:1022355723781
543:box counting dimensions
27:Figure 1. A 32-segment
1971:The Beauty of Fractals
851:Clinical Ophthalmology
535:
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428:box counting dimension
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376:box counting dimension
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279:box counting dimension
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472:multifractal analysis
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1917:Lewis Fry Richardson
1912:Hamid Naderi Yeganeh
1702:Burning Ship fractal
1634:Weierstrass function
1347:10.1167/iovs.07-1306
985:10.1364/boe.1.000268
934:Mathematical Geology
547:multifractal scaling
523:
491:
450:
424:relevant information
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285:
215:
193:
173:
153:
133:
111:
1675:Space-filling curve
1652:Multifractal system
1535:Space-filling curve
1520:Sierpinski triangle
1241:1989PhRvL..62.1327C
1079:1996PhRvE..53.5461P
824:2009PatRe..42.2460L
812:Pattern Recognition
744:Mandelbrot (1983).
648:2003BpJ....85.4041L
636:Biophysical Journal
56:size of the element
1902:Aleksandr Lyapunov
1882:Desmond Paul Henry
1846:Self-avoiding walk
1841:Percolation theory
1485:Iterated function
1426:Fractal dimensions
1369:Karperien (2002),
1313:Karperien (2004).
1184:Journal of Anatomy
1122:10.1007/BF00125351
864:10.2147/OPTH.S1579
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328:
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72:fundamental method
37:
1999:
1998:
1945:Coastline paradox
1922:Wacław Sierpiński
1907:Benoit Mandelbrot
1831:Fractal landscape
1739:Misiurewicz point
1644:Strange attractor
1525:Apollonian gasket
1515:Sierpinski carpet
1235:(12): 1327–1330.
1145:Landscape Ecology
1110:Landscape Ecology
1067:Physical Review E
907:(13): 2749–2755.
818:(11): 2460–2469.
791:978-0-8493-7636-8
759:978-0-7167-1186-5
596:Fractal dimension
438:Sliding box scans
275:fractal dimension
241:relevant features
2024:
2017:Dimension theory
1862:Michael Barnsley
1729:Lyapunov fractal
1587:Sierpiński curve
1540:Blancmange curve
1405:
1398:
1391:
1382:
1376:
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1359:
1349:
1340:(4): 1398–1406.
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1217:
1207:
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1098:
1073:(5): 5461–5468.
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642:(6): 4041–4046.
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591:Fractal analysis
569:Grid orientation
561:Scaling box size
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76:fractal analysis
48:complex patterns
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1877:Felix Hausdorff
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1814:Brownian motion
1799:
1770:
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1629:Pythagoras tree
1486:
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1475:Self-similarity
1419:Characteristics
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29:quadric fractal
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1751:Newton fractal
1748:
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1734:Mandelbrot set
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1717:Newton fractal
1714:
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1670:Fractal canopy
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1505:Koch snowflake
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1361:
1320:
1305:
1278:(2): 151–157.
1262:
1219:
1190:(3): 353–359.
1170:
1151:(4): 313–321.
1135:
1116:(3): 201–211.
1100:
1057:
1028:(2): 154–166.
1008:
979:(1): 268–277.
959:
940:(7): 919–932.
930:Cheng, Qiuming
918:
888:
857:(1): 109–122.
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731:
704:(2): 123–136.
