65:, fill space, where patterns having more or larger gaps generally have higher lacunarity. Beyond being an intuitive measure of gappiness, lacunarity can quantify additional features of patterns such as "rotational invariance" and more generally, heterogeneity. This is illustrated in Figure 1 showing three fractal patterns. When rotated 90°, the first two fairly homogeneous patterns do not appear to change, but the third more heterogeneous figure does change and has correspondingly higher lacunarity. The earliest reference to the term in geometry is usually attributed to
34:
22:
1663:
instances, it has been demonstrated that fractal dimensions and values of lacunarity were correlated, but more recent research has shown that this relationship does not hold for all types of patterns and measures of lacunarity. Indeed, as
Mandelbrot originally proposed, lacunarity has been shown to be useful in discerning amongst patterns (e.g., fractals, textures, etc.) that share or have similar
158:. Similar to looking at a slide through a microscope with changing levels of magnification, box counting algorithms look at a digital image from many levels of resolution to examine how certain features change with the size of the element used to inspect the image. Basically, the arrangement of pixels is measured using traditionally square (i.e., box-shaped) elements from an arbitrary set of
1329:
37:
The same images as above, rotated 90°. Whereas the first two images appear essentially the same as they do above, the third looks different from its unrotated original. This feature is captured in measures of lacunarity listed across the top of the figures, as calculated using standard biological
1662:
Lacunarity analyses using the types of values discussed above have shown that data sets extracted from dense fractals, from patterns that change little when rotated, or from patterns that are homogeneous, have low lacunarity, but as these features increase, so generally does lacunarity. In some
1145:
1168:
644:
2278:
997:
1543:
1639:
85:
In many patterns or data sets, lacunarity is not readily perceivable or quantifiable, so computer-aided methods have been developed to calculate it. As a measurable quantity, lacunarity is often denoted in scientific literature by the Greek letters
2942:
860:
1020:
1324:{\displaystyle \lambda _{\varepsilon }={\frac {\sum _{i=1}^{B}{m_{i,\varepsilon }^{2}p_{i,\varepsilon }-\mu _{\varepsilon }^{2}}}{\mu _{\varepsilon }^{2}}}={\frac {\sigma _{\varepsilon }^{2}}{\mu _{\varepsilon }^{2}}}}
1945:
527:
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758:
517:
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343:
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3663:
Valous, N.A.; Sun, D.-W.; Allen, P.; Mendoza, F. (January 2010). "The use of lacunarity for visual texture characterization of pre-sliced cooked pork ham surface intensities".
2844:
370:
2538:
1790:
478:
454:
3121:
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".
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2001:
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is placed successively on the image, in the end covering it completely, and each time it is laid down, the number of pixels that fall within the box is recorded. In
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410:
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the box is slid over the image so that it overlaps itself and the "Sliding Box
Lacunarity" or SLac is calculated. Figure 2 illustrates both types of box counting.
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2514:
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725:
705:
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2965:
Below is a list of some fields where lacunarity plays an important role, along with links to relevant research illustrating practical uses of lacunarity.
3583:
Vannucchi, P.; Leoni, L. (30 October 2007). "Structural characterization of the Costa Rica decollement: Evidence for seismically-induced fluid pulsing".
1140:{\displaystyle \sigma _{\varepsilon }^{2}={\mathit {v}}_{\varepsilon }=\sum _{i=1}^{B}{m_{i,\varepsilon }^{2}p_{i,\varepsilon }-\mu _{\varepsilon }^{2}}}
73:. Lacunarity analysis is now used to characterize patterns in a wide variety of fields and has application in multifractal analysis in particular (see
639:{\displaystyle \lambda _{\varepsilon ,g}=(CV_{\varepsilon ,g})^{2}=\left({\frac {\sigma _{\varepsilon ,g}}{\mu _{\varepsilon ,g}}}\right)^{2}}
3504:
3225:
1889:
2273:{\displaystyle {\mathcal {L}}(r)={\frac {\sum _{i=1}^{r^{2}}S_{i}^{2}Q(S_{i},r)}{\left(\sum _{i=1}^{r^{2}}S_{i}Q(S_{i},r)\right)^{2}}}.}
2079:, homogeneous images have a slope of 0, corresponding intuitively to the idea of no rotational or translational invariance and no gaps.
412:), for pixels per box. Because the way an image is sampled will depend on the arbitrary starting location, for any image sampled at any
2756:
992:{\displaystyle {\mathit {v}}_{\varepsilon }=\sum _{i=1}^{B}{\left(m_{i,\varepsilon }-\mu _{\varepsilon }\right)^{2}p_{i,\varepsilon }}}
126:
but it is important to note that there is no single standard and several different methods exist to assess and interpret lacunarity.
141:
135:
3251:
1538:{\displaystyle {\overline {\lambda _{g}}}={\frac {\sum _{\varepsilon =1}^{\mathrm {E} }\lambda _{\varepsilon ,g}}{\mathrm {E} }}}
2680:
3215:
3491:
Rievra-Virtudazo, R.V.; Tapia, A.K.G; Valenzuela, J.F.B.; Cruz, L.D.; Mendoza, H.D.; Castriciones, E.V. (23 November 2008).
1832:
regression line. Because they tend to behave in certain ways for respectively mono-, multi-, and non-fractal patterns,
1675:
Other methods of assessing lacunarity from box counting data use the relationship between values of lacunarity (e.g.,
480:, that the data can be gathered over, which can have varying effects on the measured distribution of pixels. Equation
3275:
1875:
To make the plots for this type of analysis, the data from box counting first have to be transformed as in
Equation
3036:
286:
is placed as though it were part of a grid overlaid on the image so that the box does not overlap itself, but in
3522:"Accuracy of Lacunarity Algorithms in Texture Classification of High Spatial Resolution Images from Urban Areas"
2046:
3694:
3171:
Plotnick, R. E.; Gardner, R. H.; O'Neill, R. V. (1993). "Lacunarity indices as measures of landscape texture".
1678:
1634:{\displaystyle \Lambda ={\frac {\sum _{{\mathit {g}}=1}^{\mathit {G}}{\overline {\lambda _{g}}}}{\mathit {G}}}}
1391:
1358:
730:
489:
2453:
321:
3403:
3343:
McIntyre, N. E.; Wiens, J. A. (2000). "A novel use of the lacunarity index to discern landscape function".
1768:
plot of these values. According to this method, the curve itself can be analyzed visually, or the slope at
2937:{\displaystyle P\Lambda ={\frac {\sum _{g=1}^{G}{\left({\frac {A_{g}}{\overline {A}}}-1\right)^{2}}}{G}}}
2640:
is particularly affected by image size and the way data are gathered, especially by the lower limit of
3699:
An online guide to lacunarity theory and analysis using free, open source biological imaging software.
667:
Alternatively, some methods sort the numbers of pixels counted into a probability distribution having
3592:
3434:
3130:
1963:
This transformation avoids undefined values, which is important because homogeneous images will have
3384:
348:
2519:
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459:
435:
3628:
Yaşar, F.; Akgünlü, F. (2005). "Fractal dimension and lacunarity analysis of dental radiographs".
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2543:
1986:
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267:
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3565:
3529:
The
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences
3521:
3360:
3306:
3188:
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2325:
150:
One well-known method of determining lacunarity for patterns extracted from digital images uses
33:
21:
3073:"Fractal methods and results in cellular morphology — dimensions, lacunarity and multifractals"
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3500:
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3146:
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has been assessed in several ways including by using the variation in or the average value of
109:
89:
66:
1966:
375:
3672:
3637:
3608:
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3557:
3556:. Wavelet Applications in Signal and Image Processing IV. Vol. 2825. pp. 109–119.
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155:
70:
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2006:
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855:{\displaystyle \mu _{\varepsilon }=\sum _{i=1}^{B}{m_{i,\varepsilon }p_{i,\varepsilon }}}
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69:, who, in 1983 or perhaps as early as 1977, introduced it as, in essence, an adjunct to
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2583:
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710:
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One box counting technique using a "gliding" box calculates lacunarity according to:
3706:
3569:
3493:"47. Lacunarity analysis of TEM Images of Heat-Treated Hybrid Organosilica Materials"
3364:
3310:
3104:
2403:
method, is based on the value obtained from box counting for the fractal dimension (
1872:
lacunarity plots have been used to supplement methods of classifying such patterns.
3192:
3015:
151:
3676:
43:
3604:
3477:
3356:
2989:
3294:
3142:
3641:
3462:"Practical quantitative lithic use-wear analysis using multiple classifiers"
3649:
3302:
3276:"Texture Analysis of Aggressive and non-Aggressive Lung Tumor CE CT Images"
3314:
3150:
3096:
1728:
in different ways from the ones noted above. One such method looks at the
58:
1447:) and by using the variation in or average over all grids (see Equation
54:
3613:
3389:
8th
European Conference on Mathematical and Theoretical Biology, Kraków
3184:
140:
134:
62:
3561:
1940:{\displaystyle f\lambda _{\varepsilon ,g}=\lambda _{\varepsilon ,g}+1}
318:
are manipulated to calculate lacunarity. One measure, denoted here as
3546:
Gorsich, D. J.; Tolle, C. R.; Karlsen, R. E.; Gerhart, G. R. (1996).
3423:"Lacunarity definition for ramified data sets based on optimal cover"
3385:"Lacunarity Analysis and Classification of Microglia in Neuroscience"
3383:
Jelinek, Herbert; Karperien, Audrey; Milosevic, Nebojsa (June 2011).