679:
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149:s. Then these
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50:by breaking a
46:for analyzing
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1964:
1960:
1958:
1955:
1953:
1950:
1946:
1943:
1942:
1940:
1936:
1935:
1933:
1929:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1859:
1857:
1853:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1825:
1822:
1820:
1819:Brownian tree
1817:
1816:
1815:
1812:
1811:
1809:
1806:
1802:
1796:
1793:
1791:
1788:
1786:
1783:
1782:
1780:
1777:
1773:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1747:
1746:Multibrot set
1744:
1740:
1737:
1736:
1735:
1732:
1730:
1727:
1723:
1722:Douady rabbit
1720:
1718:
1715:
1713:
1710:
1709:
1708:
1705:
1703:
1700:
1699:
1697:
1695:
1689:
1681:
1678:
1677:
1676:
1673:
1671:
1668:
1667:
1665:
1663:
1659:
1653:
1650:
1649:
1647:
1645:
1641:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1593:
1592:Z-order curve
1590:
1588:
1585:
1583:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1562:Hilbert curve
1560:
1558:
1555:
1551:
1548:
1547:
1546:
1545:De Rham curve
1543:
1541:
1538:
1537:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1510:Menger sponge
1508:
1506:
1503:
1501:
1498:
1496:
1495:Barnsley fern
1493:
1492:
1490:
1488:
1482:
1476:
1473:
1471:
1468:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1442:
1439:
1438:
1437:
1434:
1432:
1429:
1428:
1427:
1424:
1423:
1421:
1417:
1413:
1406:
1401:
1399:
1394:
1392:
1387:
1386:
1383:
1374:
1373:
1365:
1362:
1357:
1353:
1348:
1343:
1339:
1335:
1331:
1324:
1321:
1316:
1309:
1306:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1266:
1263:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1223:
1220:
1215:
1211:
1206:
1201:
1197:
1193:
1189:
1185:
1181:
1174:
1171:
1166:
1162:
1158:
1154:
1150:
1146:
1139:
1136:
1131:
1127:
1123:
1119:
1115:
1111:
1104:
1101:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1061:
1058:
1053:
1049:
1044:
1039:
1035:
1031:
1027:
1023:
1019:
1012:
1009:
1004:
1000:
995:
990:
986:
982:
978:
974:
970:
963:
960:
955:
951:
947:
943:
939:
935:
931:
925:
923:
919:
914:
910:
906:
902:
895:
893:
889:
884:
880:
875:
870:
865:
860:
856:
852:
848:
841:
838:
833:
829:
825:
821:
817:
813:
806:
804:
802:
798:
793:
787:
783:
776:
774:
772:
770:
766:
761:
755:
750:
749:
740:
738:
736:
732:
727:
723:
719:
715:
711:
707:
703:
699:
695:
688:
686:
684:
680:
675:
671:
666:
661:
657:
653:
649:
645:
641:
637:
633:
626:
623:
617:
613:
610:
607:
604:
602:
599:
597:
594:
592:
589:
588:
584:
582:
578:
576:
568:
566:
560:
558:
552:
550:
548:
544:
514:
512:
510:
494:
486:
477:
475:
473:
469:
453:
445:
437:
435:
433:
429:
425:
409:
401:
385:
377:
373:
369:
361:
354:
350:
344:
340:
332:
324:
318:
314:
312:
305:
303:
288:
280:
277:known as the
276:
271:
267:
258:
256:
250:
246:
244:
242:
238:
234:
233:scanning plan
218:
176:
156:
136:
106:
101:
97:
89:
87:
85:
81:
77:
73:
69:
68:digital media
65:
61:
57:
53:
49:
45:
41:
34:
30:
25:
21:
19:
1991:Chaos theory
1986:Kaleidoscope
1977:
1969:
1961:
1887:Gaston Julia
1867:Georg Cantor
1692:Escape-time
1624:Gosper curve
1572:LĂ©vy C curve
1557:Dragon curve
1436:Box-counting
1372:Box Counting
1371:
1364:
1337:
1333:
1323:
1314:
1308:
1275:
1271:
1265:
1232:
1228:
1222:
1187:
1183:
1173:
1148:
1144:
1138:
1113:
1109:
1103:
1070:
1066:
1060:
1025:
1021:
1011:
976:
972:
962:
937:
933:
904:
900:
854:
850:
840:
815:
811:
781:
747:
701:
697:
639:
635:
625:
606:Multifractal
579:
572:
564:
556:
553:Edge effects
518:
481:
441:
365:
349:
339:
313:
309:
262:
255:
245:
236:
104:
93:
84:multifractal
40:Box counting
39:
38:
20:
18:
1982:(1987 book)
1974:(1986 book)
1966:(1982 book)
1952:Fractal art
1872:Bill Gosper
1836:LĂ©vy flight
1582:Peano curve
1577:Moore curve
1463:Topological
1448:Correlation
2006:Categories
1790:Orbit trap
1785:Buddhabrot
1778:techniques
1766:Mandelbulb
1567:Koch curve
1500:Cantor set
618:References
612:Lacunarity
468:lacunarity
306:Scan types
90:The method
86:analysis.