3256:
Defining
Microglial Morphology: Form, Function, and Fractal Dimension
3039:
for an explanation of methods to address variation with grid location
2399:
Another proposed way of assessing lacunarity using box counting, the
3551:
3547:
3422:
3072:
3461:
139:
133:
32:
20:
2816:{\displaystyle {\overline {A}}={\frac {\sum _{g=1}^{G}A_{g}}{G}}}
3550:. In Unser, Michael A; Aldroubi, Akram; Laine, Andrew F (eds.).
154:, the same essential algorithm typically used for some types of
2660:
s used. The final measure is calculated as shown in
Equations
144:
Figure 2b. Boxes slid over an image in an overlapping pattern.
2710:
2525:
2096:
1777:
1625:
1600:
1586:
1045:
890:
465:
441:
357:
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in a variety of scientific fields including neuroscience.
2728:{\displaystyle A_{g}={\frac {1}{e^{{\mathit {y}}_{g}}}}}
3548:"Wavelet and fractal analysis of ground-vehicle images"
3553:
2580:) of boxes that had any pixels at all in them or else
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2301:
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2029:
2009:
1989:
1969:
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1858:
1838:
1818:
1798:
1774:
1754:
1734:
1714:
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1427:
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783:
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138:
Figure 2a. Boxes laid over an image as a fixed grid.
112:
92:
57:, meaning "gap" or "lake", is a specialized term in
2322:is the number of filled data points in the box and
61:referring to a measure of how patterns, especially
3460:Stevens, N. E.; Harro, D. R.; Hicklin, A. (2010).
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2727:
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2632:
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2552:
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2508:
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2422:
2383:
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2043:regression line would be impossible to find. With
2035:
2015:
1995:
1975:
1939:
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118:
98:
3071:Smith, T. G.; Lange, G. D.; Marks, W. B. (1996).
456:) of possible orientations, each denoted here by
3269:
3267:
3265:
3020:total number of boxes that contained any pixels
25:Figure 1. Basic fractal patterns increasing in
345:, is found from the coefficient of variation (
3116:
3114:
8:
3206:
3204:
3202:
3022:is counted to determine a fractal dimension.
3283:IEEE Transactions on Biomedical Engineering
3245:
3243:
3241:
3239:
3237:
3378:
3376:
3374:
3252:"Chapter 8 Multifractality and Lacunarity"
3166:
3164:
3162:
3160:
2072:{\displaystyle f\lambda _{\varepsilon ,g}}
3612:
2921:
2894:
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2414:
2408:
2375:
2369:
2364:the normalized frequency distribution of
2339:
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2223:
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2195:
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2153:
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2120:
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2095:
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2008:
1988:
1968:
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1900:
1891:
1857:
1837:
1817:
1797:
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1775:
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1753:
1733:
1713:
1701:{\displaystyle \lambda _{\varepsilon ,g}}
1686:
1680:
1624:
1612:
1606:
1599:
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1585:
1584:
1583:
1576:
1568:
1528:
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1503:
1492:
1485:
1471:
1465:
1463:
1426:
1414:{\displaystyle \lambda _{\varepsilon ,g}}
1399:
1393:
1381:{\displaystyle \lambda _{\varepsilon ,g}}
1366:
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1313:
1308:
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1287:
1276:
1271:
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1185:
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1125:
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919:
908:
895:
889:
888:
885:
839:
823:
818:
812:
801:
788:
782:
753:{\displaystyle \lambda _{\varepsilon ,g}}
738:
732:
712:
707:) and their corresponding probabilities (
692:
672:
630:
612:
596:
590:
576:
560:
535:
529:
512:{\displaystyle \lambda _{\varepsilon ,g}}
497:
491:
464:
463:
461:
440:
439:
437:
417:
397:
377:
372:), calculated as the standard deviation (
353:
352:
350:
329:
323:
303:
271:
269:
249:
225:
205:
185:
165:
163:
111:
91:
2489:{\displaystyle N=A\varepsilon ^{-D_{B}}}
3338:
3336:
3334:
3066:
3064:
3062:
3060:
3058:
3056:
3052:
3007:
338:{\displaystyle \lambda _{\varepsilon }}
486:shows the basic method of calculating
1658:Relationship to the fractal dimension
687:bins, and use the bin sizes (masses,
16:Term in geometry and fractal analysis
7:
2838:
2750:
2674:
2516:is calculated from the y-intercept (
2430:). This statistic uses the variable
2084:
2003:equal to 0 so that the slope of the
1883:
1560:
1455:
1162:
1014:
877:
774:
521:
3585:Earth and Planetary Science Letters
3032:
2540:) of the ln-ln regression line for
39:
3520:Filho, M.B.; Sobreira, F. (2008).
3274:Al-Kadi, O.S.; Watson, D. (2008).
2851:
1570:
1529:
1505:
272:
166:
93:
14:
3466:Journal of Archaeological Science
74:
3497:Innovations in Chemical Biology
3077:Journal of Neuroscience Methods
3499:. Springer. pp. 397–404.
3427:Physica D: Nonlinear Phenomena
3217:The Fractal Geometry of Nature
2351:
2332:
2251:
2232:
2181:
2162:
2107:
2101:
573:
550:
365:{\displaystyle {\mathit {CV}}}
294:Calculations from box counting
180:sizes, conventionally denoted
38:imaging box counting software
1:
3677:10.1016/j.foodres.2009.10.018
3447:10.1016/S0167-2789(03)00029-0
3089:10.1016/S0165-0270(96)00080-5
2533:{\displaystyle {\mathit {y}}}
1785:{\displaystyle {\mathit {g}}}
473:{\displaystyle {\mathit {g}}}
449:{\displaystyle {\mathit {G}}}
3630:Dentomaxillofacial Radiology
2904:
2765:
2668:
2662:
2653:{\displaystyle \varepsilon }
2553:{\displaystyle \varepsilon }
1996:{\displaystyle \varepsilon }
1721:{\displaystyle \varepsilon }
1618:
1477:
1434:{\displaystyle \varepsilon }
425:{\displaystyle \varepsilon }
311:{\displaystyle \varepsilon }
279:{\displaystyle \mathrm {E} }
257:{\displaystyle \varepsilon }
233:{\displaystyle \varepsilon }
213:{\displaystyle \varepsilon }
193:{\displaystyle \varepsilon }
173:{\displaystyle \mathrm {E} }
3665:Food Research International
3258:. Charles Sturt University.
3018:fractal analysis where the
1877:
1792:can be calculated from the
1449:
1443:
768:
762:
482:
432:there will be some number (
298:The data gathered for each
3729:
3605:10.1016/j.epsl.2007.07.056
2357:{\displaystyle Q(S_{i},r)}
3495:. In Sener, Bilge (ed.).
3478:10.1016/j.jas.2010.06.004
3404:"Interpreting Lacunarity"
2391:for different box sizes.
663:Probability distributions
3295:10.1109/TBME.2008.919735
3143:10.1103/physreve.53.5461
119:{\displaystyle \lambda }
99:{\displaystyle \Lambda }
3597:2007E&PSL.262..413V
3357:10.1023/A:1008148514268
1976:{\displaystyle \sigma }
760:according to Equations
392:) divided by the mean (
385:{\displaystyle \sigma }
130:Box counting lacunarity
3695:"FracLac User's Guide"
2938:
2880:
2817:
2796:
2729:
2654:
2634:
2614:
2594:
2574:
2560:and either the count (
2554:
2534:
2510:
2490:
2450:from the scaling rule
2444:
2424:
2385:
2358:
2316:
2274:
2218:
2143:
2073:
2037:
2017:
1997:
1977:
1941:
1866:
1846:
1826:
1806:
1786:
1762:
1742:
1722:
1702:
1635:
1605:
1539:
1510:
1435:
1415:
1382:
1325:
1208:
1141:
1079:
993:
924:
856:
817:
754:
721:
701:
681:
640:
513:
474:
450:
426:
406:
386:
366:
339:
312:
288:sliding box algorithms
280:
258:
234:
214:
194:
174:
147:
145:
120:
100:
47:
30:
3642:10.1259/dmfr/85149245
2939:
2860:
2818:
2776:
2730:
2655:
2635:
2615:
2595:
2575:
2555:
2535:
2511:
2491:
2445:
2425:
2423:{\displaystyle D_{B}}
2386:
2384:{\displaystyle S_{i}}
2359:
2317:
2315:{\displaystyle S_{i}}
2275:
2191:
2116:
2074:
2038:
2018:
1998:
1978:
1942:
1867:
1847:
1827:
1807:
1787:
1763:
1743:
1723:
1703:
1636:
1579:
1540:
1488:
1436:
1416:
1383:
1326:
1188:
1142:
1059:
994:
904:
857:
797:
755:
722:
702:
682:
641:
514:
475:
451:
427:
407:
387:
367:
340:
313:
281:
259:
242:standard box counting
235:
215:
195:
175:
143:
137:
121:
101:
36:
24:
3014:This contrasts with
2845:
2757:
2681:
2644:
2624:
2604:
2584:
2564:
2544:
2520:
2500:
2454:
2434:
2407:
2395:Prefactor lacunarity
2368:
2326:
2299:
2091:
2047:
2036:{\displaystyle \ln }
2027:
2016:{\displaystyle \ln }
2007:
1987:
1967:
1890:
1865:{\displaystyle \ln }
1856:
1845:{\displaystyle \ln }
1836:
1825:{\displaystyle \ln }
1816:
1805:{\displaystyle \ln }
1796:
1772:
1761:{\displaystyle \ln }
1752:
1741:{\displaystyle \ln }
1732:
1712:
1679:
1671:Graphical lacunarity
1567:
1462:
1425:
1392:
1359:
1355:Lacunarity based on
1169:
1021:
884:
781:
731:
711:
691:
671:
528:
490:
460:
436:
416:
405:{\displaystyle \mu }
396:
376:
349:
322:
302:
268:
248:
224:
204:
184:
162:
110:
90:
81:Measuring lacunarity
3439:2003PhyD..179..129T
3135:1996PhRvE..53.5461P
2158:
1318:
1303:
1281:
1264:
1230:
1135:
1101:
1038:
244:, the box for each
29:from left to right.