80:lacunarity
1897:Paul LĂ©vy
1776:Rendering
1761:Mandelbox
1707:Julia set
1619:Hexaflake
1550:Minkowski
1470:Recursion
1453:Hausdorff
954:118918429
509:Figure 2c
495:ϵ
454:ϵ
444:Figure 2b
410:ϵ
386:ϵ
368:Figure 2a
357:counting.
289:ϵ
270:extracted
219:ϵ
177:ϵ
157:ϵ
137:ϵ
64:extracted
2012:Fractals
1807:fractals
1694:fractals
1662:L-system
1604:T-square
1412:Fractals
1356:18385056
1300:31745811
1292:10491946
1257:10039645
1214:16533317
1165:18644861
1052:20740072
1003:21258464
883:19668394
726:20175299
674:14645092
608:analysis
585:See also
575:Figure 4
422:and the
400:Figure 4
372:Figure 1
259:The data
237:counting
100:Figure 1
60:Figure 1
1756:Tricorn
1609:n-flake
1458:Packing
1441:Higuchi
1431:Assouad
1237:Bibcode
1205:2100241
1130:7112365
1095:9964879
1075:Bibcode
1043:2927230
994:3005165
913:7499097
874:2698675
820:Bibcode
718:8946315
665:1303704
644:Bibcode
96:fractal
52:dataset
1855:People
1805:Random
1712:Filled
1680:H tree
1599:String
1487:system
1354:
1298:
1290:
1255:
1212:
1202:
1163:
1128:
1093:
1050:
1040:
1001:
991:
952:
911:
881:
871:
788:
756:
724:
716:
672:
662:
1931:Other
1296:S2CID
1161:S2CID
1126:S2CID
950:S2CID
722:S2CID
105:boxes
66:from
1352:PMID
1288:PMID
1253:PMID
1210:PMID
1091:PMID
1048:PMID
999:PMID
909:PMID
879:PMID
786:ISBN
754:ISBN
714:PMID
670:PMID
82:and
44:data
1342:doi
1280:doi
1245:doi
1200:PMC
1192:doi
1188:208
1153:doi
1118:doi
1083:doi
1038:PMC
1030:doi
989:PMC
981:doi
942:doi
869:PMC
859:doi
828:doi
706:doi
660:PMC
652:doi
573:As
189:in
2008::
1941:"
1350:.
1338:49
1336:.
1332:.
1294:.
1286:.
1276:89
1274:.
1251:.
1243:.
1233:62
1231:.
1208:.
1198:.
1186:.
1182:.
1159:.
1149:15
1147:.
1124:.
1112:.
1089:.
1081:.
1071:53
1069:.
1046:.
1036:.
1024:.
1020:.
997:.
987:.
975:.
971:.
948:.
938:29
936:.
921:^
905:36
903:.
891:^
877:.
867:.
853:.
849:.
826:.
816:42
814:.
800:^
768:^
734:^
720:.
712:.
702:69
700:.
696:.
682:^
668:.
658:.
650:.
640:85
638:.
634:.
549:.
511:.
474:.
434:.
1937:"
1404:e
1397:t
1390:v
1358:.
1344::
1302:.
1282::
1259:.
1247::
1239::
1216:.
1194::
1167:.
1155::
1132:.
1120::
1114:8
1097:.
1085::
1077::
1054:.
1032::
1026:3
1005:.
983::
977:1
956:.
944::
915:.
885:.
861::
855:2
834:.
830::
822::
794:.
762:.
728:.
708::
676:.
654::
646::
528:E
198:E
116:E
35:.
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