3421:Tolle, C. (2003).
3402:Karperien (2002).
3250:Karperien (2004).
3212:Mandelbrot, Benoit
3185:10.1007/BF00125351
2934:
2813:
2725:
2650:
2630:
2610:
2590:
2570:
2550:
2530:
2506:
2486:
2440:
2420:
2381:
2354:
2312:
2270:
2144:
2069:
2033:
2013:
1993:
1973:
1937:
1862:
1842:
1822:
1802:
1782:
1758:
1738:
1718:
1698:
1665:fractal dimensions
1631:
1535:
1431:
1411:
1378:
1321:
1304:
1289:
1267:
1250:
1210:
1137:
1121:
1081:
1024:
989:
852:
750:
717:
697:
677:
636:
509:
470:
446:
422:
402:
382:
362:
335:
308:
276:
254:
230:
210:
190:
170:
148:
146:
116:
96:
48:
31:
3562:10.1117/12.255224
3506:978-1-4020-6955-0
3345:Landscape Ecology
3227:978-0-7167-1186-5
3173:Landscape Ecology
3123:Physical Review E
2958:
2957:
2932:
2908:
2907:
2837:
2836:
2811:
2768:
2749:
2748:
2723:
2633:{\displaystyle A}
2613:{\displaystyle g}
2593:{\displaystyle m}
2573:{\displaystyle N}
2509:{\displaystyle A}
2443:{\displaystyle A}
2294:
2293:
2265:
1961:
1960:
1655:
1654:
1629:
1621:
1559:
1558:
1533:
1480:
1345:
1344:
1319:
1282:
1161:
1160:
1013:
1012:
876:
875:
720:{\displaystyle p}
700:{\displaystyle m}
680:{\displaystyle B}
660:
659:
624:
67:Benoit Mandelbrot
53:, from the Latin
3720:
3698:
3681:
3680:
3660:
3654:
3653:
3625:
3619:
3618:
3616:
3591:(3–4): 413–428.
3580:
3574:
3573:
3543:
3537:
3536:
3526:
3517:
3511:
3510:
3488:
3482:
3481:
3457:
3451:
3450:
3433:(3–4): 129–201.
3418:
3412:
3411:
3399:
3393:
3392:
3380:
3369:
3368:
3340:
3329:
3328:
3326:
3325:
3319:
3313:. Archived from
3280:
3271:
3260:
3259:
3247:
3232:
3231:
3208:
3197:
3196:
3168:
3155:
3154:
3118:
3109:
3108:
3068:
3040:
3029:
3023:
3012:
2985:spatial analysis
2952:
2943:
2941:
2940:
2935:
2933:
2928:
2927:
2926:
2925:
2920:
2916:
2909:
2900:
2899:
2898:
2889:
2879:
2874:
2858:
2839:
2831:
2822:
2820:
2819:
2814:
2812:
2807:
2806:
2805:
2795:
2790:
2774:
2769:
2761:
2751:
2743:
2734:
2732:
2731:
2726:
2724:
2722:
2721:
2720:
2719:
2714:
2713:
2698:
2693:
2692:
2675:
2659:
2657:
2656:
2651:
2639:
2637:
2636:
2631:
2619:
2617:
2616:
2611:
2599:
2597:
2596:
2591:
2579:
2577:
2576:
2571:
2559:
2557:
2556:
2551:
2539:
2537:
2536:
2531:
2529:
2528:
2515:
2513:
2512:
2507:
2495:
2493:
2492:
2487:
2485:
2484:
2483:
2482:
2449:
2447:
2446:
2441:
2429:
2427:
2426:
2421:
2419:
2418:
2390:
2388:
2387:
2382:
2380:
2379:
2363:
2361:
2360:
2355:
2344:
2343:
2321:
2319:
2318:
2313:
2311:
2310:
2288:
2279:
2277:
2276:
2271:
2266:
2264:
2263:
2258:
2254:
2244:
2243:
2228:
2227:
2217:
2216:
2215:
2205:
2184:
2174:
2173:
2157:
2152:
2142:
2141:
2140:
2130:
2114:
2100:
2099:
2085:
2078:
2076:
2075:
2070:
2068:
2067:
2042:
2040:
2039:
2034:
2022:
2020:
2019:
2014:
2002:
2000:
1999:
1994:
1982:
1980:
1979:
1974:
1955:
1946:
1944:
1943:
1938:
1930:
1929:
1911:
1910:
1884:
1871:
1869:
1868:
1863:
1851:
1849:
1848:
1843:
1831:
1829:
1828:
1823:
1811:
1809:
1808:
1803:
1791:
1789:
1788:
1783:
1781:
1780:
1767:
1765:
1764:
1759:
1747:
1745:
1744:
1739:
1727:
1725:
1724:
1719:
1707:
1705:
1704:
1699:
1697:
1696:
1649:
1640:
1638:
1637:
1632:
1630:
1628:
1623:
1622:
1617:
1616:
1607:
1604:
1603:
1597:
1590:
1589:
1577:
1561:
1553:
1544:
1542:
1541:
1536:
1534:
1532:
1527:
1526:
1525:
1509:
1508:
1502:
1486:
1481:
1476:
1475:
1466:
1456:
1440:
1438:
1437:
1432:
1420:
1418:
1417:
1412:
1410:
1409:
1387:
1385:
1384:
1379:
1377:
1376:
1339:
1330:
1328:
1327:
1322:
1320:
1317:
1312:
1302:
1297:
1288:
1283:
1280:
1275:
1266:
1265:
1263:
1258:
1246:
1245:
1229:
1224:
1207:
1202:
1186:
1181:
1180:
1163:
1155:
1146:
1144:
1143:
1138:
1136:
1134:
1129:
1117:
1116:
1100:
1095:
1078:
1073:
1055:
1054:
1049:
1048:
1037:
1032:
1015:
1007:
998:
996:
995:
990:
988:
987:
986:
971:
970:
965:
961:
960:
959:
947:
946:
923:
918:
900:
899:
894:
893:
878:
870:
861:
859:
858:
853:
851:
850:
849:
834:
833:
816:
811:
793:
792:
775:
759:
757:
756:
751:
749:
748:
726:
724:
723:
718:
706:
704:
703:
698:
686:
684:
683:
678:
654:
645:
643:
642:
637:
635:
634:
629:
625:
623:
622:
607:
606:
591:
581:
580:
571:
570:
546:
545:
522:
518:
516:
515:
510:
508:
507:
479:
477:
476:
471:
469:
468:
455:
453:
452:
447:
445:
444:
431:
429:
428:
423:
411:
409:
408:
403:
391:
389:
388:
383:
371:
369:
368:
363:
361:
360:
344:
342:
341:
336:
334:
333:
317:
315:
314:
309:
285:
283:
282:
277:
275:
263:
261:
260:
255:
239:
237:
236:
231:
220:, a box of size
219:
217:
216:
211:
199:
197:
196:
191:
179:
177:
176:
171:
169:
156:fractal analysis
125:
123:
122:
117:
105:
103:
102:
97:
71:fractal analysis
3728:
3727:
3723:
3722:
3721:
3719:
3718:
3717:
3703:
3702:
3693:
3690:
3685:
3684:
3662:
3661:
3657:
3627:
3626:
3622:
3582:
3581:
3577:
3545:
3544:
3540:
3524:
3519:
3518:
3514:
3507:
3490:
3489:
3485:
3459:
3458:
3454:
3420:
3419:
3415:
3401:
3400:
3396:
3382:
3381:
3372:
3342:
3341:
3332:
3323:
3321:
3317:
3278:
3273:
3272:
3263:
3249:
3248:
3235:
3228:
3210:
3209:
3200:
3170:
3169:
3158:
3120:
3119:
3112:
3070:
3069:
3054:
3049:
3044:
3043:
3030:
3026:
3013:
3009:
3004:
2990:Seismic studies
2979:Medical imaging
2963:
2950:
2890:
2887:
2883:
2882:
2859:
2843:
2842:
2829:
2797:
2775:
2755:
2754:
2741:
2707:
2702:
2684:
2679:
2678:
2642:
2641:
2622:
2621:
2602:
2601:
2582:
2581:
2562:
2561:
2542:
2541:
2518:
2517:
2498:
2497:
2474:
2466:
2452:
2451:
2432:
2431:
2410:
2405:
2404:
2397:
2371:
2366:
2365:
2335:
2324:
2323:
2302:
2297:
2296:
2286:
2235:
2219:
2207:
2190:
2186:
2185:
2165:
2132:
2115:
2089:
2088:
2053:
2045:
2044:
2025:
2024:
2005:
2004:
1985:
1984:
1965:
1964:
1953:
1915:
1896:
1888:
1887:
1854:
1853:
1834:
1833:
1814:
1813:
1794:
1793:
1770:
1769:
1750:
1749:
1730:
1729:
1710:
1709:
1682:
1677:
1676:
1673:
1660:
1647:
1608:
1578:
1565:
1564:
1551:
1511:
1487:
1467:
1460:
1459:
1423:
1422:
1395:
1390:
1389:
1362:
1357:
1356:
1353:
1337:
1231:
1187:
1172:
1167:
1166:
1153:
1102:
1042:
1019:
1018:
1005:
972:
951:
932:
931:
927:
926:
887:
882:
881:
868:
835:
819:
784:
779:
778:
734:
729:
728:
727:) to calculate
709:
708:
689:
688:
669:
668:
665:
652:
608:
592:
586:
585:
572:
556:
531:
526:
525:
493:
488:
487:
458:
457:
434:
433:
414:
413:
394:
393:
374:
373:
347:
346:
325:
320:
319:
300:
299:
296:
266:
265:
246:
245:
222:
221:
202:
201:
182:
181:
160:
159:
132:
108:
107:
88:
87:
83:
17:
12:
11:
5:
3726:
3724:
3716:
3715:
3705:
3704:
3701:
3700:
3689:
3688:External links
3686:
3683:
3682:
3671:(1): 387–395.
3655:
3636:(5): 261–267.
3620:
3575:
3538:
3512:
3505:
3483:
3452:
3413:
3394:
3370:
3330:
3289:(7): 1822–30.
3261:
3233:
3226:
3198:
3156:
3110:
3083:(2): 123–136.
3051:
3050:
3048:
3045:
3042:
3041:
3024:
3006:
3005:
3003:
3000:
2999:
2998:
2995:
2992:
2987:
2981:
2976:
2973:
2970:
2962:
2959:
2956:
2955:
2946:
2944:
2931:
2924:
2919:
2915:
2912:
2906:
2903:
2897:
2893:
2886:
2878:
2873:
2870:
2867:
2863:
2856:
2853:
2850:
2835:
2834:
2825:
2823:
2810:
2804:
2800:
2794:
2789:
2786:
2783:
2779:
2772:
2767:
2764:
2747:
2746:
2737:
2735:
2718:
2712:
2705:
2701:
2696:
2691:
2687:
2649:
2629:
2609:
2589:
2569:
2549:
2527:
2505:
2481:
2477:
2473:
2469:
2465:
2462:
2459:
2439:
2417:
2413:
2396:
2393:
2378:
2374:
2353:
2350:
2347:
2342:
2338:
2334:
2331:
2309:
2305:
2292:
2291:
2282:
2280:
2269:
2262:
2257:
2253:
2250:
2247:
2242:
2238:
2234:
2231:
2226:
2222:
2214:
2210:
2204:
2201:
2198:
2194:
2189:
2183:
2180:
2177:
2172:
2168:
2164:
2161:
2156:
2151:
2147:
2139:
2135:
2129:
2126:
2123:
2119:
2112:
2109:
2106:
2103:
2098:
2066:
2063:
2060:
2056:
2052:
2032:
2012:
1992:
1972:
1959:
1958:
1949:
1947:
1936:
1933:
1928:
1925:
1922:
1918:
1914:
1909:
1906:
1903:
1899:
1895:
1861:
1841:
1821:
1801:
1779:
1757:
1737:
1717:
1695:
1692:
1689:
1685:
1672:
1669:
1659:
1656:
1653:
1652:
1643:
1641:
1627:
1620:
1615:
1611:
1602:
1596:
1593:
1588:
1582:
1575:
1572:
1557:
1556:
1547:
1545:
1531:
1524:
1521:
1518:
1514:
1507:
1501:
1498:
1495:
1491:
1484:
1479:
1474:
1470:
1441:(see Equation
1430:
1408:
1405:
1402:
1398:
1375:
1372:
1369:
1365:
1352:
1346:
1343:
1342:
1333:
1331:
1316:
1311:
1307:
1301:
1296:
1292:
1286:
1279:
1274:
1270:
1262:
1257:
1253:
1249:
1244:
1241:
1238:
1234:
1228:
1223:
1220:
1217:
1213:
1206:
1201:
1198:
1195:
1191:
1184:
1179:
1175:
1159:
1158:
1149:
1147:
1133:
1128:
1124:
1120:
1115:
1112:
1109:
1105:
1099:
1094:
1091:
1088:
1084:
1077:
1072:
1069:
1066:
1062:
1058:
1053:
1047:
1041:
1036:
1031:
1027:
1011:
1010:
1001:
999:
985:
982:
979:
975:
969:
964:
958:
954:
950:
945:
942:
939:
935:
930:
922:
917:
914:
911:
907:
903:
898:
892:
874:
873:
864:
862:
848:
845:
842:
838:
832:
829:
826:
822:
815:
810:
807:
804:
800:
796:
791:
787:
747:
744:
741:
737:
716:
696:
676:
664:
661:
658:
657:
648:
646:
633:
628:
621:
618:
615:
611:
605:
602:
599:
595:
589:
584:
579:
575:
569:
566:
563:
559:
555:
552:
549:
544:
541:
538:
534:
506:
503:
500:
496:
467:
443:
421:
401:
381:
359:
356:
332:
328:
307:
295:
292:
274:
253:
229:
209:
189:
168:
131:
128:
115:
95:
82:
79:
15:
13:
10:
9:
6:
4:
3:
2:
3725:
3714:
3711:
3710:
3708:
3696:
3692:
3691:
3687:
3678:
3674:
3670:
3666:
3659:
3656:
3651:
3647:
3643:
3639:
3635:
3631:
3624:
3621:
3615:
3610:
3606:
3602:
3598:
3594:
3590:
3586:
3579:
3576:
3571:
3567:
3563:
3559:
3555:
3554:
3549:
3542:
3539:
3534:
3530:
3523:
3516:
3513:
3508:
3502:
3498:
3494:
3487:
3484:
3479:
3475:
3471:
3467:
3463:
3456:
3453:
3448:
3444:
3440:
3436:
3432:
3428:
3424:
3417:
3414:
3409:
3405:
3398:
3395:
3390:
3386:
3379:
3377:
3375:
3371:
3366:
3362:
3358:
3354:
3350:
3346:
3339:
3337:
3335:
3331:
3320:on 2014-04-13
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3277:
3270:
3268:
3266:
3262:
3257:
3253:
3246:
3244:
3242:
3240:
3238:
3234:
3229:
3223:
3219:
3218:
3213:
3207:
3205:
3203:
3199:
3194:
3190:
3186:
3182:
3178:
3174:
3167:
3165:
3163:
3161:
3157:
3152:
3148:
3144:
3140:
3136:
3132:
3129:(5): 5461–8.
3128:
3124:
3117:
3115:
3111:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3078:
3074:
3067:
3065:
3063:
3061:
3059:
3057:
3053:
3046:
3038:
3034:
3028:
3025:
3021:
3017:
3011:
3008:
3001:
2996:
2993:
2991:
2988:
2986:
2982:
2980:
2977:
2974:
2971:
2968:
2967:
2966:
2960:
2954:
2947:
2945:
2929:
2922:
2917:
2913:
2910:
2901:
2895:
2891:
2884:
2876:
2871:
2868:
2865:
2861:
2854:
2848:
2841:
2840:
2833:
2826:
2824:
2808:
2802:
2798:
2792:
2787:
2784:
2781:
2777:
2770:
2762:
2753:
2752:
2745:
2738:
2736:
2716:
2703:
2699:
2694:
2689:
2685:
2677:
2676:
2673:
2671:
2670:
2665:
2664:
2647:
2627:
2607:
2587:
2567:
2547:
2503:
2479:
2475:
2471:
2467:
2463:
2460:
2457:
2437:
2415:
2411:
2402:
2394:
2392:
2376:
2372:
2348:
2345:
2340:
2336:
2329:
2307:
2303:
2290:
2283:
2281:
2267:
2260:
2255:
2248:
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2110:
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2064:
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2010:
1990:
1970:
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1948:
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1920:
1916:
1912:
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1367:
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1351:
1348:Interpreting
1347:
1341:
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1332:
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1294:
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1277:
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1177:
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1164:
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1150:
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1131:
1126:
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1107:
1103:
1097:
1092:
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1002:
1000:
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977:
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915:
912:
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901:
896:
880:
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840:
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3286:
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3122:
3080:
3076:
3037:Box Counting
3027:
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3016:box counting
3010:
2997:Food science
2964:
2961:Applications
2948:
2827:
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152:box counting
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3614:2158/257208
3535:(Part B3b).
2975:Archaeology
3351:(4): 313.
3324:2014-04-10
3179:(3): 201.
3047:References
51:Lacunarity
27:lacunarity
3570:121560110
2994:Dentistry
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3365:18644861
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3303:18595800
3214:(1983).
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3435:Bibcode
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3131:Bibcode
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2969:Ecology
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3279:(PDF)
3189:S2CID
3101:S2CID
3002:Notes
44:Image
3646:PMID
3501:ISBN
3299:PMID
3222:ISBN
3147:PMID
3093:PMID
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3638:doi
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